From the Newton equation to the wave equation : the case of shock waves

We study the macroscopic limit of a chain of atoms governed by the Newton equation. It is known from the work of Blanc, Le Bris, Lions, that this limit is the solution of a nonlinear wave equation, as long as this solution remains smooth. We show, numerically and mathematically that, if the distances between particles remain bounded, it is not the case any more when there are shocks -at least for a convex nearest-neighbour interaction potential with convex derivative

Homogenization of heat diffusion in a cracked medium

We develop in this note a homogenization method to tackle the problem of a diffusion process through a cracked medium. We show that the cracked surface of the domain induces a source term in the homogenized equation. We assume that the cracks are orthogonal to the surface of the material, where an incoming heat flux is applied. The cracks are supposed to be of depth 1, of small width, and periodically arranged.Comment: 28 pages, 10 figure

Vortex patterns in a fast rotating Bose-Einstein condensate

For a fast rotating condensate in a harmonic trap, we investigate the structure of the vortex lattice using wave functions minimizing the Gross Pitaveskii energy in the Lowest Landau Level. We find that the minimizer of the energy in the rotating frame has a distorted vortex lattice for which we plot the typical distribution. We compute analytically the energy of an infinite regular lattice and of a class of distorted lattices. We find the optimal distortion and relate it to the decay of the wave function. Finally, we generalize our method to other trapping potentials

Some variance reduction methods for numerical stochastic homogenization

We overview a series of recent works devoted to variance reduction techniques for numerical stochastic homogenization. Numerical homogenization requires solving a set of problems at the micro scale, the so-called corrector problems. In a random environment, these problems are stochastic and therefore need to be repeatedly solved, for several configurations of the medium considered. An empirical average over all configurations is then performed using the Monte-Carlo approach, so as to approximate the effective coefficients necessary to determine the macroscopic behavior. Variance severely affects the accuracy and the cost of such computations. Variance reduction approaches, borrowed from other contexts of the engineering sciences, can be useful. Some of these variance reduction techniques are presented, studied and tested here

Worldwide Influence of the French Civil Code of 1804, on the Occasion of its Bicentennial Celebration

The French Civil Code (still called the Code Napoleon) is now two hundred years old. Its bicentennial has been celebrated this year in many countries. The reason is that is Code has experienced an extraordinary expansion throughout the world during the XIXth and XXth centuries. But how influential is it today? A certain weakening of its positions is due to a number of factors : legal (the abundance of models now available) and cultural (the regression of the use of French as an international legal language as well as the declining attraction of our universities in the formation of foreign lawyers). Yet, even after the last two centuries have brought so many changes in society, it remains as a model in many countries, because of its qualities in form and in substance. In fact, it is not only a monument of the past. It still has a great capacity of influence. And the present search for a jus commune in Europe can give it a new chance on the international scene

Worldwide Influence of the French Civil Code of 1804, on the Occasion of its Bicentennial Celebration

The French Civil Code (still called the Code Napoleon) is now two hundred years old. Its bicentennial has been celebrated this year in many countries. The reason is that is Code has experienced an extraordinary expansion throughout the world during the XIXth and XXth centuries. But how influential is it today? A certain weakening of its positions is due to a number of factors : legal (the abundance of models now available) and cultural (the regression of the use of French as an international legal language as well as the declining attraction of our universities in the formation of foreign lawyers). Yet, even after the last two centuries have brought so many changes in society, it remains as a model in many countries, because of its qualities in form and in substance. In fact, it is not only a monument of the past. It still has a great capacity of influence. And the present search for a jus commune in Europe can give it a new chance on the international scene