Sur les bornes inférieures de l'écart des variétés invariantes

Let m be an integer ?3, set $?(r)=\frac{1}{2}(r_1^2+?+r_{m?1}^2)+r_m$ for $r\in {\mathbb{R}}^m$, and consider a badly approximable vector ${\overline{\omega }}^0\in {\mathbb{R}}^{m?2}$. Fix $\alpha >1$, $L>0$ and $R>1+6{\overline{\omega }}^0{6}_{\infty }$. We construct a sequence $(H_N)$ of $\text{Gevrey}\text{-}(\alpha ,L)$ Hamiltonian functions of ${\mathbb{T}}^m\times {\overline{B}}_{\infty }(0,R)$, which converges to ? when $N\to \infty$, such that for each N the system generated by $H_N$ possesses a $(m?1)$-dimensional hyperbolic invariant torus with fixed frequency vector $({\overline{\omega }}^0,1)$, which admits a homoclinic point with splitting matrix of the form $\mathrm{diag}(0,{\nu }_N,\dots ,{\nu }_N,0)\in M_m(\mathbb{R})$, with ${\nu }_N?\exp (?c{(\frac{1}{{?}_N})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2(\alpha ?1)(m?2)$}\right.})$, where ${?}_N:=6H_N??6_{\alpha ,L}$ and $c>0$. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ${?}^1({\mathbf{S}}_m)$. We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ${?}^1({\mathbf{FS}}_m)$. In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ${?}^1({\mathbf{S}}_m)$, showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even ($n?2$). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for $n?1$. A similar strategy is used for the tensor algebra of a Banach space.     

Estimation du nombre de dérivées d'un processus Gaussien

We consider a real Gaussian process X with unknown smoothness $r_0\in \mathbb{N}$ where the mean-square derivative $X^{(r_0)}$ is supposed to be Hölder continuous in quadratic mean. First, from the discrete observations $X(t_1),\dots ,X(t_n)$, we study reconstruction of $X(t)$, $t\in [0,1]$, with ${\stackrel{?}{X}}_r(t)$, a piecewise polynomial interpolation of degree $r?1$. We show that the mean-square error of interpolation is a decreasing function of r but becomes stable as soon as $r?r_0$. Next, from an interpolation-based empirical criterion, we derive an estimator $\stackrel{?}{r}$ of $r_0$ and prove its strong consistency by giving an exponential inequality for $P(\stackrel{?}{r}\ne r_0)$. Finally, we prove the strong convergence of ${\stackrel{?}{X}}_{\stackrel{?}{r}}(t)$ toward $X(t)$ with a similar rate as in the case ‘$r_0$ known’. To cite this article: D. Blanke, C. Vial, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Factorization of a class of polynomials over finite fields

We study the factorization of polynomials of the form $F_r(x)=b{x}^{q^r+1}?a{x}^{q^r}+dx?c$ over the finite field ${\mathbb{F}}_q$. We show that these polynomials are closely related to a natural action of the projective linear group $\mathrm{PGL}(2,q)$ on non-linear irreducible polynomials over ${\mathbb{F}}_q$. Namely, irreducible factors of $F_r(x)$ are exactly those polynomials that are invariant under the action of some non-trivial element $[A]\in \mathrm{PGL}(2,q)$. This connection enables us to enumerate irreducibles which are invariant under $[A]$. Since the class of polynomials $F_r(x)$ includes some interesting polynomials like $x^{q^r}?x$ or $x^{q^r+1}?1$, our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over ${\mathbb{F}}_q$. At the same time, we generalize recent results about certain invariant polynomials over the binary field ${\mathbb{F}}_2$.     

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by $X^{(n)}$ the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on $H^1(X^{(n)})$ to define an integrable connection on the trivial vector bundle on $X^{(n)}$ with fiber the universal algebra of the Lie algebra associated with the descending central series of ${\pi }_1$ of $X^{(n)}$. The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.     

Le problème des diviseurs de zéro pour les groupes de Lie nilpotents

Let G be a non-Abelian, connected, nilpotent Lie group. Then there exist $0\ne \alpha \in C_c(G)$ and $0\ne \xi \in L^2(G)$ such that $\alpha 1\xi =0$, contrary to what happens for the group ${\mathbb{R}}^n$. Moreover, the set of zero divisors is a total subset of $L^2(G)$. This result is first proven for the Heisenberg group $H_n$ where it is based on the existence of non-trivial Schwartz functions f satisfying $f1(X_k+\mathrm{i}{Y}_k)=0$ for $1?k?n$. To cite this article: J. Ludwig et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).