Related papers: Asymptotic Large Sieve
The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees $n$ are needed, the use of recursion to compute the polynomials is not a good…
By application of the theory for second-order linear differential equations with two turning points developed in \cite{Olver1975}, uniform asymptotic approximations are obtained for the Lam\'{e} and Mathieu functions with a large real…
We consider weak solutions of the adjoint equation for an elliptic operator in nondivergent form, and their asymptotic properties at an interior point. We assume that the coefficients a_{ij} are bounded, measurable, complex-valued functions…
In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of…
This article is focused on the asymptotic expansions, as time tends to infinity, of solutions of a system of ordinary differential equations with non-smooth nonlinear terms. The forcing function decays to zero in a very complicated but…
We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…
We develop a discrete spectral framework for Dirichlet $L$-functions that reveals a combinatorial structure underlying their special values and connects this to their zeros. Our approach approximates the classical Dirichlet series by finite…
We prove an essentially optimal large sieve inequality for self-dual Eisenstein series of varying levels. This bound can alternatively be interpreted as a large sieve inequality for rationals ordered by height. The method of proof is…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of $L$-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of…
Improvements of the Large Sieve for Special Sequences
This paper develops asymptotic theory of integrals of empirical quantile functions with respect to random weight functions, which is an extension of classical $L$-statistics. They appear when sample trimming or Winsorization is applied to…
The paper concerns asymptotic studies for the sixth Painlev\'e transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics…
In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The…
We generalise Gelfand-Graev characters to $\mathbb R/\mathbb Z$-graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to…
We consider asymptotics of power series coefficients of rational functions of the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a number of such cases from the literature, chiefly concerned either with positivity of…
We give an overview of basic methods that can be used for obtaining asymptotic expansions of integrals: Watson's lemma, Laplace's method, the saddle point method, and the method of stationary phase. Certain developments in the field of…
The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic…
The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As…
Normal forms for wide classes of closed IL formulas were given in [4]. Here we quantify asymptotically, in exact numbers, how wide those classes are. As a consequence, we show that the "majority" of closed IL formulas have GL-equivalents,…