Related papers: Nesterenko's criterion when the small linear forms…

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are $n$ elements and $m$ sets and each element lies in $t$ randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \log t)^{1/2})$…

The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous…

We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing…

For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear…

We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd…

The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context…

In this paper we deal with the problem of finding the smallest and the largest elements of a totally ordered set of size $n$ using pairwise comparisons if $k$ of the comparisons might be erroneous where $k$ is a fixed constant. We prove…

New type III and type N approximate solutions which are regular in the linear approximation are shown to exist. For that, we use complex transformations on self-dual Robinson-Trautman metrics rather then the classical approach. The…

Building upon ideas of the second and third authors, we prove that at least $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ values of the Riemann zeta function at odd integers between 3 and $s$ are irrational, where $\varepsilon$ is any…

This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strictly convex quadratic functions assuming that exact gradients are used and…

We use bounds on bilinear forms with Kloosterman fractions and improve the error term in the asymptotic formula of Balazard and Martin (2023) on the average value of the smallest denominators of rational numbers in short intervals.

Given an ODE and its perturbation, the Alekseev formula expresses the solutions of the latter in terms related to the former. By exploiting this formula and a new concentration inequality for martingale-differences, we develop a novel…

We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}=…

In this paper we investigate the metrical theory of Diophantine approximation associated with linear forms that are simultaneously small for infinitely many integer vectors; i.e. forms which are close to the origin. A complete…

In this paper, we present an inexact Noda iteration with inner-outer iterations for finding the smallest eigenvalue and the associated eigenvector of an irreducible monotone matrix. The proposed inexact Noda iteration contains two main…

When are asymptotic approximations using the delta-method uniformly valid? We provide sufficient conditions as well as closely related necessary conditions for uniform negligibility of the remainder of such approximations. These conditions…

Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<\gamma_2<\gamma_1<1$ are two fixed constants. In this paper, it is proved that, whenever $\alpha$…

In 2005, Li, Tao, Thiele and the author raised a general question concerning upper bounds for a class of multilinear oscillatory integral operators, and established such bounds in a few cases. Most cases remain open. The present paper is…

Given a set of $n$ points in $R^2$, the Szemer\'edi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the…

We derive bounds on the extremal singular values and the condition number of NxK, with N>=K, Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link---first…