Related papers: Nesterenko's criterion when the small linear forms…
In this paper, we present new explicit simultaneous rational approximations converging sub-exponentially to the values of Bell polynomials at the points of the form $(\gamma, 1! (2a+1)\zeta(2), 2!\zeta(3),..., (m-1)!(a+1+(-1)^ma)\zeta(m)),$…
Iterative numerical algorithms are typically equipped with a stopping criterion, where the iteration process is terminated when some error or misfit measure is deemed to be below a given tolerance. This is a useful setting for comparing…
We demonstrate a family of propositional formulas in conjunctive normal form so that a formula of size $N$ requires size $2^{\Omega(\sqrt[7]{N/logN})}$ to refute using the tree-like OBDD refutation system of Atserias, Kolaitis and Vardi…
We show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order $1$),…
We give a general framework for approximations to combinatorial assemblies, especially suitable to the situation where the number $k$ of components is specified, in addition to the overall size $n$. This involves a Poisson process, which,…
We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the…
Very recently, the papers "Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof" by Jang and Ryu, and "The Iterates of Nesterov's Accelerated Algorithm Converge in the Critical Regimes" by Bot, Fadili, and…
The primary goal of this paper is to provide a general multiplicity estimate. Our main theorem allows to reduce a proof of multiplicity lemma to the study of ideals stable under some appropriate transformation of a polynomial ring. In…
We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value…
We develop a generalization of Nesterov's accelerated gradient descent method which is designed to deal with orthogonality constraints. To demonstrate the effectiveness of our method, we perform numerical experiments which demonstrate that…
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal…
Let $\mathcal{M}\subset \mathbb{R}^n$ be a compact and sufficiently smooth manifold of dimension $d$. Suppose $\mathcal{M}$ is nowhere completely flat. Let $N_{\mathcal{M}}(\delta,Q)$ denote the number of rational vectors $\mathbf{a}/q$…
As a starting point of our research, we show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to…
We investigate the Knizhnik-Zamolodchikov linear differential system. The coefficients of this system are rational functions. We prove that the solution of the KZ system is rational when $k$ is equal to two and $n$ is equal to three. While…
We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the…
We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given real…
The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's…
We prove relative asymptotic for the ratio of two sequences of multiple orthogonal polynomials with respect to Nikishin system of measures. The first Nikishin system ${\mathcal{N}}(\sigma_1,...,\sigma_m)$ is such that for each $k$,…
We continue a previous paper to show that Mel'nikov's first order formula for part of the separatrix splitting of a pendulum under fast quasi periodic forcing holds, in special examples, as an asymptotic formula in the forcing rapidity.
In this paper, we establish an improved version of a saddle point theorem ([4]) removing a weak lower semicontinuity assumption at all. We then revisit some of the applications of that theorem in the light of such an improvement. For…