On a Hölder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on ${?}^p(\Gamma )$. We introduce a modification of mean dimension which is an obstruction for ${?}^p(\Gamma )$ and ${?}^q(\Gamma )$ to be Hölder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A Gauss sum estimate in arbitrary finite fields

We establish bounds on exponential sums ${\sum }_{x\in {\mathbb{F}}_q}\psi (x^n)$ where $q=p^m$, p prime, and ψ an additive character on ${\mathbb{F}}_q$. They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order $(m?2)$. More precisely, a non-trivial estimate is obtained provided n satisfies $\gcd (n,\frac{q?1}{p^{\nu }?1})<p^{?\nu }{q}^{1?\epsilon }$ for all $1?\nu <m$, $\nu |m$, where $\epsilon >0$ is arbitrary. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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.     

Sur les nombres de Fibonacci de la forme qkyp

We study the equation $F_n=q^k{y}^p$ where q is a prime number and k is a positive integer. We solve it for all $q?1(\mathrm{mod}4)$ and get partial results when $q\equiv 1(\mathrm{mod}4)$. In particular, we answer Ribenboim's question about $F_n=2^k{y}^p$. To cite this article: Y. Bugeaud et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Piatetski-Shapiro phenomenon in the uniqueness problem

We extend the phenomenon discovered by Piatetski-Shapiro (1954) to $l^q$ spaces. To be precise, for any $q>2$ we construct a compact K on the circle, which supports a distribution S with Fourier transform $\stackrel{?}{S}\in l^q$, but does not support such a measure. To cite this article: N. Lev, A. Olevskii, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Generalized differentiation and bi-Lipschitz nonembedding in L1

We consider Lipschitz mappings, $f:X\to V$, where X is a doubling metric measure space which satisfies a Poincaré inequality, and V is a Banach space. We show that earlier differentiability and bi-Lipschitz nonembedding results for maps, $f:X\to {\mathbb{R}}^N$, remain valid when ${\mathbb{R}}^N$ is replaced by any separable dual space. We exhibit spaces which bi-Lipschitz embed in $L^1$, but not in any separable dual V. For certain domains, including the Heisenberg group with its Carnot–Caratheodory metric, we establish a new notion of differentiability for maps into $L^1$. This implies that the Heisenberg group does not bi-Lipschitz embed in $L^1$, thereby proving a conjecture of J. Lee and A. Naor. When combined with their work, this has implications for theoretical computer science. To cite this article: J. Cheeger, B. Kleiner, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Capacitary representation of positive solutions of semilinear parabolic equations

We give a global bilateral estimate on the maximal solution ${\overline{u}}_F$ of ${?}_{t}u?\mathrm{\Delta }u+u^q=0$ in ${\mathbb{R}}^N\times (0,\infty )$, $q>1$, $N?1$, which vanishes at $t=0$ on the complement of a closed subset $F?{\mathbb{R}}^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q^{\prime }}$. We deduce from this estimate that ${\overline{u}}_F$ is σ-moderate in Dynkin's sense. To cite this article: M. Marcus, L. Véron, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Une propriété de composition dans l'espace Hs (II)

We prove an almost sharp Superposition Theorem for a wide class of Besov spaces. Indeed any function which belongs locally to $B_p^{s^{\prime },\infty }(\mathbb{R})$, and vanishes at 0, acts by left composition in $B_p^{s,q}({\mathbb{R}}^n)\cap L^{\infty }({\mathbb{R}}^n)$. The conditions on the parameters are the following: $s^{\prime }>s>1$, $s?[s]>1/p$ and $q\in [1,+\infty ]$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Powerful numbers in (1ℓ + qℓ)(2ℓ + qℓ)⋯(nℓ + qℓ)

Let q be a positive integer. Recently, Niu and Liu proved that, if $n\ge \max ?\{q,1198?q\}$, then the product $(1^3+q^3)(2^3+q^3)?(n^3+q^3)$ is not a powerful number. In this note, we prove (1) that, for any odd prime power ? and $n\ge \max ?\{q,11?q\}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number, and (2) that, for any positive odd integer ?, there exists an integer $N_{q,?}$ such that, for any positive integer $n\ge N_{q,?}$, the product $(1^{?}+q^{?})(2^{?}+q^{?})?(n^{?}+q^{?})$ is not a powerful number.     

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).     

Linear systems subject to input saturation and time delay: Global asymptotic and Lp-stabilization

This Note deals with two problems on stabilization of linear systems by static feedbacks which are bounded and time-delayed, namely global asymptotic stabilization and finite gain $L^p$-stabilization, $p\in [1,\infty ]$. Regarding the first issue, we provide, under standard necessary conditions, two types of solutions for arbitrary small bound on the control and large (constant) delay. The first solution is based on the knowledge of a static stabilizing feedback in the zero-delay case and the second solution is of nested saturation type, which extends results of Mazenc et al. [IEEE Trans. Automat. Contr. 48 (1) (2003) 57–63]. For the finite-gain $L^p$-stabilization issue, we assume that the system is neutrally stable. We show the existence of a linear feedback such that, for arbitrary small bound on the control and large (constant) delay, finite gain $L^p$-stability holds with respect to every $L^p$-norm, $p\in [1,\infty ]$. Moreover, the corresponding $L^p$-gain is delay-independent. To cite this article: K. Yakoubi, Y. Chitour, C. R. Acad. Sci. Paris, Ser. I 340 2005).