Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.     

Fast local searches and updates in bounded universes

Given a bounded universe $\{0,1,\dots ,U?1\}$, we show how to perform predecessor searches in $O(\log \log \mathrm{\Delta })$ expected time, where Δ is the difference between the element being searched for and its predecessor in the structure, while supporting updates in $O(\log \log \mathrm{\Delta })$ expected amortized time, as well. This unifies the results of traditional bounded universe structures (which support predecessor searches in $O(\log \log U)$ time) and hashing (which supports membership queries in $O(1)$ time). We also show how these results can be applied to approximate nearest neighbour queries and range searching.     

On the conjugacy relation in ergodic theory

The set of pairs of transformations on the interval $[0,1]$ can be equipped with a standard Borel structure. We prove that the relation of conjugacy is not a Borel subset of this space, in fact it is complete analytic. Moreover, our construction proves that the two sets, $\{T:T\text{is conjugate of}{T}^{\mathrm{?1}}\}$, and $\{T:\text{the centralizer of}T\text{is non-trivial}\}$ are complete analytic sets. To cite this article: M.D. Foreman et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Upper bounds for the sum of Laplacian eigenvalues of graphs

Let G be a graph with n vertices and $e(G)$ edges, and let ${\mu }_1(G)?{\mu }_2(G)???{\mu }_n(G)=0$ be the Laplacian eigenvalues of G. Let $S_k(G)={\sum }_{i=1}^k{\mu }_i(G)$, where $1?k?n$. Brouwer conjectured that $S_k(G)?e(G)+\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ for $1?k?n$. It has been shown in Haemers et al. [7] that the conjecture is true for trees. We give upper bounds for $S_k(G)$, and in particular, we show that the conjecture is true for unicyclic and bicyclic graphs.     

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 counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.     

On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Consider the Hénon equation with the homogeneous Neumann boundary condition

$?\mathrm{\Delta }u+u=|x{|}^{\alpha }{u}^p,u>0\text{in}\mathrm{\Omega },\frac{?u}{?\nu }=0\text{ on }?\mathrm{\Omega },$

where $\mathrm{\Omega }?B(0,1)?{\mathbb{R}}^N,N\ge 2$ and $?\mathrm{\Omega }\cap ?B(0,1)\ne ?$. We are concerned on the asymptotic behavior of ground state solutions as the parameter $\alpha \to \infty$. As $\alpha \to \infty$, the non-autonomous term $|x{|}^{\alpha }$ is getting singular near $|x|=1$. The singular behavior of $|x{|}^{\alpha }$ for large $\alpha >0$ forces the solution to blow up. Depending subtly on the $(N?1)?$dimensional measure $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$ and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and $|?\mathrm{\Omega }\cap ?B(0,1){|}_{N?1}$. In particular, the critical exponent $2_{?}=\frac{2(N?1)}{N?2}$ for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any $p\in (1,2^{?}?1)$ and a smooth domain Ω.