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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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Journal ArticleDOI
TL;DR: In this article, a method based on the use of certain asymptotic initial conditions together with the compound matrix method is presented for the numerical solution of the Orr-Sommerfeld equation on infinite intervals.

37 citations

Journal ArticleDOI
A. Ahmed1
TL;DR: In this article, a sufficient condition for asymptotic stability based on the state-space representation is developed for two-dimensional discrete linear systems in the case of 2-D separable systems.
Abstract: A sufficient condition for asymptotic stability based on the state-space representation is developed for two-dimensional discrete linear systems In the case of 2-D separable systems, necessary and sufficient conditions for asymptotic stability are obtained Necessary conditions for asymptotic stability are also given

36 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Fourier transform of the distribution function co(x) = X(eu) is a special example of the infinite convolutions of purely discontinuous distribution functions investigated by these authors.
Abstract: 1. The present note was suggested by recent work of H. Davenport, [3 ],t S. Bochner and B. Jessen, [2 ], and A. Wintner and B. Jessen, [6 ]. Davenport established the existence of asymptotic distribution functions for a certain class of arithmetical functions by an extension of a method previously used by the author, [8], [9], in a similar investigation. This method was based on the consideration of the moments of the distribution functions. In questions of asymptotic distribution, however, Bochner and Jessen have shown the great advantage of dealing directly with the Fourier transforms of the distribution functions. This advantage becomes again apparent if the method of Fourier transforms, whose adaptation to sequences is fully developed in ?I, is applied to Davenport's problem. This is precisely what we shall do in ?II; the result thus obtained (Theorem 1) insures the existence of the asymptotic distribution function for a; very large class of (positive and multiplicative) arithmetical functions. It includes Davenport's and the author's previous results and yields readily (by suitable specializations of the arithmetical function involved) the frequencies of certain classes of integers investigated by W. Feller and E. Tornier, [4], in an entirely different way. The connectiQn with the work of Wintner and Jessen, [6], is as follows. The distribution function co(x) =X(eu) of Theorem 1 is a special example of the infinite convolutions of purely discontinuous distribution functions investigated by these authors. They have shown (1[6], Theorem 35) that such infinite convolutions can be only either purely discontinuous or else everywhere continuous, and in the latter case either singular functions or else absolutely continuous functions. These general results apply immediately to our special situation, but new and probably difficult problems arise which may be mentioned here. Theorem 1 gives simple sufficient conditions to insure continuity or discontinuity of co(x); the problem of finding similar necessary and sufficient conditions for continuity remains unsolved. The more delicate problem of deciding whether a continuous c(x) is singular or absolutely

35 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20231
20222
20181
201725
201626
201526