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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TL;DR: In this paper, the first three coefficients of the asymptotic expansion of Zelditch were computed and it was shown that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight.
Abstract: In this paper, we computed the first three coefficients of the asymptotic expansion of Zelditch We also proved that in general, the $k$-th coefficient is a polynomial of the curvature and its derivative of weight $k$
14 citations
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TL;DR: In this article, a ray method is presented for obtaining short time asymptotic solutions of diffusion equations in several dimensions, where the solutions are valid for $t \ll N$ generation times, where t is time and N is the population size.
Abstract: A variety of stochastic models in population genetics, which lead to diffusion equations in several dimensions, are described. Because these equations are difficult to solve, a ray method is presented for obtaining short time asymptotic solutions of them. The solutions are valid for $t \ll N$ generation times, where t is time and N is the population size. The method is applied to a general two dimensional boundary value problem with densities on the boundaries and at the corners. Then the resulting asymptotic solution is specialized to cases of independent traits. For a particular equation, this asymptotic solution is shown to agree with the asymptotic expansion of the exact solution. The method permits the analysis of models with more than two alleles at a locus, and with many loci. It was previously used by Voronka and Kelley [20] on problems in one dimension, and the results were in good agreement with some known exact solutions for t as large as N generation times
13 citations
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TL;DR: In this article, the need for a complete asymptotic analysis of these functions is discussed in the context of one-dimensional diffusion with random trapping sites and the necessary analysis is carried out.
Abstract: The solution of various diffusion problems, in both continuous and discrete systems, can be expressed in terms of a Laplace transform of the exponential function of a fractional power. In this Brief Report the need for a complete asymptotic analysis of these functions is discussed in the context of one-dimensional diffusion with random trapping sites and the necessary asymptotic analysis is carried out.
13 citations
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TL;DR: In this article, the authors obtained a representation for analytic solutions to linear iterative functional equations in a sectoral neighbourhood of a repelling fixed point of the iterated map and deduced an asymptotic expansion of such solutions.
Abstract: We obtain a representation for analytic solutions to linear iterative functional equations in a sectoral neighbourhood of a repelling fixed point of the iterated map. From this representation we deduce an asymptotic expansion of such solutions. Under some additional assumptions this expansion can be used to get the asymptotic behaviour of the power series coefficients of the solution. Finally examples from combinatorics and probability theory are provided.
13 citations
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TL;DR: In this article, the authors 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.
13 citations