Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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11 Jun 2013
TL;DR: In this paper, asymptotics of Green's kernels in domains with singularly perturbed boundaries and meso-scale approximations of physical fields in non-periodic domains with many inclusions are discussed.
Abstract: Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale approximations for bodies with non-periodic multiple perforations Contains illustrations and numerical examples for a range of physically realisable configurationsThere are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary Examples include perforated domains and bodies with defects of different types The accurate direct numerical treatment of such problems remains a challenge Asymptotic approximations offer an alternative, efficient solution Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems The uniformity of the asymptotic approximations is the principal point of attention We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusionsThe main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions The novel feature of these asymptotic approximations is their uniformity with respect to the independent variablesThis book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations
68 citations
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TL;DR: Positive recurrent potential theory is reviewed, giving special attention to continuous-time Markov chains, and explicit formulas for birth-and-death processes and diffusion processes, and recursive computational procedures for skip-free chains are provided.
Abstract: The simulation run length required to achieve desired statistical precision for a sample mean in a steady-state stochastic simulation experiment is largely determined by the asymptotic variance of the sample mean and, to a lesser extent, by the second-order asymptotics of the variance and the asymptotic bias. The asymptotic variance, the second-order asymptotics of the variance, and the asymptotic bias of the sample mean of a function of an ergodic Markov process can be expressed in terms of solutions of Poisson's equation, as indicated by positive recurrent potential theory. We review this positive recurrent potential theory, giving special attention to continuous-time Markov chains. We provide explicit formulas for birth-and-death processes and diffusion processes, and recursive computational procedures for skip-free chains. These results can be used to help design simulation experiments after approximating the stochastic process of interest by one of the elementary Markov processes considered here.
66 citations
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TL;DR: In this paper, a general family of asymptotic solutions to Einstein's equation are presented, all of which satisfy the peeling theorem but do not satisfy the full peeling property.
Abstract: We present a general family of asymptotic solutions to Einstein's equation which are asymptotically flat but do not satisfy the peeling theorem. Near scri, the Weyl tensor obeys a logarithmic asymptotic flatness condition and has a partial peeling property. The physical significance of this asymptotic behavior arises from a quasi-Newtonian treatment of the radiation from a collapsing dust cloud. Practically all the scri formalism carries over intact to this new version of asymptotic flatness.
66 citations
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TL;DR: In this article, an asymptotic behavior of any possible solution of the second Painleve equation near infinity is described, and connection formulas for the parameters of the asymPTotic description are presented as well.
66 citations
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01 Jan 1985TL;DR: In this paper, a higher-order asymptotic theory of estimation is presented in the framework of the geometry of the model M and the ancillary family A associated with the estimator.
Abstract: A higher-order asymptotic theory of estimation is presented in this Chapter in the framework of the geometry of the model M and the ancillary family A associated with the estimator. Conditions for the consistency and efficiency of an estimator are given in geometrical terms of A. The higher-order terms of the covariance of an efficient estimators are decomposed into the sum of three non-negative geometrical terms. This proves that the bias corrected maximum likelihood estimator is the best estimator from the point of view of the third order asymptotic evaluation. The effect of parametrization is elucidated from the geometrical viewpoint.
64 citations