The zero-one law for a complete orthonormal system

A complete orthonormal system of functions $\Theta ={\left\{{\theta }_n\right\}}_{n=1}^{\infty }\text{,}{\theta }_n\in L_{[0\text{,}1]}^{\infty }$ is constructed such that ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ converges almost everywhere on $[0\text{,}1]$ if ${\left\{a_n\right\}}_{n=1}^{\infty }\in l^2$ and ${\sum }_{n=1}^{\infty }{a}_n{\theta }_n$ diverges a.e. for any ${\left\{a_n\right\}}_{n=1}^{\infty }?l^2$. We also show that for any complete ONS ${\left\{f_n\right\}}_{n=1}^{\infty }$ of functions defined on $[0\text{,}1]$ there exists a fixed non decreasing subsequence ${\left\{n_k\right\}}_{k=1}^{\infty }$ of natural numbers such that for any $f\in L_{[0\text{,}1]}^0$ and some sequence of coefficients ${\left\{b_n\right\}}_{n=1}^{\infty }$,

$\sum _{n=1}^{n_k}{b}_n{f}_n\to f\text{a.e. when}k\to \infty \text{.}$

To cite this article: K. Kazarian, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Régularité conditionnelle des équations de Navier–Stokes

In this Note, we give sufficient conditions for the regularity of Leray–Hopf weak solutions to the Navier–Stokes equation. We prove that, if one of three conditions (i) $?u/?x_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?2$ and $9/4?r_0?3$, (ii) $\mathrm{?}{u}_3\in L_t^{s_1}{L}_x^{r_1}$ where $2/s_1+3/r_1?11/6$ and $54/23?r_0?18/5$, or (iii) $u_3\in L_t^{s_0}{L}_x^{r_0}$ where $2/s_0+3/r_0?5/8$ and $24/5?r_0?\infty$, is satisfied, then the solution is regular. These conditions improve earlier results on the conditional regularity of the Navier–Stokes equations. To cite this article: I. Kukavica, M. Ziane, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

L'inégalité de Bohr pour les séries de Dirichlet

We extend to the setting of Dirichlet series previous results of Bohr for Taylor series in one variable, themselves generalized by Paulsen, Popescu and Singh or extended to several variables by Aizenberg, Boas and Khavinson. We show in particular that, if $f(s)={\sum }_{n=1}^{\infty }{a}_n{n}^{?s}$, with $6f{6}_{\infty }:={\sup }_{\mathfrak{R}s>0}|f(s)|<\infty$, then ${\sum }_{n=1}^{\infty }|a_n|n^{\mathrm{?2}}?6f{6}_{\infty }$ and even slightly better, and ${\sum }_{n=1}^{\infty }|a_n|n^{?1/2}?C6f{6}_{\infty }$, C being an absolute constant. To cite this article: R. Balasubramanian et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Minima of sequences of Gaussian random variables

For a given sequence of real numbers $a_1,\dots ,a_n$ we denote the k-th smallest one by $k\text{-}{\min }_{1?i?n}{a}_i$. We show that there exist two absolute positive constants c and C such that for every sequence of positive real numbers $x_1,\dots ,x_n$ and every $k?n$ one has

$c\underset{1?j?k}{\max }\frac{k+1?j}{{\sum }_{i=j}^{n}1/x_i}?\mathbb{E}k\text{-}\underset{1?i?n}{\min }|x_i{g}_i|?C\ln (k+1)\underset{1?j?k}{\max }\frac{k+1?j}{{\sum }_{i=j}^{n}1/x_i},$

where $g_i\in N(0,1)$, $i=1,\dots ,n$, are independent Gaussian random variables. Moreover, if $k=1$ then the left hand side estimate does not require independence of the $g_i$s. Similar estimates hold for $\mathbb{E}k\text{-}{\min }_{1?i?n}{|x_i{g}_i|}^p$ as well. To cite this article: Y. Gordon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

A norm inequality for positive block matrices

Any positive matrix $M={(M_{i,j})}_{i,j=1}^m$ with each block $M_{i,j}$ square satisfies the symmetric norm inequality $6M6\le 6{\sum }_{i=1}^m{M}_{i,i}+{\sum }_{i=1}^{m?1}{\omega }_{i}I6$, where ${\omega }_i$ ($i=1,\dots ,m?1$) are quantities involving the width of numerical ranges. This extends the main theorem of Bourin and Mhanna (2017) [4] to a higher number of blocks.     

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 $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

Reconstruction and subgaussian processes

This Note presents a randomized method to approximate any vector v from some set $T\subset {\mathbb{R}}^n$. The data one is given is the set T, and k scalar products ${(〈X_i,v〉)}_{i=1}^k$, where ${(X_i)}_{i=1}^k$ are i.i.d. isotropic subgaussian random vectors in ${\mathbb{R}}^n$, and $k\ll n$. We show that with high probability any $y\in T$ for which ${(〈X_i,y〉)}_{i=1}^k$ is close to the data vector ${(〈X_i,v〉)}_{i=1}^k$ will be a good approximation of v, and that the degree of approximation is determined by a natural geometric parameter associated with the set T. This extends and improves recent results by Candes and Tao. To cite this article: S. Mendelson et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).