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Showing papers on "Asymptotology published in 2003"


Journal ArticleDOI
TL;DR: A turning-point theory is developed for the second-order difference equation where the coefficients An and Bn have asymptotic expansions of the form θ≠0 being a real number and it is shown how the Airy functions arise in the uniform asym PT expansion of the solutions to this three-term recurrence relation.
Abstract: A turning-point theory is developed for the second-order difference equation $$$$ where the coefficients An and Bn have asymptotic expansions of the form $$$$ θ≠0 being a real number. In particular, it is shown how the Airy functions arise in the uniform asymptotic expansions of the solutions to this three-term recurrence relation. As an illustration of the main result, a uniform asymptotic expansion is derived for the orthogonal polynomials associated with the Freud weight exp(−x4), xℝ.

53 citations


Journal ArticleDOI
TL;DR: In this paper, the authors investigated the global asymptotic behavior of solutions of the system of difference equations where the parameters A and B are in (0, ∞) and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers.
Abstract: Dedicated to Allan Peterson on the Occasion of His 60th Birthday. We investigate the global asymptotic behavior of solutions of the system of difference equations where the parameters A and B are in (0, ∞) and the initial conditions x 0 and y 0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into the basins of attraction of two types of asymptotic behavior.

19 citations


Journal ArticleDOI
TL;DR: This tutorial paper illustrates the principle of convergence to the asymptotic results is so fast that even for moderate n they yield results close to the true values by applying it to capacity calculations of multiple-antenna systems.
Abstract: Asymptotic theorems are very commonly used in probability. For systems whose performance depends on a set of n random parameters, asymptotic analyses for n → ∞ are often used to simplify calculations and obtain results yielding useful hints at the behavior of the system for finite n. These asymptotic analyses are especially useful whenever the convergence to the asymptotic results is so fast that even for moderate n they yield results close to the true values. This tutorial paper illustrates this principle by applying it to capacity calculations of multiple-antenna systems.

18 citations


01 Jan 2003
TL;DR: In this article, an asymptotic representation for a fundamental solution matrix for scalar linear dynamic systems on time scales is given, which is a generalization of the usual exponential function.
Abstract: We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.

15 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered).
Abstract: We study the Dirichlet problem in half-space for the equation ∆u+ g(u)|∇u|2 = 0, where g is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as y → ∞, where y denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function. Introduction The phenomenon called stabilization is well known for parabolic equations both in linear (see e.g. [1] and references therein) and non-linear (see e.g. [2] and references therein) cases; it means the existence of a finite limit of the solution as t → ∞. However, there are well-posed non-isotropic elliptic boundary-value problems in unbounded domains (see e.g. [3]) for which we can talk about stabilization in the following sense: the solution has a finite limit as a selected spatial variable tends to infinity. This paper is devoted to the Dirichlet problem in half-space for elliptic equations. We present necessary and sufficient conditions for the stabilization of its solution; here the spatial variable, orthogonal to the hyperplane of boundary-value data, plays the role of time. In Section 1, the linear case is presented; Sections 2 and 3 are devoted to quasi-linear equations with the so-called Burgers-Kardar-ParisiZhang non-linearity type (see e.g. [4], [5]). Equations with such non-linearities arise, for example, in modeling of directed polymers and interface growth. They also present an independent theoretical interest because they contain second powers of the first derivatives (see e.g. [6] and references therein). Note that we deal with the stabilization problem in cylindrical domains with an unbounded base (in particular, here the base of the cylinder is the whole E ). As in the parabolic case, this problem is principally different (this refers both to the results and to the methods of research) from the stabilization problem in cylindrical domains with a bounded base. The latter problem has been investigated Received by the editors March 6, 2002. 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B40, 35J60.

13 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the asymptotic behavior of a mutation-selection genetic algorithm on the integers with finite population of size p\ge 1, defined by the steps of a simple random walk and the fitness function is linear.
Abstract: We study the asymptotic behavior of a mutation--selection genetic algorithm on the integers with finite population of size $p\ge 1$. The mutation is defined by the steps of a simple random walk and the fitness function is linear. We prove that the normalized population satisfies an invariance principle, that a large-deviations principle holds and that the relative positions converge in law. After $n$ steps, the population is asymptotically around $\sqrt{n}$ times the position at time $1$ of a Bessel process of dimension $2p-1$.

8 citations


Journal ArticleDOI
TL;DR: In this paper, asymptotic estimates for singularly perturbed boundary value problems with initial jumps were obtained for the case of single-jump boundary-value problems, where the initial jump is independent of the boundary value.
Abstract: We obtain asymptotic estimates for solutions of singularly perturbed boundary-value problems with initial jumps.

6 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that the asymptotic theory of vector fields as a class is undecidable in a strong sense, which precludes a geometric answer to certain generalizations of the Grothendieck-Katz conjecture.
Abstract: We relate the integrability of vector fields, and of the vanishing of $p$-torsion, to model-theoretic questions concerning separably closed fields, endowed canonically with a derivation. While each differential field $(F_p(t)^s,D_p)$ is known to be decidable, we show that the asymptotic theory of these fields as a class is undecidable in a strong sense. This precludes a geometric answer to certain generalizations of the Grothendieck-Katz conjecture.

6 citations


Journal ArticleDOI
TL;DR: In this article, a first-order asymptotic representation is developed for low and intermediate-degree p-modes in stars for which the lower boundary of the resonant acoustic cavity is not located close to the star's centre.
Abstract: A first-order asymptotic representation is developed for low- and intermediate-degree p-modes in stars for which the lower boundary of the resonant acoustic cavity is not located close to the star's centre. To this end, afourth-order system of differential equations in the radial parts of the divergence and the radial component of the Lagrangian displacement is adopted. The lower boundary of the resonant acoustic cavity is considered to be a turning point for one of the differential equations. As in a previous asymptotic study of low-degree p-modes with high radial orders, asymptotic expansion procedures applying to self-adjoint second-order differential equations with a large parameter are used by extension of these methods. The main result is that, in contrast with the usual first-order asymptotic theory for low-degree p-modes of high radial orders, the present first-order asymptotic representation leads to small frequency separations D n , f different from zero. The validity of the asymptotic representation is tested for p-modes of the equilibrium sphere with uniform mass density, since the modes of this model are determined by means of exact analytical solutions.

5 citations


Journal ArticleDOI
V. B. Kolmanovskii1
TL;DR: Conditions of the boundedness of the solutions, stability in the first approximation, and asymptotic equivalence for discrete Volterra-type equations are proven.
Abstract: Conditions of the boundedness of the solutions, stability in the first approximation, and asymptotic equivalence for discrete Volterra-type equations are proven. All are formulated in terms of the characteristics of the equations, using an operator approach.


Journal ArticleDOI
TL;DR: In this paper, a system of conjugate elliptic equations with a small parameter at the highest derivatives is considered in a rectangle with two sides parallel to the characteristics of the limiting equations.
Abstract: A system of two conjugate elliptic equations with a small parameter at the highest derivatives is considered in a rectangle with two sides parallel to the characteristics of the limiting equations.The method of matched asymptotic expansions is used for the construction of uniform asymptotic series for solutions of this system up to an arbitrary power of the small parameter.

Journal ArticleDOI
TL;DR: In this article, an asymptotic model from a nonlinear model with thin layers was derived, where the thickness goes to 0 and the problem is written in a functional minimisation form and the analysis is performed using epiconvergence.
Abstract: We derive an asymptotic model from a nonlinear model with thin layers letting the thickness goes to 0. Usually these problems are written in a functional minimisation form and the asymptotic analysis is performed using epiconvergence. Here we give a direct proof by extending the method used on a linear model by E. Sanchez-Palencia in [J. Math. Pures et Appl., 53, 1974, 711–740]. Finally we give an application to a water transfert project studied by the Yangtse River Scientific Research Institute in China.



Journal ArticleDOI
TL;DR: In this article, asymptotic methods for contact problems are expounded and some typical integral equations are considered, where the authors consider contact problems with respect to a set of contact problems.
Abstract: Asymptotic methods for contact problems are expounded. Some typical integral equations are considered


Journal ArticleDOI
TL;DR: The topics discussed include the quantization dimension, asymptotic distributions of sets of prototypes,Asymptotically optimal quantizations, approximations and random quantizations.
Abstract: We give a brief introduction to results on the asymptotics of quantization errors. The topics discussed include the quantization dimension, asymptotic distributions of sets of prototypes, asymptotically optimal quantizations, approximations and random quantizations.

Journal ArticleDOI
TL;DR: In this article, the asymptotic properties of functional differential equations in Banach spaces are studied and the criteria of the invariant and attracting sets are obtained, and the sufficient condition of stability of the equilibrium point is given as the system has an equilibrium point.
Abstract: The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces. The criteria of the invariant and attracting sets are obtained. Particularly, the sufficient condition of asymptotic stability of the equilibrium point is given as the system has an equilibrium point. Several examples are also worked out to demonstrate the validity of the results.


01 Jan 2003
TL;DR: In this paper, a simplified accoustic model to describe nonlinear phenomena occuring in loudspeakers was proposed, which restricted to the one-dimensional isentropic Euler equations in a slab, where on the right end a membrane is moving periodically with frequency ω and maximal displacement e œ 1.
Abstract: We consider a simplified accoustic model to describe nonlinear phenomena occuring in loudspeakers. The simplification is that we restrict to the one–dimensional isentropic Euler equations in a slab, where on the right end a membrane is moving periodically with frequency ω and maximal displacement e œ 1. The asymptotic model based on the small parameter e yields hyperbolic first order systems, which are investigated numerically for two different frequencies ω.

Journal Article
TL;DR: In this article, the authors investigated the relation between a number of premises of an implicational formula and the asymptotic probability of finding a formula with this number of bases.
Abstract: This paper presents the number of results concerning problems of asymptotic densities in the variety of propositional logics. We investigate, for propositional formulas, the proportion of tautologies of the given length n against the number of all formulas of length n. We are specially interested in asymptotic behavior of this fraction. We show what the relation between a number of premises of an implicational formula and asymptotic probability of finding a formula with this number of premises is. Furthermore we investigate the distribution of this asymptotic probabilities. Distribution for all formulas is contrasted with the same distribution for tautologies only.