De la K-théorie algébrique vers la K-théorie hermitienne

In this Note, we introduce a new morphism between algebraic and hermitian K-theory. The topological analog is the Adams operation ${\psi }^2$ in real K-theory. From this morphism, we deduce a lower bound for the higher algebraic K-theory of a ring A in terms of the classical Witt group of the ring $A?A^{\mathrm{op}}$. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the existence of solutions in weighted Sobolev spaces for the exterior Oseen problem

In this Note, we present existence and uniqueness results for the exterior Oseen problem. In order to control the behavior at infinity of functions, we use as functional framework weighted Sobolev spaces. The results rely on a $L^p$-theory for $1<p<\infty$. To cite this article: C. Amrouche, U. Razafison, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Quantitative K-theory for Banach algebras

Quantitative (or controlled) K-theory for $C^{?}$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative K-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer–Vietoris sequence in this framework.     

A propos de la propriété de Bergman

In a recent preprint, George M. Bergman has investigated the following property:for any generating set X of the group G there exists an integer n such that any element of G is a product of n elements of $\text{X}\cup {\text{X}}^{\mathrm{?1}}$. We will say in this case that G has the Bergman property.We have solved some of the questions asked in the above mentioned preprint and have found it suitable to investigate this property in a more general context, in particular for rings (essentially Boolean rings). To cite this article: A. Khelif, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Convergence to the equilibrium for the Pauli equation without detailed balance condition

Let $A?B$ be an extension of commutative rings with identity, X an analytic indeterminate over B, and $R\text{:}=A+XB[[X]]$, the subring of the formal power series ring $B[[X]]$, consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module. To cite this article: S. Hizem, A. Benhissi, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

An extension of Brouwer's fixed point theorem allowing discontinuities

In this article, we extend Brouwer's fixed point theorem – which states that every continuous mapping $\text{f}\text{:B→B}$ (a closed ball of $\text{R}{}^{\mathrm{n}}$) must have a fixed point – by allowing discontinuities of f, and we apply this extension to equilibrium theory in Economics. To cite this article: P. Bich, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Solutions globales d'énergie infinie de l'équation de Navier–Stokes 2D

We study in this Note the solutions of the 2D Navier–Stokes equations with initial data in ?BMO. For $u{|}_{t=0}$ in the closure of the Schwartz class, we obtain the existence and uniqueness of a global solution, and besides an estimate on its norm in ?BMO. To cite this article: P. Germain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A Gutzwiller type formula for a reduced Hamiltonian within the framework of symmetry

For a classical Hamiltonian with a finite group of symmetries, we give semi-classical asymptotics in a neighbourhood of an energy E of a regularized spectral density of the quantum Hamiltonian restricted to symmetry subspaces of Peter–Weyl defined by irreducible characters of the group. If we suppose that the energy level ${\Sigma }_E$ is compact, non-critical, and that its periodic orbits are non-degenerate, we get a Gutzwiller type formula for the reduced Hamiltonian, whose oscillating part involves the symmetry properties of closed trajectories of ${\Sigma }_E$. To cite this article: R. Cassanas, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

A theory of anti-selfdual Lagrangians: dynamical case

We consider the class of time-dependent anti-selfdual Lagrangians, which – just like the stationary case announced in Ghoussoub [C. R. Acad. Sci. Paris, Ser. I 340 (2005)] – enjoys remarkable permanence properties and provides variational formulations and resolutions for several initial-value parabolic equations including gradient flows and other dissipative systems. Even though these evolutions do not fit in the standard Euler–Lagrange theory, we show that their solutions can be obtained as minima – but also as zeroes – of action functionals of the form ${\int }_0^{T}L(t\text{,}u(t)\text{,}\dot{u}(t)+{\Lambda }_{t}u(t)\mathrm{d}t)$ where L is a time-dependent anti-selfdual Lagrangian and where ${\Lambda }_t$ is a flow of skew-adjoint operators. To cite this article: N. Ghoussoub, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Kähler manifolds with numerically effective Ricci class and maximal first Betti number are tori

Let M be a n-dimensional Kähler manifold with numerically effective Ricci class $c_1(M)$. In this Note we prove that, if the first Betti number $b_1(M)$ is equal to 2n, then M is biholomorphic to a n-dimensional complex torus. To cite this article: F. Fang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).