Spectral properties of a Sturm-Liouville type differential operator with a retarding argument

Spectral properties of a differential operator of Sturm-Liouville type are studied in the case of retarding argument with different boundary conditions. The asymptotics of solutions to the corresponding differential equation is studied in the case of a summable potential. An asymptotics of eigenvalues and an asymptotics of eigenfunctions of the differential operator are calculated for each considered case.     

Asymptotic normality of test statistics under alternative hypotheses

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.     

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L² convergence of Szegő asymptotics on the spectrum.     

Mixed Type Multiple Orthogonal Polynomials for Two Nikishin Systems

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.     

相依条件下滑动平均过程精确渐近性的一般规律

本文研究了相依条件下滑动平均过程完全收敛的精确渐近性问题. 利用正态分布逼近的方法及相关不等式, 获得了精确渐近性的一般规律, 推广了对数率和重对数率精确渐近性的已有结果.     

Construction of Asymptotics of Solutions of Differential Equations with Cusp-Type Degeneration in the Coefficients in the Case of Multiple Roots of the Highest-Order Symbol

The asymptotics of linear differential equations with cusp-type degeneration are studied. The problem of constructing asymptotics at infinity for equations with holomorphic coefficients can be reduced to that problem. The main result is the construction of asymptotics of solutions of such equations in the case of multiple roots of the highest-order symbol under certain additional conditions on the lower-order symbol of the differential operator.     

Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.     

Problem of focusing and the asymptotics of the spectral function of the Laplace-Beltrami operator. I

The connection between the asymptotics of the spectral function and the formal shortwave expansion of the solution of the problem of the asymptotics of the Green function near a geodesically concave boundary of a two-dimensional surface is considered in the paper.     

M/T-SPH/1排队平稳指标分布的尾部衰减特征

讨论M/T-SPH/1排队平稳队长分布和平稳逗留时间分布的尾部衰减特征,其中T-SPH表示可数状态吸收生灭过程吸收时间的分布。在分布PGF和LST的基础上,给出了两个平稳分布衰减规律的完整分析.结果表明,当参数取不同值时,平稳队长与平稳逗留时间的尾部具有三种不同类型的衰减特征.     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

Travelling wavefronts of reaction-diffusion equations in cylindrical domains

Abstract

Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This article is aimed at describing refined asymtotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.     

Spectral asymptotics for problems with integral constraints

The eigenvalue problem for differential operators of arbitrary order with integral constraints is considered. The asymptotics of the eigenvalues is obtained. The results are applied to finding the asymptotics of the probability of small deviations for some detrended processes of nth order.     

Self-adjoint Extensions for the Neumann Laplacian and Applications

A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so-called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self-adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains, The error estimates for proposed approximations of shape functionals are provided.     

The second Painlevé equation in a problem concerning nonlinear effects near caustics

In a critical case we study the asymptotics (at large time) of a solution of the nonlinear Schrödinger equation. This solution arises in a series of problems when accounting for the nonlinear effects near caustics. The asymptotics is described in terms of the second Painlevé transcendent. Bibliography: 34 titles.     

A shortwave source near a smooth,convex hypersurface and the spectral function of the Laplace operator on a Riemannian manifold

The Tauberian theorem of B. M. Levitan reduces the question of the asymptotics of the spectral function of the Laplace operator on a smooth Riemannian manifold with boundary to the problem of constructing the asymptotics of a Green function possessing certain additional properties. The paper is devoted to the construction of the appropriate Green function for the case of a geodesically concave boundary.     

Asymptotics of solutions to the time-dependent Schrödinger equation with a small Planck constant

A regularized asymptotics of the solution to the time-dependent Schrödinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function.     

Asymptotics of Eigenvalues of Differential Operator With Alternating Weight Function

We study a differential operator of the sixth order with an alternating weight function. The potential of the operator has a first-order discontinuity at some point of the segment, where the operator is being considered. The boundary conditions are separated. We study the asymptotics of solutions to the corresponding differential equations and the asymptotics of eigenvalues of the considered differential operator.     

A Periodic Problem for the Landau--Ginzburg Equation

In this paper, we consider a periodic problem for the n-dimensional complex Landau--Ginzburg equation. It is shown that in the case of small initial data there exists a unique classical solution of this problem, and an asymptotics of this solution uniform in the space variable is given. The leading term of the asymptotics is exponentially decreasing in time.     

Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

The paper studies the asymptotics of the Brownian integrals with paths restricted to a bounded domain of ? ^v , when the domain is dilated to infinity. The framework is that of the Bose-Einstein statistics with paths observed within random time intervals which are integer multiplies of some fixed β > 0. The three first terms of the asymptotics are found explicitly via the functional integrals. In the case of a gas of interacting Brownian loops an expression for the volume term of the asymptotics of the log-partition function is found and the correction term is proved to by order be the boundary area of the domain.     

On the spectrum of a vector Schrödinger operator

We consider the asymptotics of the spectrum of a Sturm-Liouville operator acting on a space of vector functions and show that this asymptotics is affected by “rotation” of eigenvectors of the potential. A similar result is obtained for a vector Schrödinger operator.