A Distributional Approach To Asymptotics

Author : Ricardo Estrada,Ram P. Kanwal
Publisher : Springer Science & Business Media
Page : 467 pages
File Size : 43,5 Mb
Release : 2012-09-08
Category : Mathematics
ISBN : 9780817681302

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A Distributional Approach to Asymptotics by Ricardo Estrada,Ram P. Kanwal Pdf

"...The authors of this remarkable book are among the very few who have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book...[it] should be on your bookshelf if you are at all interested in knowing what an asymptotic series is." -"The Bulletin of Mathematics Books" (Review of the 1st edition) ** "...The book is a valuable one, one that many applied mathematicians may want to buy. The authors are undeniably experts in their field...most of the material has appeared in no other book." -"SIAM News" (Review of the 1st edition) This book is a modern introduction to asymptotic analysis intended not only for mathematicians, but for physicists, engineers, and graduate students as well. Written by two of the leading experts in the field, the text provides readers with a firm grasp of mathematical theory, and at the same time demonstrates applications in areas such as differential equations, quantum mechanics, noncommutative geometry, and number theory. Key features of this significantly expanded and revised second edition: * addition of a new chapter and many new sections * wide range of topics covered, including the Ces.ro behavior of distributions and their connections to asymptotic analysis, the study of time-domain asymptotics, and the use of series of Dirac delta functions to solve boundary value problems * novel approach detailing the interplay between underlying theories of asymptotic analysis and generalized functions * extensive examples and exercises at the end of each chapter * comprehensive bibliography and index This work is an excellent tool for the classroom and an invaluable self-study resource that will stimulate application of asymptotic

Author : Ricardo Estrada,Ram P. Kanwal
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 45,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468400298

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Asymptotic Analysis by Ricardo Estrada,Ram P. Kanwal Pdf

Asymptotic analysis is an old subject that has found applications in vari ous fields of pure and applied mathematics, physics and engineering. For instance, asymptotic techniques are used to approximate very complicated integral expressions that result from transform analysis. Similarly, the so lutions of differential equations can often be computed with great accuracy by taking the sum of a few terms of the divergent series obtained by the asymptotic calculus. In view of the importance of these methods, many excellent books on this subject are available [19], [21], [27], [67], [90], [91], [102], [113]. An important feature of the theory of asymptotic expansions is that experience and intuition play an important part in it because particular problems are rather individual in nature. Our aim is to present a sys tematic and simplified approach to this theory by the use of distributions (generalized functions). The theory of distributions is another important area of applied mathematics, that has also found many applications in mathematics, physics and engineering. It is only recently, however, that the close ties between asymptotic analysis and the theory of distributions have been studied in detail [15], [43], [44], [84], [92], [112]. As it turns out, generalized functions provide a very appropriate framework for asymptotic analysis, where many analytical operations can be performed, and also pro vide a systematic procedure to assign values to the divergent integrals that often appear in the literature.

Author : Ricardo Estrada,Ram P. Kanwal
Publisher : Birkhauser
Page : 268 pages
File Size : 50,6 Mb
Release : 2012
Category : Mathematics
ISBN : 1468400312

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Asymptotic Analysis by Ricardo Estrada,Ram P. Kanwal Pdf

1 Basic Results in Asymptotics.- 1.1 Introduction.- 1.2 Order Symbols.- 1.3 Asymptotic Series.- 1.4 Algebraic and Analytic Operations.- 1.5 Existence of Functions with a Given Asymptotic Expansion.- 1.6 Asymptotic Power Series in a Complex Variable.- 1.7 Asymptotic Approximation of Partial Sums.- 1.8 The Euler-Maclaurin Summation Formula.- 2 Introduction to the Theory of Distributions.- 2.1 Introduction.- 2.2 The Space of Distributions $$\mathcal{D}'$$.- 2.3 Algebraic and Analytic Operations.- 2.4 Regularization, Pseudofunction and Hadamard Finite Part.- 2.5 Support and Order.- 2.6 Homogeneous Distributions.- 2.7 Distributional Derivatives of Discontinuous Functions.- 2.8 Tempered Distributions and the Fourier Transform.- 2.9 Distributions of Rapid Decay.- 2.10 Spaces of Distributions Associated with an Asymptotic Sequence.- 3 A Distributional Theory of Asymptotic Expansions.- 3.1 Introduction.- 3.2 The Taylor Expansion of Distributions.- 3.3 The Moment Asymptotic Expansion.- 3.4 Expansions in the Space $$\mathcal{P}'$$.- 3.5 Laplace's Asymptotic Formula.- 3.6 The Method of Steepest Descent.- 3.7 Expansion of Oscillatory Kernels.- 3.8 The Expansion of f(?x) as ? -> ? in Other Cases.- 3.9 Asymptotic Separation of Variables.- 4 The Asymptotic Expansion of Multidimensional Generalized Functions.- 4.1 Introduction.- 4.2 Taylor Expansion in Several Variables.- 4.3 The Multidimensional Moment Asymptotic Expansion.- 4.4 Laplace's Formula.- 4.5 Fourier Type Integrals.- 4.6 Further Examples.- 4.7 Tensor Products and Partial Asymptotic Expansions.- 4.8 An Application in Quantum Mechanics.- 5 The Asymptotic Expansion of Certain Series Considered by Ramanujan.- 5.1 Introduction.- 5.2 Basic Formulas.- 5.3 Lambert Type Series.- 5.4 Distributionally Small Sequences.- 5.5 Multiple Series.- 6 Series of Dirac Delta Functions.- 6.1 Introduction.- 6.2 Basic Notions.- 6.3 Several Problems That Lead to Series of Deltas.- 6.4 Dual Taylor Series as Asymptotics of Solutions of Differential Equations.- 6.5 Singular Perturbations.- References.

Author : Ricardo Estrada,Ram P. Kanwal
Publisher : Unknown
Page : 478 pages
File Size : 44,5 Mb
Release : 2011-04-13
Category : Electronic
ISBN : 0817681310

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A Distributional Approach to Asymptotics by Ricardo Estrada,Ram P. Kanwal Pdf

Author : Igor V. Andrianov,Jan Awrejcewicz,Leonid I. Manevitch
Publisher : Springer Science & Business Media
Page : 527 pages
File Size : 53,9 Mb
Release : 2013-06-29
Category : Science
ISBN : 9783540452461

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Asymptotical Mechanics of Thin-Walled Structures by Igor V. Andrianov,Jan Awrejcewicz,Leonid I. Manevitch Pdf

In this book a detailed and systematic treatment of asymptotic methods in the theory of plates and shells is presented. The main features of the book are the basic principles of asymptotics and their applications, traditional approaches such as regular and singular perturbations, as well as new approaches such as the composite equations approach. The book introduces the reader to the field of asymptotic simplification of the problems of the theory of plates and shells and will be useful as a handbook of methods of asymptotic integration. Providing a state-of-the-art review of asymptotic applications, this book will be useful as an introduction to the field for novices as well as a reference book for specialists.

Author : Igor V. Andrianov,Jan Awrejcewicz
Publisher : CRC Press
Page : 409 pages
File Size : 43,7 Mb
Release : 2024-05-20
Category : Mathematics
ISBN : 9781040032770

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Asymptotic Methods for Engineers by Igor V. Andrianov,Jan Awrejcewicz Pdf

Asymptotic Methods for Engineers is based on the authors’ many years of practical experience in the application of asymptotic methods to solve engineering problems. This book is devoted to modern asymptotic methods (AM), which is widely used in engineering, applied sciences, physics, and applied mathematics. Avoiding complex formal calculations and justifications, the book’s main goal is to describe the main ideas and algorithms. Moreover, not only is there a presentation of the main AM, but there is also a focus on demonstrating their unity and inseparable connection with the methods of summation and asymptotic interpolation. The book will be useful for students and researchers from applied mathematics and physics and of interest to doctoral and graduate students, university and industry professors from various branches of engineering (mechanical, civil, electro-mechanical, etc.).

Author : Igor Andrianov,Jan Awrejcewicz,Vladyslav Danishevs'kyy,Andrey Ivankov
Publisher : John Wiley & Sons
Page : 281 pages
File Size : 48,9 Mb
Release : 2014-02-06
Category : Science
ISBN : 9781118725146

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Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions by Igor Andrianov,Jan Awrejcewicz,Vladyslav Danishevs'kyy,Andrey Ivankov Pdf

Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions comprehensively covers the theoretical background of asymptotic approaches and their use in solving mechanical engineering-oriented problems of structural members, primarily plates (statics and dynamics) with mixed boundary conditions. The first part of this book introduces the theory and application of asymptotic methods and includes a series of approaches that have been omitted or not rigorously treated in the existing literature. These lesser known approaches include the method of summation and construction of the asymptotically equivalent functions, methods of small and large delta, and the homotopy perturbations method. The second part of the book contains original results devoted to the solution of the mixed problems of the theory of plates, including statics, dynamics and stability of the studied objects. In addition, the applicability of the approaches presented to other related linear or nonlinear problems is addressed. Key features: • Includes analytical solving of mixed boundary value problems • Introduces modern asymptotic and summation procedures • Presents asymptotic approaches for nonlinear dynamics of rods, beams and plates • Covers statics, dynamics and stability of plates with mixed boundary conditions • Explains links between the Adomian and homotopy perturbation approaches Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions is a comprehensive reference for researchers and practitioners working in the field of Mechanics of Solids and Mechanical Engineering, and is also a valuable resource for graduate and postgraduate students from Civil and Mechanical Engineering.

Author : Wojciech Pietraszkiewicz,Czeslaw Szymczak
Publisher : CRC Press
Page : 656 pages
File Size : 55,8 Mb
Release : 2005-09-22
Category : Technology & Engineering
ISBN : 0415383900

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Shell Structures, Theory and Applications by Wojciech Pietraszkiewicz,Czeslaw Szymczak Pdf

Shells are basic structural elements of modern technology. Examples of shell structures include automobile bodies, domes, water and oil tanks, pipelines, ship hulls, aircraft fuselages, turbine blades, laudspeaker cones, but also balloons, parachutes, biological membranes, a human skin, a bottle of wine or a beer can. This volume contains full texts of over 100 papers presented by specialists from over 20 countries at the 8th Conference "Shell Structures: Theory and Applications", 12-14 October, 2005 in Jurata (Poland). The aim of the meeting was to bring together scientists, designers, engineers and other specialists in shell structures in order to discuss important results and new ideas in this field. The goal is to pursue more accurate theoretical models, to develop more powerful and versatile methods of analysis, and to disseminate expertise in design and maintenance of shell structures. Among the authors there are many distinguished specialists of shell structures, including the authors of general lectures: I.V. Andrianov (Ukraine), V.A. Eremeyev (Russia), A. Ibrahimbegovic (France), P. Klosowski (Poland), B.H. Kröplin (Germany), E. Ramm (Germany), J.M. Rotter (UK) and D. Steigmann (USA). The subject area of the papers covers various theoretical models and numerical analyses of strength, dynamics, stability, optimization etc. of different types of shell structures, their design and maintenance, as well as modelling of some surface-related mechanical phenomena.

Author : Stevan Pilipovi?,Bogoljub Stankovi?,Jasson Vindas
Publisher : World Scientific
Page : 309 pages
File Size : 52,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814366847

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Asymptotic Behavior of Generalized Functions by Stevan Pilipovi?,Bogoljub Stankovi?,Jasson Vindas Pdf

The asymptotic analysis has obtained new impulses with the general development of various branches of mathematical analysis and their applications. In this book, such impulses originate from the use of slowly varying functions and the asymptotic behavior of generalized functions. The most developed approaches related to generalized functions are those of Vladimirov, Drozhinov and Zavyalov, and that of Kanwal and Estrada. The first approach is followed by the authors of this book and extended in the direction of the S-asymptotics. The second approach ? of Estrada, Kanwal and Vindas ? is related to moment asymptotic expansions of generalized functions and the Ces'aro behavior. The main features of this book are the uses of strong methods of functional analysis and applications to the analysis of asymptotic behavior of solutions to partial differential equations, Abelian and Tauberian type theorems for integral transforms as well as for the summability of Fourier series and integrals. The book can be used by applied mathematicians, physicists, engineers and others who use classical asymptotic methods and wish to consider non-classical objects (generalized functions) and their asymptotics now in a more advanced setting.

Author : Gaëtan Borot,Alice Guionnet,Karol K. Kozlowski
Publisher : Springer
Page : 222 pages
File Size : 44,6 Mb
Release : 2016-12-08
Category : Science
ISBN : 9783319333793

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Asymptotic Expansion of a Partition Function Related to the Sinh-model by Gaëtan Borot,Alice Guionnet,Karol K. Kozlowski Pdf

This book elaborates on the asymptotic behaviour, when N is large, of certain N-dimensional integrals which typically occur in random matrices, or in 1+1 dimensional quantum integrable models solvable by the quantum separation of variables. The introduction presents the underpinning motivations for this problem, a historical overview, and a summary of the strategy, which is applicable in greater generality. The core aims at proving an expansion up to o(1) for the logarithm of the partition function of the sinh-model. This is achieved by a combination of potential theory and large deviation theory so as to grasp the leading asymptotics described by an equilibrium measure, the Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse the equilibrium measure, the Schwinger-Dyson equations and the boostrap method to finally obtain an expansion of correlation functions and the one of the partition function. This book is addressed to researchers working in random matrices, statistical physics or integrable systems, or interested in recent developments of asymptotic analysis in those fields.

Author : Michael Oberguggenberger,Joachim Toft,Jasson Vindas,Patrik Wahlberg
Publisher : Birkhäuser
Page : 276 pages
File Size : 51,6 Mb
Release : 2017-05-06
Category : Mathematics
ISBN : 9783319519111

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Generalized Functions and Fourier Analysis by Michael Oberguggenberger,Joachim Toft,Jasson Vindas,Patrik Wahlberg Pdf

This book gives an excellent and up-to-date overview on the convergence and joint progress in the fields of Generalized Functions and Fourier Analysis, notably in the core disciplines of pseudodifferential operators, microlocal analysis and time-frequency analysis. The volume is a collection of chapters addressing these fields, their interaction, their unifying concepts and their applications and is based on scientific activities related to the International Association for Generalized Functions (IAGF) and the ISAAC interest groups on Pseudo-Differential Operators (IGPDO) and on Generalized Functions (IGGF), notably on the longstanding collaboration of these groups within ISAAC.

Author : Michał Eckstein,Bruno Iochum
Publisher : Springer
Page : 155 pages
File Size : 50,5 Mb
Release : 2018-12-18
Category : Science
ISBN : 9783319947884

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Spectral Action in Noncommutative Geometry by Michał Eckstein,Bruno Iochum Pdf

What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.

Author : R. Wong
Publisher : Academic Press
Page : 561 pages
File Size : 43,6 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483220710

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Asymptotic Approximations of Integrals by R. Wong Pdf

Asymptotic Approximations of Integrals deals with the methods used in the asymptotic approximation of integrals. Topics covered range from logarithmic singularities and the summability method to the distributional approach and the Mellin transform technique for multiple integrals. Uniform asymptotic expansions via a rational transformation are also discussed, along with double integrals with a curve of stationary points. For completeness, classical methods are examined as well. Comprised of nine chapters, this volume begins with an introduction to the fundamental concepts of asymptotics, followed by a discussion on classical techniques used in the asymptotic evaluation of integrals, including Laplace's method, Mellin transform techniques, and the summability method. Subsequent chapters focus on the elementary theory of distributions; the distributional approach; uniform asymptotic expansions; and integrals which depend on auxiliary parameters in addition to the asymptotic variable. The book concludes by considering double integrals and higher-dimensional integrals. This monograph is intended for graduate students and research workers in mathematics, physics, and engineering.

Author : George A Anastassiou
Publisher : World Scientific
Page : 424 pages
File Size : 55,7 Mb
Release : 2010-10-26
Category : Mathematics
ISBN : 9789814464345

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Advanced Inequalities by George A Anastassiou Pdf

This monograph presents univariate and multivariate classical analyses of advanced inequalities. This treatise is a culmination of the author's last thirteen years of research work. The chapters are self-contained and several advanced courses can be taught out of this book. Extensive background and motivations are given in each chapter with a comprehensive list of references given at the end.The topics covered are wide-ranging and diverse. Recent advances on Ostrowski type inequalities, Opial type inequalities, Poincare and Sobolev type inequalities, and Hardy-Opial type inequalities are examined. Works on ordinary and distributional Taylor formulae with estimates for their remainders and applications as well as Chebyshev-Gruss, Gruss and Comparison of Means inequalities are studied.The results presented are mostly optimal, that is the inequalities are sharp and attained. Applications in many areas of pure and applied mathematics, such as mathematical analysis, probability, ordinary and partial differential equations, numerical analysis, information theory, etc., are explored in detail, as such this monograph is suitable for researchers and graduate students. It will be a useful teaching material at seminars as well as an invaluable reference source in all science libraries.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 55,6 Mb
Release : 2024-06-26
Category : Electronic
ISBN : 9789814467131

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Probabilistic Inequalities by Anonim Pdf