Uniform asymptotic methods for integrals

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 asymptotic analysis will be compared with De Bruijn’s book Asymptotic Methods in Analysis. The classical methods can be modified for obtaining expansions that hold uniformly with respect to additional parameters. We give an overview of examples in which special functions, such as the complementary error function, Airy functions, and Bessel functions, are used as approximations in uniform asymptotic expansions.     

Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients

The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the asymptotic characteristic equation and the spectral properties are further obtained. We show that the spectrum-determined-growth (SDG) condition holds. Thus the stability analysis of the closed-loop system can be completely converted to the infimum estimation of the asymptotic characteristic equation. Especially, we further show that the infimum is positive so as to obtain the exponential stability by estimating the recursive expression in from-leaf-to-root order. Some numerical simulations are presented to illustrate and support the theoretical results.

     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

Concerning a class of similarity tests

A two-parameter generalization of Jaccard's index of similarity is proposed as a class of measures for testing the homogeneity of two independent multinomial samples. The power approach used in modern asymptotic theory of decomposable statistics is applied to the asymptotic analysis of these measures. The asymptotic analysis is amplified by numerical tabulation yielding an asymptotically optimal similarity test in this class of measures.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 536–546, October, 1995.     

Asymptotic analysis for linear difference equations

We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

     

Stochastic Systems with Averaging in the Scheme of Diffusion Approximation

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1235–1252, September, 2005.     

Landscape of wave focusing and localization at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.     

Asymptotic behaviour of a family of gradient algorithms in ℝ d and Hilbert spaces

The asymptotic behaviour of a family of gradient algorithms (including the methods of steepest descent and minimum residues) for the optimisation of bounded quadratic operators in ℝ^d and Hilbert spaces is analyzed. The results obtained generalize those of Akaike (1959) in several directions. First, all algorithms in the family are shown to have the same asymptotic behaviour (convergence to a two-point attractor), which implies in particular that they have similar asymptotic convergence rates. Second, the analysis also covers the Hilbert space case. A detailed analysis of the stability property of the attractor is provided.     

Investigation of micropolar fluid flow in a helical pipe via asymptotic analysis

In this paper we study the flow of incompressible micropolar fluid through a pipe with helical shape. Pipe’s thickness and the helix step are considered as the small parameter ε. Using asymptotic analysis with respect to ε, the asymptotic approximation is built showing explicitly the effects of fluid microstructure and pipe’s distortion on the velocity distribution. The error estimate for the approximation is proved rigorously justifying the obtained model.     

Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section

In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and electric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain “homogenized” telegraphist's model. This model, which includes a nonlocal in time operator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross section is not homogeneous.     

Dynamic singular perturbation problems for multi‐structures

This paper contains an overview of recent development in asymptotic analysis of fields in multi‐structures. We begin with simple examples of scalar dynamic problems in two dimensions, and then present analysis of time‐dependent fields in 1D–3D multi‐structures. The asymptotic results, presented here, are based on the method of compound asymptotic expansions. Copyright © 2000 John Wiley & Sons, Ltd.     

Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains

Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L₂ containing degrees of distance r=|x| as weight multipliers. For the Neumann conditions (stresses are prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the “elastic” Neumann problem are presented. These results are based on the weighted Korn inequality. Bibliography: 29 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 162–200     

Asymptotic Expansions for Eigenvalues of the Lamé System in the Presence of Small Inclusions

We derive a complete asymptotic expansion for eigenvalues of the Lamé system of the linear elasticity in domains with small inclusions in three dimensions. By an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation, we reduce this problem to the study of the characteristic values of integral operators in the complex planes. Generalized Rouché's theorem and other techniques from the theory of meromorphic operator-valued functions are combined with asymptotic analysis of integral kernels to obtain full asymptotic expansions for eigenvalues.     

Asymptotic mean-square stability of linear multi-step methods for SODEs

In this article we present results of a linear stability analysis of stochastic linear multi-step methods for stochastic ordinary differential equations. As in deterministic numerical analysis we use a linear time-invariant test equation and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution of that test equation. Sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods are obtained with the aide of Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams-Bashforth- and Adams-Moulton-methods and the BDF method. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary conditions and boundary layers for a multi-dimensional relaxation model

In this paper we study the linearized relaxation model of Katsoulakis and Tzavaras in a half-space with arbitrary space dimension n?1. Our main interest is to establish the asymptotic equivalence of the relaxation system and its corresponding multi-dimensional equilibrium conservation law. We identify and rigorously justify a necessary and sufficient condition (which we refer to as stiff Kreiss condition, or SKC in short) on the boundary condition to guarantee the uniform stability of the initial-boundary value problem of the relaxation system independent of the relaxation rate. The asymptotic convergence and the corresponding boundary layer behavior are studied by Fourier-Laplace transform and a detailed asymptotic analysis. The SKC is shown to be more restrictive than the classical uniform Kreiss condition for all n?1.     

Asymptotic second-order consistency for two-stage estimation methodologies and its applications

We consider fixed-size estimation for a linear function of means from independent and normally distributed populations having unknown and respective variances. We construct a fixed-width confidence interval with required accuracy about the magnitude of the length and the confidence coefficient. We propose a two-stage estimation methodology having the asymptotic second-order consistency with the required accuracy. The key is the asymptotic second-order analysis about the risk function. We give a variety of asymptotic characteristics about the estimation methodology, such as asymptotic sample size and asymptotic Fisher-information. With the help of the asymptotic second-order analysis, we also explore a number of generalizations and extensions of the two-stage methodology to such as bounded risk point estimation, multiple comparisons among components between the populations, and power analysis in equivalence tests to plan the appropriate sample size for a study.     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

Analyzing retrial queues by censoring

In this paper we analyze the M/M/c retrial queue using the censoring technique. This technique allows us to carry out an asymptotic analysis, which leads to interesting and useful asymptotic results. Based on the asymptotic analysis, we develop two methods for obtaining approximations to the stationary probabilities, from which other performance metrics can be obtained. We demonstrate that the two proposed approximations are good alternatives to existing approximation methods. We expect that the technique used here can be applied to other retrial queueing models.