Showing papers on "Asymptotology published in 2002"
Book•
08 Feb 2002
TL;DR: In this paper, a distributional theory for asymptotic expansions of generalized functions is presented, based on a series of Dirac Delta functions, which are considered by Ramanujan and others.
Abstract: Preface * 1. Basic Results in Asymptotics * 2. Introduction to the Theory of Distributions * 3. A Distributional Theory for Asymptotic Expansions * 4. Asymptotic Expansion of Multi-Dimensional Generalized Functions * 5. Asymptotic Expansion of Certain Series Considered by Ramanujan * 6. Cesaro Behavior of Distributions * 7. Series of Dirac Delta Functions * References * Index
98 citations
01 Jan 2002
TL;DR: In this paper, the Hille-Yosida type theorems settle the question of well-posedness to a great extend, many perturbation and approximation results have been established, and for a large class of problems the asymptotic behaviour can be studied on the basis of spectral theory and transform methods.
Abstract: There is a striking difference between autonomous and non-autonomous linear evolution equations. Autonomous problems are well understood in the framework of strongly continuous operator semigroups and their generalizations. The Hille-Yosida type theorems settle the question of well-posedness to a great extend, many perturbation and approximation results have been established, and for a large class of problems the asymptotic behaviour can be studied on the basis of spectral theory and transform methods. In these and many other areas semigroup theory has reached a considerable degree of maturity, and its applications thrive in plenty of fields.
71 citations
TL;DR: In this paper, the asymptotic properties of least squares estimates of Hammerstein-Wiener model structures are investigated, and a consistent procedure for the estimation of the variance of the parameter estimates is provided.
Abstract: This paper investigates the asymptotic properties of least squares estimates of Hammerstein-Wiener model structures, and in doing so establishes consistency and asymptotic normality under fairly mild conditions on the additive noise process, the inputs and the static non-linearities. In relation to the asymptotic distributional results, a consistent procedure for the estimation of the asymptotic variance of the parameter estimates is provided. A key theme of this paper is to demonstrate how recent results from the econometrics literature may be employed in an engineering setting. In this respect the Hammerstein-Wiener model structure serves as a demonstration example. A simulation study complements the theoretical findings.
38 citations
Book•
31 Jan 2002
TL;DR: The Navier-Fourier Viscous Incompressible Model and the Triple-Deck Model as discussed by the authors are three specific asymptotic models for nonlinear acoustics.
Abstract: Preface and Acknowledgments. 1. Introductory Comments and Summary. Part I: Setting the Scene. 2. Newtonian Fluid Flow: Equations and Conditions. 3. Some Basic Aspects of Asymptotic Analysis and Modelling. 4. Useful Limiting Forms of the NS-F Equations. Part II: Main Asymptotic Models. 5. The Navier-Fourier Viscous Incompressible Model. 6. The Inviscid/Nonviscous Euler Model and Some Hydro-Aerodynamics Problems. 7. Boundary-Layer Models for High-Reynolds Numbers. 8. Some Models of Nonlinear Acoustics. 9. Low-Reynolds Numbers asymptotics. Part III: Three Specific Asymptotic Models. 10. Asymptotic Modelling of Thermal Convection and Interfacial Phenomena. 11. Meteo-Fluid-Dynamics Models. 12. Singular Coupling and the Triple-Deck Model. References.
35 citations
10 Dec 2002
TL;DR: In this article, the convergence to the asymptotic results is so fat that even for moderate parameter values they yield results close to the true values, and the authors illustrate this principle through a number of examples taken from multiple-antenna systems.
Abstract: Asymptotic theorems are very commonly used in probability. For systems whose performance depends on a set of random variables, asymptotic analyses are often used to simplify calculations and obtain results yielding useful hints at the behavior of the system when the parameters take on finite values. These asymptotic analyses are especially useful whenever the convergence to the asymptotic results is so fat that even for moderate parameter values they yield results close to the true values. The goal of this paper is to illustrate this principle through a number of examples taken from multiple-antenna systems.
32 citations
Book•
30 Nov 2002
TL;DR: Asymptotic Methods and Physical Theories as mentioned in this paper have been studied in a wide range of applications in physics, from Harmonic Waves to Solitons to scaling transformations.
Abstract: Foreword. Preface. Acknowledgments. Synopsis. 1. Introduction. 2. What Are Asymptotic Methods? 3. A Little Mathematics. 4. How Asymptotic Methods Work. 5. Asymptotic Methods and Physical Theories. 6. Phenomenology and First Principles. 7. A Little History. 8. Fathers of Asymptotic Methods. 9. Conclusion. Appendices: A. Linear and Nonlinear Mathematical Physics: from Harmonic Waves to Solitons. B. Certain Mathematical Notions of Catastrophe Theory. C. Asymptotics and Scaling Transformations. D. Asymptotic Approaches: Attempt af a Definition. E. Some Web-pages. References. About the Authors. Author Index. Topic Index.
25 citations
Dissertation•
06 Mar 2002
15 citations
Posted Content•
13 citations
TL;DR: In this paper, a solution is obtained using uniform asymptotic formulae along with semiclassic methods, based on Olver's theory (Asymptotics and Special Functions, Academic Press, New York, 1974).
Abstract: This work deals with the problem of the calculation of the radial spheroidal functions. A solution is obtained using uniform asymptotic formulae along with semiclassic methods. The present uniform asymptotic method is based on Olver's theory (Asymptotics and Special Functions, Academic Press, New York, 1974) to asymptotic solutions of the differential equations. Here we show that it is possible to accurately interpolate the spheroidal radial function in the real plane using Airy's functions.
11 citations
TL;DR: In this paper, the eigenvalue problem for the Schrodinger equation with integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity was considered.
Abstract: We consider the eigenvalue problem for the two-dimensional Schrodinger equation containing an integral Hartree-type nonlinearity with an interaction potential having a logarithmic singularity. Global asymptotic solutions localized in the neighborhood of a line segment in the plane are constructed using the matching method for asymptotic expansions. The Bogoliubov and Airy polarons are used as model functions in these solutions. An analogue of the Bohr–Sommerfeld quantization rule is established to find the related series of eigenvalues.
7 citations
01 Jan 2002
TL;DR: In this article, the complex amplitude and the expected photon number for the thermal states were simultaneously measured for state estimation in two different cases: in one case we cannot use the quantum correlations between samples, and in the other case we can use them.
Abstract: Concerning state estimation, we will compare two cases. In one case we cannot use the quantum correlations between samples. In the other case, we can use them. In addition, under the later case, we will propose a method which simultaneously measures the complex amplitude and the expected photon number for the thermal states.
01 Jan 2002
TL;DR: In this article, the generalized Epstein-Hubbell integral is considered for values of the variable k close to its upper limit k = 1, and the distributional approach is used for deriving two convergent expansions of this integral in increasing powers of 1 - k 2.
Abstract: The generalized Epstein-Hubbell integral recently introduced by Kalla & Tuan (Comput. Math. Applic. 32,1996) is considered for values of the variable k close to its upper limit k = 1. Distributional approach is used for deriving two convergent expansions of this integral in increasing powers of 1 - k 2 . For certain values of the parameters, one of these expansions involves also a logarithmic term in the asymptotic variable 1 - k 2 . Coefficients of these expansions are given in terms of the Appell function and its derivative. All the expansions are accompanied by an error bound at any order of the approximation. Numerical experiments show that this bound is considerably accurate.
TL;DR: This paper provides a personal overview of the local asymptotic normality (LAN) results for linear processes, nonlinear processes, diffusion processes, long-memory processes, and locally stationary processes, etc..
Abstract: The local asymptotic normality (LAN) , introduced by LeCam, is the most fundamental concept in the statistical asymptotic theory. If LAN property for a class of statistical models is established, then the asymptotic optimality of estimator and test can be described in terms of the central sequence. This concept gives a unified view for the statistical estimation and testing theory. Recently the LAN concept has been introduced to the asymptotic theory for time series. This paper provides a personal overview of the LAN results for linear processes, nonlinear processes, diffusion processes, long-memory processes, and locally stationary processes, etc.. The results are applied to the asymptotic estimation, testing theory, and discriminant analysis in time series. Then, construction of asymptotically optimal estimator, test and discriminator is discussed.
TL;DR: In this article, asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrodinger equation in two-dimensional spaces was studied. But the authors focused on the time complexity of small solutions.
Abstract: We study asymptotic behavior in time of global small solutions to the quadratic nonlinear Schrodinger equation in two-dimensional spaces i∂tu
TL;DR: In this paper, an asymptotic analysis of a class of parabolic systems arising from singularly perturbed diffusions is presented, where the underlying system has a fast varying component and a slowly changing component.
Abstract: This work is concerned with asymptotic properties of a class of parabolic systems arising from singularly perturbed diffusions. The underlying system has a fast varying component and a slowly changing component. One of the distinct features is that the fast varying diffusion is transient. Under such a setup, this paper presents an asymptotic analysis of the solutions of such parabolic equations. Asymptotic expansions of functional satisfying the parabolic system are obtained. Error bounds are derived.
TL;DR: In this article, global asymptotic stability and behaviour of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global stability are derived, which are then used to derive the global as-ymptotics of solutions of delay difference problems.
Abstract: In this paper global asymptotic stability and asymptotic behaviour of solutions of nonlinear delay difference equations has been studied and a few sets of sufficient conditions for global asymptotic stability are derived.
10 Dec 2002
TL;DR: In this paper, a simple criterion for global asymptotic stability is established for homogeneous cooperative systems, a class of positive systems, and it is shown that these systems admit a fairly simple asymPTotic behavior, thereby generalizing the Perron-Frobenius theorem from linear to homogeneous systems.
Abstract: This paper deals with homogeneous cooperative systems, a class of positive systems. It is shown that they admit a fairly simple asymptotic behavior, thereby generalizing the well-known Perron-Frobenius theorem from linear to homogeneous systems. As a corollary a simple criterion for global asymptotic stability is established. Then these systems are subject to constant inputs and we prove that asymptotic stability of the uncontrolled system is inherited by the new equilibrium point of the controlled system. Recent results on monotone control systems indicate the importance of this property in proving small gain theorems.
TL;DR: In this article, a comparison of matching asymptotic solutions for the limiting case of the restricted three-body problem by the use of perturbation methods is presented, which is of a singular-perturbation type.
Abstract: This work presents a comparison of matching asymptotic solutions for the limiting case of the restricted three-body problem by the use of perturbation methods. The problem is of a singular-perturbation type. We investigate two alternative methods to deal with it: the classical method of matched asymptotic expansions and the improved method of matched asymptotic expansions. Two expansions, outer and inner, are involved. The outer expansion breaks down in the inner region where sharp changes occur, and the inner expansion becomes nonuniformly valid in the outer region. To obtain a uniformly valid composite solution, we need a matching procedure to relate these two expansions. Instead of straightforward matching of the outer and inner expansions to higher-order terms, in the improved technique the higher-order solutions are derived by generating perturbations between the lower-order composite solutions and the exact solutions. The perturbation equations are then integrated in the outer and inner regions, respectively, for a higher-order matching. Improved asymptotic solutions of second order are obtained for the limiting case of the restricted three-body problem. Compared to the solutions obtained by the classical method of matched asymptotic expansions and the pure numerical integration for various values of a small parameter μ, the improved asymptotic solutions are very accurate. Moreover, the asymptotic solutions obtained by use of the improved method give better accuracy than those using the classical method over wide ranges of the small parameter.
TL;DR: Analysis of Time Series Structure: SSA and Related Techniques is a wellresearched introduction to the basic methodology and theory behind SSA, and addresses issues of existence, uniqueness, rank, and separability.