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 article, a general theory for finite asymptotic expansions in real powers was developed for expansions of type (*),x → x0 where the ordered n-tuple forms an asymptic scale at x 0, i.e., as x → x 0, 1 ≤ i ≤ n − 1, and is practically assumed to be an extended complete Chebyshev system on a one-sided neighborhood of x o.
Abstract: After studying finite asymptotic expansions
in real powers, we have developed a general theory for expansions of type (*) ,x → x0 where
the ordered n-tuple forms an asymptotic scale at x0 , ie as x → x0, 1 ≤ i ≤ n – 1, and is practically assumed to be an extended
complete Chebyshev system on a one-sided neighborhood of x o As in previous papers by the author concerning polynomial, real-power
and two-term theory, the locution “factorizational theory” refers to the
special approach based on various types of factorizations of a differential
operator associated to Moreover, the guiding thread of our theory is the property of formal
differentiation and we aim at characterizing some n-tuples of asymptotic expansions formed by (*) and n -1 expansions obtained
by formal applications of suitable linear differential operators of orders 1,2,…,n-1
Some considerations lead to restrict the attention to two sets of operators
naturally associated to “canonical factorizations” This gives rise to
conjectures whose proofs build an analytic theory of finite asymptotic
expansions in the real domain which, though not elementary, parallels the
familiar results about Taylor’s formula One of the results states that to each
scale of the type under consideration it remains associated an important class
of functions (namely that of generalized convex functions) enjoying the
property that the expansion(*), if valid, is automatically formally
differentiable n-1 times in two special senses
8 citations
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TL;DR: In this paper, the authors apply the methods of the above authors to discuss asymptotic behaviour of capillarities in cusps, which is a corner with opening angle 0.
Abstract: Concus and Finn [2] discovered that capillary surfaces rise to infinity in corners with sufficiently small opening angle. They also found the leading term of an asymptotic expansion. Miersemann [5] improved this result to obtain a complete asymptotic expansion. In the present paper we will apply the methods of the above authors to discuss asymptotic behaviour of capillarities in cusps, which is a corner with opening angle 0. A large variety of asymptotic formulas will be provided. The general comparison theorem from Concus and Finn will play an important role in the proofs.
8 citations
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22 Jun 2017TL;DR: In this paper, the fixed point index theory is applied to the Banach space to obtain fixed points of the integral operator in order to obtain solutions that satisfy some certain kind of asymptotic behavior.
Abstract: In this work we will consider integral equations defined on the whole real line and look for solutions which satisfy some certain kind of asymptotic behavior. To do that, we will define a suitable Banach space which, to the best of our knowledge, has never been used before. In order to obtain fixed points of the integral operator, we will consider the fixed point index theory and apply it to this new Banach space.
8 citations
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8 citations
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TL;DR: In this paper, the authors discuss five topics of current interest in asymptotic analysis: the use of probabilistic methods to estimate the growth of combinatorial sequences, asymPTotic methods in the theory of random walks, the estimation of solutions of difference equations, asmptotic expansions in generalized scales, and the computation by asymptonotic methods of distributions whose moments are known.
8 citations