Universitext – serie

This graduate textbook offers an introduction to the spectral theory of ordinary differential equations, focusing on Sturm–Liouville equations.Sturm–Liouville theory has applications in partial differential equations and mathematical physics.

The calculus of variations has a long history of interaction with other branches of mathematics such as geometry and differential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applications in other fields such as economics and electrical engineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations.This book is an introduction to the calculus of variations for mathematicians and scientists. The reader interested primarily in mathematics will find results of interest in geometry and differential equations. I have paused at times to develop the proofs of some of these results, and discuss briefly various topics not normally found in an introductory book on this subject such as the existence and uniqueness of solutions to boundary-value problems, the inverse problem, and Morse theory. I have made “passive use” of functional analysis (in particular normed vector spaces) to place certain results in context and reassure the mathematician that a suitable framework is available for a more rigorous study. For the reader interested mainly in techniques and applications of the calculus of variations, I leavened the book with numerous examples mostly from physics. In addition, topics such as Hamilton’s Principle, eigenvalue approximations, conservation laws, and nonholonomic constraints in mechanics are discussed. More importantly, the book is written on two levels. The technical details for many of the results can be skipped on the initial reading. The student can thus learn the main results in each chapter and return as needed to the proofs for a deeper understanding.

This book provides a foundation for arithmetic topology, a new branch of mathematics that investigates the analogies between the topology of knots, 3-manifolds, and the arithmetic of number fields.

This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs).

This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research.Thesecond edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.

The topics introduced include arithmetic of rings, modules, especially principal ideal rings and the classification of modules over such rings, Galois theory, as well as an introduction to more advanced topics such as homological algebra, tensor products, and algebraic concepts involved in algebraic geometry.

The abstract branch of theoretical computer science known as Computation Theory typically appears in undergraduate academic curricula in a form that obscures both the mathematical concepts that are central to the various components of the theory and the relevance of the theory to the typical student. This regrettable situation is due largely to the thematic tension among three main competing principles for organizing the material in the course.This book is motivated by the belief that a deep understanding of, and operational control over, the few "big" mathematical ideas that underlie Computation Theory is the best way to enable the typical student to assimilate the "big" ideas of Computation Theory into her daily computational life.

This book is a primer in harmonic analysis using an elementary approach. Its first aim is to provide an introduction to Fourier analysis, leading up to the Poisson Summation Formula. Secondly, it makes the reader aware of the fact that both, the Fourier series and the Fourier transform, are special cases of a more general theory arising in the context of locally compact abelian groups. The third goal of this book is to introduce the reader to the techniques used in harmonic analysis of noncommutative groups. There are two new chapters in this new edition. One on distributions will complete the set of real variable methods introduced in the first part. The other on the Heisenberg Group provides an example of a group that is neither compact nor abelian, yet is simple enough to easily deduce the Plancherel Theorem.

Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor and what it describes. Second, one must be able to compute these things, and, often, this involves yet another language: spectral sequences. This book gives a treatment of homological algebra which motivates the subject in terms of its origins in algebraic topology.In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author also included material about homotopical algebra, alias K-theory, contrasting it with homological algebra.

If mathematics is a language, then taking a topology course at the undergraduate level is cramming vocabulary and memorizing irregular verbs: a necessary, but not always exciting exercise one has to go through before one can read great works of literature in the original language.The present book grew out of notes for an introductory topology course at the University of Alberta. It provides a concise introduction to set theoretic topology (and to a tiny little bit of algebraic topology). It is accessible to undergraduates from the second year on, but even beginning graduate students canbenefit from some parts.Great care has been devoted to the selection of examples that are not self-serving, but already accessible for students who have a background in calculus and elementary algebra, but not necessarily in real or complex analysis.In some points, the book treats its material differently than other texts on the subject: Baire's theorem is derived from Bourbaki's Mittag-Leffler theorem nets are used extensively, in particular for an intuitive proof of Tychonoff's theorem a short and elegant, but little known proof for the Stone-Weierstrass theorem is given

Lie groups, named for the 19th century mathematician Sophus Lie, has been an increasing area of focus and rich research since the middle of the 20th century. Procesi's masterful approach to Lie groups through invariants and representations gives the reader a comprehensive treatment of the classical groups along with an extensive introduction to a wide range of topics associated with Lie groups: symmetric functions, theory of algebraic forms, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic groups, group representations, invariant, Hilbert theory, and binary forms with fields ranging from pure algebra to functional analysis.Key to this unique exposition is the large amount of background material presented so the book is accessible to a reader with relatively modest mathematical background. Historical information, examples, exercises are all woven into the text.Lie Groups: An Approach through Invariants and Representations will engage a broad audience, including advanced undergraduates, graduates, mathematicians in a variety of areas from pure algebra to functional analysis and physics.

Also called Ito calculus, the theory of stochastic integration has applications in virtually every scientific area involving random functions. This introductory textbook provides a concise introduction to the Ito calculus. From the reviews: "Introduction to Stochastic Integration is exactly what the title says. I would maybe just add a 'friendly' introduction because of the clear presentation and flow of the contents." --THE MATHEMATICAL SCIENCES DIGITAL LIBRARY

From Math Reviews: "This is a charming textbook, introducing the reader to the classical parts of algebra. The exposition is admirably clear and lucidly written with only minimal prerequisites from linear algebra. The new concepts are, at least in the first part of the book, defined in the framework of the development of carefully selected problems. Thus, for instance, the transformation of the classical geometrical problems on constructions with ruler and compass in their algebraic setting in the first chapter introduces the reader spontaneously to such fundamental algebraic notions as field extension, the degree of an extension, etc... The book ends with an appendix containing exercises and notes on the previous parts of the book. However, brief historical comments and suggestions for further reading are also scattered through the text."

Sequential Dynamical Systems (SDS) are a class of discrete dynamical systems which significantly generalize many aspects of systems such as cellular automata, and provide a framework for studying dynamical processes over graphs.This text is the first to provide a comprehensive introduction to SDS. Driven by numerous examples and thought-provoking problems, the presentation offers good foundational material on finite discrete dynamical systems which leads systematically to an introduction of SDS. Techniques from combinatorics, algebra and graph theory are used to study a broad range of topics, including reversibility, the structure of fixed points and periodic orbits, equivalence, morphisms and reduction. Unlike other books that concentrate on determining the structure of various networks, this book investigates the dynamics over these networks by focusing on how the underlying graph structure influences the properties of the associated dynamical system.This book is aimed at graduate students and researchers in discrete mathematics, dynamical systems theory, theoretical computer science, and systems engineering who are interested in analysis and modeling of network dynamics as well as their computer simulations. Prerequisites include knowledge of calculus and basic discrete mathematics. Some computer experience and familiarity with elementary differential equations and dynamical systems are helpful but not necessary.

In our Springer Graduate Text [EN00] we presented this beautiful theory, together with many applications, and tried to show the progress made since the pub- cation in 1957 of the now classical monograph [HP57] by E.

This book is an introduction to constructive mathematics with an emphasis on techniques and results obtained in the last twenty years. The text covers fundamental theory of the real line and metric spaces, focusing on locatedness in normed spaces and with associated results about operators and their adjoints on a Hilbert space. The first appendix gathers together some basic notions about sets and orders, the second gives the axioms for intuitionistic logic. No background in intuitionistic logic or constructive analysis is needed in order to read the book, but some familiarity with the classical theories of metric, normed and Hilbert spaces is necessary.

This book uses a distinctly applied framework to present the most important topics in stochastic processes, including Gaussian and Markovian processes, Markov Chains, Poisson processes, Brownian motion and queueing theory.

The calculus of variations has a long history of interaction with other branches of mathematics, such as geometry and differential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applications in other fields such as economics and electrical engineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations.This book is an introductory account of the calculus of variations suitable for advanced undergraduate and graduate students of mathematics, physics, or engineering. The mathematical background assumed of the reader is a course in multivariable calculus, and some familiarity with the elements of real analysis and ordinary differential equations. The book focuses on variational problems that involve one independent variable. The fixed endpoint problem and problems with constraints are discussed in detail. In addition, more advanced topics such as the inverse problem, eigenvalue problems, separability conditions for the Hamilton-Jacobi equation, and Noether's theorem are discussed.The text contains numerous examples to illustrate key concepts along with problems to help the student consolidate the material. The book can be used as a textbook for a one semester course on the calculus of variations, or as a book to supplement a course on applied mathematics or classical mechanics. Bruce van Brunt is Senior Lecturer at Massey University, New Zealand. He is the author of The Lebesgue-Stieltjes Integral, with Michael Carter, and has been teaching the calculus of variations to undergraduate and graduate students for several years.

From reviews of the German edition: "This is an exciting text and a refreshing contribution to an area in which challenges continue to flourish and to captivate the viewer. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions. One who completes this text not only gains an appreciation of both the depth and the breadth of the theory of finite groups, but also witnesses the evolutionary development of concepts that form a basis for current investigations. This is accomplished by providing a thread that permits a natural flow from one concept to another rather than compartmentalizing. Operators on sets and groups are introduced early and used effectively throughout. The bibliography provides excellent supplemental support...The text is tight; there is no fluff. The format builds on concepts essential for later expansion and associated reading. On occasion, results are stated without proof; continuity is maintained. Several proofs are provided free of representation theory on which the originals were based. More generally the proofs are direct, perhaps at times brief.The focus is on the underlying structural components, with selected details left to the reader. As a result the reader develops the maturity required for approaching the literature with confidence. The first eight chapters have an abundance of exercises, not prorated, and some of the more challenging are addressed later in the text. Due to the nature of the material, fewer exercises appear in the remaining chapters. " (H. Bechtell, Mathematical Reviews)

Ordinary di?erential equations serve as mathematical models for many exciting “real-world” problems, not only in science and technology, but also in such diverse ?elds as economics, psychology, defense, and demography. Rapid growth in the theory of di?erential equations and in its applications to almost every branch of knowledge has resulted in a continued interest in its study by students in many disciplines. This has given ordinary di?er- tial equations a distinct place in mathematics curricula all over the world and it is now being taught at various levels in almost every institution of higher learning. Hundredsofbooksonordinarydi?erentialequationsareavailable. H- ever, the majority of these are elementary texts which provide a battery of techniquesfor?ndingexplicitsolutions. Thesizeofsomeofthesebookshas grown dramatically—to the extent that students are often lost in deciding wheretostart. Thisisallduetotheadditionofrepetitiveexamplesand- ercises, and colorful pictures. The advanced books are either on specialized topics or are encyclopedic in character. In fact, there are hardly any rig- ousandperspicuousintroductorytextsavailablewhichcanbeuseddirectly in class for students of applied sciences. Thus, in an e?ort to bring the s- ject to a wide audience we provide a compact, but thorough, introduction to the subject in An Introduction to Ordinary Di?erential Equations. This book is intended for readers who have had a course in calculus, and hence it canbeusedforaseniorundergraduatecourse. Itshouldalsobesuitablefor a beginning graduate course, because in undergraduate courses, students do not have any exposure to various intricate concepts, perhaps due to an inadequate level of mathematical sophistication.

From Math Reviews: This is Volume II of a two-volume introductory text in classical algebra. The text moves carefully with many details so that readers with some basic knowledge of algebra can read it without difficulty. The book can be recommended either as a textbook for some particular algebraic topic or as a reference book for consultations in a selected fundamental branch of algebra. The book contains a wealth of material. Amongst the topics covered in Volume II the reader can find: the theory of ordered fields (e.g., with reformulation of the fundamental theorem of algebra in terms of ordered fields, with Sylvester's theorem on the number of real roots), Nullstellen-theorems (e.g., with Artin's solution of Hilbert's 17th problem and Dubois' theorem), fundamentals of the theory of quadratic forms, of valuations, local fields and modules. The book also contains some lesser known or nontraditional results; for instance, Tsen's results on solubility of systems of polynomial equations with a sufficiently large number of indeterminates. These two volumes constitute a very good, readable and comprehensive survey of classical algebra and present a valuable contribution to the literature on this subject.

Class field theory, the study of abelian extensions of algebraic number fields, is one of the largest branches of algebraic number theory. It brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. Some of its consequences (e.g., the Chebotarev density theorem) apply even to nonabelian extensions.This book is an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof (global to local), but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included exercises throughout the text.

Thomas Cecil is a math professor with an unrivalled grasp of Lie Sphere Geometry. Here, he provides a clear and comprehensive modern treatment of the subject, as well as its applications to the study of Euclidean submanifolds.

Nature and the world around us that we ourselves design, furnish, and build contain many geometric patterns and structures. This is one of the reasons that geometry should be studied at school. At ?rst, the study of geometry is experimental. Results are taught and used in numerous examples. Only later do proofs come into play. But are these proofs truly necessary, or can we do without them? A natural answer is that every statement must be provided with a proof, because we want to know whether it is true. However, it is clear that the less experienced student may become frustrated by the pr- ence of too many proofs. Only later will the student understand that proofs not only show the correctness of a statement, but also provide better insight into the relationsamong various propertiesof the objects that arebeing st- ied. Learning statements without proofs, you risk not being able to see the forest for the trees. For this reason, we will pay much attention to a careful presentation of proofs in this book. In the development of the theory of plane geometry there are, however, many tricky questions, especially at the beg- ning. The presentation of proofs at that stage is in general more concealing than revealing. My ?rst objective in writing this book has been to give an accessible - position of the most common notions and properties of elementary Euclidean geometry in dimensions two and three.

This is intended for a graduate course on Siegel modular forms, Hecke operators, and related zeta functions. The author's aim is to present a concise and self-contained introduction to an important and developing area of number theory that will serve to attract young researchers to this beautiful field. Topics include: * analytical properties of radial Dirichlet series attached to modular forms of genuses 1 and 2; * the abstract theory of Hecke--Shimura rings for symplectic and related groups; * action of Hecke operators on Siegel modular forms; * applications of Hecke operators to a study of multiplicative properties of Fourier coefficients of modular forms; * Hecke zeta functions of modular forms in one variable and to spinor (or Andrianov) zeta functions of Siegel modular forms of genus two; * the proof of analytical continuation and functional equation (under certain assumptions) for Euler products associated with modular forms of genus two. This text contains a number of exercises and the only prerequisites are standard courses in Algebra and Calculus (one and several variables).

Topics in Hyperplane Arrangements, Polytopes and Box-Splines brings together many areas of research that focus on methods to compute the number of integral points in suitable families or variable polytopes. The topics introduced expand upon differential and difference equations, approximation theory, cohomology, and module theory. This book, written by two distinguished authors, engages a broad audience by proving the a strong foudation. This book may be used in the classroom setting as well as a reference for researchers.

"Mirrors and Reflections" presents a systematic and elementary introduction to the properties of finite groups generated by reflections. The approach is based on fundamental geometric considerations in Coxeter complexes, and emphasizes the intuitive geometric aspects of the theory of reflection groups.

In this undergraduate/graduate textbook, the authors introduce ODEs and PDEs through 50 class-tested lectures. Mathematical concepts are explained with clarity and rigor, using fully worked-out examples and helpful illustrations. Exercises are provided at the end of each chapter for practice. The treatment of ODEs is developed in conjunction with PDEs and is aimed mainly towards applications. The book covers important applications-oriented topics such as solutions of ODEs in form of power series, special functions, Bessel functions, hypergeometric functions, orthogonal functions and polynomials, Legendre, Chebyshev, Hermite, and Laguerre polynomials, theory of Fourier series.Undergraduate and graduate students in mathematics, physics and engineering will benefit from this book. The book assumes familiarity with calculus.

The tread of this book is formed by two fundamental principles of Harmonic Analysis: the Plancherel Formula and the Poisson S- mation Formula. We ?rst prove both for locally compact abelian groups. For non-abelian groups we discuss the Plancherel Theorem in the general situation for Type I groups. The generalization of the Poisson Summation Formula to non-abelian groups is the S- berg Trace Formula, which we prove for arbitrary groups admitting uniform lattices. As examples for the application of the Trace F- mula we treat the Heisenberg group and the group SL (R). In the 2 2 former case the trace formula yields a decomposition of the L -space of the Heisenberg group modulo a lattice. In the case SL (R), the 2 trace formula is used to derive results like the Weil asymptotic law for hyperbolic surfaces and to provide the analytic continuation of the Selberg zeta function. We ?nally include a chapter on the app- cations of abstract Harmonic Analysis on the theory of wavelets. The present book is a text book for a graduate course on abstract harmonic analysis and its applications. The book can be used as a follow up of the First Course in Harmonic Analysis, [9], or indep- dently, if the students have required a modest knowledge of Fourier Analysis already. In this book, among other things, proofs are given of Pontryagin Duality and the Plancherel Theorem for LCA-groups, which were mentioned but not proved in [9].