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We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on…
We present general prescriptions for the asymptotic expansion of massive multi-loop Feynman integrals near threshold. As in the case of previously known prescriptions for various limits of momenta and masses, the terms of the threshold…
In this paper we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. Applications to classical and free probability theory are discussed.
A semilinear singularly perturbed reaction-diffusion equation with Dirichlet boundary conditions is considered in a convex unbounded sector. The singular perturbation parameter is arbitrarily small, and the "reduced equation" may have…
We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the…
Point processes are finding growing applications in numerous fields, such as neuroscience, high frequency finance and social media. So classic problems of classification and clustering are of increasing interest. However, analytic study of…
We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the…
Given a Dirichlet series $L(s) = \sum a_n n^{-s}$, the asymptotic growth rate of $\sum_{n\le X} a_n$ can be determined by a Tauberian theorem. Bounds on the error term are typically controlled by the size of $|L(\sigma+it)|$ for fixed real…
Many optimization algorithms$\unicode{x2013}$including gradient descent, proximal methods, and operator splitting techniques$\unicode{x2013}$can be formulated as fixed-point iterations (FPI) of continuous operators. When these operators are…
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
We consider the rigorous derivation of asymptotic formulas for initial-boundary value problems using the nonlinear steepest descent method. We give detailed derivations of the asymptotics in the similarity and self-similar sectors for the…
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function…
We study the asymptotics of the moments of arithmetic functions that have a limit distribution, not necessarily normal, defined on a subset of the natural series that satisfies certain requirements. Several assertions are proved on…
We present a practical framework to prove, in a simple way, two-terms asymptotic expansions for Fourier integrals $$ {\mathcal I}(t) = \int_{\mathbb R}({\rm e}^{it\phi(x)}-1) {\rm d} \mu(x) $$ where $\mu$ is a probability measure on…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…
In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified…
We give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at…
Asymptotic expansion of a variation with anticipative weights is derived by the theory of asymptotic expansion for Skorohod integrals having a mixed normal limit. The expansion formula is expressed with the quasi-torsion, quasi-tangent and…
We obtain the asymptotic expansion of the Voigt functions $K(x,y)$ and $L(x,y)$ for large (real) values of the variables $x$ and $y$, paying particular attention to the exponentially small contributions. A Stokes phenomenon is encountered…