Perturbation Theory for Linear Operators

Author: Tosio Kato

Publisher: Springer Science & Business Media

ISBN: 364266282X

Category: Mathematics

Page: 623

View: 6590

From the reviews: "[...] An excellent textbook in the theory of linear operators in Banach and Hilbert spaces. It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory. [...] I can recommend it for any mathematician or physicist interested in this field." Zentralblatt MATH
Posted in Mathematics

Perturbation Theory for Linear Operators

Author: Tosio Kato

Publisher: Springer Science & Business Media

ISBN: 9783540586616

Category: Mathematics

Page: 623

View: 7669

From the reviews: "[...] An excellent textbook in the theory of linear operators in Banach and Hilbert spaces. It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory. [...] I can recommend it for any mathematician or physicist interested in this field." Zentralblatt MATH
Posted in Mathematics

Spectral Approximation of Linear Operators

Author: Fran?oise Chatelin

Publisher: SIAM

ISBN: 0898719992

Category: Mathematics

Page: 480

View: 6603

Originally published: New York: Academic Press, 1983.
Posted in Mathematics

Perturbation Bounds for Matrix Eigenvalues

Author: Rajendra Bhatia

Publisher: SIAM

ISBN: 0898716314

Category: Mathematics

Page: 191

View: 3145

For the SIAM Classics edition, the author has added over 60 pages of material covering recent results and discussing the important advances made in the last two decades. It is an excellent research reference for all those interested in operator theory, linear algebra, and numerical analysis.
Posted in Mathematics

Invariant Subspaces of Matrices with Applications

Author: Israel Gohberg,Peter Lancaster,Leiba Rodman

Publisher: SIAM

ISBN: 089871608X

Category: Mathematics

Page: 692

View: 5030

This unique book addresses advanced linear algebra using invariant subspaces as the central notion and main tool. It comprehensively covers geometrical, algebraic, topological, and analytic properties of invariant subspaces, laying clear mathematical foundations for linear systems theory with a thorough treatment of analytic perturbation theory for matrix functions.
Posted in Mathematics

Linear Differential Operators

Author: Cornelius Lanczos

Publisher: SIAM

ISBN: 9781611971187

Category: Calculus, Operational

Page: 564

View: 9798

Originally published in 1961, this Classics edition continues to be appealing because it describes a large number of techniques still useful today. Although the primary focus is on the analytical theory, concrete cases are cited to forge the link between theory and practice. Considerable manipulative skill in the practice of differential equations is to be developed by solving the 350 problems in the text. The problems are intended as stimulating corollaries linking theory with application and providing the reader with the foundation for tackling more difficult problems. Lanczos begins with three introductory chapters that explore some of the technical tools needed later in the book, and then goes on to discuss interpolation, harmonic analysis, matrix calculus, the concept of the function space, boundary value problems, and the numerical solution of trajectory problems, among other things. The emphasis is constantly on one question: "What are the basic and characteristic properties of linear differential operators?" In the author's words, this book is written for those "to whom a problem in ordinary or partial differential equations is not a problem of logical acrobatism, but a problem in the exploration of the physical universe. To get an explicit solution of a given boundary value problem is in this age of large electronic computers no longer a basic question. But of what value is the numerical answer if the scientist does not understand the peculiar analytical properties and idiosyncrasies of the given operator? The author hopes that this book will help in this task by telling something about the manifold aspects of a fascinating field."
Posted in Calculus, Operational

Linear Operators and their Spectra

Author: E. Brian Davies

Publisher: Cambridge University Press

ISBN: 1139464337

Category: Mathematics

Page: N.A

View: 7002

This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, and contains many illustrative examples and exercises. Fredholm theory, Hilbert-Schmidt and trace class operators are discussed, as are one-parameter semigroups and perturbations of their generators. Two chapters are devoted to using these tools to analyze Markov semigroups. The text also provides a thorough account of the new theory of pseudospectra, and presents the recent analysis by the author and Barry Simon of the form of the pseudospectra at the boundary of the numerical range. This was a key ingredient in the determination of properties of the zeros of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non-self-adjoint Schrodinger operators are described. The text concludes with a description of the surprising spectral properties of the non-self-adjoint harmonic oscillator.
Posted in Mathematics

Advanced Mathematical Methods for Scientists and Engineers I

Asymptotic Methods and Perturbation Theory

Author: Carl M. Bender,Steven A. Orszag

Publisher: Springer Science & Business Media

ISBN: 1475730691

Category: Mathematics

Page: 593

View: 5544

A clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.
Posted in Mathematics

Mathematics of Classical and Quantum Physics

Author: Frederick W. Byron,Robert W. Fuller

Publisher: Courier Corporation

ISBN: 0486135063

Category: Science

Page: 672

View: 3835

Graduate-level text offers unified treatment of mathematics applicable to many branches of physics. Theory of vector spaces, analytic function theory, theory of integral equations, group theory, and more. Many problems. Bibliography.
Posted in Science

Matrix Polynomials

Author: I. Gohberg,P. Lancaster,L. Rodman

Publisher: SIAM

ISBN: 0898716810

Category: Mathematics

Page: 184

View: 4190

This book is the definitive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener-Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials. The book is appropriate for students, instructors, and researchers in linear algebra, operator theory, differential equations, systems theory, and numerical analysis. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex analysis.
Posted in Mathematics

Szegő's Theorem and Its Descendants

Spectral Theory for L2 Perturbations of Orthogonal Polynomials

Author: Barry Simon

Publisher: Princeton University Press

ISBN: 9781400837052

Category: Mathematics

Page: 664

View: 7447

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line. In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC.
Posted in Mathematics

Spectra and Pseudospectra

The Behavior of Nonnormal Matrices and Operators

Author: Lloyd Nicholas Trefethen,Mark Embree

Publisher: Princeton University Press

ISBN: 9780691119465

Category: Mathematics

Page: 606

View: 6410

Pure and applied mathematicians, physicists, scientists, and engineers use matrices and operators and their eigenvalues in quantum mechanics, fluid mechanics, structural analysis, acoustics, ecology, numerical analysis, and many other areas. However, in some applications the usual analysis based on eigenvalues fails. For example, eigenvalues are often ineffective for analyzing dynamical systems such as fluid flow, Markov chains, ecological models, and matrix iterations. That's where this book comes in. This is the authoritative work on nonnormal matrices and operators, written by the authorities who made them famous. Each of the sixty sections is written as a self-contained essay. Each document is a lavishly illustrated introductory survey of its topic, complete with beautiful numerical experiments and all the right references. The breadth of included topics and the numerous applications that provide links between fields will make this an essential reference in mathematics and related sciences.
Posted in Mathematics

Dynamics of Linear Operators

Author: Frédéric Bayart,Étienne Matheron

Publisher: Cambridge University Press

ISBN: 0521514967

Category: Mathematics

Page: 337

View: 1610

The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.
Posted in Mathematics

The Symmetric Eigenvalue Problem

Author: Beresford N. Parlett

Publisher: SIAM

ISBN: 9781611971163

Category: Algebras, Linear

Page: 398

View: 5985

According to Parlett, "Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for eigenvalue calculations in an ever richer variety of contexts." Anyone who performs these calculations will welcome the reprinting of Parlett's book (originally published in 1980). In this unabridged, amended version, Parlett covers aspects of the problem that are not easily found elsewhere. The chapter titles convey the scope of the material succinctly. The aim of the book is to present mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few. The author explains why the selected information really matters and he is not shy about making judgments. The commentary is lively but the proofs are terse. The first nine chapters are based on a matrix on which it is possible to make similarity transformations explicitly. The only source of error is inexact arithmetic. The last five chapters turn to large sparse matrices and the task of making approximations and judging them.
Posted in Algebras, Linear

Ten Mathematical Essays on Approximation in Analysis and Topology

Ten Mathematical Essays

Author: Juan Ferrera,J. Lopez-Gomez,F.R. Ruiz del Portal

Publisher: Elsevier

ISBN: 9780080459196

Category: Mathematics

Page: 284

View: 8485

This book collects 10 mathematical essays on approximation in Analysis and Topology by some of the most influent mathematicians of the last third of the 20th Century. Besides the papers contain the very ultimate results in each of their respective fields, many of them also include a series of historical remarks about the state of mathematics at the time they found their most celebrated results, as well as some of their personal circumstances originating them, which makes particularly attractive the book for all scientist interested in these fields, from beginners to experts. These gem pieces of mathematical intra-history should delight to many forthcoming generations of mathematicians, who will enjoy some of the most fruitful mathematics of the last third of 20th century presented by their own authors. This book covers a wide range of new mathematical results. Among them, the most advanced characterisations of very weak versions of the classical maximum principle, the very last results on global bifurcation theory, algebraic multiplicities, general dependencies of solutions of boundary value problems with respect to variations of the underlying domains, the deepest available results in rapid monotone schemes applied to the resolution of non-linear boundary value problems, the intra-history of the the genesis of the first general global continuation results in the context of periodic solutions of nonlinear periodic systems, as well as the genesis of the coincidence degree, some novel applications of the topological degree for ascertaining the stability of the periodic solutions of some classical families of periodic second order equations, the resolution of a number of conjectures related to some very celebrated approximation problems in topology and inverse problems, as well as a number of applications to engineering, an extremely sharp discussion of the problem of approximating topological spaces by polyhedra using various techniques based on inverse systems, as well as homotopy expansions, and the Bishop-Phelps theorem. Key features: - It contains a number of seminal contributions by some of the most world leading mathematicians of the second half of the 20th Century. - The papers cover a complete range of topics, from the intra-history of the involved mathematics to the very last developments in Differential Equations, Inverse Problems, Analysis, Nonlinear Analysis and Topology. - All contributed papers are self-contained works containing rather complete list of references on each of the subjects covered. - The book contains some of the very last findings concerning the maximum principle, the theory of monotone schemes in nonlinear problems, the theory of algebraic multiplicities, global bifurcation theory, dynamics of periodic equations and systems, inverse problems and approximation in topology. - The papers are extremely well written and directed to a wide audience, from beginners to experts. An excellent occasion to become engaged with some of the most fruitful mathematics developed during the last decades.
Posted in Mathematics

Spectral Theory of Block Operator Matrices and Applications

Author: Christiane Tretter

Publisher: World Scientific

ISBN: 1860947689

Category: Science

Page: 264

View: 4967

This book presents new concepts in operator theory and covers classes of operators (in particular, non-selfadjoint operators) which exhibit various interesting phenomena. Special attention is paid to applications in many areas of mathematical physics, including quantum mechanics, fluid mechanics, and magnetohydrodynamics.The author also discusses an operator theoretic approach to spectral problems for linear operators admitting a certain block structure. The results apply to bounded or finite-dimensional operators like block matrices as well to unbounded operators describing systems of differential equations. New concepts of numerical range are developed.
Posted in Science

Numerical Methods for Large Eigenvalue Problems

Revised Edition

Author: Yousef Saad

Publisher: SIAM

ISBN: 9781611970739

Category: Eigenvalues

Page: 276

View: 5602

This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices. It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific applications. Each chapter was updated by shortening or deleting outdated topics, adding topics of more recent interest, and adapting the Notes and References section. Significant changes have been made to Chapters 6 through 8, which describe algorithms and their implementations and now include topics such as the implicit restart techniques, the Jacobi-Davidson method, and automatic multilevel substructuring.
Posted in Eigenvalues

Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators

Author: Tailen Hsing,Randall Eubank

Publisher: John Wiley & Sons

ISBN: 0470016914

Category: Mathematics

Page: 480

View: 5501

Functional data is data in the form of curves that is becoming a popular method for interpreting scientific data. Statistical Analysis of Functional Data provides an authoritative account of function data analysis covering its foundations, theory, methodology, and practical implementation. It also contains examples taken from a wide range of disciplines, including finance, medicine, and psychology. The book includes a supporting Web site hosting the real data sets analyzed in the book and related software. Statistical researchers or practitioners analyzing functional data will find this book useful.
Posted in Mathematics

Functional Analysis

Author: Kôsaku Yosida

Publisher: Springer Science & Business Media

ISBN: 3662117916

Category: Mathematics

Page: 468

View: 9571

Posted in Mathematics