Paradoxe de Klein pour l'équation de Klein–Gordon chargée : superradiance et opérateur de diffusion

We develop the scattering theory for the charged Klein–Gordon equation on ${\mathbb{R}}_t\times {\mathbb{R}}_x$, when the electrostatic potential $A(x)$ has different asymptotics $a^{\pm }$ as $x\to \pm \infty$. In this case, the conserved energy is not positive definite (Klein Paradox). We construct the spectral representation for the harmonic equation, and we establish the existence of a Scattering Operator the symbol of which has a norm strictly larger than 1, for the frequencies in $(a^{?}\text{,}{a}^{+})$. These results can be applied to the DeSitter–Reissner–Nordstrøm metric, to justify the notion of superradiance of the charged black-holes. To cite this article: A. Bachelot, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Low minor faces in 3-polytopes

It is trivial that every 3-polytope has a face of degree at most 5, called minor. The height $h\left(f\right)$ of a face $f$ is the maximum degree of the vertices incident with $f$. It follows from the partial double $n$-pyramids that $h\left(f\right)$ can be arbitrarily large for each $f$ if a 3-polytope is allowed to have faces of types $(4,4,\infty )$ or $(3,3,3,\infty )$.In 1996, M. Horňák and S. Jendrol’ proved that every 3-polytope without faces of types $(4,4,\infty )$ and$(3,3,3,\infty )$ has a minor face of height at most 39 and constructed such a 3-polytope satisfying $h\left(f\right)\ge 30$ for all minor faces $f$.The purpose of this paper is to prove that every 3-polytope without faces of types $(4,4,\infty )$ and$(3,3,3,\infty )$ has a minor face of height at most 30, which bound is tight due to the Horňák–Jendrol’ construction.     

On the Frobenius complexity of determinantal rings

We compute the Frobenius complexity for the determinantal ring of prime characteristic p obtained by modding out the $2\times 2$ minors of an $m\times n$ matrix of indeterminates, where $m>n?2$. We also show that, as $p\to \infty$, the Frobenius complexity approaches $m?1$.     

On global attraction to solitary waves. Klein–Gordon equation with mean field interaction at several points

We consider the $\mathbf{U}(1)$-invariant Klein–Gordon equation in dimension $n?3$, self-interacting via the mean field mechanism in finitely many regions. We prove that, under certain generic assumptions, each solution converges as $t\to \pm \infty$ to the two-dimensional set of all “nonlinear eigenfunctions” of the form $?(x)e^{?i\omega t}$. The proof is based on the analysis of omega-limit trajectories. The Titchmarsh Convolution Theorem allows us to prove that the time spectrum of any omega-limit trajectory of each finite energy solution consists of a single point. This proves the convergence to the attractor in local sub-energy norms.     

Random version of Dvoretzky’s theorem in ℓpn

We study the dependence on $\epsilon$ in the critical dimension $k(n,p,\epsilon )$ for which one can find random sections of the ${?}_p^n$-ball which are $(1+\epsilon )$-spherical. We give lower (and upper) estimates for $k(n,p,\epsilon )$ for all eligible values $p$ and $\epsilon$ as $n\to \infty$, which agree with the sharp estimates for the extreme values $p=1$ and $p=\infty$. Toward this end, we provide tight bounds for the Gaussian concentration of the ${?}_p$-norm.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.     

Existence of positive periodic solutions to nonlinear second order differential equations

In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation $u^{″}\left(t\right)+a\left(t\right)u\left(t\right)=f(t,u\left(t\right))\text{,}t\in R\text{,}$ where $a:R\to [0,+\infty )$ is an $\omega$-periodic continuous function with $a\left(t\right)⁄\equiv 0$, $f:R\times [0,+\infty )\to [0,+\infty )$ is continuous and $f(\cdot ,u):R\to [0,+\infty )$ is also an $\omega$-periodic function for each $u\in [0,+\infty )$. Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation.     

Extensions of the disc algebra and of Mergelyanʼs theorem

We investigate the uniform limits of the set of polynomials on the closed unit disc $\overline{D}$ with respect to the chordal metric χ. More generally, we examine analogous questions replacing $\mathbb{C}\cup \{\infty \}$ by other metrizable compactifications of $\mathbb{C}$.     

Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

Erd?s and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter $\kappa \in (1,\infty ]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $\kappa \in (2,\infty ]$, tight in the case $\kappa =2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $\kappa \in (1,2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $\kappa \in (2,\infty ]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.