Showing papers on "Asymptotology published in 1981"
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216 citations
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05 Jun 1981
113 citations
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TL;DR: In this paper, a single-phase soliton-form solution of the Korteweg-de Vries equation with variable coefficients is presented, and the Dirichlet series for constructing asymptotic expansions for general equations.
Abstract: CONTENTSIntroductionChapter I. The Korteweg-de Vries equation with variable coefficients § 1. Basic definitions. A single-phase soliton-form solution of the Korteweg-de Vries equation with variable coefficients § 2. Construction of an asymptotic single-phase soliton-form solution § 3. Conservation lawsChapter II. The Kadomtsev-Petviashvili equation and the sine-Gordon equation § 1. The Kadomtsev-Petviashvili equation § 2. The sine-Gordon equation with variable coefficientsChapter III. Multi-phase asymptotic solutions of non-linear equations and Dirichlet series § 1. Multi-phase asymptotic solutions § 2. Dirichlet series for constructing asymptotic expansions for general equationsReferences
102 citations
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94 citations
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TL;DR: The good cut equation for a specific asymptotic shear was solved in this article, and the metric of the associated H•space was found to be type N, asmptotically flat and positive frequency.
Abstract: The good cut equation for a specific asymptotic shear is solved and the metric of the associated H‐space is found to be type N, asymptotically flat and positive frequency.
33 citations
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TL;DR: In this article, a new asymptotic expansion algorithm related to the Chapman-Enskog expansion in kinetic theory is applied to systems of linear evolution equations, and the uniform convergence of the asymPT solution to the exact one is shown.
Abstract: A new asymptotic expansion algorithm related to the Chapman-Enskog expansion in kinetic theory is applied to systems of linear evolution equations. The uniform convergence of the asymptotic solution to the exact one is shown. The algorithm is applied to the linearized Carleman model of the Boltzmann equation, to the neutron transport equation, and to the Fokker-Planck equation.
22 citations
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TL;DR: In this article, the asymptotic behavior of both the physical and unphysical metric is obtained in a Bondi-type coordinate system, and the assumptions made in this paper and the resultant metrics are compared with those of Persides in his recent paper.
Abstract: Starting with a few basic assumptions on the local asymptotic behavior of space-time and using Penrose's conformal technique the asymptotic behavior of both the physical and unphysical metric is obtained in a Bondi-type coordinate system. The space-times under consideration are not necessarily empty in the asymptotic region nor are they necessarily asymptotically flat. However, they do have the usual “falloff” behavior as one goes out toward infinity in a given null direction. The assumptions made in this paper and the resultant metrics are compared with those of Persides in his recent paper on the definition of asymptotic flatness.
19 citations
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TL;DR: The Lagrangian analysis as mentioned in this paper is a special case of the Lagrangians, which was introduced by the W.K.B. method for solving the Dirac equation.
Abstract: H. Poincare defined asymptotic expansions. Their use by the W. K. B. method introduced a new kind of solution of linear differential equations. Maslov showed their singularities to be merely apparent. The clarification of those results leads to the introduction of "Lagrangian functions", of their scalar product and of "Lagrangian operators", which constitutes a new structure: the "Lagrangian analysis". The last step of its definition requires the choice of a constant. That constant has to be Planck's constant, when the equation is the Schrodinger or the Dirac equation describing the hydrogen atom-the study of atoms with several electrons is very incomplete. 1. Henri Poincare's main field, more precisely the one where the number of his publications is the highest, happens to be celestial mechanics. For instance, he tried to establish the convergence of the series by means of which the motion of the solar system is computed; it was a failure. He proved indeed the opposite: the divergence of those series, whose numerical values furnished the most impressive, precise and famous predictions in science during the last century! Henri Poincare explained that paradox: those series give a very good approximation of the wanted result, provided only their first terms, namely, a reasonable number of them, are taken into account. Of course, demanding mathematicians to be reasonable is dubious but Henri Poincare [4] made it clear by defining the asymptotic expansion ^mi0anx n of a function of x at the origin: it is a, formal series such that for each natural number N there exists a positive number cN such that I N I ƒ(•*) 2 <*n* < cN\x\ * for x near 0. (1.1) I "° I Thus an asymptotic expansion of ƒ is a formal series able to give a very good approximation of ƒ(*), when x is small, but unable to supply the exact value of/(x). 2. The W.K.B. method constructs asymptotic solutions of a linear differential equation n(x, ~ -j£ W *) 0 ( x G l = R ^ 6 i [0, oo[), (2.1) whose unknown is the function u and whose parameter v tends to /oo. Presented at the Symposium on the Mathematical Heritage of Henry Poincare in April, 1980; received by the editors June 15, 1980. 1980 Mathematics Subject Classification. Primary 47B99, 81C99; Secondary 35S99, 42B99.
16 citations
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15 citations
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TL;DR: By formulating quantum field theory with an extended object in terms of the asymptotic condition, the correlation between the extended object and the quanta is studied and several new results are derived.
Abstract: By formulating quantum field theory with an extended object in terms of the asymptotic condition, we derive several new results such as the asymptotic Hamiltonian, the asymptotic field, the generalized coordinates, etc. In this derivation no approximation (such as the tree approximation, etc.) is used. The correlation between the extended object and the quanta is studied.
11 citations
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TL;DR: In this paper, a general dimensional analysis is applied to derive, in Euclidean space, the asymptotic behaviour of renormalised (subtracted-out) Feynman amplitudes A. The maximising subspaces for the bond of A are constructed and an explicit expression for the behaviour of A is given including both power and logarithmic coefficients.
Abstract: A general dimensional analysis is applied to derive, in Euclidean space, the asymptotic behaviour of renormalised (subtracted-out) Feynman amplitudes A. The maximising subspaces for the bond of A are constructed and an explicit expression for the behaviour of A is given including both power and logarithmic asymptotic coefficients. The analysis provides rules for obtaining the asymptotic expression for A when some (or all) of the external momenta become large, some (or all) become small, and some of the masses are led to approach zero. Examples are then worked out as illustrations of these rules.
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TL;DR: In this article, the singularly perturbed linear evolution equations of resonance type are considered in a Banach space and the Hilbert and Chapman-Enskog algorithms for generating asymptotic solutions are presented and shown to lead to different results at each finite order of approximation.
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TL;DR: In this article, the Mises functional was extended for the two-sample problem and it was shown that it also has the asymptotic property given by von Mises (1947,Ann. Math. Statist.,18, 309-348) and by Filippova (1962,Theory Prob. Appl.,7, 24-57) in the one-sample case.
Abstract: Mises functional is extended for the two-sample problem. It is shown that the extended Mises functional also has the asymptotic property given by von Mises (1947,Ann. Math. Statist.,18, 309–348) and by Filippova (1962,Theory Prob. Appl.,7, 24–57) in the one-sample case. Asymptotic behavior ofU-statistic in the two-sample case, the statistic of Cramer-von Mises type for testing homogeneity and so forth are investigated as important examples of the theory.
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TL;DR: In this paper, various categories of problems are distinguished with different first approximation formulations, showing different degrees of non-linearity, through the asymptotic approach to elastic theory of beam-like structural elements.
Abstract: Through the asymptotic approach to elastic theory of beam-like structural elements, various categories of problems are distinguished with different first approximation formulations, showing different degrees of non-linearity.
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TL;DR: In this article, the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for non-autonomous dynamical systems, and sufficient conditions are given for uniform asymptotic eventual strongly partial stability.
Abstract: In this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.
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