Least squares cross-validation for the kernel deconvolution density estimatorValidation croisée pour l'estimateur à noyau de la déconvolution d'une densité

Assume we have i.i.d. replications from the corrupted random variable Y=X+ε, where X and ε are independent. We propose a data-driven bandwidth based on cross-validation ideas, for the kernel deconvolution estimator of the density of X. The proposed method is shown to be asymptotically optimal. To cite this article: É. Youndjé, M.T. Wells, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 509–513.     

Justification de la théorie non linéaire de Kirchhoff–Love,comme application d'une nouvelle méthode d'inversion singulièreJustification of the nonlinear Kirchhoff–Love theory,as the application of a new singular inverse method

In the framework of nonlinear elasticity, we consider a three-dimensional plate made of a St Venant–Kirchhoff isotropic and homogeneous material of thickness 2ε and periodic in the two other directions. By a change of scales, the problem can be mapped on a fixed open set, and seen as a nonlinear singular perturbation problem. We introduce a new singular inverse method. Applying this method, we prove that for fixed and small enough exterior forces, the three-dimensional displacement converges to the solution of the nonlinear Kirchhoff–Love theory of plate as the thickness 2ε tends to zero. The limit plate model contains in particular that of von Kármán. We also give a quantitative estimate of the convergence. To cite this article: R. Monneau, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 615–620.     

Vanishing viscosity approximation to hyperbolic conservation laws

We study high order convergence of vanishing viscosity approximation to scalar hyperbolic conservation laws in one space dimension. We prove that, under suitable assumptions, in the region where the solution is smooth, the viscous solution admits an expansion in powers of the viscosity parameter ε. This allows an extrapolation procedure that yields high order approximation to the non-viscous limit as ε→0. Furthermore, an integral across a shock also admits a power expansion of ε, which allows us to construct high order approximation to the location of the shock. Numerical experiments are presented to justify our theoretical findings.     

Asymptotic modelling of conductive thin sheets

We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H ¹-modelling error for an expansion with N terms is bounded by O(ε ^N+1) in the exterior of the sheet and by O(ε ^N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.     

Positive solutions for slightly super-critical elliptic equations in contractible domainsSolutions positives pour l'équation Δu+u(n+2)/(n−2)+ε=0 en ouverts contractibles

We give examples of bounded domains $\text{Ω}$, even contractible, having the following property: there exists $\text{k}\text{?}\text{(Ω)}$ such that, for every integer $\text{k?}\text{k}\text{?}\text{(Ω)}$, problem $\text{P(ε,Ω)}$ below, for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. We also prove that the blow-up points tend to some points of $\text{?Ω}$ as k→∞. To cite this article: R. Molle, D. Passaseo, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 459–462.     

Asymptotic modelling of a fluid–structure coupling in the case of a prestressed inflated orthotropic membrane shell

In this Note, we consider an inflated orthotropic linearly elastic generalized membrane shell submitted to an outer surface perturbation. We obtain the strong convergence towards the solution of a well-posed “2D” problem of the mean value in the membrane thickness 2ε of the “3D” scaled displacements, as ε approaches zero. To cite this article: R. Luce et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The motion of a particle on a light viscoelastic bar: asymptotic analysis of its quasilinear parabolic–hyperbolic equation

We study the large longitudinal motion of a nonlinearly viscoelastic bar with one end fixed and the other end attached to a heavy particle. This problem is a precise continuum-mechanical analog of the basic discrete mechanical problem of the motion of a particle on a (massless) spring. This motion is governed by an initial-boundary-value problem for a class of third-order quasilinear parabolic–hyperbolic partial differential equations subject to a nonstandard boundary condition, which is the equation of motion of the particle. The ratio of the mass of the bar to that of the tip mass is taken to be a small parameter ε. We prove that this problem has a unique globally defined solution that admits a valid asymptotic expansion, including an initial-layer expansion, in powers of ε for ε near 0. The validity of the expansion gives a precise meaning to the solution of the reduced problem, obtained by setting ε=0, which curiously is seldom governed by the expected ordinary differential equation. The fundamental constitutive hypothesis that the tension be a uniformly monotone function of the strain rate plays a critical role in a delicate proof that each term of the initial-layer expansion decays exponentially in time. These results depend on new decay estimates for the solution of quasilinear parabolic equations.     

Weak hyperbolicity on periodic orbits for polynomials

We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n^5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations. To cite this article: J. Rivera-Letelier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1113–1118.     

Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters 0?<?ε?≤?μ?≤?1, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have boundary layers which overlap and interact, based on the relative size of ε and μ. We show how one can construct full asymptotic expansions together with error bounds that cover the complete range 0?<?ε?≤?μ?≤?1. For the present case of analytic input data, we present derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.     

A note on the inverse-partitioned-matrix method in linear regression analysis

Let the classical regression model y = Xβ + σε, E(ε) = 0, Cov ε = V be given. It is shown that the inverse-partitioned-matrix (IPM) method due to C. R. Rao (1973) can easily be understood if the prediction problem for linear models is considered.     

Solutions concentrating at curves for some singularly perturbed elliptic problems

We study positive solutions of the equation ?ε²Δu+u=u^p, where p>1 and ε>0 is small, with Neumann boundary conditions in a three-dimensional domain $\text{Ω}$. We prove the existence of solutions concentrating along some closed curve on $\text{?Ω}$. To cite this article: A. Malchiodi, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Étude asymptotique d'un modèle simple de flammes prémélangées dans un domaine extérieur de RN

We consider a semilinear elliptic equation ?ΔT_ε+u·?T_ε=f_ε(T_ε)(1?T_ε) in outer domains of $\text{R}{}^{\mathrm{N}}$ with Dirichlet's boundary conditions. This Note deals with the questions of existence, uniqueness and the asymptotic behavior of solutions T_ε as ε tends to 0 and the reaction term behaves as a Dirac distribution. Such problems arise in the modelling of premixed flames in the limit of high activations energies. To cite this article: G. Sagon, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Derivation of the k–ε model for locally homogeneous turbulence by homogenization techniques

We derive the incompressible and compressible k–ε model for locally homogeneous turbulence. The model is rigorously derived on formal mathematical grounds using the MPP modelling technique. This lets us calculate by either analytical or numerical means the closure constants of the model. To cite this article: T. Chacón Rebollo, D. Franco Coronil, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Secondary characteristic classes of super-foliations

We construct secondary classes for super-foliations of codimension $0+\mathrm{\epsilon }1$ and $1+\mathrm{\epsilon }1$. We indicate how to generalize this construction for any regular super-foliations on super-manifolds. We interpret the secondary classes as classes of foliated flat connections. To cite this article: C. Laurent-Gengoux, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés

We introduce a generalized finite-difference method for anisotropic diffusion operators on distorted grids. We calculate the second-order derivatives in space using a Taylor expansion. The resulting global matrix associated to the scheme is an M-matrix. Thanks to a certain assumption on the grid properties, we show the convergence of the scheme. We show the robustness of the method in comparison with analytical solutions and results obtained by other numerical schemes. To cite this article: C. Le Potier, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Insensitizing controls for a large-scale ocean circulation model

We consider here a linear quasi-geostrophic ocean model. We look for controls insensitizing (resp. ε-insensitizing) an observation function of the state. The existence of such controls is equivalent to a null controllability property (resp. an approximate controllability property) for a cascade Stokes-like system. Under reasonable assumptions on the spatial domains where the observation and the control are performed, we are able to prove these properties. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).