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Showing papers on "Asymptotology published in 1972"


Journal ArticleDOI
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




Journal ArticleDOI
TL;DR: Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account as discussed by the authors.
Abstract: Miller's recurrence algorithm for tabulating the subdominant solution of a second-order difference equation is modified so as to take the asymptotic behaviour of the solution into account. The asymptotic solutions of various types of equations are listed, and a method is given for estimating the error in the tabulated solution.

17 citations


Journal ArticleDOI
TL;DR: In this paper, the rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur are considered in the same way as nonzero ones.
Abstract: The paper deals with problems of rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur. Zero observations are considered in the same way as nonzero ones. Two ways of treatment of ties are used in the paper, randomization of ties and the method of averaged scores. The asymptotic distributions of the statistics are derived under hypothesis of symmetry and under contiguous alternatives of location. The asymptotic power and efficiency of tests are established.

16 citations


Journal ArticleDOI
01 Feb 1972
TL;DR: In this article, a variation of the variation of constants formula for nonlinear systems is used to study the comparative asymptotic behavior of the systems x' = f(t, x) and y'=f(T, y)+g(t, y).
Abstract: A version of the variation of constants formula for nonlinear systems is used to study the comparative asymptotic behavior of the systems x'=f(t, x) and y'=f(t, y)+g(t, y).

11 citations


Journal ArticleDOI
TL;DR: The asymptotic behavior at large times for certain dynamical systems arising in the Hamiltonian formulation of classical mechanics is investigated in this paper, where it is shown that for potentials which die out sufficiently fast at large distances, the unbounded states of the system are asyptotically free.
Abstract: The asymptotic behavior at large times for certain dynamical systems arising in the Hamiltonian formulation of classical mechanics is investigated. It is shown that for potentials which die out sufficiently fast at large distances the unbounded states of the system are asymptotically free. This result complements the corresponding result for quantum mechanical systems, and is obtained by analogous methods. In addition, the existence, differentiability, and asymptotic completeness of the associated wave mappings is established under appropriate further assumptions by classical methods.

9 citations


Journal ArticleDOI
TL;DR: In this paper, the density of eigenvalues of a potential well is calculated in an asymptotic expansion for large geometrical size and explicit, readily calculable expressions for volume and surface contributions are obtained for the case of a spherical Woods-Saxon well.

8 citations



Journal ArticleDOI
TL;DR: In this article, a method for the asymptotic estimation of integrals with a kernel of δ-function type is presented, where the kernel is defined as a linear combination of two functions.
Abstract: A METHOD for the asymptotic estimation of integrals with a kernel of δ-function type is presented.

5 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived a Timelike asymptotic series for many-particle matrix elements of products of almost local fields, which generalize and extend the Araki-Haag series of quasilocal operators.
Abstract: Timelike asymptotic series for many‐particle matrix elements of products of almost local fields are derived which generalize and extend the Araki‐Haag series of quasilocal operators. An interpretation of the asymptotic leading terms in the form of contributions from disconnected intermediate particle states is given. A discussion of the dependence of the asymptotic leading terms on the smearing in the space variables is presented.

Journal ArticleDOI
TL;DR: In this article, the regularization of a singularity with respect to a parameter is derived by means of an extension of the original operator and subsequent application of perturbation theory in an unbounded space, and used to solve an extended problem asymptotically.
Abstract: In this paper the regularization of a singularity with respect to a parameter is derived by means of an extension of the original operator and subsequent application of perturbation theory in an unbounded space, and used to solve an extended problem asymptotically. It is proved that this asymptotic solution is unique. An appropriate restriction of the asymptotic solution thus obtained will be an asymptotic solution of the original problem; this restriction is also unique. The theory of this method is illustrated by an example of an ordinary linear system of general form.

Book ChapterDOI
31 Jan 1972

Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of constructing asymptotic expansions of the solutions of integral equations of the second kind depending on a small parameter ϵ 〉 0 in such a way that the kernel of the integral operator has the nature of a δ-function.
Abstract: WE consider the problem of constructing asymptotic expansions of the solutions of integral equations of the second kind depending on a small parameter ϵ 〉 0 in such a way that as ϵ 〉 0 the kernel of the integral operator has the nature of a δ-function. It is proved that under certain conditions a boundary layer effect arises, and an iterative process is given for constructing the asymptotic expansion of the solution.