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: This work tries to show how to put different asymptotic problems, what is known about their solutions, and how algebraic-geometric codes influence the situation.
29 citations
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TL;DR: In this article, the asymptotic expansion of an integral of the type [formula omitted] is derived in terms of the large parameter t. The result is used to find the leading term of the double integral.
Abstract: The asymptotic expansion of an integral of the type [formula omitted], is derived in terms of the large parameter t. Functions Φ(k) and ψ(k) are assumed analytic, and ψ(k) may have zeros at a stationary phase point. The usual one dimensional stationary phase and Airy integral terms are found as special cases of a more general result. The result is used to find the leading term of the asymptotic expansion of the double integral. A particular two dimensional Φ(k) relevant to surface water wave problems is considered in detail, and the order of magnitude of the integral is shown to depend on the nature of ψ(k) at the stationary phase point.
29 citations
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TL;DR: This paper considers classes of finite structures where the authors have good control over the sizes of the definable sets, and a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.
Abstract: In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs (d, μ) ∈ ω × Q>0 so that in any finite field F and for any ā ∈ Fm the size |o(Fn,ā)| is “approximately” μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.
29 citations
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TL;DR: In this paper, the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one are obtained.
Abstract: Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re-expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions, as well being considerably more accurate and providing a smooth interpretation of the Stokes phenomenon. They are also of strikingly simple form. In addition, explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained.
29 citations