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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7 citations
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TL;DR: In this paper, an asymptotic perturbation of transfer operators and a method that avoids resolvent's perturbations was proposed to investigate the Hausdorff dimension of a perturbed cookie-cutter set.
Abstract: We consider an asymptotic behaviour of the topological pressure, the Gibbs measure and the measure-theoretic entropy concerning a potential defined on a subshift. Our results are obtained by considering asymptotic perturbation of transfer operators and by using a method that avoids resolvent’s perturbation. In application, we investigate an asymptotic behaviour of the Hausdorff dimension of a perturbed cookie-cutter set.
7 citations
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7 citations
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TL;DR: In this paper, it was shown that the Lommel function can be expressed as a linear combination of the confluent hypergeometric function (a, b, c, c; \\x) and Lommel functions.
Abstract: and for a, ß, a i, b¡ suitably restricted. (Our analysis will reveal that many of these restrictions may be dropped.) Since fi\\x) has an asymptotic representation in descending powers of \\x, (2) may be interpreted as a summation process which converts the generally divergent expansion into a convergent one. Important special cases of (2) yield expansions for the confluent hypergeometric function \\p(a, c; \\x) and Lommel functions. We will treat only the case Q — P — 1 > 0 since the case P + 1 ^ Q may be handled by an elementary analysis. In the former, n(M), as we shall see, has the unusual behavior of exponential decay as n —-> °o f in contrast to the latter case, where $>AM) behaves as inverse powers of w!, or at worst (P + 1 = Q), algebraically in n. In Section II, we first prove three lemmas; the first establishes an integral representation for $„(ö)(x); the second estimates for large n a closely related integral, and the third gives the desired asymptotic formula for $n(e)(X). Our main theorem follows when we find we can express <Ên(A/)(A) as a linear combination of the functions $„(e)[X exp (wi(Q M 2k))]. Section III is devoted to examples. There are quantities and assumptions about them which occur frequently in this paper, and they will always be as below :
7 citations
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TL;DR: In this paper, the eigenvalue problem for the Schrodinger equation with integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity was considered.
Abstract: We consider the eigenvalue problem for the two-dimensional Schrodinger equation containing an integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity. Global asymptotic solutions localized in the neighborhood of a line segment in the plane are constructed using the matching method for asymptotic expansions. The Bogoliubov and Airy polarons are used as model functions in these solutions. An analogue of the Bohr–Sommerfeld quantization rule is established to find the related series of eigenvalues.
7 citations