Showing papers on "Asymptotology published in 1982"
TL;DR: Maslov asymptotic ray theory is used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases as discussed by the authors.
Abstract: Asymptotic ray theory is widely used to describe body waves in inhomogeneous media, but caustics, shadows, critical points, etc. have to be treated as special cases. Unfortunately, these singularities are often the points of greatest interest as they are caused by inhomogeneities in the model. Transform methods, e.g., the reflectivity method and WKBJ seismograms, are used to investigate waves at these singular points but are restricted to laterally homogeneous media. Maslov asymptotic theory uses the ideas of asymptotic ray theory and transform methods, combining the advantages—simplicity and generality—of both techniques. In this paper, Maslov asymptotic theory is developed for the computation of body-wave seismograms. The eikonal equation of asymptotic ray theory is equivalent to Hamilton9s canonical equations, and the ray trajectories can be considered in the phase space of position and slowness. Normal asymptotic ray theory gives the wave solution in the spatial domain. However, the asymptotic solution for other generalized coordinates in phase space can also be found. For instance, normal transform methods find the solution in the mixed domain where the horizontal slowness replaces the coordinate. Maslov asymptotic theory extends this idea to inhomogeneous media, and the asymptotic solution in a mixed domain (position and slowness) is obtained by a canonical transformation from the spatial domain. The method is useful as the singularities in the mixed and spatial domains are at different locations, and Maslov theory provides a uniform result, combining the solutions in the different domains. These transforms between the mixed-frequency and spatial-time domains are evaluated exactly using the WKBJ seismogram algorithm. This avoids the oscillatory integrals of asymptotic theory and stabilizes the numerical solution by providing the smoothed, discrete seismograms directly. The result is a rigorous but simple method for computing body-wave seismograms in inhomogeneous media. The theory is developed in outline, and numerical examples are included.
260 citations
116 citations
49 citations
TL;DR: In this paper, the authors considered natural mechanical systems that approach their equilibrium positions with unlimited increase of time and showed that the increase in time increases the probability that the system will reach its equilibrium.
25 citations
01 Jan 1982
TL;DR: In this article, the authors propose a method to solve the problem of the problem: this article... ]..,.. )].. [1].
Abstract: ii
18 citations
TL;DR: In this article, with the aid of the branching theory of nonlinear equations, a coarse asymptotics of the probabilities of large deviations for integral statistics of the form======¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯) were derived for the two-sampled variants of these statistics.
Abstract: Part I has been published in the collection “Studies in the Theory of Probability Distributions. IV” (Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst., Vol. 85), Leningrad, 1979, pp. 175–187. With the aid of the methods of the branching theory of nonlinear equations, one finds a coarse asymptotics of the probabilities of large deviations for integral statistics of the form
, which are generalizations of the Cramer-von Mises-Smirnov
statistic, and also for the twosample variants of these statistics. The obtained results allow us to compute the local exact Bahadur relative asymptotic efficiency. One establishes that the latter coincides with both the Bahadur approximate and the Pitman efficiencies.
17 citations
11 citations
TL;DR: In this article, the fundamental solution of a divergence operator of the following form is considered, and two types of asymptotics of are considered, i.e. as and at infinity.
Abstract: Let be the fundamental solution of a divergence operator of the following form: Two types of asymptotics of are considered in the paper: the asymptotic behavior at infinity, i.e. as , and the asymptotic behavior of at . In the first case, for operators with smooth, quasiperiodic coefficients the principal term of the asymptotic expression is found, and a power estimate of the remainder term is established. In the second case the principal term in the asymptotic expression for as is found for an operator with arbitrary bounded and measurable coefficients . These results are obtained by means of the concept of the -convergence of elliptic differential operators. Further, applications of the results are given to the asymptotics of the spectrum of the operator in a bounded domain .Bibliography: 13 titles.
7 citations
TL;DR: In this article, the authors compared the results obtained from the exact solution, from those found by Reiss and from the matched asymptotic expansion solution developed here, show that the last is far more accurate than the second.
Abstract: Traditional singular perturbation methods are employed to develop a solution to a differential equation considered by Reiss [SIAM J. Appl. Math., 39 (1980), pp. 440–455] which models an elementary chemical process. The results are compared with those found by Reiss, who used a novel asymptotic method to construct solutions which exhibit rapid transient behavior. It is shown that Reiss’ jump solution corresponds to the asymptotic (large time) representation of the more complete solution found from a formal matched asymptotic expansion procedure. A comparison of results in the rapid transition region obtained from the exact solution, from those found by Reiss and from the matched asymptotic expansion solution developed here, show that the last is far more accurate than the second.
7 citations
TL;DR: In this paper, a three-dimensional differential system is proposed, in which the motion becomes chaotic at some parameter value, and the asymptotic solution is calculated by using Kuzmak's method.
TL;DR: In this article, the authors used the renormalization-group equation and the loop expansion to obtain the asymptotic behavior of the effective potential in the classical field variable, and applied it to √ √ ε = 0.
Abstract: We use the renormalization-group equation and the loop expansion to obtain the asymptotic behavior of the effective potential in the classical field variable. This is applied to ${\ensuremath{\varphi}}^{4}$ theory as an explicit example. Some remarks on possible uses and extensions are given.
TL;DR: In this article, it was shown that under suitable conditions (primarily of a symmetry nature) an adaptive L-sta-tistic has the same asymptotic distribution as a non-adaptive L-statistic.
Abstract: The purpose of this note is to give a simple demonstration of the apparently widely-known principle that, under suitable conditions (primarily of a symmetry nature) an adaptive L-sta-tistic has the same asymptotic distribution as a non-adaptive L-statistic.
01 Aug 1982
TL;DR: In this paper, a first order approximation to the solution of a singularly perturbed second order system is obtained using a matched asymptotic expansion (MAE) method.
Abstract: Matched asymptotic expansions (MAE) are used to obtain a first order approximation to the solution of a singularly perturbed second order system. A special case is considered in which the uniform asymptotic solution obtained by MAE is shown to converge to the exact solution. Ways in which the method can be used to sole higher-order linear systems, including those which are not singularly perturbed, are also discussed.