Recovery of sparsest signals via ℓq-minimization

In this paper, it is proved that every s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ can be exactly recovered from the measurement vector $\mathbf{z}=\mathbf{Ax}\in {\mathbb{R}}^m$ via some ${?}^q$-minimization with $0<q?1$, as soon as each s-sparse vector $\mathbf{x}\in {\mathbb{R}}^n$ is uniquely determined by the measurement z. Moreover it is shown that the exponent q in the ${?}^q$-minimization can be so chosen to be about $0.6796\times (1?{\delta }_{2s}(\mathbf{A}))$, where ${\delta }_{2s}(\mathbf{A})$ is the restricted isometry constant of order 2s for the measurement matrix A.     

Necessary and sufficient conditions of some strong optimal solutions to the interval linear programming

This paper considers optimal solutions of general interval linear programming problems. Necessary and sufficient conditions of (A,b)$(\mathbf{A},\mathbf{b})$-strong and (A,b,c)$(\mathbf{A},\mathbf{b},\mathbf{c})$-strong optimal solutions to the interval linear programming with inequality constraints are proposed. The features of the proposed methods are illustrated by some examples.     

A negative answer to a question about leading terms ideals of polynomial ideals

We construct an example of a finitely generated ideal $I$ of $R\left[X\right]$, where $R$ is a one-dimensional domain, whose leading terms ideal is not finitely generated. This gives a negative answer to the open question of whether if $R$ is a domain with Krull dimension ≤1, then for any finitely generated ideal $I$ of $R\left[X\right]$, the leading terms ideal of $I$ is also finitely generated. Moreover, as a positive part of our answer, we prove that for any one-dimensional domain $R$ and any $a,b\in R$, the ideal of $R\left[X\right]$ generated by the leading terms of $〈1+aX,b〉$ is finitely generated.     

Beltrami's solutions of general equilibrium equations in continuum mechanics

M. Gurtin has proved that the Beltrami representation, $\mathbf{S}=\mathrm{rot}\mathrm{rot}\mathbf{A}$, of a smooth, divergence-free stress tensor in a smooth domain, is verified if and only if S is self-equilibrated. Here, Gurtin's conditions are extended to the case of a bounded domain with a Lipschitz-continuous boundary, for a tensor field $\mathbf{S}\in L^2(\Omega ;M_{\mathrm{sym}}^3)$. We apply this result to obtain an extension of the Saint Venant's equations of compatibility to non necessarily simply-connected domains. To cite this article: G. Geymonat, F. Krasucki, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Eléments finis nodaux pour les équations de Maxwell

An original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: $\mathbf{E}={\mathbf{E}}^R+\lambda {\mathbf{x}}_P$, where ${\mathbf{E}}^R\in H^1{\left(\omega \right)}^2$, λ depends on the data and domain and ${\mathbf{x}}_P$ is known explicitly. The same splitting can be used for the magnetic field. No cut-off function is needed and improved error estimates are derived. To cite this article: E. Jamelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).