Inverse Scattering Problems And Their Application To Nonlinear Integrable Equations
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Inverse Scattering Problems And Their Application To Nonlinear Integrable Equations
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Author : Pham Loi Vu
language : en
Publisher: CRC Press
Release Date : 2019-11-11
Inverse Scattering Problems And Their Application To Nonlinear Integrable Equations written by Pham Loi Vu and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-11 with Mathematics categories.
Inverse Scattering Problems and Their Application to Nonlinear Integrable Equations is devoted to inverse scattering problems (ISPs) for differential equations and their application to nonlinear evolution equations (NLEEs). The book is suitable for anyone who has a mathematical background and interest in functional analysis, partial differential equations, equations of mathematical physics, and functions of a complex variable. This book is intended for a wide community working with inverse scattering problems and their applications; in particular, there is a traditional community in mathematical physics. In this monograph, the problems are solved step-by-step, and detailed proofs are given for the problems to make the topics more accessible for students who are approaching them for the first time. Features • The unique solvability of ISPs are proved. The scattering data of the considered inverse scattering problems (ISPs) are described completely. • Solving the associated initial value problem or initial-boundary value problem for the nonlinear evolution equations (NLEEs) is carried out step-by-step. Namely, the NLEE can be written as the compatibility condition of two linear equations. The unknown boundary values are calculated with the help of the Lax (generalized) equation, and then the time-dependent scattering data (SD) are constructed from the initial and boundary conditions. • The potentials are recovered uniquely in terms of time-dependent SD, and the solution of the NLEEs is expressed uniquely in terms of the found solutions of the ISP. • Since the considered ISPs are solved well, then the SPs generated by two linear equations constitute the inverse scattering method (ISM). The application of the ISM to solving the NLEEs is consistent and is effectively embedded in the schema of the ISM.
Direct And Projective Limits Of Geometric Banach Structures
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Author : Patrick Cabau
language : en
Publisher: CRC Press
Release Date : 2023-10-06
Direct And Projective Limits Of Geometric Banach Structures written by Patrick Cabau and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-10-06 with Mathematics categories.
This book describes in detail the basic context of the Banach setting and the most important Lie structures found in finite dimension. The authors expose these concepts in the convenient framework which is a common context for projective and direct limits of Banach structures. The book presents sufficient conditions under which these structures exist by passing to such limits. In fact, such limits appear naturally in many mathematical and physical domains. Many examples in various fields illustrate the different concepts introduced. Many geometric structures, existing in the Banach setting, are "stable" by passing to projective and direct limits with adequate conditions. The convenient framework is used as a common context for such types of limits. The contents of this book can be considered as an introduction to differential geometry in infinite dimension but also a way for new research topics. This book allows the intended audience to understand the extension to the Banach framework of various topics in finite dimensional differential geometry and, moreover, the properties preserved by passing to projective and direct limits of such structures as a tool in different fields of research.
Separate And Joint Continuity
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Author : Jiling Cao
language : en
Publisher: CRC Press
Release Date : 2024-07-09
Separate And Joint Continuity written by Jiling Cao and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-07-09 with Mathematics categories.
Separate and Joint Continuity presents and summarises the main ideas and theorems that have been developed on this topic, which lies at the interface between General Topology and Functional Analysis (and the geometry of Banach spaces in particular). The book offers detailed, self-contained proofs of many of the key results. Although the development of this area has now slowed to a point where an authoritative book can be written, many important and significant problems remain open, and it is hoped that this book may serve as a springboard for future and emerging researchers into this area. Furthermore, it is the strong belief of the authors that this area of research is ripe for exploitation. That is to say, it is their belief that many of the results contained in this monograph can, and should be, applied to other areas of mathematics. It is hoped that this monograph may provide an easily accessible entry point to the main results on separate and joint continuity for mathematicians who are not directly working in this field, but who may be able to exploit some of the deep results that have been developed over the past 125 years. Features Provides detailed, self-contained proofs of many of the key results in the area Suitable for researchers and postgraduates in topology and functional analysis Is the first book to offer a detailed and up-to-date summary of the main ideas and theorems on this topic
Fractional Integrals Potentials And Radon Transforms
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Author : Boris Rubin
language : en
Publisher: CRC Press
Release Date : 2024-08-14
Fractional Integrals Potentials And Radon Transforms written by Boris Rubin and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-08-14 with Mathematics categories.
Fractional Integrals, Potentials, and Radon Transforms, Second Edition presents recent developments in the fractional calculus of functions of one and several real variables, and shows the relation of this field to a variety of areas in pure and applied mathematics. In this thoroughly revised new edition, the book aims to explore how fractional integrals occur in the study of diverse Radon type transforms in integral geometry. Beyond some basic properties of fractional integrals in one and many dimensions, this book also contains a mathematical theory of certain important weakly singular integral equations of the first kind arising in mechanics, diffraction theory and other areas of mathematical physics. The author focuses on explicit inversion formulae that can be obtained by making use of the classical Marchaud’s approach and its generalization, leading to wavelet type representations. New to this Edition Two new chapters and a new appendix, related to Radon transforms and harmonic analysis of linear operators commuting with rotations and dilations have been added. Contains new exercises and bibliographical notes along with a thoroughly expanded list of references. This book is suitable for mathematical physicists and pure mathematicians researching in the area of integral equations, integral transforms, and related harmonic analysis.
Aspects Of Integration
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Author : Ronald B. Guenther
language : en
Publisher: CRC Press
Release Date : 2023-08-24
Aspects Of Integration written by Ronald B. Guenther and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-24 with Mathematics categories.
Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals is comprised of two parts. The first part is devoted to the Riemann integral, and provides not only a novel approach, but also includes several neat examples that are rarely found in other treatments of Riemann integration. Historical remarks trace the development of integration from the method of exhaustion of Eudoxus and Archimedes, used to evaluate areas related to circles and parabolas, to Riemann’s careful definition of the definite integral, which is a powerful expansion of the method of exhaustion and makes it clear what a definite integral really is. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory. Our approach is novel in part because it uses integrals of continuous functions rather than integrals of step functions as its starting point. This is natural because Riemann integrals of continuous functions occur much more frequently than do integrals of step functions as a precursor to Lebesgue integration. In addition, the approach used here is natural because step functions play no role in the novel development of the Riemann integral in the first part of the book. Our presentation of the Riesz-Nagy approach is significantly more accessible, especially in its discussion of the two key lemmas upon which the approach critically depends, and is more concise than other treatments. Features Presents novel approaches designed to be more accessible than classical presentations A welcome alternative approach to the Riemann integral in undergraduate analysis courses Makes the Lebesgue integral accessible to upper division undergraduate students How completion of the Riemann integral leads to the Lebesgue integral Contains a number of historical insights Gives added perspective to researchers and postgraduates interested in the Riemann and Lebesgue integrals
Classical Clifford Algebras
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Author : Ilwoo Cho
language : en
Publisher: CRC Press
Release Date : 2024-04-08
Classical Clifford Algebras written by Ilwoo Cho and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-04-08 with Mathematics categories.
Classical Clifford Algebras: Operator-Algebraic and Free-Probabilistic Approaches offers novel insights through operator-algebraic and free-probabilistic models. By employing these innovative methods, the author sheds new light on the intrinsic connections between Clifford algebras and various mathematical domains. This monograph should be an essential addition to the library of any researchers interested in Clifford Algebras or Algebraic Geometry more widely. Features Includes multiple examples and applications Suitable for postgraduates and researchers working in Algebraic Geometry Takes an innovative approach to a well-established topic
Constructive Analysis Of Semicircular Elements
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Author : Ilwoo Cho
language : en
Publisher: CRC Press
Release Date : 2023-05-12
Constructive Analysis Of Semicircular Elements written by Ilwoo Cho and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-12 with Mathematics categories.
This book not only examines the constructions and free-probabilistic properties of semicircular elements, as defined within the text, but also considers certain Banach-space operators acting on these semicircular elements and shows how they deform (i.e., preserve-or-distort) the semicircular law induced by orthogonal projections. FEATURES Suitable for graduate students and professional researchers in operator theory and/or analysis Numerous applications in related scientific fields and areas
Solitons Nonlinear Evolution Equations And Inverse Scattering
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Author : Mark J. Ablowitz
language : en
Publisher: Cambridge University Press
Release Date : 1991-12-12
Solitons Nonlinear Evolution Equations And Inverse Scattering written by Mark J. Ablowitz and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-12-12 with Mathematics categories.
This book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.
A Qualitative Approach To Inverse Scattering Theory
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Author : Fioralba Cakoni
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-10-28
A Qualitative Approach To Inverse Scattering Theory written by Fioralba Cakoni and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-10-28 with Mathematics categories.
Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006
Integrable Hamiltonian Hierarchies
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Author : Vladimir Gerdjikov
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-06-02
Integrable Hamiltonian Hierarchies written by Vladimir Gerdjikov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-06-02 with Science categories.
This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their hierarchies.