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In this paper we determine all Padovan numbers that are palindromic concatenations of two distinct repdigits.

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; This paper concerns the simulation of a class of nonlinear discrete-time systems under a set of initial conditions described by an ellipsoid. We derive a procedure allowing the propagation of such ellipsoids through time, which makes it possible to set a guaranteed hard bound on the evolution of the state of the system for all the possible initial conditions. At the end of the paper, we show an application of the method through three academics examples, two of which are taken from the theory of fractals.

International audience; The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].

In this paper we study the moderate deviations for the magnetization of critical Curie-Weiss model. Chen, Fang and Shao considered a similar problem for non-critical model by using Stein method. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cram\'er-type result., Comment: 12 pages, accepted for publication in Electronic Communication in Probability

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

International audience; In this paper, we prove a stability estimate of the conductivity parameters identification problem in cardiac electrophysiology. The propagation of the electrical wave in the heart is described by the monodomain model coupled to an elliptic equation describing the diffusion of the electrical wave in the whole body. Our result concerns both heart and torso conductivity parameters. The main difficulty that we solve in this paper is related to the transmission conditions between the heart and the torso. We first, establish Carleman estimates for the coupled heart-torso system. Then, using these estimates and the Bukhgeim and Klibanov approach, we prove a Lipschitz stability estimate of cardiac and torso conductivity parameters.

International audience; We study change-points tests based on U-statistics for absolutely regular observations. Our method avoids some technical assumptions on the data and the kernel. The asymptotic properties of the U-statistics are studied under the null hypothesis, under fixed alternatives and under a sequence of local alternatives. The asymptotic distributions of the test statistics under the null hypothesis and under the local alternatives are given explicitly and the tests are shown to be consistent. A small set of simulations is done for evaluating the performance of the tests in detecting changes in the mean, variance and autocorrelation of some simple time series.

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).

In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann–Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.

International audience; We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback.

This paper proposes two novel families of non-stationary subdivision schemes with a shape parameter of hierarchically and efficiently generating mixed hyperbolic/trigonometric curves of order 3 and 4. An analysis of convergence and smoothness of the proposed schemes is established by using the asymptotic equivalence method. This paper also discusses the bivariate tensor-product subdivision scheme for the surface modeling on the regular meshes. The numerical results they produce are very encouraging. In this context, the performance of our algorithms has been exposed by considering examples, which illustrate how the shape parameter and the control points are assigned to reproduce such analytic curves and surfaces using tensor-product notion.

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

There are two major routes to address the ubiquitous family of inverse problems appearing in signal and image processing, such as denoising or deblurring. A first route relies on Bayesian modeling, where prior probabilities are used to embody models of both the distribution of the unknown variables and their statistical dependence with the observed data. The estimation process typically relies on the minimization of an expected loss (e.g. minimum mean squared error, or MMSE). The second route has received much attention in the context of sparse regularization and compressive sensing: it consists in designing (often convex) optimization problems involving the sum of a data fidelity term and a penalty term promoting certain types of unknowns (e.g., sparsity, promoted through an 1 norm). Well known relations between these two approaches have lead to some widely spread misconceptions. In particular, while the so-called Maximum A Posterori (MAP) estimate with a Gaussian noise model does lead to an optimization problem with a quadratic data-fidelity term, we disprove through explicit examples the common belief that the converse would be true. It has already been shown [7, 9] that for denoising in the presence of additive Gaussian noise, for any prior probability on the unknowns, MMSE estimation can be expressed as a penalized least squares problem, with the apparent characteristics of a MAP estimation problem with Gaussian noise and a (generally) different prior on the unknowns. In other words, the variational approach is rich enough to build all possible MMSE estimators associated to additive Gaussian noise via a well chosen penalty. We generalize these results beyond Gaussian denoising and characterize noise models for which the same phenomenon occurs. In particular, we prove that with (a variant of) Poisson noise and any prior probability on the unknowns, MMSE estimation can again be expressed as the solution of a penalized least squares optimization problem. For additive scalar denois-ing the phenomenon holds if and only if the noise distribution is log-concave. In particular, Laplacian denoising can (perhaps surprisingly) be expressed as the solution of a penalized least squares problem. In the multivariate case, the same phenomenon occurs when the noise model belongs to a particular subset of the exponential family. For multivariate additive denoising, the phenomenon holds if and only if the noise is white and Gaussian., Comment: This work and the companion paper [10] are dedicated to the memory of Mila Nikolova, who passed away prematurely in June 2018. Mila dedicated much of her energy to bring the technical content to completion during the spring of 2018. The first author did his best to finalize the papers as Mila would have wished. He should be held responsible for any possible imperfection in the final manuscript

International audience; In this paper, we consider the design of wavelet filters based on the Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed using the common-factor approach are then approximately analytic. Thus, it is of interest to control their analyticity. The purpose of this paper is to first provide precise and explicit expressions as well as easily exploitable bounds for quantifying the analytic approximation of this complex wavelet. Then, we prove the existence of such filters enjoying the classical perfect reconstruction conditions, with arbitrarily many vanishing moments.

Fully revised. To appear in Nonlinearity; International audience; This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincar\'e integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint classical-quantum density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

In a previous paper Chevallier et al. (2018), it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u ( t , x ) of a neural field equation (NFE). The value u ( t , x ) represents the membrane potential at time t of a typical neuron located in position x , embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u ( t , x ) . Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u ( t , x ) . To the best of our knowledge, this result appears to be new in the literature.

International audience; The paper deals with output feedback regulation of exponentially stable systems by an integral controller. We have recently proposed an appropriate Lyapunov functional to prove exponential stability of the closed-loop system. The approach is dedicated in this paper to hyperbolic systems and especially to the de Saint-Venant equations giving explicitly the gains to ensure an exponentially stabilized integral controller: the parameters expression is deduced directly of the Lyapunov functional based on the Forwarding approach. Numerical simulations illustrate this approach.

In view of studying incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscid fluids. A natural analogue of Brenier's problem is introduced, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. It turns out that this new variational problem is an entropy minimization problem with marginal constraints entering the class of convex minimization problems. This paper explores the connection between this variational problem and Brenier's original problem. Its dual problem is derived and the general shape of its solution is described. Under the restrictive assumption that the pressure is a nice function, the kinematics of its solution is made explicit and its connection with the Navier-Stokes equation is established.

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 study the problem of constructing flat inputs for multi-output dynamical systems. The notion of flat inputs has been introduced by Waldherr and Zeitz in [30, 31] and can be seen as dual to that of flat outputs. In the single-output case, a flat input can be constructed if and only if the original dynamical system together with its output is observable. In the multi-output case, the observability is not necessary for the existence of flat inputs. The observable case has been treated in [31], where a system of linear algebraic equations has been proposed in order to determine the control vector fields associated to the flat inputs. The goal of this paper is to treat the unobservable case for multi-output dynamical systems. We start by discussing the case when the dynamical system together with the given output is observable and we present a generalization of the results of [31] by relating them with the notion of minimal differential weight. Then we give our main results. We consider the unobservable case for which locally, on an open and dense subset of $\mathbb R^n$ , we construct control vector fields $g_1 , \ldots, g_m$ such that the associated control system is flat. Finally, we explain how our results can be applied to private communication.

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.

Hall and Marron (1987) introduced kernel estimators of integrals of the squared m -order derivatives of a probability density. Mokkadem and Pelletier (2020) gave recursive versions of their estimators, but the main drawback of these estimators is that their update requires the use of all past data. The aim of this paper is the study of online versions of the estimators introduced by Hall and Marron (1987), that is of estimators which are not only recursive, but which also have the property that their update uses only the last available data. Rates of convergence in mean squared error (MSE) are calculated. Similarly to the estimators of Hall and Marron (1987) and of Mokkadem and Pelletier (2020), our online estimators achieve the parametric rate n − 1 when m = 0 or when higher order kernels are used. For the case when the parametric rate is not obtained, we also study an online version of the estimator proposed by Jones and Sheather (1991). Finally, we provide recursive estimators of the optimal bandwidth in the framework of density estimation.

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_n)$ which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from smooth Wasserstein distances to total variation in a quantitative way. This is a well studied topic and one can consult for instance Bally and Caramellino [Electron. J. Probab. 2014], Bogachev, Kosov and Zelenov [Trans. Amer. Math. Soc. 2018], Hu, Lu and Nualart [J. Funct. Anal. 2014], Nourdin and Poly [Stoch. Proc. Appl. 2013] and the references therein for an overview of this issue. Each of the aforementioned references share the fact that some non-degeneracy is required along the whole sequence. We provide here the first result removing this costly assumption as we require only non-degeneracy at the limit. The price to pay is to control the smooth Wasserstein distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limit as their Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main findings of Nourdin, Peccati and Swan [J. Funct. Anal. 2014]. Another application concerns the approximation of the semi-group of a diffusion process by the Euler scheme in a quantitative way and under the H\"ormander condition.

This paper is devoted to the study of reflected Stochastic Differential Equations with jumps when the constraint is not on the paths of the solution but acts on the law of the solution. This type of reflected equations have been introduced recently by Briand, Elie and Hu [BEH18] in the context of BSDEs, when no jumps occur. In [BCdRGL16], the authors study a numerical scheme based on particle systems to approximate these reflected SDEs. In this paper, we prove existence and uniqueness of solutions to this kind of reflected SDEs with jumps and we generalize the results obtained in [BCdRGL16] to this context., Comment: arXiv admin note: substantial text overlap with arXiv:1612.06886

International audience; In this paper, we address an instance of uniquely solvable mean-field game with a common noise whose corresponding counterpart without common noise has several equilib-ria. We study the selection problem for this mean-field game without common noise via three approaches. A common approach is to select, amongst all the equilibria, those yielding the minimal cost for the representative player. Another one is to select equilibria that are included in the support of the zero noise limit of the mean-field game with common noise. A last one is to select equilibria supported by the limit of the mean-field component of the corresponding N-player game as the number of players goes to infinity. The contribution of this paper is to show that, for the class under study, the last two approaches select the same equilibria, but the first approach selects another one.

In the present paper we prove an homogenisation result for a locally perturbed transport stochastic equation. The model is similar to the stochastic Burgers' equation and it is inspired by the LWR model. Therefore, the interest in studying this equation comes from it's application for traffic flow modelling. In the first part of paper we study the inhomogeneous equation. More precisely we give an existence and uniqueness result for the solution. The technical difficulties of this part come from the presence of the function \begin{document}$ \varphi $\end{document} under assumptions coherent for the model, which is giving the inhomogeneity with respect to the space variable, not present in the classical results. The second part of the paper is the homogenisation result in space.

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

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}) ).

The paper is dedicated to the asymptotic investigation of textiles as an elasticity problem on beam structures. The structure is subjected to a simultaneous homogenization and dimension reduction with respect to the asymptotic behavior of the beams' thickness and periodicity. Important for the problem are the contact conditions between the beams, which yield multiple limits depending on the order. In this paper two limiting cases are presented: a linear case and a Leray-Lions-type problem.

International audience; This paper presents both a recursive scheme to perform Geometric Algebra operations over a prefix tree, and Garamon, a C++ library generator implementing these recursive operations. While for low dimension vector spaces, precomputing all the Geometric Algebra products is an efficient strategy, it fails for higher dimensions where the operation should be computed at run time. This paper describes how a prefix tree can be a support for a recursive formulation of Geometric Algebra operations. This recursive approach presents a much better complexity than the usual run time methods. This paper also details how a prefix tree can represent efficiently the dual of a multivector. These results constitute the foundations for Garamon, a C++ library generator synthesizing efficient C++ / Python libraries implementing Geometric Algebra in both low and higher dimensions, with any arbitrary metric. Garamon takes advantage of the prefix tree formulation to implement Geometric Algebra operations on high dimensions hardly accessible with state-of-the-art software implementations. Garamon is designed to produce easy to install , easy to use, effective and numerically stable libraries. The design of the libraries is based on a data structure using precomputed functions for low dimensions and a smooth transition to the new recursive products for higher dimensions. Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.

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

In this paper, we first provide a unified algorithm for computing simple forms for systems of pseudo-linear equations. We prove that the existing methods for linear differential and difference systems can be extended to handle more general pseudo-linear systems. We explain how the reduction to a simple form can be used to compute efficiently local data for a system of pseudo-linear equations. We then propose an alternative, again based on simple forms, to previous algorithms for computing rational solutions of pseudo-linear systems. Moreover we develop a new algorithm for computing rational solutions of systems in two variables composed of linear differential and difference equations. Finally, we show that this algorithm can be generalized to the case of a system of partial pseudo-linear equations. All the algorithms described in this paper have been implemented in Maple and some examples of computations are provided.

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to $Lu=1$, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, $1/u$, acts as an effective potential. Hence from the single measurement of $u$, we obtain, via $1/u$, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum. (This version strengthens and simplifies the results of the first one by replacing a global bi-Lipschitz hypothesis on the domain with a local bi-Lipschitz hypothesis. It improves on the second version by adding pictures and numerical examples. This version is identical to the third version; all that is changed is to correct some tex mistakes in symbols in this abstract. There are no changes to the paper itself.), Comment: 33 pages

In a recent paper by the first two authors, existence of martingale solutions to a stochastic nonlinear Schrodinger equation driven by a Levy noise was proved. In this paper, we prove pathwise uniqueness, uniqueness in law and existence of strong solutions to this problem using an abstract uniqueness result of Kurtz.

International audience; In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits.

Let Ω be a domain of R n , n ≥ 3 . The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ ( n − 2 ) 2 4 , there exists a constant K ( Ω , γ , s ) > 0 such that ( H S ) ∫ Ω | u | 2 ⋆ ( s ) | x | s d x 2 2 ⋆ ( s ) ≤ K ( Ω , γ , s ) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 ( Ω ) (the completion of C c ∞ ( Ω ) for the relevant norm). When 0 ∈ Ω is an interior point, the range ( − ∞ , ( n − 2 ) 2 4 ) for γ cannot be improved: moreover, the optimal constant K ( Ω , γ , s ) is independent of Ω and there is no extremal for ( H S ) . But when 0 ∈ ∂ Ω , the situation turns out to be drastically different since the geometry of the domain impacts : • the range of γ ’s for which ( H S ) holds. • the value of the optimal constant K ( Ω , γ , s ) ; • the existence of extremals for ( H S ) . When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider nonsmooth domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω . When γ is small, we introduce a new geometric object at the conical singularity that generalizes the ”mean curvature”: this allows to get extremals for ( H S ) . The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own.

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate S B D p functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy G S B D p functions by S B V p ones.

Remainder Padé Approximation (RPA) consists in adding to the n-th partial sum of a series a suitable Padé approximant of the asymptotic expansion in the variable 1/n of the remainder term. In a previous paper, we proved the non-trivial property that the RPA of the exponential function is identical to the Padé approximant to the exponential function. In this paper, we extend this property to the hypergeometric series 1F1(1; a; z) and 2F0(b, 1; z).

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998). In a recent paper (Chambolle and Crismale in J Eur Math Soc (JEMS), 2018) we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti et al. (Ann Inst H Poincare Anal Non Lineaire 36:455–474, 2019) and Chambolle et al. (J Math Pures Appl, 2019. https://doi.org/10.1016/j.matpur.2019.02.001 ).

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 α≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.

We present a systematic method for the construction of discrete Painlev\'e equations. The method, dubbed `restoration', allows one to obtain all discrete Painlev\'e equations that share a common autonomous limit, up to homographic transformations, starting from any one of those limits. As the restoration process crucially depends on the classification of canonical forms for the mappings in the QRT family, it can in principle only be applied to mappings that belong to that family. However, as we show in this paper, it is still possible to obtain the results of the restoration even when the initial mapping is not of QRT type (at least for the system at hand, but we believe our approach to be of much wider applicability). For the equations derived in this paper we also show how, starting from a form where the independent variable advances one step at a time, one can obtain versions corresponding to multistep evolutions., Comment: 17 pages, 1 figure

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; This paper proposes a representation theory for any finite lattice via set-colored posets, in the spirit of Birkhoff for distributive lattices. The notion of colored posets was introduced in Nourine (2000) [34] and the generalization to set-colored posets was given in Nourine (2000) [35]. In this paper, we give a characterization of set-colored posets for general lattices, and show that set-colored posets capture the order induced by join-irreducible elements of a lattice as Birkhoff’s representation does for distributive lattices. We also give a classification for some lattices according to the coloring property of their set-colored representation including upper locally distributive, upper locally distributive, meet-extremal and semidistributive lattices.

International audience; In the continuity of a recent paper ([6]), dealing with finite Markov chains, this paper proposes and analyzes a recursive algorithm for the approximation of the quasi-stationary distribution of a general Markov chain living on a compact metric space killed in finite time. The idea is to run the process until extinction and then to bring it back to life at a position randomly chosen according to the (possibly weighted) empirical occupation measure of its past positions. General conditions are given ensuring the convergence of this measure to the quasi-stationary distribution of the chain. We then apply this method to the numerical approximation of the quasi-stationary distribution of a diffusion process killed on the boundary of a compact set and to the estimation of the spectral gap of irreducible Markov processes. Finally, the sharpness of the assumptions is illustrated through the study of the algorithm in a non-irreducible setting.

International audience; In this paper, we consider a nonconvex differential inclusion with constant delay. We study the existence of viable solutions when the state is constrained to the closure of an open subset of R n. The main contribution is a relaxation result stating that, under some assumptions, each "viable solution" of the convexified inclusion can be approximated by "viable solutions" of the original one. This result is obtained thanks to an extension of the celebrated Filippov's theorem to the case of delay differential inclusions.

International audience; This paper deals with the stability analysis of time delay systems based on continuous-time approach. The originality of the present paper relies on the construction of several models for a same time-delay systems using the interconnection of an ordinary differential equation and a transport partial differential equation. The stability analysis is then performed using a Lyapunov functional. These models are constructed in order to first reduce potentially the complexity of the resulting stability conditions. Second several models are build in order to be interpreted as a discretization scheme as the one usually used in the Lyapunov functional. The proposed result can be seen as a generalized (N − M) discretization which consists in both a time-discretization of the delay interval into M sub-intervals, and the projection of the state function within each sub-interval on the Legendre polynomials of degree less than N. The efficiency of this novel approach is illustrated on an academic example.

For the final version of the paper, seehttps://hal.archives-ouvertes.fr/hal-01943863v1; In this paper we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi– Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterizes it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure. A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?

In this paper, we describe in detail a model of geometric-functional variability between fshapes. These objects were introduced for the first time by the authors in [Charlier et al. 2015] and are basically the combination of classical deformable manifolds with additional scalar signal map. Building on the aforementioned work, this paper's contributions are several. We first extend the original $L^2$ model in order to represent signals of higher regularity on their geometrical support with more regular Hilbert norms (typically Sobolev). We describe the bundle structure of such fshape spaces with their adequate geodesic distances, encompassing in one common framework usual shape comparison and image metamorphoses. We then propose a formulation of matching between any two fshapes from the optimal control perspective, study existence of optimal controls and derive Hamiltonian equations and conservation laws describing the dynamics of geodesics. Secondly, we tackle the discrete counterpart of these problems and equations through appropriate finite elements interpolation schemes on triangular meshes. At last, we show a few results of metamorphosis matchings on synthetic and several real data examples in order to highlight the key specificities of the approach., 46 pages, 12 figures