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Showing papers on "Asymptotology published in 1974"


Journal ArticleDOI
TL;DR: In this article, a survey of tools used in asymptotic analysis for enumeration problems is presented, focusing on tools which are general, are easily applied, and give estimates of the form $a_n \sim f(n)$.
Abstract: This is an expository paper dealing with those tools in asymptotic analysis which are especially useful in obtaining asymptotic results in enumeration problems. Emphasis is on tools which are general, are easily applied, and give estimates of the form $a_n \sim f(n)$. Many examples are given to illustrate the usage of the various tools. It is assumed that a summation or a generating function for $a_n $ is explicitly or implicitly given.

443 citations


Journal ArticleDOI
TL;DR: The stability and asymptotic behavior of solutions of an autonomous linear differential system x' =,4x are determined by the spectrum of the constant matrix A as discussed by the authors, which is the limiting system X' = Ax.

68 citations




Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of finding the asymptotic stability of solutions of (1.1) when the kernel a(t) is known to have a specific (asymptotically stable) representation.
Abstract: and a(t) * a(O+), then the solution u(t) of (1.1) satisfies (1.2). In this paper, we are not concerned with sufficient conditions for the asymptotic stability of (1.1) but rather with the rate of decay of solutions of (1.1) when it is known that the equation is asymptotically stable. Thus, when the kernel a(t) is known to have a specific asymptotic representation, one seeks for asymptotic representation of the solution of (1.1). This approach has been investigated in a recent paper of

12 citations



Journal ArticleDOI
TL;DR: In this paper, a one-to-one correspondence between properly defined scaling, the leading light-cone singularity and the asymptotic behavior of the corresponding Jost-Lehmann spectral function in the sense of distribution theory is established.
Abstract: For a local amplitude we prove a one-to-one correspondence between properly defined scaling, the leading light-cone singularity and the asymptotic behaviour of the corresponding Jost-Lehmann spectral function in the sense of distribution theory. The cases of canonical and non-canonical scaling are considered.

9 citations


01 Jan 1974

6 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that even for some external lines on the mass shell, the procedure of dropping the mass-insertion term in the Callan-Symanzik equation is justified for the form factor at high squared momentum transfer in a certain class of models.
Abstract: We show that even for some external lines on the mass shell, the procedure of dropping the mass-insertion term in the Callan-Symanzik equation is justified for the form factor at high squared momentum transfer in a certain class of models. This provides a very quick method of summing leading contributions in perturbation theory, as well as summing the next-to-leading terms.

6 citations


Journal ArticleDOI
TL;DR: In this paper, a formal, uniformly valid, asymptotic expansion of the Klein-Gordon equation with spatially varying coefficients is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which are called the diffraction functions.
Abstract: The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call the diffraction functions. The validity of the expansion is established when the coefficients in the Klein-Gordon equation are constants, and the results are applied to a signaling problem for a class of acoustic wave guides.

4 citations


Journal ArticleDOI
Mitsuo Kono1
TL;DR: The wave packet formalism is applied to the Vlasov-Poisson equations to derive a K-dV equation and also to the problem of the wave modulation by taking an example of the Bussinesque equation as discussed by the authors.
Abstract: Some asymptotic methods except for the reductive perturbation method are presented. The wave packet formalism is applied to the Vlasov-Poisson equations to derive a K-dV equation and also to the problem of the wave modulation by taking an example of the Bussinesque equation. The derivative expansion method and the extended Krylov­ Bogoliubov-Mitropolsky method are also discussed.




Book ChapterDOI
TL;DR: In this paper, a deductive asymptotic theory of non-linear oscillations and wave propagation is developed, which uses from the outset concepts and methods of asymptic analysis.
Abstract: Publisher Summary This chapter focuses on the asymptotic theory of non-linear oscillations. New approaches to the asymptotic theory of non-linear oscillations and wave propagation are reported and the asymptotic method of Krilov–Bogolioubov–Mitropolski is discussed. A deductive asymptotic theory is developed, which uses from the outset concepts and methods of asymptotic analysis. The necessary preliminaries are given and the fundamental tool of our method of analysis is introduced: in a suitable (asymptotic) sense, a local average value of the function Y (t , e) is defined. With the aid of this concept, a deductive procedure establishes the fundamental theory of Krilov–Bogolioubov–Mitropolski under the most general conditions. For a class of problems validity of the asymptotic approximation on 0 ≤ t