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).     

Un théorème de Liouville pour l'opérateur de Schrödinger avec dérive

Let $(M,g)$ be a complete Riemannian manifold without boundary of dimension n and V be a $C^2$ vector field on M such that $r(x)|V(x)|$ is bounded. Suppose that ${\mathrm{Ric}}_g(x)??\min \{\lambda (r(x))?{\mu }_{\mathrm{?}V}(x),\beta (r(x))\}$ outside a compact set of M, where ${\mu }_{\mathrm{?}V}$ denotes the upper eigenvalue of ?V and $\lambda ,\beta$ are non-negative decreasing functions such that ${\lim }_{t\to +\infty }{t}^2\lambda (t)=0$. There exists positive numbers $b_n$ and $c_n$ which depend only on n and $6V{6}_{\infty }$ such that if h is a $C^2$ function defined on M with $\mathrm{\Delta }h??c_n{a}^2$ and ${\mathrm{lim?sup}}_{R\to \infty }{R}^{\mathrm{?2}}{\min }_{x\in B_p(3R)?B_p(R)}h(x)??b_n{a}^2$, where $0?a<{\mathrm{lim?inf}}_{j\to \infty }h(z_j)$, where $(z_j)$ is a sequence of M such that $r(z_j)\to \infty$, then the equation $\mathrm{\Delta }u(x)+V(x)?\mathrm{?}u(x)+h(x)u(x)=0$ has no positive $C^2$ solution on M. To cite this article: S. Asserda, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n?k?1)\text{st}$ homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions $[B_1,\dots ,B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete k-uniform hypergraph H and where $1\in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

The explicit characterization of coefficients of a.e. convergent orthogonal series

We characterize sequences of numbers $(a_n)$ such that ${\sum }_{n?1}{a}_n{\Phi }_n$ converges a.e. for any orthonormal system $({\Phi }_n)$ in any $L_2$-space. To cite this article: A. Paszkiewicz, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

On the Hochschild homology of quantum SL(N)

We show that the quantized coordinate ring $A:=k_q[\mathit{SL}(N)]$ satisfies van den Bergh's analogue of Poincaré duality for Hochschild (co)homology with dualizing bimodule being $A_{\sigma }$, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism σ of the Haar functional. This implies that ${\mathrm{H}}_{N^2?1}(A,A_{\sigma })?k$, generalizing our previous result for $k_q[\mathit{SL}(2)]$. To cite this article: T. Hadfield, U. Krähmer, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On the proportion of rank 0 twists of elliptic curves

Let E be an elliptic curve defined over Q, let $E_d$ denote its dth quadratic twist, and $r_{E_d}:=\mathrm{rank}{E}_d(\mathbf{Q})$. We prove, that, for any positive integer k there are pairwise non-isogenous elliptic curves $E^1,\dots ,E^k$ such that $r_{{E^1}_p}=\cdots =r_{{E^k}_p}=0$ for a positive proportion of primes p. To cite this article: A. Dąbrowski, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On sub-harmonic bifurcations

Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family $h_u$ of transformations near $(u_0,x_0)$ when $h_{u_0}(x_0)=x_0$ and $\mathrm{d}{h}_{u_0}(x_0)$ has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Matched pairs approach to set theoretic solutions of the Yang–Baxter equation

We study set-theoretic solutions $(X,r)$ of the Yang–Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterisation of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that r extends as a solution $r_S$ on $S(X,r)$ as a set. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products $S?S?S$ equivalent to $r_S$ a solution, and extensions $(S?T,r_{S?T})$. Examples include a general ‘double’ construction $(S?S,r_{S?S})$ and some concrete extensions, their actions and graphs based on small sets.     

Modularity of hypertetrahedral representations

Let F be a number field, $G_F$ its absolute Galois group, and $\rho :G_F\to {\mathrm{GL}}_4(\mathbb{C})$ an irreducible continuous Galois representation. Let $\overline{G}$ denote the projective image of ρ in ${\mathrm{PGL}}_4(\mathbb{C})$. We say that ρ is hypertetrahedral if $\overline{G}$ is an extension of $A_4$ by the Klein group $V_4$. In this case, we show that ρ is modular, i.e., ρ corresponds to an automorphic representation π of ${\mathrm{GL}}_4({\mathbb{A}}_F)$ such that their L-functions are equal. This gives new examples of irreducible 4-dimensional monomial representations which are modular, but are not induced from normal extensions and are not essentially self-dual. To cite this article: K. Martin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).