Related papers: On optimal truncation of divergent series solution…
This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the…
This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of…
In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid…
Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this…
For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the…
We obtain Berry-Esseen-type bounds for the sum of random variables with a dependency graph and uniformly bounded moments of order $\delta \in (2,\infty]$ using a Fourier transform approach. Our bounds improve the state-of-the-art in the…
This paper studies the asymptotic behavior of the constant step Stochastic Gradient Descent for the minimization of an unknown function F , defined as the expectation of a non convex, non smooth, locally Lipschitz random function. As the…
Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.
Given a graph where vertices are partitioned into $k$ terminals and non-terminals, the goal is to compress the graph (i.e., reduce the number of non-terminals) using minor operations while preserving terminal distances approximately.The…
The notions of bounded expansion and nowhere denseness have been applied very successfully in algorithmic graph theory. We study the corresponding notions of directed bounded expansion and nowhere crownfulness on directed graphs. We show…
In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the…
We consider nonparametric inference of finite dimensional, potentially non-pathwise differentiable target parameters. In a nonparametric model, some examples of such parameters that are always non pathwise differentiable target parameters…
Stokes phenomenon refers to the fact that the asymptotic expansion of complex functions can differ in different regions of the complex plane, and that beyond the so-called Stokes lines has an unphysical divergence. An important special case…
Bregman proximal point algorithm (BPPA) has witnessed emerging machine learning applications, yet its theoretical understanding has been largely unexplored. We study the computational properties of BPPA through learning linear classifiers…
The Landau-Lifshitz equation is the first in an infinite series of approximations to the Lorentz-Abraham-Dirac equation obtained from `reduction of order'. We show that this series is divergent, predicting wildly different dynamics at…
High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it…
This paper deals with non-Lipschitz nonlinear systems. Such systems can be approximated by a linear map of so-called signatures, which play a crucial role in the theory of rough paths and can be interpreted as collections of iterated…
Recently, Weniger (delta sequence) method has been proposed by the authors of Ref. [1] (Jentschura et al.) for resummation of truncated perturbation series in quantum field theories. Those authors presented numerical evidence suggesting…
A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the…
On a smooth variety, Serre's intersection formula computes intersection multiplicities via an alternating sum of the lengths of Tor groups. When the variety is singular, the corresponding sum can be a divergent series. But there are…