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299
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, 1995
"... High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combin ..."
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Cited by 102 (8 self)
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High order differences of simple number sequences may be analysed asymptotically by means of integral representations, residue calculus, and contour integration. This technique, akin to Mellin transform asymptotics, is put in perspective and illustrated by means of several examples related to combinatorics and the analysis of algorithms like digital tries, digital search trees, quadtrees, and distributed leader election.
The number of new species, and the increase in population coverage, when a sample is increased
 Biometrika
, 1956
"... A sample of size N is drawn at random from a population of animals of various species. Methods are given for estimating, knowing only the contents of this sample, the number of species which will be represented r times in a second sample of size AN; these also enable us to estimate the number of dif ..."
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Cited by 43 (1 self)
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A sample of size N is drawn at random from a population of animals of various species. Methods are given for estimating, knowing only the contents of this sample, the number of species which will be represented r times in a second sample of size AN; these also enable us to estimate the number of different species and the proportion of the whole population represented in the second sample. A formula is found for the variance of the estimate; when A> 2, this variance becomes in general very large, so that the estimate is useless without some modification. This difficulty can be partly overcome, at least for A < 5, by using Euler's method with a suitable parameter or the methods described by Shanks (1955) to hasten the convergence of the series by which the estimate is expressed. The methods are applied to samples of words from Our Mutual Friend, to an entomological sample, and to a sample of nouns from Macaulay's essay on Bacon. 1.
THE GENERALIZED ABELPLANA FORMULA. APPLICATIONS TO BESSEL FUNCTIONS AND CASIMIR EFFECT
, 2000
"... One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on using the AbelPlana summation formula. This allows to derive the regularized quantities by manifestly cutoff independent way and to present them in the form of st ..."
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Cited by 34 (25 self)
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One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on using the AbelPlana summation formula. This allows to derive the regularized quantities by manifestly cutoff independent way and to present them in the form of strongly convergent integrals. However the application of AbelPlana formula in usual form is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the AbelPlana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It have been shown that these results generalize the special cases existing in literature. Further the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows to extract the divergent parts from the vacuum expectation values for the local physical observables in the manifestly cutoff independent way. Present paper reviews these results. Some new considerations are added as well.
On the Detection of Singularities of a Periodic Function
 ADV. COMPUT. MATH
"... We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometri ..."
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Cited by 31 (16 self)
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We discuss the problem of detecting the location of discontinuities of derivatives of a periodic function, given either finitely many Fourier coefficients of the function, or the samples of the function at uniform or scattered data points. Using the general theory, we develop a class of trigonometric polynomial frames suitable for this purpose. Our methods also help us to analyze the capabilities of periodic spline wavelets, trigonometric polynomial wavelets, and some of the classical summability methods in the theory of Fourier series.
The Devil’s Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
 ACTA APPLICANDAE MATHEMATICAE 56: 1–98, 1999
, 1999
"... Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain ..."
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Cited by 28 (3 self)
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Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a ‘hyperasymptotic’ approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescaleandadd process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.
LogarithmicExponential Power Series
 J. London Math. Soc
"... . We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain nonelementary and improper integrals. We also use this model to answer a question o ..."
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Cited by 27 (6 self)
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. We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain nonelementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x)(log log x) is not asymptotic as x ! +1 to a composition of semialgebraic functions, log and exp. x1 Introduction and preliminaries Let RfX 1 ; : : : ; Xm g denote the ring of all real power series in X 1 ; : : : ; Xm that converge in a neighborhood of I m , where I = [\Gamma1; 1]. For f 2 RfX 1 ; : : : ; Xm g we let e f : R m ! R be given by: e f(x) = ae f(x); for x 2 I m , 0; x 62 I m . We call the e f 's restricted analytic functions. Let L an be the language of ordered rings f!; 0; 1; +; \Gamma; \Deltag augmented by a new function symbol for each function e f . We let R an be the reals with its natur...
Extremal problems for polynomials with exponential weights
 Trans. Amer. Math. Soc
, 1984
"... in recognition of his contributions to the study of orthogonal polynomials and his motivating influence on the present work. Abstract. For the extremal problem: E„r(a): = minexp(W«)(x+ ■■■)\\L „ a> 0, where U (0 < r < oo) denotes the usual integral norm over R, and the minimum is taken ..."
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Cited by 27 (3 self)
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in recognition of his contributions to the study of orthogonal polynomials and his motivating influence on the present work. Abstract. For the extremal problem: E„r(a): = minexp(W«)(x+ ■■■)\\L „ a> 0, where U (0 < r < oo) denotes the usual integral norm over R, and the minimum is taken over all monic polynomials of degree n, we describe the asymptotic form of the error E „ r(a) (as n » oo) as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case r = 2 yields new information regarding the polynomials {p„(a; x) y„(a)x " + • • • } which are orthonormal on R with respect to exp(2A°). In particular, it is shown that a conjecture of Freud concerning the leading coefficients y „(a) is true in a Cesaro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud's conjecture. For r = oo, a> 0 we also prove that, if deg P„(x) < n, the norm exp(x")/:'„(x)i«> is attained on the finite interval [(«/A0)1/a,(z,A„)1/a], whereAa = r(«)/22{r(a/2)}2. Extensions of Nikolskiitype inequalities are also given.
Sukochev Spectral flow and Dixmier traces
, 2002
"... We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semifinite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞)s ..."
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Cited by 24 (7 self)
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We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semifinite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞)summable BreuerFredholm module in terms of a Hochschild 1cycle. We explain how to derive a Wodzicki residue for pseudodifferential operators along the orbits of an ergodic R n action on a compact space X. Finally we give a short proof an index theorem of Lesch for generalised Toeplitz operators.