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Perturbation Methods, Instability, Catastrophe and Chaos

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Perturbation Methods, Instability, Catastrophe and Chaos by C. F. Chan Man Fong Book Summary:

This important book introduces perturbation and qualitative methods for differential equations in terms understandable to students with only a basic knowledge of calculus and ordinary linear differential equations. Theorems are stated clearly with their limitations and restrictions and are applied to solve examples from various disciplines. The writing style is informal and new ideas are introduced gradually via concepts already familiar to the reader.

A Short Introduction to Perturbation Theory for Linear Operators

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A Short Introduction to Perturbation Theory for Linear Operators by Tosio Kato Book Summary:

This book is a slightly expanded reproduction of the first two chapters (plus Introduction) of my book Perturbation Theory tor Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Springer 1980. Ever since, or even before, the publication of the latter, there have been suggestions about separating the first two chapters into a single volume. I have now agreed to follow the suggestions, hoping that it will make the book available to a wider audience. Those two chapters were intended from the outset to be a comprehen­ sive presentation of those parts of perturbation theory that can be treated without the topological complications of infinite-dimensional spaces. In fact, many essential and. even advanced results in the theory have non­ trivial contents in finite-dimensional spaces, although one should not forget that some parts of the theory, such as those pertaining to scatter­ ing. are peculiar to infinite dimensions. I hope that this book may also be used as an introduction to linear algebra. I believe that the analytic approach based on a systematic use of complex functions, by way of the resolvent theory, must have a strong appeal to students of analysis or applied mathematics, who are usually familiar with such analytic tools.

Perturbation Theory for Linear Operators (Classics in Mathematics)

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Perturbation Theory for Linear Operators (Classics in Mathematics) by Tosio Kato Book Summary:

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

Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory

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Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory by Barry G. Adams Book Summary:

This book provides an introduction to the use of algebraic methods and sym­ bolic computation for simple quantum systems with applications to large order perturbation theory. It is the first book to integrate Lie algebras, algebraic perturbation theory and symbolic computation in a form suitable for students and researchers in theoretical and computational chemistry and is conveniently divided into two parts. The first part, Chapters 1 to 6, provides a pedagogical introduction to the important Lie algebras so(3), so(2,1), so(4) and so(4,2) needed for the study of simple quantum systems such as the D-dimensional hydrogen atom and harmonic oscillator. This material is suitable for advanced undergraduate and beginning graduate students. Of particular importance is the use of so(2,1) in Chapter 4 as a spectrum generating algebra for several important systems such as the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This approach provides an interesting and important alternative to the usual textbook approach using series solutions of differential equations.

Perturbation Theory and the Nuclear Many Body Problem (Dover Books on Physics)

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Perturbation Theory and the Nuclear Many Body Problem (Dover Books on Physics) by Kailash Kumar Book Summary:

This introductory treatment begins with an overview of the basic ideas of perturbation theory, addressing the conditions under which the theory may be set up and the various forms of perturbation expansions. Subsequent chapters explore diagrammatic methods in terms of linked cluster theorem and general formulas as well as rearrangement methods. Techniques of solving the t-matrix equation and other equations that arise in the nuclear many body problem are examined in terms of approximate methods, and the intuitive reasoning behind each of them is given. The text's final chapter collects other methods of approaching the many body problem and shows how they may be compared with those of previous chapters.Suitable for advanced undergraduates and graduate students, this volume features many helpful citations to literature on the subject, and a list of main symbols has been appended to each chapter for easy reference.

Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory

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Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory by Carl M. Bender Book Summary:

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.

Intermolecular Forces and Their Evaluation by Perturbation Theory (Lecture Notes in Chemistry)

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Intermolecular Forces and Their Evaluation by Perturbation Theory (Lecture Notes in Chemistry) by P. Arrighini Book Summary:

The aim of these notes is to offer a modern picture of the pertur­ bative approach to the calculation of intermolecular forces. The point of view taken is that a perturbative series truncated at a low order can provide a valuable way for ~valuating interaction energies, especial­ ly if one limits oneself to the case of intermediate- and long-range distances between the interacting partners. Although the situation corresponding to short distances is essen­ tially left out from our presentation, the problems which are within the range of the theory form a vast and important class: a large var­ iety of phenomena of matter, in fact, depends on the existence of in­ teractions among atoms or molecules, which over a substantial range of distances should be classified as weak in comparison to the interactions occurring inside atoms or molecules. We are aware of the omission of some topics, which in principle could have been included in our review. For instance, a very scarce at­ tention has been paid to the analysis of problems involving interacting partners in degenerate states, which is of particular relevance in the case of interactions between excited atoms (only a rather quick presen­ tation of the formal apparatus of degenerate perturbation theory is in­ cluded in Chap. III). Interactions involving the simultaneous presence of more than two atoms (or mOlecules) have not been considered, with the consequent non-necessity of considering nonadditive effects which characterize the general N-body problem.

Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory

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Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory by Carl M. Bender Book Summary:

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.

Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences)

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Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences) by K. W. Chang Book Summary:

Our purpose in writing this monograph is twofold. On the one hand, we want to collect in one place many of the recent results on the exist­ ence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems. On the other, we hope to raise along the way a number of questions for further study, mostly ques­ tions we ourselves are unable to answer. The presentation involves a study of both scalar and vector boundary value problems for ordinary dif­ ferential equations, by means of the consistent use of differential in­ equality techniques. Our results for scalar boundary value problems obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior. The linear theory for such problems is incredibly complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory. Our results for vector boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equa­ tions. The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.

Perturbation Theory for the Schrödinger Operator with a Periodic Potential (Lecture Notes in Mathematics)

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Perturbation Theory for the Schrödinger Operator with a Periodic Potential (Lecture Notes in Mathematics) by Yulia E. Karpeshina Book Summary:

The book is devoted to perturbation theory for the Schrödinger operator with a periodic potential, describing motion of a particle in bulk matter. The Bloch eigenvalues of the operator are densely situated in a high energy region, so regular perturbation theory is ineffective. The mathematical difficulties have a physical nature - a complicated picture of diffraction inside the crystal. The author develops a new mathematical approach to this problem. It provides mathematical physicists with important results for this operator and a new technique that can be effective for other problems. The semiperiodic Schrödinger operator, describing a crystal with a surface, is studied. Solid-body theory specialists can find asymptotic formulae, which are necessary for calculating many physical values.

Perturbation Analysis of Optimization Problems (Springer Series in Operations Research and Financial Engineering)

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Perturbation Analysis of Optimization Problems (Springer Series in Operations Research and Financial Engineering) by J.Frederic Bonnans Book Summary:

A presentation of general results for discussing local optimality and computation of the expansion of value function and approximate solution of optimization problems, followed by their application to various fields, from physics to economics. The book is thus an opportunity for popularizing these techniques among researchers involved in other sciences, including users of optimization in a wide sense, in mechanics, physics, statistics, finance and economics. Of use to research professionals, including graduate students at an advanced level.

Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory (Mathematics and Its Applications)

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Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics: Generalized Solitons and Hyperasymptotic Perturbation Theory (Mathematics and Its Applications) by John P. Boyd Book Summary:

This is the first thorough examination of weakly nonlocal solitary waves, which are just as important in applications as their classical counterparts. The book describes a class of waves that radiate away from the core of the disturbance but are nevertheless very long-lived nonlinear disturbances.

Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences) Softcover reprint of edition by Chang, K. W., Howes, F. A. (2013) Paperback

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Nonlinear Singular Perturbation Phenomena: Theory and Applications (Applied Mathematical Sciences) Softcover reprint of edition by Chang, K. W., Howes, F. A. (2013) Paperback Book Summary:

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