Stable products of spheres in the non-linear coupling of oscillators or quasi-periodic motions

For generic families of vector fields or transformations, normally hyperbolic invariant products of spheres appear near partially elliptic rest points. To cite this article: M. Kammerer-Colin de Verdière, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Trois formulations d'un modèle de plasma quasi-neutre avec courant non nul

In this Note, we propose three formulations of a model describing a quasi-neutral plasma with non-vanishing current. In order to study and compare the numerical efficiency of each formulation, two test-problems are implemented in one dimension. The first is a periodic perturbation of a uniform stationary plasma. The second is a case of plasma expansion in vacuum between two electrodes. To cite this article: P. Crispel et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Weak density of smooth maps for the Dirichlet energy between manifolds

We prove that given a simply connected compact manifold M and a closed manifold N, any map in the Sobolev space W ^1,2(M,N) can be approximated weakly by smooth maps between M and N. Submitted: September 2002, Final version: November 2002.     

Genericity of the Morse-Smale Property for Damped Wave Equations

We prove that for damped hyperbolic equations the Morse-Smale property (hyperbolicity of equilibria and transversal intersection of stable and unstable manifolds) is generic. More precisely, we prove that in an appropriate functional space of nonlinear terms in the equation, the set of functions for which the latter has the Morse-Smale property is residual, i.e., it is a countable intersection of open dense sets. The result extends a similar result proved in [1] for reaction diffusion equations. However, because of the absence of knowledge about nodal sets of polutions new ideas were needed in the proof.     

Lagrangian skeletons,periodic geodesic flows and symplectic cuttings

In this paper, we investigate symplectic manifolds endowed with a Morse–Bott function with only two critical submanifolds, one of which is Lagrangian while the other one is symplectic.     

Corner spontaneous magnetization

Details are given of a new method allowing an exact calculation of the spontaneous magnetization in the corner as well as along the edge at an arbitrary distance of the corner for a rectangular planar Ising ferromagnet.     

Thermal and chemical nanofractal relaxation

We studied shape relaxation of nano-fractal islands, during annealing, after their growth from antimony cluster deposition on graphite surface. Annealing at 180^°C shows evidence of an increase of the fractal branch width with time followed by branch fragmentation, without changing the fractal dimension. The time evolution of the width of the arm suggests the surface self-diffusion mechanism as the main relaxation process. With Monte Carlo simulations, we confirmed the observed behavior. Comparison is done with our previous results on fragmentation of nano-fractal silver islands when impurity added to the incident cluster promotes rapid fragmentation by surface self-diffusion enhancement [1].     

Infinite dimensional duality and applications

The usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear. In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty. We apply this theory to solve the until now unsolved problem of finding, in the infinite dimensional case, the Lagrange multipliers associated to optimization problems or to variational inequalities. As an example, we find the Lagrange multiplier associated to a general elastic–plastic torsion problem.