Bases d'ondelettes adaptées au règlement de la divergence infra-rouge

We propose bi-orthogonal wavelet bases that solve the problem of infrared divergence phenomenon for usual wavelet expansions in the homogeneous Sobolev spaces ${\dot{H}}^s({\mathbb{R}}^n)$. These bases remove the divergence in the case $s?\frac{n}{2}?\mathbb{N}$ since they are also bases of the realization of ${\dot{H}}^s({\mathbb{R}}^n)$. In the critical case $s?\frac{n}{2}\in \mathbb{N}$, they provide a confinement of the infrared divergence in a ‘small’ space. This method of confinement is also applied to the Mumford process. To cite this article: B. Vedel, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A canonical frame for nonholonomic rank two distributions of maximal class

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in ${\mathbb{R}}^5$. We solve the analogous problems for rank 2 distributions in ${\mathbb{R}}^n$ for arbitrary $n>5$. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence. To cite this article: B. Doubrov, I. Zelenko, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Regularity and decay of solutions of nonlinear harmonic oscillators

We prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators. Namely, we show holomorphic extension to a sector in the complex domain, with a corresponding Gaussian decay, according to the basic properties of the Hermite functions in ${\mathbb{R}}^d$. Our results apply, in particular, to nonlinear eigenvalue problems for the harmonic oscillator associated to a real-analytic scattering, or asymptotically conic, metric in ${\mathbb{R}}^d$, as well as to certain perturbations of the classical harmonic oscillator.     

Enumerating invariant subspaces of Rn

In this article, we develop an algorithm to calculate the set of all integers m for which there exists a linear operator T on ${\mathbb{R}}^n$ such that ${\mathbb{R}}^n$ has exactly m T-invariant subspaces. Moreover, the algorithm gives a method for the explicit construction of such a linear operator T. A brief discussion is included as how these methods can be extended to vector spaces over arbitrary fields.