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Showing papers on "Asymptotology published in 2006"


Book
01 Jan 2006
TL;DR: Asymptotic analysis of exponential integrals has been studied extensively in the literature, see as discussed by the authors for a survey of asymptotics of exponential and weakly nonlinear waves.
Abstract: Fundamentals: Themes of asymptotic analysis The nature of asymptotic approximations Asymptotic analysis of exponential integrals: Fundamental techniques for integrals Laplace's method for asymptotic expansions of integrals The method of steepest descents for asymptotic expansions of integrals The method of stationary phase for asymptotic analysis of oscillatory integrals Asymptotic analysis of differential equations: Asymptotic behavior of solutions of linear second-order differential equations in the complex plane Introduction to asymptotics of solutions of ordinary differential equations with respect to parameters Asymptotics of linear boundary-value problems Asymptotics of oscillatory phenomena Weakly nonlinear waves Appendix: Fundamental inequalities Bibliography Index of names Subject index.

343 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic theory for a large class of Boltzmann type equations suitable to model the evolution of multicellular systems in biology with special attention to the onset of nonlinear phenomena was developed.
Abstract: This paper develops an asymptotic theory for a large class of Boltzmann type equations suitable to model the evolution of multicellular systems in biology with special attention to the onset of nonlinear phenomena. The mathematical method shows how various levels of diffusion phenomena, linear and non-linear, can be obtained by suitable asymptotic limits. The time scaling corresponding to different speeds related to cell movement and biological evolution plays a crucial role and different macroscopic equations corresponds to different scaling.

52 citations


Journal ArticleDOI
TL;DR: In this article, a new technique is proposed for the analysis of shape optimization problems using the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains.
Abstract: A new technique is proposed for the analysis of shape optimization problems. The technique uses the asymptotic analysis of boundary value problems in singularly perturbed geometrical domains. The asymptotics of solutions are derived in the framework of compound and matched asymptotics expansions. The analysis involves the so–called interior topology variations. The asymptotic expansions are derived for a model problem, however the technique applies to general elliptic boundary value problems. The self–adjoint extensions of elliptic operators and the weighted spaces with detached asymptotics are exploited for the modelling of problems with small defects in geometrical domains. The error estimates for proposed approximations of shape functionals are provided.

46 citations


Journal ArticleDOI
TL;DR: A novel technique unifies different approaches to asymptotic integration and addresses a new type of asymPTotic behavior in a class of second-order nonlinear differential equations locally near infinity.

35 citations


Journal ArticleDOI
TL;DR: The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed in this paper, where the leading terms of the expansion for eigenelements are constructed.
Abstract: The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues

30 citations


Journal ArticleDOI
TL;DR: In this paper, a space asymptotic model for the study of pulsed experiments in neutron multiplying systems is presented, where the Laplace transformed one-group transport equation is derived.

25 citations


Journal ArticleDOI
TL;DR: The notion of non-uniform Robust Global Asymptotic Stability (RGAS) presented in this paper generalizes the notion of uniform in time RGAS for finiteor infinite-dimensional discrete-time systems to finitedimensional discrete- time systems obtained by time discretization of continuous-time Systems by the explicit Euler method.
Abstract: The notion of non-uniform Robust Global Asymptotic Stability (RGAS) presented in this paper generalizes the notion of non-uniform in time RGAS for finiteor infinite-dimensional discrete-time systems. Lyapunov characterizations for this stability notion are provided. The results are applied to finitedimensional discrete-time systems obtained by time discretization of continuous-time systems by the explicit Euler method.

23 citations


Journal Article
TL;DR: In this article, the stability of a class of linear systems including gamma-distributed delay with a gap is studied. And a complete characterization of stability regions is given in the corresponding (delay, mean-delay) parameter space.
Abstract: This paper focuses on the stability of a class of linear systems including gamma-distributed delay with a gap. More precisely, a complete characterization of stability regions is given in the corresponding (delay, mean-delay) parameter-space. Optimal delay intervals are explicitly computed. The stabilizing/destabilizing delay effect will be explicitly outlined, and discussed. Several illustrative examples complete the paper.

21 citations


Journal ArticleDOI
TL;DR: In this paper, a general reaction-diffusion food-limited population model with time-delay is proposed, and the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered.

15 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of testing the hypothesis about the sub-mean vector and showed that the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-^1.

13 citations


Journal ArticleDOI
TL;DR: In this paper, the authors deal with the asymptotic stability analysis of a discrete dynamical inclusion whose right-hand side is a convex process and provide necessary and sufficient conditions for weak stability.

Journal ArticleDOI
TL;DR: For a class of nonlinear parabolic equations and inequalities, the authors established conditions guaranteeing that any solution tends to 0 as t → ∞, where t is the length of the shortest path.
Abstract: For a class of nonlinear parabolic equations and inequalities, we establish conditions guaranteeing that any solution tends to 0 as t → ∞.


01 Jan 2006
TL;DR: In this paper, uniform asymptotic approximations of Green's kernels for boundary value problems of elasticity in singularly perturbed domains containing small holes were presented for two and three dimensions.
Abstract: We present uniform asymptotic approximations of Green’s kernels for boundary value problems of elasticity in singularly perturbed domains containing small holes. We consider the case of an isotropic Lam e operator with the Dirichlet boundary conditions for two and three dimensions, in addition we shall also study the situation of anti-plane shear. The main feature of the asymptotic approximations mentioned is their uniformity with respect to the independent spatial variables. We also oer examples, where results of asymptotic approximations are compared with independent accurate numerical simulations, and demonstrate the superior features of the asymptotic method. The motivation for the talk, came from the asymptotic formulae derived

Journal ArticleDOI
TL;DR: In this article, the authors give an outline of mathematical models of multi-structures based on an asymptotic analysis of a class of elliptic boundary value problems posed in singularly perturbed domains.
Abstract: This review paper gives an outline of mathematical models of multi-structures. The approach is based on an asymptotic analysis of a class of elliptic boundary value problems posed in singularly perturbed domains. A particular attention is given to formulation of the “junction conditions” to link different parts of the multi-structure. The paper also discusses a range of applications of this asymptotic theory in physics and mechanics.

Journal ArticleDOI
TL;DR: In this paper, the conditions of asymptotic stability of second-order linear dynamic equations on time scales were examined, and the stability estimates were established by using integral representations of the solutions via asymPTotic solutions, error estimates, and calculus.
Abstract: We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales. To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.


Journal ArticleDOI
TL;DR: In this paper, the authors improved the second-order asymptotic theory of higher-order nonradial p-modes in spherically symmetric stars that was developed by Smeyers et al. as an alternative for Tassoul's approach.
Abstract: Aims. We improve the second-order asymptotic theory of higher-order non-radial p-modes in spherically symmetric stars that was developed by Smeyers et al. (1996) as an alternative for Tassoul’s approach (1990). Methods. Like the previous authors, we use asymptotic methods appropriate for singular perturbation problems, i.e. expansion procedures in terms of two variables and boundary-layer theory. However, in contrast with them, we no longer adopt boundary-layer coordinates near the singular boundary points that are identical to the fast variable used in the asymptotic expansions at larger distances. Results. By our definitions of the boundary-layer coordinates, the matchings of the boundary-layer expansions to the asymptotic expansions valid at larger distances from the boundary points, and the constructions of the uniformly valid asymptotic expansions are more transparent. Conclusions. The present asymptotic theory confirms that the application of expansions in terms of two variables and boundary-layer theory to the fourth-order system of differential equations established by Pekeris (1938, ApJ, 88, 189) is particularly appropriate for the construction of the asymptotic representation of higher-order p-modes in spherically symmetric stars. For these modes, the divergence of the Lagrangian displacement is the basic function, and the radial component of the Lagrangian displacement is of one order higher in the small expansion parameter. In the lowest-order asymptotic approximation, the divergence of the Lagrangian displacement obeys a second-order differential equation of the Sturm-Liouville type. This property explains that the eigenfunction that is associated with the nth eigenfrequency displays n − 1 nodes, with n = 1, 2, 3 ,. . .

Journal ArticleDOI
TL;DR: In this paper, the authors studied the asymptotic behavior of solutions of a Schrodinger-type equation with oscillating potential, which was studied by A. Its.
Abstract: We are interested in the asymptotic behavior of solutions of a Schrodinger-type equation with oscillating potential which was studied by A. Its. Here we use a different technique, based on Levinson's Fundamental Lemma, to analyze the asymptotic behavior, and our approach leads to a complete asymptotic representation of the solutions. We also discuss formal simplifications for differential equations with what might be called “regular/irregular singular points with periodic coefficients”. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Journal ArticleDOI
TL;DR: The asymPTotic expansion technique is applied to verify the order of accuracy of asymptotic expansion of linear and nonlinear initial value problems.

Journal ArticleDOI
TL;DR: Diverse problems arising in economics, engineering, the social sciences, medicine, physics, chemistry, and other areas can be modelled in such a way that the central limit theorem comes into play.
Abstract: 1. INTRODUCTION. In his 1974 book entitled \" The Life and Times of the Central Limit Theorem, \" Adams [1] describes this theorem as \" one of the most remarkable results in all of mathematics \" and \" a dominating personality in the world of probability and statistics. \" More than three decades later, his description is not only still pertinent but has also been corroborated and reinforced by developments in different branches of knowledge. In fact, the central limit theorem owes much of its importance to its proven application well beyond the field of probability. Diverse problems arising in economics, engineering, the social sciences, medicine, physics, chemistry, and other areas can be modelled in such a way that the central limit theorem comes into play. Empirically, one observes that a great many natural phenomena, such as the heights of individuals in a given population, obey an approximately normal distribution, that is, a symmetric bell-shaped distribution with scores more concentrated in the middle than in the tails (see Figure 1). One explanation suggested for this is that these phenomena are sums of a large number of independent random effects, none of which is predominant. Actually, the classical version of the central limit theorem asserts that the sum of many independent random variables is asymptotically normally distributed provided that each summand is small with high probability.

Journal Article
TL;DR: In this article, a sufficient condition is derived based on matrix inequality, in which the system is found to be impulse-free and asymptotic stable, and based on this result, the problem of stabilization is considered by using the Riccati equation corresponding to this Lyapunov equation.
Abstract: In this paper,the problem of asymptotic stability and the problem of stabilization for singular time-varying systems with time-delay are studied.First of all,a sufficient condition is derived based on matrix inequality,in which the system is found to be impulse-free and asymptotic stable.furthermore,based on this result,the problem of stabilization is considered by using the Riccati equation corresponding to this Lyapunov equation.Finally an example is given to illustrate the validity of the main results proposed.



Journal ArticleDOI
TL;DR: An asymptotic surrogate constraints method for the minimax semi-infinite programming problem is presented by making use of two general discrete approximation methods.

Journal ArticleDOI
TL;DR: In this paper, the authors derived expansions with exact asymptotic expressions for the remainders for solutions of multi-dimensional renewal equations, taking into account the effect of the roots of the characteristic equation on the representation of solutions.
Abstract: We derive expansions with exact asymptotic expressions for the remainders for solutions of multi-dimensional renewal equations. The effect of the roots of the characteristic equation on the asymptotic representation of solutions is taken into account. The resulting formulae are used to investigate the asymptotic behaviour of the average number of particles in age-dependent branching processes having several types of particles.

Journal ArticleDOI
TL;DR: In this paper, a perturbation method, the Lindstedt-Poincare method, was used to obtain the asymptotic expansions of the solutions of a nonlinear differential equation arising in general relativity.
Abstract: A perturbation method, the Lindstedt-Poincare method, is used to obtain the asymptotic expansions of the solutions of a nonlinear differential equation arising in general relativity. The asymptotic solutions contain no secular term, which overcomes a defect in Khuri’s paper. A technique of numerical order verification is applied to demonstrate that the asymptotic solutions are uniformly valid for small parameter.

Journal ArticleDOI
TL;DR: In this article, a workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and an other fields of applied mathematics such as shape optimization.
Abstract: This workshop focused on asymptotic analysis and its fundamental role in the derivation and understanding of the nonlinear structure of mathematical models in various fields of applications, its impact on the development of new numerical methods and an other fields of applied mathematics such as shape optimization. This was complemented by a review as well as the presentation of some of the latest developments of singular perturbation methods.

Journal ArticleDOI
TL;DR: In this article, sufficient conditions for singularly perturbed systems with piecewise continuous perturbations were obtained for an arbitrary infinite interval [t0,+∞] of the positive half-line not containing the initial time.
Abstract: with piecewise continuous perturbation Q(t), ‖Q(t)‖ ≤ δ, t ≥ 0. Starting from the fundamental papers by Tikhonov, numerous papers by Butuzov, Vasil’eva, Fedoryuk, Lomov, Rozov, Mishchenko, Vazov, Shishkin, et al. dealt with the analysis of singularly perturbed systems of a more general form. Necessary and sufficient conditions for all solutions y (t, y0, ε), y (0, y0, ε) = y0 ∈ R, of system (1(A+Q)/ε) with continuous matrix A(t) = diag [a1(t), . . . , an(t)] and with all possible perturbations Q(t) of sufficiently small norm to tend to zero as ε → +0 (for fixed t) on any finite interval [t0, t1] of the positive half-line not containing the initial time were obtained in [1]. These conditions are as follows: ∫ t 0 maxi {ai(τ)} dτ < 0, i = 1, . . . , n, for all t ∈ [t0, t1] ⊂ (0, t1]. In the present paper, similar conditions are obtained for an arbitrary infinite interval [t0,+∞) ⊂ (0,+∞). Note that, in this case, the condition ∫ t 0 maxi {ai(τ)} dτ < 0, t ∈ [t0,+∞), does not guarantee that ‖y (t, y0, ε)‖ → 0 as ε → +0 for arbitrary perturbations of sufficiently small norm on the entire interval (0,+∞). Let us illustrate this by an example. Example. For all solutions x (t, x0, ε) of the scalar equation εẋ = −(t + 1)−1x, x ∈ R, ε ∈ (0, 1], t ≥ 0, we have |x (t, x0, ε)| = |x0| exp [−ε−1 ln(t + 1)] → 0 as ε → +0 for all t ∈ [t0,+∞) ⊂ (0,+∞) and for an arbitrary x0 ∈ R. But if we consider the singularly perturbed equation εẏ = −(t + 1)−1y + δy, y ∈ R, ε ∈ (0, 1], δ > 0, t ≥ 0, whose solutions have the form y (t, x0, ε) = y0 exp [ε−1(δt − ln(t + 1))], then for an arbitrarily small δ > 0, there exists a sufficiently large time T = T (δ), determined by the relation δT ≥ ln(T + 1), such that |y (t, y0, ε)| → +∞ as ε → +0 for all t > T . Theorem. The solutions y (t, y0, ε) of the linear system (1(A+Q)/ε) with a continuous matrix A(t) = diag[a1(t), . . . , an(t)] satisfy the relation limε→+0 y (t, y0, ε) = 0 (uniformly with respect to t ∈ [t0,+∞) ⊂ (0,+∞)) for all y0 = y (0, y0, ε) ∈ R and for arbitrary piecewise continuous

Journal ArticleDOI
TL;DR: In this article, the authors considered the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root AR(1) with fixed, non-zero startup.