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Showing papers on "Asymptotology published in 1989"


Book
01 Jan 1989
TL;DR: In this paper, some basic limiting procedures for multivariate asymptotic expansions of conditional distributions have been discussed, including Edgeworth and allied expansions, as well as a general discussion on multivariate distributions.
Abstract: Preliminary notions.- Some basic limiting procedures.- Asymptotic expansions.- Edgeworth and allied expansions.- Miscellany on multivariate distributions.- Multivariate asymptotic expansions.- Expansions for conditional distributions.- Postscript.

655 citations


Book ChapterDOI
01 Jan 1989
TL;DR: An asymptotic formula is a form of a function that approximates the value of the integrand and its derivatives at a finite number of points, or in terms of some simpler integral as discussed by the authors.
Abstract: An asymptotic formula or asymptotic form for a function f(x) is the name usually given to an approximate formula f(x) ≈ g(x) in some domain of values of x, where g(x) is ‘simpler’ then f(x). For example, if f(x) is an integral, then g(x) must either be given in terms of the values of the integrand and its derivatives at a finite number of points, or in terms of some simpler integral. If f(x) is a solution of an ordinary differential equation, then g(x) must either be expressed in quadratures or be the solution of a ‘simpler’ differential equation. This list can be extended—there is an unwritten heirarchy of asymptotic formulae. Of course all these definitions are very blurred. ‘“What is asymptotics?” This question is about as difficult to answer as the question “What is mathematics?”’

99 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigate the local asymptotic stabilizability of a real analytic control system around an equilibrium point and give necessary and sufficient conditions for local stabilization using continuous feedback.

47 citations


Journal ArticleDOI
TL;DR: In this paper, asymptotic behavior of the eigenvalues of the φ-laplacian was studied. But the π-la placian is not a convex polytope.
Abstract: (1989). Asymptotic behaviour of the eigenvalues of the φ—laplacian. Communications in Partial Differential Equations: Vol. 14, No. 8-9, pp. 1059-1069.

47 citations


Journal ArticleDOI
Jun Shao1
TL;DR: In this paper, asymptotic properties of statistics obtained by evaluating some functionals at the empirical distribution function were studied using a functional calculus approach, and the results were applied to robust M-estimation and L -estimation problems.

9 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the various terms of the common solution of MAE can be generated as polynomials in stretched variables, without actually solving them from the outer solution, as is currently the practice.
Abstract: The present evaluation of the method of asymptotic expansions (MAE) indicates that the various terms of the common solution of MAE can be generated as polynomials in stretched variables, without actually solving them from the outer solution, as is currently the practice. It is also noted that the common solution of the MAE and the intermediate solution of the singular-perturbation method are the same; these methods therefore yield identical results for a certain class of problems. Two illustrative problems are treated.

7 citations


Journal ArticleDOI
TL;DR: In this article, Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain, and these inequalities are then used to prove a generalisation of the Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.
Abstract: Korn's inequalities are proved for star-shaped domains and it is shown how the constants in these inequalities depend on the dimensions of the domain. These inequalities are then used to prove a generalisation of Saint-Venant's Principle for nonlinear elasticity and additionally to establish the asymptotic behaviour of solutions to the traction boundary value problem for a non-prismatic cylinder.

6 citations



Book ChapterDOI
01 Jun 1989

5 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the numerical solution of a reaction-diffusion system involving a reaction term of integral type arising from biological models and showed the existence and the asymptotic behavior of nonnegative numerical solutions.
Abstract: This paper presents the study of the numerical solution of a reaction-diffusion system involving a reaction term of integral type arising from biological models. By means of a monotone approach we introduce upper and lower solutions and then we show the existence and the asymptotic behavior of nonnegative numerical solutions. To this end, we require the positivity of the numerical scheme and so we can use some properties of positive and M-matrices. Finally we give some sufficient conditions to verify the asymptotic stability of the numerical solution.

4 citations



Journal ArticleDOI
TL;DR: In this paper, an algorithm for the approximate solution (in the asymptotic sense) of a singularly perturbed linear time-optimal control problem is proposed, and a computational procedure is outlined, which permits the use of the resulting approximation for.

Journal ArticleDOI
TL;DR: In this article, the asymptotic equivalence of the Enskog equation with the Boltzmann equation and the hydrodynamic limit was analyzed for the case when the radius of the gas particles and the Knudsen number tend to zero.
Abstract: This paper deals with the analysis of some mathematical results on the asymptotic behaviour of the solutions to the initial value problem for the Enskog equation when the radius of the gas particles and the Knudsen number tend to zero, that is, respectively, analysis of the asymptotic equivalence with the Boltzmann equation and hydrodynamic limit.


Journal ArticleDOI
TL;DR: In this paper, Lagrangian analysis on dissipative-asymptotic manifolds is considered and applied to the asymptotic theory of partial differential equations with complex germ.
Abstract: Lagrangian analysis on dissipative-asymptotic manifolds is considered and applied to the asymptotic theory of partial differential equations. The special case of the theory of asymptotic manifolds with complex germ is presented.


Journal ArticleDOI
TL;DR: In this paper, a family of jump Markov processes, depending on a real parameter, is considered and a test is suggested for such a problem and its second order asymptotic efficiency is proved.
Abstract: A family of jump Markov processes, depending on a real parameter, is considered. For this parameter, on the basis of an increasing number of observations, one tests the simple hypothesis against a one-sided alternative. A test is suggested for such a problem and its second order asymptotic efficiency is proved. This proof does not require the construction of the asymptotic expansions of the corresponding power functions.




01 Jan 1989
TL;DR: In this article, the use of auxiliary functions which can be computed more efficiently than their corresponding asymptotic epansions is discussed, and applications are discussed for incomplete gamma functions and Bessel functions, with error function and Airy functions as basic approximants.
Abstract: The use of asymptotic representations of some special functions is discussed. Instead of using the asymptotic series expansions of the functions we consider auxiliary functions which can be computed more efficiently than their corresponding asymptotic epansions. Applications are discussed for incomplete gamma functions and Bessel functions, with, respectively, error function and Airy functions as basic approximants.

Journal ArticleDOI
Wu Chi-kuang1
TL;DR: In this article, a new asymptotic method for singular perturbation problems of difference equation with a small parameter was constructed, and the method was used to solve the problem.
Abstract: In this paper, we constructed a new asymptotic method for singular perturbation problems of difference equation with a small parameter.


Journal ArticleDOI
01 Aug 1989
TL;DR: In this paper, the authors considered the initial boundary value problem for nonlinear scalar parabolic equations and derived a Hopf-Cole transformation for the unique classical solution, and studied the asymptotic behavior of the solution as time goes to ∞.
Abstract: We consider initial boundary value problems for certain nonlinear scalar parabolic equations. A formula for the unique classical solution by Hopf-Cole transformations is obtained and the asymptotic behaviour of the solution as time goes to ∞ is studied.