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).     

Note sur les corps différentiellement clos valués

In the paper by Guzy and Point, Differential topological fields, the model-completion ${(\mathit{OVF})}_D^1$ of the theory of ordered valued differential fields ${\mathit{OVF}}_D$ is established. Models of this theory are closed ordered differential fields (the theory CODF was studied by Singer) which have a non-trivial convex (for the order) subring as valuation ring. Here we prove the valued analogue of a result of Singer: if K is a model of ${(\mathit{OVF})}_D^1$ then $K(\mathrm{i})$ (${\mathrm{i}}^2=?1$) is a model of the theory of differentially closed valued fields which is the model-completion of the theory of non-trivially valued differential fields of characteristic zero. To cite this article: N. Guzy, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On some problems related to Berezin symbols

The following problem was formulated by Zorboska [Proc. Amer. Math. Soc. 131 (2003) 793–800]: It is not known if the Berezin symbols of a bounded operator on the Bergman space $L_a^2(\mathbf{D})$ must have radial limits almost everywhere on the unit circle. In this Note we solve this problem in the negative, showing that there is a concrete class of diagonal operators for which the Berezin symbol does not have radial boundary values anywhere on the unit circle. A similar result is also obtained in case of the Hardy space $H^2(\mathbf{D})$ over the unit disk D. Moreover, we give an alternative proof to the famous theorem of Beurling on z-invariant subspaces in the Hardy space $H^2(\mathbf{D})$, using the concepts of reproducing kernels and Berezin symbols. To cite this article: M.T. Karaev, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Conditional p-adic probability logic

In this paper we present the proof-theoretical approach to p-adic valued conditional probabilistic logics. We introduce two such logics denoted by ${\mathit{CPL}}_{{\mathbf{Z}}_{\mathbf{p}}}$ and ${\mathit{CPL}}_{{\mathbf{Q}}_{\mathbf{p}}}^{\mathit{fin}}$. Each of these logics extends classical propositional logic with a list of binary (conditional probability) operators. Formulas are interpreted in Kripke-like models that are based on p-adic probability spaces. Axiomatic systems with infinitary rules of inference are given and proved to be sound and strongly complete. The decidability of the satisfiability problem for each logic is proved.     

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.     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

Théorème de Liouville et propriété de la moyenne biharmonique restreinte

We prove, under some conditions, that a bounded Lebesgue measurable function satisfying the restricted mean value for the biharmonic functions in ${\mathbf{R}}^n$, $n\ne 2$, or in an open set of ${\mathbf{R}}^2$ with polar complement, is constant. To cite this article: M. El Kadiri, C. R. Acad. Sci. Paris, Ser. I 340 (2005).