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Showing papers on "Asymptotology published in 2007"


Journal ArticleDOI
TL;DR: In this article, a vector-valued version of the asymptotic expansion is constructed, which allows us to determine the order of a Levin-type method with highly oscillatory kernels, such as Airy functions or Bessel functions.
Abstract: We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order.

46 citations


Journal ArticleDOI
TL;DR: In this article, asymptotic methods of nonlinear mechanics and the method of Fokker-Planck-Kolmogorov equations were applied to stochastic oscillations in quasilinear oscillating systems with random perturbations.
Abstract: We study applications of asymptotic methods of nonlinear mechanics and the method of Fokker-Planck-Kolmogorov equations to stochastic oscillations in quasilinear oscillating systems with random perturbations.

44 citations


Journal ArticleDOI
TL;DR: In this article, a complete asymptotic expansion for eigenvalues of the Lame system of the linear elasticity in domains with small inclusions in three dimensions was derived by an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation.
Abstract: We derive a complete asymptotic expansion for eigenvalues of the Lame system of the linear elasticity in domains with small inclusions in three dimensions. By an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation, we reduce this problem to the study of the characteristic values of integral operators in the complex planes. Generalized Rouche's theorem and other techniques from the theory of meromorphic operator-valued functions are combined with asymptotic analysis of integral kernels to obtain full asymptotic expansions for eigenvalues.

32 citations


Journal ArticleDOI
TL;DR: This paper considers classes of finite structures where the authors have good control over the sizes of the definable sets, and a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.
Abstract: In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs (d, μ) ∈ ω × Q>0 so that in any finite field F and for any ā ∈ Fm the size |o(Fn,ā)| is “approximately” μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.

29 citations


Journal ArticleDOI
Bernard Garel1
TL;DR: An overview of the very recent asymptotic results in the problem of testing homogeneity against a two-component mixture is provided and illustrations of new and known results are presented.

29 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary and showed that the leading terms of the expansion for the eigenelements of the eigenvalue of the limit problem can be obtained from the spectral equation.
Abstract: We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.

27 citations


Journal ArticleDOI
TL;DR: Equations for the Edgeworth expansion of the distributions of the estimators in exploratory factor analysis and structural equation modeling and results show that asymptotic expansion gives substantial improvement of approximation to the exact distribution constructed by simulations over the usual normal approximation.
Abstract: Equations for the Edgeworth expansion of the distributions of the estimators in exploratory factor analysis and structural equation modeling are given. The equations cover the cases of non-normal data, as well as normal ones with and without known first-order asymptotic standard errors. When the standard errors are unknown, the distributions of the Studentized statistics are expanded. Methods of constructing confidence intervals of population parameters with arbitrary asymptotic confidence coefficients are given using the Cornish-Fisher expansion. Simulations are performed to see the usefulness of the asymptotic expansions in exploratory factor analysis with rotated solutions and confirmatory factor analysis. The results show that asymptotic expansion gives substantial improvement of approximation to the exact distribution constructed by simulations over the usual normal approximation.

19 citations


Journal Article
TL;DR: After developing the tools required for application of the fixed point theory in the investigation, some general results about the long-time behavior of solutions of n-th order nonlinear differential equations are presented.
Abstract: We discuss a number of issues important for the asymptotic integration of ordinary differential equations. After developing the tools required for application of the fixed point theory in the investigation, we present some general results about the long-time behavior of solutions of n-th order nonlinear differential equations with an emphasis on the existence of polynomial-like solutions, the asymptotic representation for the derivatives and the effect of perturbations upon the asymptotic behavior of solutions.

16 citations


Journal ArticleDOI
TL;DR: A Finite Element Method implementation for the asymptotic partial decomposition is considered and the relation with the “mixed formulation” is discussed.
Abstract: We consider a Finite Element Method (F.E.M.) implementation for the asymptotic partial decomposition. The advantage of this approach is an important reduction of the number of nodes. The convergence is proved for some model problems. Finally the relation with the “mixed formulation” is discussed.

15 citations


Journal ArticleDOI
TL;DR: In this article, the authors obtained a representation for analytic solutions to linear iterative functional equations in a sectoral neighbourhood of a repelling fixed point of the iterated map and deduced an asymptotic expansion of such solutions.
Abstract: We obtain a representation for analytic solutions to linear iterative functional equations in a sectoral neighbourhood of a repelling fixed point of the iterated map. From this representation we deduce an asymptotic expansion of such solutions. Under some additional assumptions this expansion can be used to get the asymptotic behaviour of the power series coefficients of the solution. Finally examples from combinatorics and probability theory are provided.

13 citations


Proceedings ArticleDOI
07 Sep 2007
TL;DR: In this article, an asymptotic expansion of the well-known Szasz-Mirakyan operators is presented, which is based on an expansion of a well known Hungarian algorithm.
Abstract: We obtain an asymptotic expansion of the well‐known Szasz‐Mirakyan operatorsWe obtain an asymptotic expansion of the well‐known Szasz‐Mirakyan operators


Journal ArticleDOI
TL;DR: A review of some asymptotic methods for construction of nonlinear normal modes for continuous system (NNMCS) has been provided in this article, which deals with both traditional approaches and, less widely used, new approaches.
Abstract: This paper provides a review of some asymptotic methods for construction of nonlinear normal modes for continuous system (NNMCS). Asymptotic methods of solving problems relating to NNMCS have been developed by many authors. The main features of this paper are that (i) it is devoted to the basic principles of asymptotic approaches for constructing of NNMCS; (ii) it deals with both traditional approaches and, less widely used, new approaches; and (iii) it pays a lot of attention to the analysis of widely used simplified mechanical models for the analysis of NNMCS. The author has paid special attention to examples and discussion of results.

Journal ArticleDOI
TL;DR: In this article, a system of integro-differential equations with rapidly varying kernels, one of which has an unstable spectral value, is considered and an algorithm based on the method of normal forms is proposed for finding asymptotic solutions (of an arbitrary order).
Abstract: A system of integro-differential equations with rapidly varying kernels, one of which has an unstable spectral value, is considered. An algorithm based on the method of normal forms is proposed for finding asymptotic solutions (of an arbitrary order). The contrast structures (internal transition layers) in solutions to the problem are investigated by analyzing the leading term of the asymptotic expansion. It is shown that the contrast structures are caused by the instability of the spectral value and by the presence of inhomogeneity. The role of the kernels of the integral operators in the formation of contrast structures is clarified.


Posted Content
TL;DR: In this paper, the authors considered the asymptotic expansion of the heat-kernel for singular differential operators on manifolds with conical singularities and showed that the heat kernel admits an anomalous expansion in powers of its argument whose exponents depend on external parameters.
Abstract: The asymptotic expansion of the heat-kernel for small values of its argument has been studied in many different cases and has been applied to 1-loop calculations in Quantum Field Theory. In this thesis we consider this asymptotic behavior for certain singular differential operators which can be related to quantum fields on manifolds with conical singularities. Our main result is that, due to the existence of this singularity and of infinitely many boundary conditions of physical relevance related to the admissible behavior of the fields on the singular point, the heat-kernel has an "unusual" asymptotic expansion. We describe examples where the heat-kernel admits an asymptotic expansion in powers of its argument whose exponents depend on "external" parameters. As far as we know, this kind of asymptotics had not been found and therefore its physical consequences are still unexplored.

Journal ArticleDOI
TL;DR: A review of asymptotic methods in the theory of plates and shells can be found in this paper, where the authors have paid special attention to examples and discussion of results rather than to burying the ideas in formalism, notation, and technical details.
Abstract: This paper provides a state-of-the-art review of asymptotic methods in the theory of plates and shells. Asymptotic methods of solving problems related to theory of plates and shells have been developed by many authors. The main features of our paper are: (i) it is devoted to the fundamental principles of asymptotic approaches, and (ii) it deals with both traditional approaches, and less widely used, new approaches. The authors have paid special attention to examples and discussion of results rather than to burying the ideas in formalism, notation, and technical details.

Proceedings ArticleDOI
01 Jan 2007
TL;DR: In this paper, the Sturm-Liouville differential equation (r(t)x′)−1 p p−1, where p is a special case of a general half-linear second order differential equation.
Abstract: p−1 p p−1 . This equation is a special case of a general half-linear second order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2) where Φ(x) := |x| sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear Sturm-Liouville differential equation (r(t)x′)′ + c(t)x = 0,


Journal ArticleDOI
TL;DR: Battezzati and Magnasco as mentioned in this paper showed that this triple integral can be reduced to a single integral from which the asymptotic behaviour is readily obtained using Laplace's method.
Abstract: The asymptotic evaluation and expansion of the Keesom integral, K(a), is discussed at some length in Battezzati and Magnasco (2004 J. Phys. A: Math. Gen. 37 9677; 2005 J. Phys. A: Math. Gen. 38 6715). Here, using standard identities, it is shown that this triple integral can be reduced to a single integral from which the asymptotic behaviour is readily obtained using Laplace's method.

Journal ArticleDOI
TL;DR: In this article, the authors derived uniform asymptotic expansions of solutions to the fourth-order differential equation where x is a real variable and λ is a large positive parameter.
Abstract: In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x.



Journal ArticleDOI
TL;DR: An asymptotic formula for the solution of the ordinary differential Abel's equation of the first kind, which is uniform in the x-variable, is constructed and substantiated in this article.
Abstract: An asymptotic formula as for the solution of the ordinary differential Abel's equation of the first kind , which is uniform in the x-variable, is constructed and substantiated. Bibliography: 13 titles.



Journal ArticleDOI
TL;DR: In this paper, conditions under which asymptotic solutions of the Cauchy problem for the wave equation can be applied to the computation of a piston model of tsunami in a basin of constant depth are derived.
Abstract: Conditions are derived under which asymptotic solutions of the Cauchy problem for the wave equation can be applied to the computation of a piston model of tsunami in a basin of constant depth. The derivation is carried out for the initial displacement of the bottom in the form of Gaussian exponentials by comparing the results of numerical integration with computations using asymptotic formulas.