Differential calculus and equations Books

Book SynopsisMany nonlinear systems around us can generate a very complex and counter-intuitive dynamics that contrasts with their simplicity, but their understanding requires concepts that are outside the basic training of most science students. This textbook, which is the fruit of graduate courses that the authors have taught at their respective universities, provides a richly illustrated introduction to nonlinear dynamical systems and chaos and a solid foundation for this fascinating subject. It will satisfy those who want discover this field, including at the undergraduate level, but also those who need a compact and consistent overview, gathering the concepts essential to nonlinear scientists.The first and second chapters describe the essential concepts needed to describe nonlinear dynamical systems as well as their stability. The third chapter introduces the concept of bifurcation, where the qualitative dynamical behavior of a system changes. The fourth chapter deals with oscillations, from their birth to their destabilization, and how they respond to external driving. The fifth and sixth chapters discuss complex behaviors that only occur in state spaces of dimension three and higher: quasi-periodicity and chaos, from their general properties to quantitative methods of characterization. All chapters are supplemented by exercises ranging from direct applications of the notions introduced in the corresponding chapter to elaborate problems involving concepts from different chapters, as well as numerical explorations.

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Book SynopsisThis volume, the third of a series, consists of applications of Mathematica® to a potpourri of more advanced topics. These include differential geometry of curves and surfaces, differential equations and special functions and complex analysis. Some of the newest features of Mathematica® are demonstrated and explained and some problems with the current implementation pointed out and possible future improvements suggested. Contains a large number of worked out examples. Explains some of the most recent mathematical features of Mathematica®. Considers topics discussed rarely or not at all in the context of Mathematica®. Can be used to supplement several different courses. Based on actual university courses.

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Book SynopsisThe present book is based on the extensive lecture notes of the author and contains a concise course on Linear Algebra. The sections begin with an intuitive presentation, aimed at the beginners, and then often include rather non-trivial topics and exercises. This makes the book suitable for introductory as well as advanced courses on Linear Algebra.The first part of the book deals with the general idea of systems of linear equations, matrices and eigenvectors. Linear systems of differential equations are developed carefully and in great detail. The last chapter gives an overview of applications to other areas of Mathematics, like calculus and differential geometry. A large number of exercises with selected solutions make this a valuable textbook for students of the topic as well as lecturers, preparing a course on Linear Algebra.

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Book SynopsisThis book is intended to be used as a rather informal, and surely not complete, textbook on the subjects indicated in the title. It collects my Lecture Notes held during three academic years at the University of Siena for a one semester course on "Basic Mathematical Physics", and is organized as a short presentation of few important points on the arguments indicated in the title. It aims at completing the students' basic knowledge on Ordinary Differential Equations (ODE) - dealing in particular with those of higher order - and at providing an elementary presentation of the Partial Differential Equations (PDE) of Mathematical Physics, by means of the classical methods of separation of variables and Fourier series. For a reasonable and consistent discussion of the latter argument, some elementary results on Hilbert spaces and series expansion in othonormal vectors are treated with some detail in Chapter 2. Prerequisites for a satisfactory reading of the present Notes are not only a course of Calculus for functions of one or several variables, but also a course in Mathematical Analysis where - among others - some basic knowledge of the topology of normed spaces is supposed to be included. For the reader's convenience some notions in this context are explicitly recalled here and there, and in particular as an Appendix in Section 1.4. An excellent reference for this general background material is W. Rudin's classic Principles of Mathematical Analysis. On the other hand, a complete discussion of the results on ODE and PDE that are here just sketched are to be found in other books, specifically and more deeply devoted to these subjects, some of which are listed in the Bibliography. In conclusion and in brief, my hope is that the present Notes can serve as a second quick reading on the theme of ODE, and as a first introductory reading on Fourier series, Hilbert spaces, and PDE

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Book SynopsisThis textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject. No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solutions, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements.Trade Review“This textbook furnishes the basis for a 1-year introductory course in partial differential equations for advanced undergraduates. … The book is written with great care and great attention to detail throughout. At the end of every chapter there are well-chosen exercises that genuinely add depth to the concepts treated in the text. … this book can be wholeheartedly recommended.” (M. Kunzinger, Monatshefte für Mathematik, Vol. 181, 2016)“This book easily covers all the material one might want in a course aimed at first-time students of PDEs. … I recommend this one highly: It provides the best first-course introduction to a vast and ever-more relevant and active area. Students, and perhaps instructors too, will learn much from it. If they wish to go beyond the material taught in a first course, this text will prepare them better than any other I know.” (SIAM Review, Vol. 56 (3), September, 2014)“Introduction to Partial Differential Equations is a complete, well-written textbook for upper-level undergraduates and graduate students. Olver … thoroughly covers the topic in a readable format and includes plenty of examples and exercises, ranging from the typical to independent projects and computer projects. … Instructors teaching an introduction to partial differential equations course will want to consider this textbook as a viable option for their students. Summing Up: Highly Recommended. Upper-division undergraduates, graduate students, and faculty.” (S. L. Sullivan, Choice, Vol. 51 (11), July, 2014)“This introduction to partial differential equations is addressed to advanced undergraduates or graduate students … . an imposing book that includes plenty of material for two semesters even at the graduate level. … The author succeeds at maintaining a good balance between solution methods, mathematical rigor, and applications. With appropriate selection of topics this could serve for a one semester introductory course for undergraduates or a full year course for graduate students. … the author has clearly taken pains to make it readable and accessible.” (William J. Satzer, MAA Reviews, January, 2014)Table of ContentsWhat are Partial Differential Equations?.- Linear and Nonlinear Waves.- Fourier Series.- Separation of Variables.- Finite Differences.- Generalized Functions and Green’s Functions.- Complex Analysis and Conformal Mapping.- Fourier Transforms.- Linear and Nonlinear Evolution Equations.- A General Framework for Linear Partial Differential Equations.- Finite Elements and Weak Solutions.- Dynamics of Planar Media.- Partial Differential Equations in Space​.

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Book SynopsisThis research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter.Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has become an important subject of Mathematics and Applied Mathematics, because of its mathematical richness and its importance for applications in many areas of Physics, Biology, Economics and Finance, where random processes play an increasingly important role. One important aspect is the connection between diffusion processes and linear partial differential equations of second order, which is in particular the basis for Monte Carlo numerical methods for linear PDEs. Since the pioneering work of Peng and Pardoux in the early 1990s, a new type of SDEs called backward stochastic differential equations (BSDEs) has emerged. The two main reasons why this new class of equations is important are the connection between BSDEs and semilinear PDEs, and the fact that BSDEs constitute a natural generalization of the famous Black and Scholes model from Mathematical Finance, and thus offer a natural mathematical framework for the formulation of many new models in Finance.Trade Review“This 668-page magnum opus of stochastic ODEs and PDEs belongs on the shelf of every researcher in these areas, as well as any mathematician or scientist who wants to learn more about the subject. … my opinion is that this book accomplished a Herculean task of making an arguably technical subject that is daunting to a beginner accessible. This book wants to be read!” (Mark A. McKibben, Mathematical Reviews, April, 2016)“The present monograph gives a rather complete treatment of backward stochastic differential equations as tool for the stochastic interpretation of second order PDEs. As the reader is guided from basic knowledge on stochastic analysis through the Itō calculus and the theory of stochastic differential equations to that of the backward equations, the monograph represents in my eyes a precious textbook for Master students, PhD students, but also specialists in this domain.” (Rainer Buckdahn, zbMATH 1321.60005, 2015)Table of ContentsIntroduction.- Background of Stochastic Analysis.- Ito’s Stochastic Calculus.- Stochastic Differential Equations.- SDE with Multivalued Drift.- Backward SDE.- Annexes.- Bibliography.- Index. ​ ​

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Book SynopsisThe purpose of the volume is to provide a support for a first course in Mathematics. The contents are organised to appeal especially to Engineering, Physics and Computer Science students, all areas in which mathematical tools play a crucial role. Basic notions and methods of differential and integral calculus for functions of one real variable are presented in a manner that elicits critical reading and prompts a hands-on approach to concrete applications. The layout has a specifically-designed modular nature, allowing the instructor to make flexible didactical choices when planning an introductory lecture course. The book may in fact be employed at three levels of depth. At the elementary level the student is supposed to grasp the very essential ideas and familiarise with the corresponding key techniques. Proofs to the main results befit the intermediate level, together with several remarks and complementary notes enhancing the treatise. The last, and farthest-reaching, level requires the additional study of the material contained in the appendices, which enable the strongly motivated reader to explore further into the subject. Definitions and properties are furnished with substantial examples to stimulate the learning process. Over 350 solved exercises complete the text, at least half of which guide the reader to the solution. This new edition features additional material with the aim of matching the widest range of educational choices for a first course of Mathematics.Trade ReviewFrom the book reviews:“I enjoyed reading the present textbook. It provides a good coverage of the material, a very good choice of exercises, and an impeccable graphical presentation. The textbook is written for first-year students whose interest is not mainly in mathematics, yet mathematics plays an important role in their curricula. The authors attained their goal.” (George Stoica, zbMATH, Vol. 1305, 2015)Table of Contents1 Basic notions.- 2 Functions.- 3 Limits and continuity I.- 4 Limits and continuity II.- 5 Local comparison of functions. Numerical sequences and series.- 6 Differential calculus.- 7 Taylor expansions and applications.- 8 Geometry in the plane and in space.- 9 Integral calculus I.- 10 Integral calculus II.- 11 Ordinary differential equations.- 12 A.1 The Principle of Mathematical Induction.- 13 A.2 Complements on limits and continuity.- 14 A.3 Complements on the global features of continuous maps.- 15 A.4 Complements on differential calculus.- 16 A.5 Complements on integral calculus.- 17 Tables and Formulas.

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Book SynopsisThe purpose of this book is to give a quick and elementary, yet rigorous, presentation of the rudiments of the so-called theory of Viscosity Solutions which applies to fully nonlinear 1st and 2nd order Partial Differential Equations (PDE). For such equations, particularly for 2nd order ones, solutions generally are non-smooth and standard approaches in order to define a "weak solution" do not apply: classical, strong almost everywhere, weak, measure-valued and distributional solutions either do not exist or may not even be defined. The main reason for the latter failure is that, the standard idea of using "integration-by-parts" in order to pass derivatives to smooth test functions by duality, is not available for non-divergence structure PDE.Trade Review“In this small book, the author, after introducing basic and non-basic concepts of the theory of viscosity solutions for first and second order PDEs, applies the theory to two specific problems such as existence of viscosity solution for the Euler-Lagrange PDE and for the ∞-Laplacian. … The book can be certainly used as text for an advanced course and also as manual for researchers.” (Fabio Bagagiolo, zbMATH, Vol. 1326.35006, 2016)“The book under review is a nice introduction to the theory of viscosity solutions for fully nonlinear PDEs … . The book, which is addressed to a public having basic knowledge in PDEs, is based on a course given by the author … . The explanations are very clear, and the reader is introduced to the theory step by step, the author taking the time to explain several technical details, but without making the exposition too heavy.” (Enea Parini, Mathematical Reviews, November, 2015)Table of Contents1 History, Examples, Motivation and First Definitions.- 2 Second Definitions and Basic Analytic Properties of the Notions.- 3 Stability Properties of the Notions and Existence via Approximation.- 4 Mollification of Viscosity Solutions and Semi convexity.- 5 Existence of Solution to the Dirichlet Problem via Perron’s Method.- 6 Comparison results and Uniqueness of Solution to the Dirichlet Problem.- 7 Minimisers of Convex Functionals and Viscosity Solutions of the Euler-Lagrange PDE.- 8 Existence of Viscosity Solutions to the Dirichlet Problem for the Laplacian.- 9 Miscellaneous topics and some extensions of the theory.

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Book SynopsisThis book offers readers a primer on the theory and applications of Ordinary Differential Equations. The style used is simple, yet thorough and rigorous. Each chapter ends with a broad set of exercises that range from the routine to the more challenging and thought-provoking. Solutions to selected exercises can be found at the end of the book. The book contains many interesting examples on topics such as electric circuits, the pendulum equation, the logistic equation, the Lotka-Volterra system, the Laplace Transform, etc., which introduce students to a number of interesting aspects of the theory and applications. The work is mainly intended for students of Mathematics, Physics, Engineering, Computer Science and other areas of the natural and social sciences that use ordinary differential equations, and who have a firm grasp of Calculus and a minimal understanding of the basic concepts used in Linear Algebra. It also studies a few more advanced topics, such as Stability Theory and Boundary Value Problems, which may be suitable for more advanced undergraduate or first-year graduate students. The second edition has been revised to correct minor errata, and features a number of carefully selected new exercises, together with more detailed explanations of some of the topics.A complete Solutions Manual, containing solutions to all the exercises published in the book, is available. Instructors who wish to adopt the book may request the manual by writing directly to one of the authors.Trade Review“This is the second edition of an undergraduate introduction to ordinary differential equations suitable for mathematicians and engineers. … The style is clean and concise with many examples and exercises. Basic results are proven, more involved results are only stated. The new edition features some new exercises and better explanations at various points. So if you are looking for an application oriented introduction which is still concise and rigorous, this book might be just right for you.” (G. Teschl, Monatshefte für Mathematik, 2016)Table of Contents1 First order linear differential equations.- 2 Theory of first order differential equations.- 3 First order nonlinear differential equations.- 4 Existence and uniqueness for systems and higher order equations.- 5 Second order equations.- 6 Higher order linear equations.- 7 Systems of first order equations.- 8 Qualitative analysis of 2x2 systems and nonlinear second order equations.- 9 Sturm Liouville eigenvalue theory.- 10 Solutions by infinite series and Bessel functions.- 11 Laplace transform.- 12 Stability theory.- 13 Boundary value problems.- 14 Appendix A. Numerical methods.- 15 Answers to selected exercises.

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Book SynopsisAuthor received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equationsTrade ReviewFrom the reviews: "...these volumes are excellently written and make for greatly profitable reading. For years to come they will surely be a main reference for anyone wishing to study partial differential operators."-- MATHEMATICAL REVIEWS "This volume focuses on linear partial differential operators with constant coefficients … . Each chapter ends with notes on the literature, and there is a large bibliography. … The binding of this softcover reprint seems quite good … . Overall, it is great to have this book back at an affordable price. It really does deserve to be described as a classic." (Fernando Q. Gouvêa, MathDL, January, 2005)Table of ContentsExistence and Approximation of Solutions of Differential Equations.- Interior Regularity of Solutions of Differential Equations.- The Cauchy and Mixed Problems.- Differential Operators of Constant Strength.- Scattering Theory.- Analytic Function Theory and Differential Equations.- Convolution Equations.

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Book SynopsisThere has recently been a renewal of interest in Fokker-Planck operators, motivated by problems in statistical physics, in kinetic equations, and differential geometry. Compared to more standard problems in the spectral theory of partial differential operators, those operators are not self-adjoint and only hypoelliptic. The aim of the analysis is to give, as generally as possible, an accurate qualitative and quantitative description of the exponential return to the thermodynamical equilibrium. While exploring and improving recent results in this direction, this volume proposes a review of known techniques on: the hypoellipticity of polynomial of vector fields and its global counterpart, the global Weyl-Hörmander pseudo-differential calculus, the spectral theory of non-self-adjoint operators, the semi-classical analysis of Schrödinger-type operators, the Witten complexes, and the Morse inequalities.Trade ReviewFrom the reviews of the first edition: "The aim of this text is to give an account of how the known techniques from partial differential equations and spectral theory can be applied for the analysis of Fokker-Plank operators or Witten Laplacians … . This synthetic text is very challenging and useful for researchers in partial differential equations, probability theory and mathematical physics." (Viorel Iftimie, Zentralblatt MATH, Vol. 1072, 2005)Table of Contents1. Introduction.- 2. Kohn's Proof of the Hypoellipticity of the Hörmander Operators.- 3. Compactness Criteria for the Resolvent of Schrödinger Operators.- 4. Global Pseudo-differential Calculus.- 5. Analysis of some Fokker-Planck Operator.- 6. Return to Equillibrium for the Fokker-Planck Operator.- 7. Hypoellipticity and nilpotent groups.- 8. Maximal Hypoellipticity for Polynomial of Vector Fields and Spectral Byproducts.- 9. On Fokker-Planck Operators and Nilpotent Techniques.- 10. Maximal Microhypoellipticity for Systems and Applications to Witten Laplacians.- 11. Spectral Properties of the Witten-Laplacians in Connection with Poincaré inequalities for Laplace Integrals.- 12. Semi-classical Analysis for the Schrödinger Operator: Harmonic Approximation.- 13. Decay of Eigenfunctions and Application to the Splitting.- 14. Semi-classical Analysis and Witten Laplacians: Morse Inequalities.- 15. Semi-classical Analysis and Witten Laplacians: Tunneling Effects.- 16. Accurate Asymptotics for the Exponentially Small Eigenvalues of the Witten Laplacian.- 17. Application to the Fokker-Planck Equation.- 18. Epilogue.- References.- Index.

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Book SynopsisChoice Outstanding Title! (January 2006) This richly illustrated text covers the Cauchy and Neumann problems for the classical linear equations of mathematical physics. A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging! What makes the book unique is Arnold's particular talent at holding a topic up for examination from a new and fresh perspective. He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject. No other mathematical writer does this quite so well as Arnold.Trade ReviewFrom the reviews of the German edition: "This book provides an introductory text (in German) to basic partial differential equations, based on the author's lectures at Moscow University. […] Most of the standard themes are treated (see list below), but some unusual topics are covered as well. For instance, in chapter 10 double layer potentials are considered, and chapters 11 and 13 deal (among others) with Maxwell's theorem on the multipole expansion of spherical functions. The style of the book is quite non-technical (it contains almost no estimates), taking a mainly geometric viewpoint. [...]" Markus Kunze, Zentralblatt für Mathematik 1076.35001 From the reviews: "[...] This excellent and stimulating textbook gives a beautiful first view on some basic aspects of the theory of partial differential equations and can be warmly recommended to any graduate student in mathematics and physics." M.Günther, Zeitschrift für Angewandte Analysis und Ihre Anwendungen, Vol. 24, Issue 4, 2005 "…..Arnold .. has long held a reputation as one of the world's leaders in dynamics and geometry. His Lectures survey big ideas; accordingly, he largely suppresses both the functional analytic formalism and the delicate estimates so characteristic of the subject. He takes the viewpoint that the most important PDEs arise in physics and the most important mathematical ideas contributing to their solution derive from physical principles. Amold concentrates on the simplest equations of a given type and shows how the key ideas play out. For example, he attacks the general theory of one first-order equation, first via wave-particle duality, then via Hamiltonian dynamics. .... The author's stature and the book's lucidity make this an essential acquisition for all College libraries. …." D.V.Feldman, CHOICE, January 2005 Vol. 42 No. 05 "... Professor Arnold's Lectures on Partial Differential Equations is an ambitious, intensely personal effort to reconnect the subject with some of its roots in modeling physical processes. He does so in a lively lecture-style format, resulting in a book that would complement almost any course in PDEs. ... As can be gleaned from the previous paragraph, we bouth found the book by V.I.Arnold most stimulating and thought provoking, leading to statements such as, "It has been years since I enjoyed a book so much" by RBG and "I cannot point to any other book in mathematics written with the same intensity" by EAT. ... ... what follows [...] is a beautiful book on PDEs, interwoven with the exposition of deep physical, geometrical, and topological insights that contribute to both the understanding and history of PDEs. Prof. Arnold's book ... connects with the roots of the field and brings in concepts from geometry, continuum mechanics, and analysis. It can be used together with any book on PDEs and students will welcome its directness and freshness. We know of no other book like it on the market and highly recommend it for individual reading and as an accompaniment to any course in PDEs. ..." R.B. Guenther, E.A.Thomann, SIAM Review, Vol. 47, No. 1, 2005 "This book contains the transcripts of twelve lectures on partial differential equations … . The presentation gives a vivid sense of what was actually said and discussed in the lecture course, and in this fashion the book differs markedly from many text books with similar titles. … The author uses physical intuition to derive the various mathematical theories, and is thus able to explain the ideas … in a fashion which is clear and helpful to both novice and expert." M. Groves, ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 85 (4), 2005 "[...] In brief, this book contains beautifully structured lectures on classical theory of linear partial differential equations of mathematical physics. Professor Arnold stresses the importance of physical intuitions and offers in his lecture a deep geometric insight into these equations. The book is highly recommended to anybody interested in partial differential equations as well as those involved in lecturing on these topics. I encourage readers of this book to take note of the Preface which contains very interesting comments on the role of Bourbaki's group in mathematics, a theme which resurfaces many times in these lectures." J.Chabrowski, Gazette, Australian Mathematical Society, Vol. 31, Issue 5, 2004 "... As a result the author has aimed to impart to students with pre-knowledge of only a basic kind (linear algebra, basic analysis, ordinary differential equations, ...) the essence of the theory and applications of the subject of partial differential equations. Of course the subject is fundamental in mathematics and in physics and the author is an evangelist for keeping the subject mainstream for mathematicians and for physicists. He has attempted, he writes, to adhere to the principle of minimal generality, according to which every idea should first be clearly understood in the simplest situation! This is successfully done, so that this book should prove attractive in length and in scope to its target readership. ... In this new excellent text are included a large number of interesting problems; at the end of the book there is a full set of problems from examinations given in Moscow. ..." F.H.Berkshire, Imperial College London, Contemporary Physics 2004, Vol. 45, Issue 6 "Like all Vladimir Arnold’s books, this one is full of geometric insight. Arnold illustrates every principle with a figure. This book aims to cover the most basic parts of the subject … . A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging!" L’Enseignement Mathematique, Vol. 50 (1-2), 2004 "Dieses Buch betont geometrische Einsicht und physikalische Intuition. Die Prinzipien werden an Bildern erläutert, und das Buch enthält mehr Text als Formeln und Sätze. […]. Neben einer großen Anzahl von Übungsaufgaben, die im Buch verstreut sind, finden sich interessante Prüfungsbeispiele der Moskauer Universität." J. Hertling, Internationale Mathematische Nachrichten, 2004, Issue 197, p. 47-48 "The book is based on a short course of lectures delivered to the third year mathematics students of the Independent University of Moscow … . The book can serve as a nonstandard, geometrically motivated introduction to PDEs for students … . It is, probably, worth mentioning that the introduction contains some general philosophical views of the author on the subject of PDEs and modern mathematics as a whole and will be of interest to a broad mathematical audience." (Victor Shubov, MathDL, January, 2001) "Like other books of Arnold, this is a very original introduction to the subject. It is … based on a course delivered to third-year students of mathematics. The aim of this book is to teach the fundamental ideas of partial differential equations and mathematical physics. … Not only students but also professional mathematicians from other fields of mathematics can learn the basic and simple ideas of partial differential equations from this unique book." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 74, 2008)Table of Contents1. The General Theory for One First-Order Equation.- 2. The General Theory for One First-Order Equation (Continued).- 3. Huygens’ Principle in the Theory of Wave Propagation.- 4. The Vibrating String (d’Alembert’s Method).- 5. The Fourier Method (for the Vibrating String).- 6. The Theory of Oscillations. The Variational Principle.- 7. The Theory of Oscillations. The Variational Principle (Continued).- 8. Properties of Harmonic Functions.- 9. The Fundamental Solution for the Laplacian. Potentials.- 10. The Double-Layer Potential.- 11. Spherical Functions. Maxwell’s Theorem. The Removable Singularities Theorem.- 12. Boundary-Value Problems for Laplace’s Equation. Theory of Linear Equations and Systems.- A. The Topological Content of Maxwell’s Theorem on the Multifield Representation of Spherical Functions.- A.1. The Basic Spaces and Groups.- A.2. Some Theorems of Real Algebraic Geometry.- A.3. From Algebraic Geometry to Spherical Functions.- A.4. Explicit Formulas.- A.6. The History of Maxwell’s Theorem.- Literature.- B. Problems.- B.1. Material from the Seminars.- B.2. Written Examination Problems.

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Book SynopsisFrom the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student." --New Zealand Mathematical Society, 1985Trade Review“This book is a bibliographical monument to the theory of both theoretical and applied PDEs that has not acquired any flaws due to its age. On the contrary, it remains a crucial and essential tool for the active research in the field.” (Francesco Petitta, SIAM Review, Vol. 61 (4), December, 2019)From the reviews:"The aim of the book is to present "the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process". The book is divided into two parts. The first (Chapters 2-8) is devoted to the linear theory, the second (Chapters 9-15) to the theory of quasilinear partial differential equations. These 14 chapters are preceded by an Introduction (Chapter 1) which expounds the main ideas and can serve as a guide to the book. ...The authors have succeeded admirably in their aims; the book is a real pleasure to read".Mathematical Reviews,1986 "Advanced students and professionals are snapping up this paperback text on linear and quasilinear partial differential equations. Whether you use their book as textbook or reference, the authors give you plenty to think about and work on, including an epilogue summarizing the latest research."Amazon.com delivers Mathematics and Statistics e-bulletin, July 2001Table of ContentsChapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation 2.5 The Poisson Integral 2.6 Convergence Theorems 2.7 Interior Estimates of Derivatives 2.8 The Dirichlet Problem; the Method of Subharmonic Functions 2.9 Capacity Problems Chapter 3. The Classical Maximum Principle 3.1 The Weak Maximum Principle 3.2 The Strong Maximum Principle 3.3 Apriori Bounds 3.4 Gradient Estimates for Poisson's Equation 3.5 A Harnack Inequality 3.6 Operators in Divergence Form Notes Problems Chapter 4. Poisson's Equation and Newtonian Potential 4.1 Hölder Continuity 4.2 The Dirichlet Problem for Poisson's Equation 4.3 Hölder Estimates for the Second Derivatives 4.4 Estimates at the Boundary 4.5 Hölder Estimates for the First Derivatives Notes Problems Chapter 5. Banach and Hilbert Spaces 5.1 The Contraction Mapping 5.2 The Method of Cintinuity 5.3 The Fredholm Alternative 5.4 Dual Spaces and Adjoints 5.5 Hilbert Spaces 5.6 The Projection Theorem 5.7 The Riesz Representation Theorem 5.8 The Lax-Milgram Theorem 5.9 The Fredholm Alternative in Hilbert Spaces 5.10 Weak Compactness Notes Problems Chapter 6. Classical Solutions; the Schauder Approach 6.1 The Schauder Interior Estimates 6.2 Boundary and Global Estimates 6.3 The Dirichlet Problem 6.4 Interior and Boundary Regularity 6.5 An Alternative Approach 6.6 Non-Uniformly Elliptic Equations 6.7 Other Boundary Conditions; the Obliue Derivative Problem 6.8 Appendix 1: Interpolation Inequalities 6.9 Appendix 2: Extension Lemmas Notes Problems Chapter 7. Sobolev Spaces 7.1 L^p spaces 7.2 Regularization and Approximation by Smooth Functions 7.3 Weak Derivatives 7.4 The Chain Rule 7.5 The W^(k,p) Spaces 7.6 DensityTheorems 7.7 Imbedding Theorems 7.8 Potential Estimates and Imbedding Theorems 7.9 The Morrey and John-Nirenberg Estimes 7.10 Compactness Results 7.11 Difference Quotients 7.12 Extension and Interpolation Notes Problems Chapter 8 Generalized Solutions and Regularity 8.1 The Weak Maximum Principle 8.2 Solvability of the Dirichlet Problem 8.3 Diferentiability of Weak Solutions 8.4 Global Regularity 8.5 Global Boundedness of Weak Solutions 8.6 Local Properties of Weak Solutions 8.7 The Strong Maximum Principle 8.8 The Harnack Inequality 8.9 Hölder Continuity 8.10 Local Estimates at the Boundary 8.11 Hölder Estimates for the First Derivatives 8.12 The Eigenvalue Problem Notes Problems Chapter 9. Strong Solutions 9.1 Maximum Princiles for Strong Solutions 9.2 L^p Estimates: Preliminary Analysis 9.3 The Marcinkiewicz Interpolation Theorem 9.4 The Calderon-Zygmund Inequality 9.5 L^p Estimates 9.6 The Dirichlet Problem 9.7 A Local Maximum Principle 9.8 Hölder and Harnack Estimates 9.9 Local Estimates at the Boundary Notes Problems Part II: Quasilinear Equations Chapter 10. Maximum and Comparison Principles 10.1 The Comparison Principle 10.2 Maximum Principles 10.3 A Counterexample 10.4 Comparison Principles for Divergence Form Operators 10.5 Maximum Principles for Divergence Form Operators Notes Problems Chapter 11. Topological Fixed Point Theorems and Their Application 11.1 The Schauder Fixes Point Theorem 11.2 The Leray-Schauder Theorem: a Special Case 11.3 An Application 11.4 The Leray-Schauder Fixed Point Theorem 11.5 Variational Problems Notes Chapter 12. Equations in Two Variables 12.1 Quasiconformal Mappings 12.2 hölder Gradient Estimates for Linear Equations 12.3 The Dirichlet Problem for Uniformly Elliptic Equations 12.4 Non-Uniformly Elliptic Equations Notes Problems Chapter 13. Hölder Estimates for

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Book SynopsisThese notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997. The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to s- plectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT - formal symplectic seminar 93-96, I would like to acknowledge the contributions of Allen Knutson, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa Prato, Eugene Lerman, Jonathan Weitsman, Lisa Jeffrey, Reyer Sjamaar, Shaun Martin, Stephanie Singer, Sue Tolman and, last but not least, Yael Karshon. Thanks to everyone sitting in Math 242 in the Fall of 1997 for all the c- ments they made, and especially to those who wrote notes on the basis of which I was better able to reconstruct what went on: Alexandru Scorpan, Ben Davis, David Martinez,DonBarkauskas,EzraMiller,HenriqueBursztyn,John-PeterLund,Laura De Marco, Olga Radko, Peter P? rib' ?k, Pieter Collins, Sarah Packman, Stephen Bigelow, Susan Harrington, Tolga Etgu .. and Yi Ma.Trade Review“I find this to be both the best introduction to symplectic geometry as well as a model for how to introduce any field of study. … one feels the hand of a master in the text’s homework sets: concrete, illustrative, and enhancing the material presented. … For an upper-level undergraduate or beginning graduate student, Lectures on Symplectic Geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics.” (Andrew McInerney, MAA Reviews, June, 2018)From the reviews of the first printing Over the years, there have been several books written to serve as an introduction to symplectic geometry and topology, […] The text under review here fits well within this tradition, providing a useful and effective synopsis of the basics of symplectic geometry and possibly serving as the springboard for a prospective researcher. The material covered here amounts to the "usual suspects" of symplectic geometry and topology. From an introductory chapter of symplectic forms and symplectic algebra, the book moves on to many of the subjects that serve as the basis for current research:symplectomorphisms, Lagrangian submanifolds, the Moser theorems, Darboux-Moser-Weinstein theory, almost complex structures, Kãhler structures, Hamiltonian mechanics, symplectic reduction, etc. The text is written in a clear, easy-to-follow style, that is most appropriate in mathematical sophistication for second-year graduate students; […]. This text had its origins in a 15-week course that the author taught at UC Berkeley. There are some nice passages where the author simply lists some known results and some well-known conjectures, much as one would expect to see in a good lecture on the same subject. Particularly eloquent is the author’s discussion of the compact examples and counterexamples of symplectic, almost complex, complex and Kähler manifolds. Throughout the text, she uses specific, well-chosen examples to illustrate the results. In the initial chapter, she provides a detailed section on the classical example of the syrnplectic structure of the cotangent bundle of a manifold. Showing a good sense of pedagogy, the author often leaves these examples as well-planned homework assignments at the end of some of the sections. […] In all of these cases, the author gives the reader a chance to illustrate and understand the interesting results of each section, rather than relegating the tedious but needed results to the reader. Mathematical Reviews 2002iTable of ContentsSymplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kähler Manifolds.- Complex Manifolds.- Kähler Forms.- Compact Kähler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.

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Book SynopsisFrom 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 MATHTrade Review"The monograph by T. Kato is 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.In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced. Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4). Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8). The fundamentals of semigroup theory are given in chapter 9. The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10.The first edition is now 30 years old. The revised edition is 20 years old. Nevertheless it is a standard textbook for the theory of linear operators. It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located. In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory. However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field.Zentralblatt MATH, 836Table of ContentsOne Operator theory in finite-dimensional vector spaces.- § 1. Vector spaces and normed vector spaces.- 1. Basic notions.- 2. Bases.- 3. Linear manifolds.- 4. Convergence and norms.- 5. Topological notions in a normed space.- 6. Infinite series of vectors.- 7. Vector-valued functions.- § 2. Linear forms and the adjoint space.- 1. Linear forms.- 2. The adjoint space.- 3. The adjoint basis.- 4. The adjoint space of a normed space.- 5. The convexity of balls.- 6. The second adjoint space.- § 3. Linear operators.- 1. Definitions. Matrix representations.- 2. Linear operations on operators.- 3. The algebra of linear operators.- 4. Projections. Nilpotents.- 5. Invariance. Decomposition.- 6. The adjoint operator.- § 4. Analysis with operators.- 1. Convergence and norms for operators.- 2. The norm of Tn.- 3. Examples of norms.- 4. Infinite series of operators.- 5. Operator-valued functions.- 6. Pairs of projections.- § 5. The eigenvalue problem.- 1. Definitions.- 2. The resolvent.- 3. Singularities of the resolvent.- 4. The canonical form of an operator.- 5. The adjoint problem.- 6. Functions of an operator.- 7. Similarity transformations.- § 6. Operators in unitary spaces.- 1. Unitary spaces.- 2. The adjoint space.- 3. Orthonormal families.- 4. Linear operators.- 5. Symmetric forms and symmetric operators.- 6. Unitary, isometric and normal operators.- 7. Projections.- 8. Pairs of projections.- 9. The eigenvalue problem.- 10. The minimax principle.- Two Perturbation theory in a finite-dimensional space.- § 1. Analytic perturbation of eigenvalues.- 1. The problem.- 2. Singularities of the eigenvalues.- 3. Perturbation of the resolvent.- 4. Perturbation of the eigenprojections.- 5. Singularities of the eigenprojections.- 6. Remarks and examples.- 7. The case of T(x) linear in x.- 8. Summary.- § 2. Perturbation series.- 1. The total projection for the ?-group.- 2. The weighted mean of eigenvalues.- 3. The reduction process.- 4. Formulas for higher approximations.- 5. A theorem of Motzkin-Taussky.- 6. The ranks of the coefficients of the perturbation series.- § 3. Convergence radii and error estimates.- 1. Simple estimates.- 2. The method of majorizing series.- 3. Estimates on eigenvectors.- 4. Further error estimates.- 5. The special case of a normal unperturbed operator.- 6. The enumerative method.- § . Similarity transformations of the eigenspaces and eigenvectors.- 1. Eigenvectors.- 2. Transformation functions.- 3. Solution of the differential equation.- 4. The transformation function and the reduction process.- 5. Simultaneous transformation for several projections.- 6. Diagonalization of a holomorphic matrix function.- § 5. Non-analytic perturbations.- 1. Continuity of the eigenvalues and the total projection.- 2. The numbering of the eigenvalues.- 3. Continuity of the eigenspaces and eigenvectors.- 4. Differentiability at a point.- 5. Differentiability in an interval.- 6. Asymptotic expansion of the eigenvalues and eigenvectors.- 7. Operators depending on several parameters.- 8. The eigenvalues as functions of the operator.- § 6. Perturbation of symmetric operators.- 1. Analytic perturbation of symmetric operators.- 2. Orthonormal families of eigenvectors.- 3. Continuity and differentiability.- 4. The eigenvalues as functions of the symmetric operator.- 5. Applications. A theorem of Lidskii.- Three Introduction to the theory of operators in Banach spaces.- § 1. Banach spaces.- 1. Normed spaces.- 2. Banach spaces.- 3. Linear forms.- 4. The adjoint space.- 5. The principle of uniform boundedness.- 6. Weak convergence.- 7. Weak* convergence.- 8. The quotient space.- § 2. Linear operators in Banach spaces.- 1. Linear operators. The domain and range.- 2. Continuity and boundedness.- 3. Ordinary differential operators of second order.- § 3. Bounded operators.- 1. The space of bounded operators.- 2. The operator algebra ?(X).- 3. The adjoint operator.- 4. Projections.- § 4. Compact operators.- 1. Definition.- 2. The space of compact operators.- 3. Degenerate operators. The trace and determinant.- § 5. Closed operators.- 1. Remarks on unbounded operators.- 2. Closed operators.- 3. Closable operators.- 4. The closed graph theorem.- 5. The adjoint operator.- 6. Commutativity and decomposition.- § 6. Resolvents and spectra.- 1. Definitions.- 2. The spectra of bounded operators.- 3. The point at infinity.- 4. Separation of the spectrum.- 5. Isolated eigenvalues.- 6. The resolvent of the adjoint.- 7. The spectra of compact operators.- 8. Operators with compact resolvent.- Four Stability theorems.- §1. Stability of closedness and bounded invertibility.- 1. Stability of closedness under relatively bounded perturbation.- 2. Examples of relative boundedness.- 3. Relative compactness and a stability theorem.- 4. Stability of bounded in vertibility.- § 2. Generalized convergence of closed operators.- 1. The gap between subspaces.- 2. The gap and the dimension.- 3. Duality.- 4. The gap between closed operators.- 5. Further results on the stability of bounded in vertibility.- 6. Generalized convergence.- § 3. Perturbation of the spectrum.- 1. Upper semicontinuity of the spectrum.- 2. Lower semi-discontinuity of the spectrum.- 3. Continuity and analyticity of the resolvent.- 4. Semicontinuity of separated parts of the spectrum.- 5. Continuity of a finite system of eigenvalues.- 6. Change of the spectrum under relatively bounded perturbation.- 7. Simultaneous consideration of an infinite number of eigenvalues.- 8. An application to Banach algebras. Wiener’s theorem.- § 4. Pairs of closed linear manifolds.- 1. Definitions.- 2. Duality.- 3. Regular pairs of closed linear manifolds.- 4. The approximate nullity and deficiency.- 5. Stability theorems.- § 5. Stability theorems for semi-Fredholm operators.- 1. The nullity, deficiency and index of an operator.- 2. The general stability theorem.- 3. Other stability theorems.- 4. Isolated eigenvalues.- 5. Another form of the stability theorem.- 6. Structure of the spectrum of a closed operator.- § 6. Degenerate perturbations.- 1. The Weinstein-Aronszajn determinants.- 2. The W-A formulas.- 3. Proof of the W-A formulas.- 4. Conditions excluding the singular case.- Five Operators in Hilbert spaces.- § 1. Hilbert space.- 1. Basic notions.- 2. Complete orthonormal families.- § 2. Bounded operators in Hilbert spaces.- 1. Bounded operators and their adjoints.- 2. Unitary and isometric operators.- 3. Compact operators.- 4. The Schmidt class.- 5. Perturbation of orthonormal families.- § 3. Unbounded operators in Hilbert spaces.- 1. General remarks.- 2. The numerical range.- 3. Symmetric operators.- 4. The spectra of symmetric operators.- 5. The resolvents and spectra of selfadjoint operators.- 6. Second-order ordinary differential operators.- 7. The operators T*T.- 8. Normal operators.- 9. Reduction of symmetric operators.- 10. Semibounded and accretive operators.- 11. The square root of an m-accretive operator.- § 4. Perturbation of self adjoint operators.- 1. Stability of selfadjointness.- 2. The case of relative bound 1.- 3. Perturbation of the spectrum.- 4. Semibounded operators.- 5. Completeness of the eigenprojections of slightly non-selfadjoint operators.- § 5. The Schrödinger and Dirac operators.- 1. Partial differential operators.- 2. The Laplacian in the whole space.- 3. The Schrödinger operator with a static potential.- 4. The Dirac operator.- Six Sesquilinear forms in Hilbert spaces and associated operators.- § 1. Sesquilinear and quadratic forms.- 1. Definitions.- 2. Semiboundedness.- 3. Closed forms.- 4. Closable forms.- 5. Forms constructed from sectorial operators.- 6. Sums of forms.- 7. Relative boundedness for forms and operators.- § 2. The representation theorems.- 1. The first representation theorem.- 2. Proof of the first representation theorem.- 3. The Friedrichs extension.- 4. Other examples for the representation theorem.- 5. Supplementary remarks.- 6. The second representation theorem.- 7. The polar decomposition of a closed operator.- § 3. Perturbation of sesquilinear forms and the associated operators.- 1. The real part of an m-sectorial operator.- 2. Perturbation of an m-sectorial operator and its resolvent.- 3. Symmetric unperturbed operators.- 4. Pseudo-Friedrichs extensions.- § 4. Quadratic forms and the Schrödinger operators.- 1. Ordinary differential operators.- 2. The Dirichlet form and the Laplace operator.- 3. The Schrödinger operators in R3.- 4. Bounded regions.- § 5. The spectral theorem and perturbation of spectral families.- 1. Spectral families.- 2. The selfadjoint operator associated with a spectral family.- 3. The spectral theorem.- 4. Stability theorems for the spectral family.- Seven Analytic perturbation theory.- § 1. Analytic families of operators.- 1. Analyticity of vector- and operator-valued functions.- 2. Analyticity of a family of unbounded operators.- 3. Separation of the spectrum and finite systems of eigenvalues.- 4. Remarks on infinite systems of eigenvalues.- 5. Perturbation series.- 6. A holomorphic family related to a degenerate perturbation.- § 2. Holomorphic families of type (A).- 1. Definition.- 2. A criterion for type (A).- 3. Remarks on holomorphic families of type (A).- 4. Convergence radii and error estimates.- 5. Normal unperturbed operators.- § 3. Selfadjoint holomorphic families.- 1. General remarks.- 2. Continuation of the eigenvalues.- 3. The Mathieu, Schrödinger, and Dirac equations.- 4. Growth rate of the eigenvalues.- 5. Total eigenvalues considered simultaneously.- § 4. Holomorphic families of type (B).- 1. Bounded-holomorphic families of sesquilinear forms.- 2. Holomorphic families of forms of type (a) and holomorphic families of operators of type (B).- 3. A criterion for type (B).- 4. Holomorphic families of type (B0).- 5. The relationship between holomorphic families of types (A) and (B).- 6. Perturbation series for eigenvalues and eigenprojections.- 7. Growth rate of eigenvalues and the total system of eigenvalues.- 8. Application to differential operators.- 9. The two-electron problem.- § 5. Further problems of analytic perturbation theory.- 1. Holomorphic families of type (C).- 2. Analytic perturbation of the spectral family.- 3. Analyticity of |H(x)| and |H(x)|?.- § 6. Eigenvalue problems in the generalized form.- 1. General considerations.- 2. Perturbation theory.- 3. Holomorphic families of type (A).- 4. Holomorphic families of type (B).- 5. Boundary perturbation.- Eight Asymptotic perturbation theory.- § 1. Strong convergence in the generalized sense.- 1. Strong convergence of the resolvent.- 2. Generalized strong convergence and spectra.- 3. Perturbation of eigenvalues and eigenvectors.- 4. Stable eigenvalues.- § 2. Asymptotic expansions.- 1. Asymptotic expansion of the resolvent.- 2. Remarks on asymptotic expansions.- 3. Asymptotic expansions of isolated eigenvalues and eigenvectors.- 4. Further asymptotic expansions.- § 3. Generalized strong convergence of sectorial operators.- 1. Convergence of a sequence of bounded forms.- 2. Convergence of sectorial forms “from above”.- 3. Nonincreasing sequences of symmetric forms.- 4. Convergence from below.- 5. Spectra of converging operators.- § 4. Asymptotic expansions for sectorial operators.- 1. The problem. The zeroth approximation for the resolvent.- 2. The 1/2-order approximation for the resolvent.- 3. The first and higher order approximations for the resolvent.- 4. Asymptotic expansions for eigenvalues and eigenvectors.- § 5. Spectral concentration.- 1. Unstable eigenvalues.- 2. Spectral concentration.- 3. Pseudo-eigenvectors and spectral concentration.- 4. Asymptotic expansions.- Nine Perturbation theory for semigroups of operators.- § 1. One-parameter semigroups and groups of operators.- 1. The problem.- 2. Definition of the exponential function.- 3. Properties of the exponential function.- 4. Bounded and quasi-bounded semigroups.- 5. Solution of the inhomogeneous differential equation.- 6. Holomorphic semigroups.- 7. The inhomogeneous differential equation for a holomorphic semigroup.- 8. Applications to the heat and Schrödinger equations.- § 2. Perturbation of semigroups.- 1. Analytic perturbation of quasi-bounded semigroups.- 2. Analytic perturbation of holomorphic semigroups.- 3. Perturbation of contraction semigroups.- 4. Convergence of quasi-bounded semigroups in a restricted sense.- 5. Strong convergence of quasi-bounded semigroups.- 6. Asymptotic perturbation of semigroups.- § 3. Approximation by discrete semigroups.- 1. Discrete semigroups.- 2. Approximation of a continuous semigroup by discrete semigroups.- 3. Approximation theorems.- 4. Variation of the space.- Ten Perturbation of continuous spectra and unitary equivalence.- §1. The continuous spectrum of a selfadjoint operator.- 1. The point and continuous spectra.- 2. The absolutely continuous and singular spectra.- 3. The trace class.- 4. The trace and determinant.- § 2. Perturbation of continuous spectra.- 1. A theorem of Weyl-von Neumann.- 2. A generalization.- § 3. Wave operators and the stability of absolutely continuous spectra.- 1. Introduction.- 2. Generalized wave operators.- 3. A sufficient condition for the existence of the wave operator.- 4. An application to potential scattering.- § 4. Existence and completeness of wave operators.- 1. Perturbations of rank one (special case).- 2. Perturbations of rank one (general case).- 3. Perturbations of the trace class.- 4. Wave operators for functions of operators.- 5. Strengthening of the existence theorems.- 6. Dependence of W± (H2, H1) on H1 and H2.- § 5. A stationary method.- 1. Introduction.- 2. The ? operations.- 3. Equivalence with the time-dependent theory.- 4. The ? operations on degenerate operators.- 5. Solution of the integral equation for rank A = 1.- 6. Solution of the integral equation for a degenerate A.- 7. Application to differential operators.- Supplementary Notes.- Supplementary Bibliography.- Notation index.- Author index.

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Book SynopsisIn this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.Table of ContentsPreface.- Contents.- Chap. I: Ordinary differential equations.- Chap. II: Scalar first order equations with one space variable.- Chap. III: Scalar first order equations with several variables.- Chap. IV: First order systems of conservation laws with one space.- Chap. V: Compensated compactness.- Chap. VI: Nonlinear perturbations of the wave equation.- Chap. VII: Nonlinear perturbations of the Klein-Gordon equation.- Chap. VIII: Microlocal analysis.- Chap. IX: Pseudo-differential operators of type 1,1.- Chap. X: Paradifferential calculus.- Chap. XI: Propagation of singularities.- Appendix on pseudo-Riemannian geometry.- References.- Index of notations.- Index.

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Book SynopsisThe main characteristic of this classic exposition of the inverse scattering method and its applications to soliton theory is its consistent Hamiltonian approach to the theory. The nonlinear Schrödinger equation is considered as a main example, forming the first part of the book. The second part examines such fundamental models as the sine-Gordon equation and the Heisenberg equation, the classification of integrable models and methods for constructing their solutions.Trade Review Table of ContentsThe Nonlinear Schrödinger Equation (NS Model).- Zero Curvature Representation.- The Riemann Problem.- The Hamiltonian Formulation.- General Theory of Integrable Evolution Equations.- Basic Examples and Their General Properties.- Fundamental Continuous Models.- Fundamental Models on the Lattice.- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models.- Conclusion.- Conclusion.

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Book SynopsisAt the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject. Trade ReviewFrom the reviews:"The book is aimed to old and new problems of optimal transport. … This meticulous work is based on very large bibliography … that is converted into a very valuable monograph that presents many statements and theorems written specifically for this approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes." (Mihail Voicu, Zentralblatt MATH, Vol. 1156, 2009)“This book wins the challenge to give a new and broad perspective on the multifacet topic of the optimal mass transport. … Besides extensive and accurate references therein the reader will find comments on related questions barely touched upon in the main text as well as lively presentations on how ideas and results have developed. This book should prove useful both to the expert and to the beginner looking for a reference text on the subject.” (Dario Cordero Erausquin, Mathematical Reviews, Issue 2010 f)“The book is an in-depth, modern, clear exposition of the advanced theory of optimal transport, and it tries to put together in a unified way almost all the recent developments of the theory. … the book is extremely well written and very pleasant to read. … I strongly recommend this excellent book to every researcher or graduate student in the field of optimal transport. … of interest to many mathematicians in different areas, who are simply interested in having an overview of the subject.” (Alessio Figalli, Bulletin of the American Mathematical Society, Vol. 47 (4), February, 2010)Table of ContentsCouplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge—Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

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Book SynopsisThis volume covers the stability of nonautonomous differential equations in Banach spaces in the presence of nonuniform hyperbolicity. Topics under discussion include the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, and the construction and regularity of topological conjugacies. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.Trade ReviewFrom the reviews: “In this book, the authors give a unified presentation of a substantial body of work which they have carried out and which revolves around the concept of nonuniform exponential dichotomy. … This is a well-written book which contains many interesting results. The reader will find significant generalizations of the standard invariant manifold theories, of the Hartman-Grobman theorem … . Anyone interested in these topics will profit from reading this book.” (Russell A. Johnson, Mathematical Reviews, Issue 2010 b)Table of ContentsExponential dichotomies.- Exponential dichotomies and basic properties.- Robustness of nonuniform exponential dichotomies.- Stable manifolds and topological conjugacies.- Lipschitz stable manifolds.- Smooth stable manifolds in Rn.- Smooth stable manifolds in Banach spaces.- A nonautonomous Grobman–Hartman theorem.- Center manifolds, symmetry and reversibility.- Center manifolds in Banach spaces.- Reversibility and equivariance in center manifolds.- Lyapunov regularity and stability theory.- Lyapunov regularity and exponential dichotomies.- Lyapunov regularity in Hilbert spaces.- Stability of nonautonomous equations in Hilbert spaces.

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Book SynopsisLectures: A. Jeffrey: Lectures on nonlinear wave propagation.- Y. Choquet-Bruhat: Ondes asymptotiques.- G. Boillat: Urti.- Seminars: D. Graffi: Sulla teoria dell’ottica non-lineare.- G. Grioli: Sulla propagazione del calore nei mezzi continui.- T. Manacorda: Onde nei solidi con vincoli interni.- T. Ruggeri: "Entropy principle" and main field for a non linear covariant system.- B. Straughan: Singular surfaces in dipolar materials and possible consequences for continuum mechanicsTable of ContentsLectures: A. Jeffrey: Lectures on nonlinear wave propagation.- Y. Choquet-Bruhat: Ondes asymptotiques.- G. Boillat: Urti.- Seminars: D. Graffi: Sulla teoria dell’ottica non-lineare.- G. Grioli: Sulla propagazione del calore nei mezzi continui.- T. Manacorda: Onde nei solidi con vincoli interni.- T. Ruggeri: "Entropy principle" and main field for a non linear covariant system.- B. Straughan: Singular surfaces in dipolar materials and possible consequences for continuum mechanics.

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Book SynopsisFractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations. Trade ReviewFrom the reviews:“This book treats a fast growing field of fractional differential equations, i.e., differential equations with derivatives of non-integer order. … The book consists of two parts, eight chapters, an appendix, references and an index. … The book is well written and easy to read. It could be used for, a course in the application of fractional calculus for students of applied mathematics and engineering.” (Teodor M. Atanacković, Mathematical Reviews, Issue 2011 j)“This monograph is intended for use by graduate students, mathematicians and applied scientists who have an interest in fractional differential equations. The Caputo derivative is the main focus of the book, because of its relevance to applications. … The monograph may be regarded as a fairly self-contained reference work and a comprehensive overview of the current state of the art. It contains many results and insights brought together for the first time, including some new material that has not, to my knowledge, appeared elsewhere.” (Neville Ford, Zentralblatt MATH, Vol. 1215, 2011)Table of ContentsFundamentals of Fractional Calculus.- Riemann-Liouville Differential and Integral Operators.- Caputo’s Approach.- Mittag-Leffler Functions.- Theory of Fractional Differential Equations.- Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- Multi-Term Caputo Fractional Differential Equations.

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Book SynopsisIn recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.Trade ReviewFrom the reviews:“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)Table of ContentsPreface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.

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Book SynopsisIn this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction. Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure. The book is arranged in four sections, devoted to realizing the universal principle force equals curvature: Part I: The Euclidean Manifold as a ParadigmPart II: Ariadne's Thread in Gauge TheoryPart III: Einstein's Theory of Special RelativityPart IV: Ariadne's Thread in Cohomology For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum. Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos). Trade ReviewFrom the reviews:“This book is the third volume of a complete exposition of the important mathematical methods used in modern quantum field theory. It presents the very basic formalism, important results, and the most recent advances emphasizing the applications to gauge theory. … the book’s greatest strength is Zeidler’s zeal to help students understand fundamental mathematics better. I thus find the book extremely useful since it signifies the role of mathematics for the road to reality … .” (Gert Roepstorff, Zentralblatt MATH, Vol. 1228, 2012)“The present book is a good companion to the literature on the subject of the volume title, especially for those already familiar with it. … the book touches upon a large number of subjects on the interface between mathematics and physics, providing a good overview of gauge theory in both fields. It contains lots of background material, many historical remarks, and an extensive bibliography that helps the interested reader to continue his or her more thorough studies elsewhere.” (Walter D. van Suijlekom, Mathematical Reviews, Issue 2012 m)Table of ContentsPrologue.- Part I. The Euclidean Manifold as a Paradigm: 1. The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure).- 2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Cli_ord Algebra, Lie Algebra).- 3. Representations of Symmetries in Mathematics and Physics.- 4. The Euclidean Manifold E3.- 5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry.- 6. Infinitesimal Rotations and Constraints in Physics.- 7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin.- 8. Changing Observers - A Glance at Invariant Theory Based on the Principle of the Correct Index Picture.- 9. Applications of Invariant Theory to the Rotation Group.- 10. Temperature Fields on the Euclidean Manifold E3.- 11. Velocity Vector Fields on the Euclidean Manifold E3.- 12. Covector Fields and Cartan's Exterior Differential - the Beauty of Differential Forms.- Part II. Ariadne's Thread in Gauge Theory: 13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field.- 14. Symmetry Breaking.- 15. The Noncommutative Yang{Mills SU(N)-Gauge Theory.- 16. Cocycles and Observers.- 17. The Axiomatic Geometric Approach to Bundles.- Part III. Einstein's Theory of Special Relativity: 18. Inertial Systems and Einstein's Principle of Special Relativity.- 19. The Relativistic Invariance of the Maxwell Equations.- 20. The Relativistic Invariance of the Dirac Equation and the Electron Spin.- Part IV. Ariadne's Thread in Cohomology: 21. The Language of Exact Sequences.- 22. Electrical Circuits as a Paradigm in Homology and Cohomology.- 23. The Electromagnetic Field and the de Rham Cohomology.- Appendix.- Epilogue.- References.- List of Symbols.- Index

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Book SynopsisIn recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.Trade ReviewFrom the reviews:“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)Table of ContentsPreface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.

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Book SynopsisThis self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix.The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition.Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.Trade Review“Every line of the book reflects that the author is the leading expert for hierarchical matrices. … Hierarchical matrices: algorithms and analysis is without a doubt a beautiful, comprehensive introduction to hierarchical matrices that can serve as both a graduate level textbook and a valuable resource for future research.” (Thomas Mach, Mathematical Reviews, April, 2017)“The book ‘Hierarchical matrices: algorithms and analysis’ is a self-contained monograph which presents an efficient possibility to handle the numerical treatment of fully populated large scale matrices appearing in scientific computations, and therefore it is of interest to scientists in computational mathematics, physics, chemistry and engineering.” (Constantin Popa, zbMATH 1336.65041, 2016)Table of ContentsPreface.- Part I: Introductory and Preparatory Topics.- 1. Introduction.- 2. Rank-r Matrices.- 3. Introductory Example.- 4. Separable Expansions and Low-Rank Matrices.- 5. Matrix Partition.- Part II: H-Matrices and Their Arithmetic.- 6. Definition and Properties of Hierarchical Matrices.- 7. Formatted Matrix Operations for Hierarchical Matrices.- 8. H2-Matrices.- 9. Miscellaneous Supplements.- Part III: Applications.- 10. Applications to Discretised Integral Operators.- 11. Applications to Finite Element Matrices.- 12. Inversion with Partial Evaluation.- 13. Eigenvalue Problems.- 14. Matrix Functions.- 15. Matrix Equations.- 16. Tensor Spaces.- Part IV: Appendices.- A. Graphs and Trees.- B. Polynomials.- C. Linear Algebra and Functional Analysis.- D. Sinc Functions and Exponential Sums.- E. Asymptotically Smooth Functions.- References.- Index.

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Book SynopsisThis book provides a clear and easy-to-understand introduction to higher mathematics with numerous examples. The author shows how to solve typical problems in a recipe-like manner and divides the material into short, easily digestible learning units.Have you ever cooked a 3-course meal based on a recipe? That generally works quite well, even if you are not a great cook. What does this have to do with mathematics? Well, you can solve a lot of math problems recipe-wise: Need to solve a Riccati's differential equation or the singular value decomposition of a matrix? Look it up in this book, you'll find a recipe for it here. Recipes are available for problems from the· Calculus in one and more variables,· linear algebra,· Vector Analysis,· Theory on differential equations, ordinary and partial,· Theory of integral transformations,· Function theory.Other features of this book include:· The division of Higher Mathematics into approximately 100 chapters of roughly equal length. Each chapter covers approximately the material of a 90-minute lecture.· Many tasks, the solutions to which can be found in the accompanying workbook.· Many problems in higher mathematics can be solved with computers. We always indicate how it works with MATLAB®.For the present 3rd edition, the book has been completely revised and supplemented by a section on the solution of boundary value problems for ordinary differential equations, by the topic of residue estimates for Taylor expansions and by the characteristic method for partial differential equations of the 1st order, as well as by several additional problems.Table of ContentsPreface.- 1 Ways of speaking, symbols and quantities.- 2 The natural, whole and rational numbers.- 3 The real numbers.- 4 Machine numbers.- 5 Polynomials.- 6 Trigonometric functions.- 7 Complex numbers - Cartesian coordinates.- 8 Complex numbers - Polar coordinates.- 9 Systems of linear equations.- 10 Calculating with matrices.- 11 LR-decomposition of a matrix.- 12 The determinant.- 13 Vector spaces.- 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear balancing problem.- 14 The linear balancing problem. 14 Generating systems and linear (in)dependence.- 15 Bases of vector spaces.- 16 Orthogonality I.- 17 Orthogonality II.- 18 The linear compensation problem.- 19 The QR-decomposition of a matrix.- 20 Sequences.- 21 Computation of limit values of sequences.- 22 Series.- 23 Illustrations.- 24 Power series.- 25 Limit values and continuity.- 26 Differentiation.- 27 Applications of differential calculus I.- 28 Applications of differential calculus I.- 28 Applications of differential calculus II.- 28 Applications of differential calculus I.- 28 Applications of differential calculus II. 28 Applications of differential calculus II.- 29 Polynomial and spline interpolation.- 30 Integration I.- 31 Integration II.- 32 Improper integrals.- 33 Separable and linear differential equations of the 1st order.- 34 Linear differential equations with constant coefficients.- 35 Some special types of differential equations.- 36 Numerics of ordinary differential equations I.- 37 Linear mappings and representation matrices.- 38 Basic transformation.- 39 Diagonalization - Eigenvalues and eigenvectors.- 40 Numerical computation of eigenvalues and eigenvectors.- 41 Quadrics.- 42 Schurzdecomposition and singular value decomposition.- 43 Jordan normal form I.- 44 Jordan normal form II.- 45 Definiteness and matrix norms.- 46 Functions of several variables.- 47 Partial differentiation - gradient, Hessian matrix, Jacobian matrix.- 48 Applications of partial derivatives.- 49 Determination of extreme values.- 50 Determination of extreme values under constraints.- 51 Total differentiation, differential operators.- 52 Implicit functions.- 53 Coordinate transformations.- 54 Curves I.- 55 Curves II.- 56 Curve integrals.- 57 Gradient fields.- 58 Domain integrals.- 59 The transformation formula.- 60 Areas and area integrals.- 61 Integral theorems I.- 62 Integral theorems II.- 63 General about differential equations.- 64 The exact differential equation.- 65 Systems of linear differential equations I.- 66 Systems of linear differential equations II.- 67 Systems of linear differential equations II.- 68 Boundary value problems.- 69 Basic concepts of numerics.- 70 Fixed point iteration.- 71 Iterative methods for systems of linear equations.- 72 Optimization.- 73 Numerics of ordinary differential equations II.- 74 Fourier series - Calculation of Fourier coefficients.- 75 Fourier series - Background, theorems and application.- 76 Fourier transform I.- 77 Fourier transform II.- 78 Discrete Fourier transform.- 79 The Laplacian transform.- 80 Holomorphic functions.- 81 Complex integration.- 82 Laurent series.- 83 The residue calculus.- 84 Conformal mappings.- 85 Harmonic functions and Dirichlet's boundary value problem.- 86 Partial differential equations 1st order.- 87 Partial differential equations 2nd order - General.- 88 The Laplace or Poisson equation.- 89 The heat conduction equation.- 90 The wave equation.- 91 Solving pDGLs with Fourier and Laplace transforms.- Index.

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Book SynopsisThe Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization or perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. This volume deals with the application of this method to many problems of physics, including some frontier problems which have previously required much more computationally-intensive approaches. The opening chapters deal with various fundamental aspects of the decomposition method. Subsequent chapters deal with the application of the method to nonlinear oscillatory systems in physics, the Duffing equation, boundary-value problems with closed irregular contours or surfaces, and other frontier areas. The potential application of this method to a wide range of problems in diverse disciplines such as biology, hydrology, semiconductor physics, wave propagation, etc., is highlighted. For researchers and graduate students of physics, applied mathematics and engineering, whose work involves mathematical modelling and the quantitative solution of systems of equations. Trade Review`I recommend Adomian's new book to all researchers in the area of mathematical modeling and solving complex dynamical systems.' Foundations of Physics, 1994 Table of ContentsPreface. Foreword. 1. On Modelling Physical Phenomena. 2. The Decomposition Method for Ordinary Differential Equations. 3. The Decomposition Method in Several Dimensions. 4. Double Decomposition. 5. Modified Decomposition. 6. Applications of Modified Decomposition. 7. Decomposition Solutions for Neumann Boundary Conditions. 8. Integral Boundary Conditions. 9. Boundary Conditions at Infinity. 10. Integral Equations. 11. Nonlinear Oscillations in Physical Systems. 12. Solution of the Duffing Equation. 13. Boundary-Value Problems with Closed Irregular Contours or Surfaces. 14. Applications in Physics. Appendix I: Padé and Shanks Transform. Appendix II: On Staggered Summation of Double Decomposition Series. Appendix III: Cauchy Products of Infinite Series. Index.

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Book SynopsisChapter 1  Distributions.- Chapter 2  Fundamental Solutions of Linear Differential Operators.- Chapter 3  Boundary Value Problems of the Laplace Equations.- Chapter 4  Boundary Value Problem of Modified Helmholtz Equation.- Chapter 5  Boundary Value Problems of Helmholtz Equation.- Chapter 6  Boundary Value Problems of the Navier Equations.- Chapter 7  Boundary Value Problems of the Stokes Equations.- Chapter 8  Some Nonlinear Problems.- Chapter 9  Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method.

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Book SynopsisGlobally integrable quantum systems and their perturbations.- On two-dimensional Dirac operators with $delta$-shell interactions supported on unbounded curves with straight ends.- Attractor Subspace and Decoherence-Free Algebra of Quantum Dynamics.- Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension.- Hearing the boundary conditions of the one-dimensional Dirac operator Bosonized Momentum Distribution of a Fermi Gas via Friedrichs Diagrams.- Self-adjointness and Domain of Generalized Spin–Boson Models with Mild Ultraviolet Divergences.- Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back.- New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D.- Resolvent limits of exterior boundary value problems and singular perturbation of Laplace operator in 3D.- The Search for NLS Ground States on a hybrid domain: motivations, methods, and results.- From microscopic to macroscopic: the large number dynamics of agents and cells, possibly interacting with a chemical background.- Open problems and perspectives on solving Friedrichs systems by Krylov approximation.- Singularity: a Seventh Memo.

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Book SynopsisChapter 1 A comparison of the Coco-Russo scheme and -FEM for elliptic equations in arbitrary domains.- Chapter 2 A semi-implicit method for a degenerating convection-diffusion-reaction problem modeling secondary settling tanks.- Chapter 3 Multidimensional approximate Riemann solvers for hyperbolic nonconservative systems: a review.- Chapter 4 Challenges in Stochastic Galerkin Methods for Nonlinear Hyperbolic Systems with Uncertainty.- Chapter 5 On the role of momentum correction factor and general tube law in one-dimensional blood flow models for networks of vessels.- Chapter 6 Numerical modelling of the hemodynamic changes in the inferior vena cava in response to the Valsalva maneuver.

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Book SynopsisPreface.- The Autonomous Duffing Equation.- The Periodically Forced Duffing Equation.- Chaos in the Duffing Equation: With Some Simulations.- Topological Methods for the Detection of Chaos.- Applications to the Superlinear Duffing Equation.- Laser.- The Forced Pendulum.- Chaos in the Duffing-type Equation related to Tides.- Index.

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Book SynopsisTrade Review"Overall, the reviewer considers this text to offer a good and useful coverage of advanced mathematics for engineers. It gives useful and succinct coverage of the topics included." --IEEE PulseTable of Contents1. Prologue 2. Ordinary Differential Equations 3. Laplace and Fourier Transform Methods 4. Matrices and Linear Systems of Equations 5. Analytical Methods for Solving Partial Differential Equations 6.Difference Numerical Methods for Differential Equations 7. Finite Element Technique 8. Treatment of Experimental Results 9. Numerical Analysis 10. Introduction to Complex Analysis 11. Nondimensionalisation 12. Nonlinear Differential Equations 13. Integral Equations 14. Calculus of Variations

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Book SynopsisLinear non-autonomous equations arise as mathematical models in mechanics, chemistry, and biology. This book explores the preservation of invariant tori of dynamic systems under perturbation. It is a useful contribution to the literature on stability theory and provides a source of reference for postgraduates and researchers.Trade Review"This volume will be of great interest to researchers and students dealing with nonautonomous systems." - Zentralblatt fur Mathematik, Vol. 1026Table of ContentsExponentially Dichotomous Linear Systems of Differential Equations and Lyapunov Functions of Variable Sign. Exponential Dichotomy Criterion for Linear Systems in Terms of Quadratic Forms. Decomposition Over the Whole R Axis of Linear Systems of Differential Equations Exponentially Dichotomous on Semiaxes R+ and R_. Degeneracy of the Quadratic Form Possessing a Definite-Sign Derivative Along the Solutions of the System (1.1.1). Integral Representation of Weakly Regular Systems Bounded on the Whole R Axis. Complement to the Exponentially Dichotomous of Weakly Regular on R Linear Systems. Regularity of Linear Systems of the Block-Triangular Form. Perturbation of the Block-Triangular Form Linear Systems which are Regular and Weakly Regular on the Whole R Axis. Exponentially Dichotomous Linear Systems with Parameters. Comments and References. Linear Extension of Dynamical Systems on a Torus. Necessary Existence Conditions for Invariant Tori. The Green Function. Sufficient Existence Conditions for an Invariant Torus. Existence Conditions for an Exponentially Stable Invariant Torus. Uniqueness Conditions for the Green Function and its Properties. Sufficient Conditions for Exponential Dichotomy of the Invariant Torus. Necessary Conditions for Exponential Dichotomy of the Invariant Torus. Existence Criterion for the Green Function. The Non-Unique Green Function and the Properties of the System Implied by its Existence. Invariant Tori of Linear Extensions with Slowly Changing Phase. Preserving the Green Function Under Small Perturbations of Linear Expansions on a Torus. On the Smoothness of an Exponentially Stable Invariant Torus. On the Dependence of Green Functions on Parameters. Continuity and Differentiability of the Green Function. Invariant Tori of Linear Extensions with a Degenerate Matrix at the Derivatives. Bounded Invariant Manifolds of Dynamical Systems and their Smoothness. Comments and References. Splitability of Linear Extensions of Dynamical Systems on a Torus. Sufficient Conditions for Splitability of Linear Extensions of Dynamical Systems on a Torus. Reversibility of the Theorem on Splitability. On Triangulation and the Relationship of C'-Block Splitability of a Linear System with the Problem on r-frame Complementability up to the Periodic Basis in R^Tn. Reducing on Linearized Systems to a Diagonal Form. On the Relationship of Exponentially Dichotomous Linear Expansions with the Algebraic System Solvability. Three Block Divisibility of Linear Extensions and Lyapunov Functions of Variable Sign. Algebraic Problems of the K-Blocked Divisibility of Linear Extensions on a Torus. Comments and References. Problems of Perturbation Theory of Smooth Invariant Tori of Dynamical Systems. Solution Variations on the Manifold M. Exponential Stability and Dichotomy Conditions for Linear Extensions of Dynamical Systems on a Torus. Roughness Conditions for the Green Function of the Linear Extension of a Dynamical System on a Torus with the Index of Smoothness. A Theorem of Perturbation Theory of an Invariant Torus of a Dynamical System. Green Function for a Linear Matrix Equation. On the Problem of Structure of Some Regular Linear Extensions of Dynamical Systems on a Torus. Invariant Manifolds of Autonomous Differential Equations and Lyapunov Functions with Alternating Signs. Comments and References. Index.

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Book SynopsisNonlinear systems with random structures arise quite frequently as mathematical models in diverse disciplines. This monograph presents a systematic treatment of stability theory and the theory of stabilization of nonlinear systems with random structure in terms of new developments in the direct Lyapunov's method. The analysis focuses on dynamic systems with random Markov parameters. This high-level research text is recommended for all those researching or studying in the fields of applied mathematics, applied engineering, and physics-particularly in the areas of stochastic differential equations, dynamical systems, stability, and control theory.Trade Review"This volume will be of interest to researchers and students in stochastic stability theory." - Zentralblatt fur Mathematik, Vol. 1026Table of ContentsIntroductory Remarks. Random Variables and Probability Distributions. Probability Processes and their Mathematical Description. Random Differential Equations. System with Random Structure. Stability Analysis Using Scalar Lyapunov Functions. Stability Concepts for Stochastic Systems. Random Scalar Lyapunov Functions. Conditions of Stability in Probability. Converse Theorems. Stability in Mean Square. Stability in Mean Square of Linear Systems. Stability Analysis Using Multi-component Lyapunov Functions. Vector Lyapunov Functions. Stochastic Matrix-Valued Lyapunov Functions. Stability Analysis in General. Stability Analysis of Systems in Ito's Form. Stochastic Singularly Perturbed Systems. Large-Scale Singularly Perturbed Systems. Stability Analysis by the First-Order Approximation. Stability Criterion by the First-Order Approximation. Stability with Respect to the First-Order Approximation. Stability by First-Order Approximation of Systems with Random Delay. Convergence of Stochastic Approximation Procedure. Stabilization of Controlled Systems with Random Structure. Problems of Stabilization. Optimal Stabilization. Linear-Quadratic Optimal Stabilization. Sufficient Stabilization Conditions for Linear Systems. Optimal Solution Existence. The Small Parameter Method Algorithm. Applications. A Stochastic Version of the Lefschetz Problem. Stability in Probability of Oscillating Systems. Stability in Probability of Regulation Systems. Price Stability in a Stochastic Market Model. References. Index. Lyapunov Functions. Stability Analysis Using Multicomponent Lyapunov Functions. Stability Analysis by the First-order Approximation. Stabilization of Controlled Systems with Random Structure. Applications; References; Index.

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Book SynopsisEquations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case.Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.Table of ContentsSome Preliminaries from Analysis and the Theory of Wave Processes. Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium. Wave Fields in a Layer of Constant Thickness. Analytical Methods for Simply Connected Bounded Domains. Integral Equations in Diffraction by Linear Obstacles. Short-Wave Asymptotic Methods on the Basis of Multiple Integrals. Inverse Problems of the Short-Wave Diffraction. Ill-Posed Equations of Inverse Diffraction Problems for Arbitrary Boundary. Numerical Methods for Irregular Operator Equations.

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Book SynopsisThis second edition of the book, Nonlinear Random Vibration: Analytical Techniques and Applications, expands on the original edition with additional detailed steps in various places in the text. It is a first systematic presentation on the subject. Its features include:â a concise treatment of Markovian and non- Markovian solutions of nonlinear stochastic differential equations,â exact solutions of Fokker-Planck-Kolmogorov equations,â methods of statistical linearization,â statistical nonlinearization techniques,â methods of stochastic averaging,â truncated hierarchy techniques, andâ an appendix on probability theory.A special feature is its incorporation of detailed steps in many examples of engineering applications.Targeted audience: Graduates, research scientists and engineers in mechanical, aerospace, civil and environmental (earthquake, wind and transportation), automobile, naval, architectural, and mining engineering.Trade ReviewIn summary, the technical material in Prof. To’s 2012 second edition of Nonlinear Random Vibration: Analytical Techniques and Applications is well presented, of sufficient depth, detail, and quality, and supported by a good number of solved example problems.Robert M. KochNaval Undersea Warfare Center, Newport, RI, USAIn: Noise Control Engr. J. 61 (2), March-April 2013, pp 251-252Table of ContentsIntroduction. Markovian and Non-Markovian Solutions of Stochastic Nonlinear Differential Equations. Exact Solution of the Fokker-Planck-Kolmogorov Equation. Methods of Statistical Linearization. Statistical Nonlinearization Techniques. Methods of Stochastic Averaging. Truncated Hierarchy and other Techniques.

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