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).     

A nonlinear Korn inequality on a surface

Let ω be a domain in ${\mathbb{R}}^2$ and let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion. The main purpose of this paper is to establish a “nonlinear Korn inequality on the surface $\mathit{\theta }(\overline{\omega })$”, asserting that, under ad hoc assumptions, the ${\mathit{H}}^1(\omega )$-distance between the surface $\mathit{\theta }(\overline{\omega })$ and a deformed surface is “controlled” by the ${\mathit{L}}^1(\omega )$-distance between their fundamental forms. Naturally, the ${\mathit{H}}^1(\omega )$-distance between the two surfaces is only measured up to proper isometries of ${\mathbb{R}}^3$.This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ${\mathit{\theta }}^k:\omega \to {\mathbb{R}}^3$, $k⩾1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$, $k⩾1$, stay uniformly away from zero; and finally, the fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the fundamental forms of the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\overline{\omega })$ as $k\to \infty$.Such results have potential applications to nonlinear shell theory, the surface $\mathit{\theta }(\overline{\omega })$ being then the middle surface of the reference configuration of a nonlinearly elastic shell.     

On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

Continuity in H1-norms of surfaces in terms of the L1-norms of their fundamental forms

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in ${\mathbb{R}}^2$, let $\mathit{\theta }:\overline{\omega }\to {\mathbb{R}}^3$ be a smooth immersion, and let ${\mathit{\theta }}^k:\overline{\omega }\to {\mathbb{R}}^3$, $k?1$, be mappings with the following properties: They belong to the space ${\mathit{H}}^1(\omega )$; the vector fields normal to the surfaces ${\mathit{\theta }}^k(\omega )$, $k?1$, are well defined a.e. in ω and they also belong to the space ${\mathit{H}}^1(\omega )$; the principal radii of curvature of the surfaces ${\mathit{\theta }}^k(\omega )$ stay uniformly away from zero; and finally, the three fundamental forms of the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{L}}^1(\omega )$ toward the three fundamental forms of the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. Then, up to proper isometries of ${\mathbb{R}}^3$, the surfaces ${\mathit{\theta }}^k(\omega )$ converge in ${\mathit{H}}^1(\omega )$ toward the surface $\mathit{\theta }(\omega )$ as $k\to \infty$. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

New constructions of SSPDs and their applications

We present a new optimal construction of a semi-separated pair decomposition (i.e., SSPD) for a set of n points in ${\mathbb{R}}^d$. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties.As an application of the new construction, for a fixed $t>1$, we present a new construction of a t-spanner with $O(n)$ edges and maximum degree $O({\log }^{2}n)$ that has a separator of size $O(n^{1?1/d})$.     

A note on plane pointless curves

Let $d(q)$ denote the minimal degree of a smooth projective plane curve that is defined over the finite field ${\mathbb{F}}_q$ and does not contain ${\mathbb{F}}_q$ rational points. We are interested in the asymptotic behavior of $d(q)$ for $q\to \infty$. To the best of the author's knowledge the problem of estimating the asymptotic behavior of $d(q)$ was not considered previously. In this note we establish the following bounds:(1)$\frac{1}{4}⩽\underset{q\to \infty }{\underline{\lim }}{\log }_{q}d(q)⩽\frac{1}{3}.$ More specifically, for every characteristic $p>3$ we construct a sequence of pointless Fermat curves$x^{d_k}+y^{d_k}+z^{d_k}=0,\text{over }{\mathbb{F}}_{p^{m_k}},$ such that ${\lim }_{k\to \infty }{\log }_{p^{m_k}}{d}_k=1/3$.