Calculus and mathematical analysis Books

Book SynopsisThis book is a text on mathematical analysis suitable for graduate students and advanced undergraduates. It provides an extensive introduction to proof and to rigorous mathematical thinking. It contains many remarks and examples and 500 exercises designed to provide motivation, test understanding, help practice mathematical writing and explore additional topics.Trade Review'The self-contained text, suitable for advanced undergraduates, provides an extensive introduction into mathematical analysis, from the fundamentals to more advanced material.' Zentralblatt fur Didaktik der MathematikTable of Contents1. Introduction; 2. The real and complex numbers; 3. Real and complex sequences; 4. Series; 5. Power series; 6. Metric spaces; 7. Continuous functions; 8. Calculus; 9. Some special functions; 10. Lebesgue measure on the line; 11. Lebesgue integration on the line; 12. Function spaces; 13. Fourier series; 14. Applications of Fourier series; 15. Ordinary differential equations; Appendix: the Banach-Tarski paradox; Hints for some exercises.

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Book SynopsisThe subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.Trade Review'The book is a welcome extension of the existing literature about this important topic … It is recommended to students of mathematics and physics interested in the applications of the theory and the theory itself.' European Mathematical Society'With an easy mind the reviewer can recommend this book to those who want to become acquainted with the subject and to those who look for a book which can serve as guide for a course on the subject … the exemplary way in which Elliptic Curves is written, made reviewing a pleasure.' Niew Archief voor WiskundeTable of Contents1. First ideas: complex manifolds, Riemann surfaces, and projective curves; 2. Elliptic functions and elliptic integrals; 3. Theta functions; 4. Modular groups and molecular functions; 5. Ikosaeder and the quintic; 6. Imaginary quadratic fields; 7. The arithmetic of elliptic fields.

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Book SynopsisMathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard university course on the subject.Table of ContentsPreface; Introduction: calculus and analysis; 1. Numbers; 2. Sequences; 3. Series; 4. Continuity; 5. Limits and continuity; 6. Differentiation; 7. Integration; 8. Power series; Appendix 1. Sets, functions and proofs; Appendix 2. Standard derivatives and primitives; Appendix 3. The first 1,000 decimal places of the square root of 2, e and pi; Appendix 4. Solutions to the problems; Index.

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Book SynopsisA self-contained and elementary presentation of Lie group theory, concentrating on analysis on Lie groups. The author describes in detail many interesting examples with topics ranging from Haar measure to harmonic functions. With numerous exercises and worked examples, it's ideal for a graduate course on analysis on Lie groups.Trade Review"The main themes are carefully explained and illustrated by well-chosen examples. He succeeds in putting a remarkable wealth of material into a 300-page book which will certainly serve as a basis for many courses on the subject." Joachim Hilgert, Mathematical ReviewsTable of ContentsPreface; 1. The linear group; 2. The exponential map; 3. Linear Lie groups; 4. Lie algebras; 5. Haar measure; 6. Representations of compact groups; 7. The groups SU(2) and SO(3), Haar measure; 8. Analysis on the group SU(2); 9. Analysis on the sphere; 10. Analysis on the spaces of symmetric and Hermitian matrices; 11. Irreducible representations of the unitary group; 12. Analysis on the unitary group; Bibliography; Index.

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Book SynopsisIntegration has a long history: its roots can be traced as far back as the ancient Greeks. The first genuinely rigorous definition of an integral was that given by Riemann, and further (more general, and so more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory, and it presents a unified yet elementary approach that is suitable for beginning graduate and final year undergraduate students.Trade Review'… already it is worthy of a place in our standard curriculum … The book of Lee and Vyborny serves well as an introduction and reference for anyone interested in this topic.' J. Alan Alewine and Eric Schechter, American Mathematical Monthly'… the authors do an excellent job of presenting their material. The book is written with clarity and enthusiasm.' Brian Jefferies'This is a valuable addition to the literature …'. Jean Mawhin, Bulletin of the Belgian Mathematical SocietyTable of ContentsPreface; 1. Introduction; 2. Basic theory; 3. Theory development; 4. The SL-integral; 5. Generalized AC function; 6. Integration in several dimensions; 7. Some applications; 8. List of symbols; Appendices.

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Book SynopsisSpecial functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and applicaTrade Review'Occasionally there is published a mathematics book that one is compelled to describe as, well, let us say, special. Special Functions is certainly one of those rare books. … this treatise … should become a classic. Every student, user, and researcher in analysis will want to have it close at hand as she/he works.' The Mathematical Intelligencer' … the material is written in an excellent manner … I recommend this book warmly as a rich source of information to everybody who is interested in 'Special Functions'.' Zentralblatt MATH' … this book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too … the book can be warmly recommended and should be in all good libraries.' Adam McBride, The Mathematical Gazette' … it comes into the range of affordable books that you want to (and probably should have on your desk'. Jean Mawhin, Bulletin of the Belgian Mathematical Society'The book is full of beautiful and interesting formulae, as was always the case with mathematics centred around special functions. It is written in the spirit of the old masters, with mathemtics developed in terms of formulas. There are many historical comments in the book. It can be recommended as a very useful reference.' European Mathematical Society'… full of beautiful and interesting formulae … It can be recommended as a very useful reference.' EMS Newsletter'a very erudite text and reference in special functions' Allen Stenger, MAA ReviewsTable of Contents1. The Gamma and Beta functions; 2. The hypergeometric functions; 3. Hypergeometric transformations and identities; 4. Bessel functions and confluent hypergeometric functions; 5. Orthogonal polynomials; 6. Special orthogonal transformations; 7. Topics in orthogonal polynomials; 8. The Selberg integral and its applications; 9. Spherical harmonics; 10. Introduction to q-series; 11. Partitions; 12. Bailey chains; Appendix 1. Infinite products; Appendix 2. Summability and fractional integration; Appendix 3. Asymptotic expansions; Appendix 4. Euler-Maclaurin summation formula; Appendix 5. Lagrange inversion formula; Appendix 6. Series solutions of differential equations.

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Book SynopsisThe problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed in this book, first published in 2004, are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting, rather than the shortest, path to the results. Along the way, they illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.Trade Review'I recommend this book highly as a source of rewarding projects for undergraduates (and others) to home their analytic skills and gain an appreciation for this area of mathematics. The authors clearly had great love for the material and their enthusiasm comes through in an infectious manner.' SIAM Review'The authors have managed to write a very readable account about integrals, accessible even to advanced undergraduates. Some of the topics of the book could be used for undergraduate reading and research projects. This way the book could serve as a 'springboard to many unexpected investigations and discoveries in mathematics.' Zentralblatt MATHTable of Contents1. Introduction; 2. Factorials and binomial coefficients; 3. The method of partial fractions; 4. A simple rational function; 5. A review of power series; 6. The exponential and logarithm functions; 7. The trigonometric functions and pi; 8. A quartic integral; 9. The normal integral; 10. Euler's constant; 11. Eulerian integrals: the Gamma and Beta functions; 12. The Riemann zeta function; 13. Logarithmic integrals; 14. A master formula; 15. Appendix: the revolutionary WZ method.

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Book SynopsisThe author presents an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A self-contained presentation of 'exotic' functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a 'Coulomb moving frame' is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces.Trade Review'The book is very well written and it contains truly beautiful geometrical analysis. It also contains a quick, direct introduction to the current research.' EMS NewsletterTable of ContentsPreface; Introduction; Acknowledgements; Notations; 1. Geometric and analytic setting; 2. Harmonic maps with symmetries; 3. Compensations and exotic function spaces; 4. Harmonic maps without symmetries; 5. Surfaces with mean curvature in L2; References.

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Book SynopsisThis book, first published in 2003, is a self-contained introduction to the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, plus some of their main applications. An indispensable introduction to the theory of operator spaces for graduate students and experts alike.Trade Review'This book has been written by one of the leading figures in the field. the choice of the presented material has been done in a masterly manner … an excellent introduction to this theory for graduate students. It should also provide a valuable reference source for researchers in the field.' Zentralblatt für Mathematik'The book is carefully written, proofs are often accompanied with notes helping to explain the situation.' EMS Newsletter'Paulsen's book has the advantage of still being concise and staying close to the origins of the theory … the subject of operator spaces is now very well covered and has been made accessible to both the newcomer to the subject, and the specialist looking for concise references, alike. In conclusion, we quote from the cover text of [2]: 'This will be an indispensable introduction to the theory of operator spaces for all who want to know more.' We add: you surely will want to know more.' Martin Mathieu, Queen's University BelfastTable of Contents1. Introduction; 2. Positive maps; 3. Completely positive maps; 4. Dilation theorems; 5. Commuting contractions; 6. Completely positive maps into Mn; 7. Arveson's extension theorems; 8. Completely bounded maps; 9. Completely bounded homomorphisms; 10. Polynomially bounded operators; 11. Applications to K-spectral sets; 12. Tensor products and joint spectral sets; 13. Operator systems and operator spaces; 14. An operator space bestiary; 15. Injective envelopes; 16. Multipliers and operator algebras; 17. Completely bounded multilinear maps; 18. Applications of operator algebras; 19. Similarity and factorization degree.

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Book SynopsisThis revised and expanded new edition will continue to meet the needs for an authoritative, up-to-date, self contained and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. Some sections and exercises have been added and the bibliography has been revised to maintain its comprehensiveness.Trade Review'I love this book! It is great! This really is a book you can learn the subject from. The plentiful exercises vary from elementary to challenging with lots of each. Congratulations and thanks are due the authors.' George Andrews, American Math. Monthly'The book is remarkable in many ways. It is comprehensive, at least, comprehensive to date. As is typical of most works on the subject, it is clearly and carefully written. While no book can conceivably incorporate all the important results, particularly those obtained in the last decade, many of them are included as exercises. And this is the feature all other books on the subject lack: a set of exercises. Each chapter is topped off by a challenging series of problems which lead the reader to recreate recent discoveries. Anyone who works even a small percentage of them will soon be an expert. A generous series of historical notes concludes each chapter. The book is user friendly in every respect. The book has two excellent Appendices which summarize the identities and summation formulas derived in the text, an exhaustive 25 page list of references, and a nontrivial index. Now anyone working in combinatorics, group representation theory, coding theory, and related fields will want to own it. Many physicists will find it bears directly on matters of interest to them. Computer scientists may find the book increasingly timely. Those who have refrained from entering the field because of the tortuous notation can now have untroubled access to its mysteries. I say, come in, the water's fine.' Jet Wimp, SIAM Review'This is an excellent and very informative book on the subject. After a gentle introduction to basic series and some special cases (such as the 'q'-binomial theorem) the authors bring the reader up to the latest results on the general theory and its extensions, many such results are due to them. The exercises are utilized to include results that found no room in the detailed treatment. In addition to these exercises, notes at the end of each chapter point the reader to related topics. This alone makes the book an invaluable reference to those who are interested in basic series.' Waleed A. Al-Salam, Math. Reviews'Thus the present book, devoted to 'q'-hypergeometric series, appears at a very timely moment. The result is excellent. The first chapter presents a clear and elementary introduction to the subject. At the end of the book there are excellent indices and compendia of formulas.' Tom H. Koornwinder, Bulletin of London Mathematical Society'… a very modern, self-contained, comprehensive and successful monograph, interesting and useful, for physicists as well as for mathematicians from various branches, who wish to learn about the subject.' European Mathematical Society NewsletterTable of ContentsForeword; Preface; 1. Basic hypergeometric series; 2. Summation, transformation, and expansion formulas; 3. Additional summation, transformation, and expansion formulas; 4. Basic contour integrals; 5. Bilateral basic hypergeometric series; 6. The Askey-Wilson q-beta integral and some associated formulas; 7. Applications to orthogonal polynomials; 8. Further applications; 9. Linear and bilinear generating functions for basic orthogonal polynomials; 10. q-series in two or more variables; 11. Elliptic, modular, and theta hypergeometric series; Appendices; References; Author index; Subject index; Symbol index.

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Using the Cauchy-Schwarz inequality as a guide, this 2004 book presents a fascinating collection of problems related to inequalities and coaches readers through solutions. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book perfect for self-study or as a supplement to probability and analysis courses.

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Book SynopsisThis authoritative text presents a broad view of the spectral theory of non-self-adjoint linear operators and contains many illustrative examples and exercises. Topics discussed include Fredholm theory, Hilbert-Schmidt and trace class operators, one-parameter semigroups, perturbations of their generators and a thorough account of the new theory of pseudospectra.Trade Review'One will look in vain for the notions of pseudospectrum, hull and numerical range in standard functional analysis texts, so Davies has done us a great service by explaining them through beautiful theorems and examples. More generally, his book is the first to offer a comprehensive survey of the spectral theory of non-self-adjoint operators, including both 'classical' and 'cutting edge' results, showing that this theory holds as much promise as the self-adjoint theory in both foundations and application. The scope of the book is truly enormous and is only partly reflected by listing the chapter titles … [a] beautiful volume, which has no competitors.' The Mathematical IntelligencerTable of ContentsPreface; 1. Elementary operator theory; 2. Function spaces; 3. Fourier transforms and bases; 4. Intermediate operator theory; 5. Operators on Hilbert space; 6. One-parameter semigroups; 7. Special classes of semigroup; 8. Resolvents and generators; 9. Quantitative bounds on operators; 10. Quantitative bounds on semigroups; 11. Perturbation theory; 12. Markov chains and graphs; 13. Positive semigroups; 14. NSA Schrödinger operators.

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Book SynopsisIn this book, first published in 2006, the authors' main aims are first to present classical results in a way that's accessible to non-specialists. Second, to describe results of Smirnov in conformal invariance. It is essential reading for all working in this exciting area.Trade Review'This book contains a complete account of most of the important results in the fascinating area of percolation. Elegant and straightforward proofs are given with minimal background in probability or graph theory. It is self-contained, accessible to a wide readership and widely illustrated with numerous examples. It will be of considerable interest for both beginners and advanced searchers alike.' Zentralblatt MATHTable of ContentsPreface; 1. Basic concepts; 2. Probabilistic tools; 3. Percolation on Z2 - the Harris-Kesten theorem; 4. Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky; 5. Uniqueness of the infinite open cluster and critical probabilities; 6. Estimating critical probabilities; 7. Conformal invariance - Smirnov's theorem; 8. Continuum percolation; Bibliography; Index; List of notation.

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Book SynopsisThe essential text and reference for modern scientific computing now also covers computational geometry, classification and inference, and much more.Trade Review'This monumental and classic work is beautifully produced and of literary as well as mathematical quality. It is an essential component of any serious scientific or engineering library.' Computing Reviews'… an instant 'classic,' a book that should be purchased and read by anyone who uses numerical methods …' American Journal of Physics'… replete with the standard spectrum of mathematically pretreated and coded/numerical routines for linear equations, matrices and arrays, curves, splines, polynomials, functions, roots, series, integrals, eigenvectors, FFT and other transforms, distributions, statistics, and on to ODE's and PDE's … delightful.' Physics in Canada'… if you were to have only a single book on numerical methods, this is the one I would recommend.' EEE Computational Science & Engineering'This encyclopedic book should be read (or at least owned) not only by those who must roll their own numerical methods, but by all who must use prepackaged programs.' New Scientist'These books are a must for anyone doing scientific computing.' Journal of the American Chemical Society'The authors are to be congratulated for providing the scientific community with a valuable resource.' The Scientist'I think this is an incredibly valuable book for both learning and reference and I recommend it for any scientists or student in a numerate discipline who need to understand and/or program numerical algorithms.' International Association for Pattern Recognition'The attractive style of the text and the availability of the codes ensured the popularity of the previous editions and also recommended this recent volume to different categories of readers, more or less experienced in numerical computation.' Octavian Pastravanu, Zentralblatt MATHTable of Contents1. Preliminaries; 2. Solution of linear algebraic equations; 3. Interpolation and extrapolation; 4. Integration of functions; 5. Evaluation of functions; 6. Special functions; 7. Random numbers; 8. Sorting and selection; 9. Root finding and nonlinear sets of equations; 10. Minimization or maximization of functions; 11. Eigensystems; 12. Fast Fourier transform; 13. Fourier and spectral applications; 14. Statistical description of data; 15. Modeling of data; 16. Classification and inference; 17. Integration of ordinary differential equations; 18. Two point boundary value problems; 19. Integral equations and inverse theory; 20. Partial differential equations; 21. Computational geometry; 22. Less-numerical algorithms; References.

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Book SynopsisBased on a lecture course given at Chalmers University of Technology, this 2002 book is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before applying it to study a range of randomized algorithms with important applications in optimization and other problems in computing. Amongst the algorithms covered are the Markov chain Monte Carlo method, simulated annealing, and the recent Propp-Wilson algorithm. This book will appeal not only to mathematicians, but also to students of statistics and computer science. The subject matter is introduced in a clear and concise fashion and the numerous exercises included will help students to deepen their understanding.Trade Review'Has climbing up onto the MCMC juggernaut seemed to require just too much effort? This delightful little monograph provides an effortless way in. The chapters are bite-sized with helpful, do-able exercises (by virtue of strategically placed hints) that complement the text.' Publication of the International Statistical Institute'… a very nice introduction to the modern theory of Markov chain simulation algorithms.' R. E. Maiboroda, Zentralblatt MATH' … extremely elegant. I am sure that students will find great pleasure in using the book - and that teachers will have the same pleasure in using it to prepare a course on the subject.' Mathematics of Computation'This elegant little book is a beautiful introduction to the theory of simulation algorithms, using (discrete) Markov chains (on finite state spaces) … highly recommended to anyone interested in the theory of Markov chain simulation algorithms.' Nieuw Archief voor WiskundeTable of Contents1. Basics of probability theory; 2. Markov chains; 3. Computer simulation of Markov chains; 4. Irreducible and aperiodic Markov chains; 5. Stationary distributions; 6. Reversible Markov chains; 7. Markov chain Monte Carlo; 8. Fast convergence of MCMC algorithms; 9. Approximate counting; 10. Propp-Wilson algorithm; 11. Sandwiching; 12. Propp-Wilson with read once randomness; 13. Simulated annealing; 14. Further reading.

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Book SynopsisProfessor Zygmund's Trigonometric Series, first published in Warsaw in 1935, established itself as a classic. It presented a concise account of the main results then known, but was on a scale which limited the amount of detailed discussion possible. A greatly enlarged second edition published by Cambridge in two volumes in 1959 took full account of developments in trigonometric series, Fourier series and related branches of pure mathematics since the publication of the original edition. The two volumes are here bound together with a foreword from Robert Fefferman outlining the significance of this text. Volume I, containing the completely rewritten material of the original work, deals with trigonometric series and Fourier series. Volume II provides much material previously unpublished in book form.Trade Review'... much material previously unpublished in book form.' Zentralblatt MATHTable of ContentsPart I: 1. Trigonometric series and Fourier series, auxilliary results; 2. Fourier coefficients, elementary theorems on the convergence of S[f] and \tilde{S}[f]; 3. Summability of Fourier series; 4. Classes of functions and Fourier series; 5. Special trigonometric series; 6. The absolute convergence of trigonometric series; 7. Complex methods in Fourier series; 8. Divergence of Fourier series; 9. Riemann's theory of trigonometric series; Part II: 10. Trigonometric interpolation; 11. Differentiation of series, generalised derivatives; 12. Interpolation of linear operations, more about Fourier coefficients; 13. Convergence and summability almost everywhere; 14. More about complex methods; 15. Applications of the Littlewood-Paley function to Fourier series; 16. Fourier integrals; 17. A topic in multiple Fourier series.

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Book SynopsisThis book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson''s work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson'Table of ContentsThe infinitesimal rule of threeNonstandard methods in the precalculus curruculumDifference quotients and smoothnessContinuous maps with special propertiesSome nonstandard methods in geometric topologyDelayed bifurcations in perturbed systems analysis of slow passage of Suhl-thresholdFunctional analysis and NSANear-standard compact internal linear operatorsDiscrete Fredholm's equationsNonstandard theory of generalized functionsRepresenting distributions by nonstandard polynomialsContributions of nonstandard analysis to partial differential equationsLoeb measure theoryUnions of Loeb nullsets: the contextGredient lines and distributions of functionals in infinite dimensional Euclidean spacesNonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interactionNonstandard analysis in selective uniersesLattices and monadsA neometric surveyLong sequences and neocompact sets

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Book SynopsisThis volume constitutes the proceedings of a conference on functional analysis and its applications, which took place in India during December 1996. Topics include topological vector spaces, Banach algebras, meromorphic functions, partial differential equations, variational equations and inequalities, optimization, wavelets, elastroplasticity, numerical integration, fractal image compression, reservoir simulation, forest management, and industrial maths.Table of ContentsStructural results; bariational methods; applications in science and industry.

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Book SynopsisThe theory of automorphic forms is playing increasingly important roles in several branches of mathematics, even in physics, and is almost ubiquitous in number theory. This book introduces the reader to the subject and, in particular, to elliptic modular forms with emphasis on their number-theoretical aspects.Table of Contents* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

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Book SynopsisAn elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Developing, with many examples, the elementary theory of elliptic curves, this book goes on to the subject of modular forms and the first connections with elliptic curves.

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Book SynopsisAddressing the topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. It includes fixed point theorems and applications to functional equations and optimization theory.Trade Review"The book is intended as a textbook on real analysis for graduate students in economics. It is largely graduate level mathematics, and the students should have a solid undergraduate real analysis background... The author's writing style is ... in general quite attractive. The book should be quite successful for its intended purpose."--Gerald A. Heuer, Zentralblatt MATH "Important and commendable, this indispensable resource should be highly prized by all concerned with courses on mathematics for economists and by graduate students working on economic theory. Rarely do books meet such high aspirations and carry out their aims, yet this one certainly does. Well written in an engaging style and impressively researched in the requirements of graduate students of economics and finance, Real Analysis with Economic Applications is sure to become the definitive work for its intended audience. Real Analysis with Economic Applications with its large number of economics applications and variety of exercises represents the single most important mathematical source for students of economics applications and it will be the book, for a long time to come, to which they will turn with confidence, as well as pleasure, in all questions of economic applications."--Current Engineering PracticeTable of ContentsPreface xvii Prerequisites xxvii Basic Conventions xxix Part I: SET THEORY 1 Chapter A: Preliminaries of Real Analysis 3 A.1 Elements of Set Theory 4 A.1.1 Sets 4 A.1.2 Relations 9 A.1.3 Equivalence Relations 11 A.1.4 Order Relations 14 A.1.5 Functions 20 A.1.6 Sequences, Vectors, and Matrices 27 A.1.7* A Glimpse of Advanced Set Theory: The Axiom of Choice 29 A.2 Real Numbers 33 A.2.1 Ordered Fields 33 A.2.2 Natural Numbers, Integers, and Rationals 37 A.2.3 Real Numbers 39 A.2.4 Intervals and R 44 A.3 Real Sequences 46 A.3.1 Convergent Sequences 46 A.3.2 Monotonic Sequences 50 A.3.3 Subsequential Limits 53 A.3.4 Infinite Series 56 A.3.5 Rearrangement of Infinite Series 59 A.3.6 Infinite Products 61 A.4 Real Functions 62 A.4.1 Basic Definitions 62 A.4.2 Limits, Continuity, and Differentiation 64 A.4.3 Riemann Integration 69 A.4.4 Exponential, Logarithmic, and Trigonometric Functions 74 A.4.5 Concave and Convex Functions 77 A.4.6 Quasiconcave and Quasiconvex Functions 80 Chapter B: Countability 82 B.1 Countable and Uncountable Sets 82 B.2 Losets and Q 90 B.3 Some More Advanced Set Theory 93 B.3.1 The Cardinality Ordering 93 B.3.2* The Well-Ordering Principle 98 B.4 Application: Ordinal Utility Theory 99 B.4.1 Preference Relations 100 B.4.2 Utility Representation of Complete Preference Relations 102 B.4.3* Utility Representation of Incomplete Preference Relations 107 Part II: ANALYSIS ON METRIC SPACES 115 Chapter C: Metric Spaces 117 C.1 Basic Notions 118 C.1.1 Metric Spaces: Definition and Examples 119 C.1.2 Open and Closed Sets 127 C.1.3 Convergent Sequences 132 C.1.4 Sequential Characterization of Closed Sets 134 C.1.5 Equivalence of Metrics 136 C.2 Connectedness and Separability 138 C.2.1 Connected Metric Spaces 138 C.2.2 Separable Metric Spaces 140 C.2.3 Applications to Utility Theory 145 C.3 Compactness 147 C.3.1 Basic Definitions and the Heine-Borel Theorem 148 C.3.2 Compactness as a Finite Structure 151 C.3.3 Closed and Bounded Sets 154 C.4 Sequential Compactness 157 C.5 Completeness 161 C.5.1 Cauchy Sequences 161 C.5.2 Complete Metric Spaces: Definition and Examples 163 C.5.3 Completeness versus Closedness 167 C.5.4 Completeness versus Compactness 171 C.6 Fixed Point Theory I 172 C.6.1 Contractions 172 C.6.2 The Banach Fixed Point Theorem 175 C.6.3* Generalizations of the Banach Fixed Point Theorem 179 C.7 Applications to Functional Equations 183 C.7.1 Solutions of Functional Equations 183 C.7.2 Picard's Existence Theorems 187 C.8 Products of Metric Spaces 192 C.8.1 Finite Products 192 C.8.2 Countably Infinite Products 193 Chapter D: Continuity I 200 D.1 Continuity of Functions 201 D.1.1 Definitions and Examples 201 D.1.2 Uniform Continuity 208 D.1.3 Other Continuity Concepts 210 D.1.4* Remarks on the Differentiability of Real Functions 212 D.1.5 A Fundamental Characterization of Continuity 213 D.1.6 Homeomorphisms 216 D.2 Continuity and Connectedness 218 D.3 Continuity and Compactness 222 D.3.1 Continuous Image of a Compact Set 222 D.3.2 The Local-to-Global Method 223 D.3.3 Weierstrass' Theorem 225 D.4 Semicontinuity 229 D.5 Applications 237 D.5.1* Caristi's Fixed Point Theorem 238 D.5.2 Continuous Representation of a Preference Relation 239 D.5.3* Cauchy's Functional Equations: Additivity on Rn 242 D.5.4* Representation of Additive Preferences 247 D.6 CB(T) and Uniform Convergence 249 D.6.1 The Basic Metric Structure of CB(T) 249 D.6.2 Uniform Convergence 250 D.6.3* The Stone-Weierstrass Theorem and Separability of C(T) 257 D.6.4* The Arzela-Ascoli Theorem 262 D.7* Extension of Continuous Functions 266 D.8 Fixed Point Theory II 272 D.8.1 The Fixed Point Property 273 D.8.2 Retracts 274 D.8.3 The Brouwer Fixed Point Theorem 277 D.8.4 Applications 280 Chapter E: Continuity II 283 E.1 Correspondences 284 E.2 Continuity of Correspondences 287 E.2.1 Upper Hemicontinuity 287 E.2.2 The Closed Graph Property 294 E.2.3 Lower Hemicontinuity 297 E.2.4 Continuous Correspondences 300 E.2.5* The Hausdorff Metric and Continuity 302 E.3 The Maximum Theorem 306 E.4 Application: Stationary Dynamic Programming 311 E.4.1 The Standard Dynamic Programming Problem 312 E.4.2 The Principle of Optimality 315 E.4.3 Existence and Uniqueness of an Optimal Solution 320 E.4.4 Application: The Optimal Growth Model 324 E.5 Fixed Point Theory III 330 E.5.1 Kakutani's Fixed Point Theorem 331 E.5.2* Michael's Selection Theorem 333 E.5.3* Proof of Kakutani's Fixed Point Theorem 339 E.5.4* Contractive Correspondences 341 E.6 Application: The Nash Equilibrium 343 E.6.1 Strategic Games 343 E.6.2 The Nash Equilibrium 346 E.6.3* Remarks on the Equilibria of Discontinuous Games 351 Part III: ANALYSIS ON LINEAR SPACES 355 Chapter F: Linear Spaces 357 F.1 Linear Spaces 358 F.1.1 Abelian Groups 358 F.1.2 Linear Spaces: Definition and Examples 360 F.1.3 Linear Subspaces, Affine Manifolds, and Hyperplanes 364 F.1.4 Span and Affine Hull of a Set 368 F.1.5 Linear and Affine Independence 370 F.1.6 Bases and Dimension 375 F.2 Linear Operators and Functionals 382 F.2.1 Definitions and Examples 382 F.2.2 Linear and Affine Functions 386 F.2.3 Linear Isomorphisms 389 F.2.4 Hyperplanes, Revisited 392 F.3 Application: Expected Utility Theory 395 F.3.1 The Expected Utility Theorem 395 F.3.2 Utility Theory under Uncertainty 403 F.4* Application: Capacities and the Shapley Value 409 F.4.1 Capacities and Coalitional Games 410 F.4.2 The Linear Space of Capacities 412 F.4.3 The Shapley Value 415 Chapter G: Convexity 422 G.1 Convex Sets 423 G.1.1 Basic Definitions and Examples 423 G.1.2 Convex Cones 428 G.1.3 Ordered Linear Spaces 432 G.1.4 Algebraic and Relative Interior of a Set 436 G.1.5 Algebraic Closure of a Set 447 G.1.6 Finitely Generated Cones 450 G.2 Separation and Extension in Linear Spaces 454 G.2.1 Extension of Linear Functionals 455 G.2.2 Extension of Positive Linear Functionals 460 G.2.3 Separation of Convex Sets by Hyperplanes 462 G.2.4 The External Characterization of Algebraically Closed and Convex Sets 471 G.2.5 Supporting Hyperplanes 473 G.2.6* Superlinear Maps 476 G.3 Reflections on Rn 480 G.3.1 Separation in Rn 480 G.3.2 Support in Rn 486 G.3.3 The Cauchy-Schwarz Inequality 488 G.3.4 Best Approximation from a Convex Set in Rn 489 G.3.5 Orthogonal Complements 492 G.3.6 Extension of Positive Linear Functionals, Revisited 496 Chapter H: Economic Applications 498 H.1 Applications to Expected Utility Theory 499 H.1.1 The Expected Multi-Utility Theorem 499 H.1.2* Knightian Uncertainty 505 H.1.3* The Gilboa-Schmeidler Multi-Prior Model 509 H.2 Applications to Welfare Economics 521 H.2.1 The Second Fundamental Theorem of Welfare Economics 521 H.2.2 Characterization of Pareto Optima 525 H.2.3* Harsanyi's Utilitarianism Theorem 526 H.3 An Application to Information Theory 528 H.4 Applications to Financial Economics 535 H.4.1 Viability and Arbitrage-Free Price Functionals 535 H.4.2 The No-Arbitrage Theorem 539 H.5 Applications to Cooperative Games 542 H.5.1 The Nash Bargaining Solution 542 H.5.2* Coalitional Games without Side Payments 546 Part IV: ANALYSIS ON METRIC/NORMED LINEAR SPACES 551 Chapter I: Metric Linear Spaces 553 I.1 Metric Linear Spaces 554 I.2 Continuous Linear Operators and Functionals 561 I.2.1 Examples of (Dis-)Continuous Linear Operators 561 I.2.2 Continuity of Positive Linear Functionals 567 I.2.3 Closed versus Dense Hyperplanes 569 I.2.4 Digression: On the Continuity of Concave Functions 573 I.3 Finite-Dimensional Metric Linear Spaces 577 I.4* Compact Sets in Metric Linear Spaces 582 I.5 Convex Analysis in Metric Linear Spaces 587 I.5.1 Closure and Interior of a Convex Set 587 I.5.2 Interior versus Algebraic Interior of a Convex Set 590 I.5.3 Extension of Positive Linear Functionals, Revisited 594 I.5.4 Separation by Closed Hyperplanes 594 I.5.5* Interior versus Algebraic Interior of a Closed and Convex Set 597 Chapter J: Normed Linear Spaces 601 J.1 Normed Linear Spaces 602 J.1.1 A Geometric Motivation 602 J.1.2 Normed Linear Spaces 605 J.1.3 Examples of Normed Linear Spaces 607 J.1.4 Metric versus Normed Linear Spaces 611 J.1.5 Digression: The Lipschitz Continuity of Concave Maps 614 J.2 Banach Spaces 616 J.2.1 Definition and Examples 616 J.2.2 Infinite Series in Banach Spaces 618 J.2.3* On the "Size" of Banach Spaces 620 J.3 Fixed Point Theory IV 623 J.3.1 The Glicksberg-Fan Fixed Point Theorem 623 J.3.2 Application: Existence of the Nash Equilibrium, Revisited 625 J.3.3* The Schauder Fixed Point Theorems 626 J.3.4* Some Consequences of Schauder's Theorems 630 J.3.5* Applications to Functional Equations 634 J.4 Bounded Linear Operators and Functionals 638 J.4.1 Definitions and Examples 638 J.4.2 Linear Homeomorphisms, Revisited 642 J.4.3 The Operator Norm 644 J.4.4 Dual Spaces 648 J.4.5* Discontinuous Linear Functionals, Revisited 649 J.5 Convex Analysis in Normed Linear Spaces 650 J.5.1 Separation by Closed Hyperplanes, Revisited 650 J.5.2* Best Approximation from a Convex Set 652 J.5.3 Extreme Points 654 J.6 Extension in Normed Linear Spaces 661 J.6.1 Extension of Continuous Linear Functionals 661 J.6.2* Infinite-Dimensional Normed Linear Spaces 663 J.7* The Uniform Boundedness Principle 665 Chapter K: Differential Calculus 670 K.1 Frechet Differentiation 671 K.1.1 Limits of Functions and Tangency 671 K.1.2 What Is a Derivative? 672 K.1.3 The Frechet Derivative 675 K.1.4 Examples 679 K.1.5 Rules of Differentiation 686 K.1.6 The Second Frechet Derivative of a Real Function 690 K.1.7 Differentiation on Relatively Open Sets 694 K.2 Generalizations of the Mean Value Theorem 698 K.2.1 The Generalized Mean Value Theorem 698 K.2.2* The Mean Value Inequality 701 K.3 Frechet Differentiation and Concave Maps 704 K.3.1 Remarks on the Differentiability of Concave Maps 704 K.3.2 Frechet Differentiable Concave Maps 706 K.4 Optimization 712 K.4.1 Local Extrema of Real Maps 712 K.4.2 Optimization of Concave Maps 716 K.5 Calculus of Variations 718 K.5.1 Finite-Horizon Variational Problems 718 K.5.2 The Euler-Lagrange Equation 721 K.5.3* More on the Sufficiency of the Euler-Lagrange Equation 733 K.5.4 Infinite-Horizon Variational Problems 736 K.5.5 Application: The Optimal Investment Problem 738 K.5.6 Application: The Optimal Growth Problem 740 K.5.7* Application: The Poincare-Wirtinger Inequality 743 Hints for Selected Exercises 747 References 777 Glossary of Selected Symbols 789 Index 793

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Book SynopsisPresents the analytic foundations to the theory of the hypoelliptic Laplacian. This book shows that the hypoelliptic Laplacian provides a geometric version of the Fokker-Planck equations. It gives the proper functional analytic setting in order to study this operator and develop a pseudodifferential calculus.Table of Contents*Frontmatter, pg. i*Contents, pg. v*Introduction, pg. 1*Chapter 1. Elliptic Riemann-Roch-Grothendieck and flat vector bundles, pg. 11*Chapter 2. The hypoelliptic Laplacian on the cotangent bundle, pg. 25*Chapter 3. Hodge theory, the hypoelliptic Laplacian and its heat kernel, pg. 44*Chapter 4. Hypoelliptic Laplacians and odd Chern forms, pg. 62*Chapter 5. The limit as t --> + and b --> 0 of the superconnection forms, pg. 98*Chapter 6. Hypoelliptic torsion and the hypoelliptic Ray-Singer metrics, pg. 113*Chapter 7. The hypoelliptic torsion forms of a vector bundle, pg. 131*Chapter 8. Hypoelliptic and elliptic torsions: a comparison formula, pg. 162*Chapter 9. A comparison formula for the Ray-Singer metrics, pg. 171*Chapter 10. The harmonic forms for b --> 0 and the formal Hodge theorem, pg. 173*Chapter 11. A proof of equation (8.4.6), pg. 182*Chapter 12. A proof of equation (8.4.8), pg. 190*Chapter 13. A proof of equation (8.4.7), pg. 194*Chapter 14. The integration by parts formula, pg. 214*Chapter 15. The hypoelliptic estimates, pg. 224*Chapter 16. Harmonic oscillator and the J0 function, pg. 247*Chapter 17. The limit of A'2phib,+-H as b --> 0, pg. 264*Bibliography, pg. 353*Subject Index, pg. 359*Index of Notation, pg. 361

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Book SynopsisThe hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula.Table of Contents*FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325

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Book SynopsisOffers an introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. This book traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter.Trade Review"Each chapter ends with a rich and useful section of notes and references. The exercises are merely problems or even theorems. The author of the book presents a large list of references and a detailed index of notions, names, and symbols. The graphical presentation of the book is pleasant... [T]his book is well written, it fully deserves all its goals mentioned at the beginning of the review, and is a pleasure to read it."--Marian Muresan, Mathematica "This is an extremely well-crafted textbook. If you plan to teach a first course to advanced students on the calculus of variations and optimal control and you like the selection of topics that the author has chosen to present (and I do), it is the text you need. What impresses me most is the careful balance between the formal derivations and the explanations that precede or accompany the statements and proofs... All in all, it is a first-rate, enjoyable text."--Zvi Artstein, Mathematical Reviews Clippings

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Book SynopsisHybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. This title unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems.Trade Review"The book is carefully written and contains many examples. It will be a good resource for both researchers already familiar with hybrid systems and those starting from scratch."--Daniel Liberzon, Mathematical Reviews Clippings "The book presents a clean and self-contained exposition of hybrid systems, starting from the elementary definitions, continuing with the basic tools and finishing with more recent contributions in the literature."--Marco Castrillon Lopez, European Mathematical SocietyTable of ContentsPreface ix Chapter 1: Introduction 1 1.1 The modeling framework 1 1.2 Examples in science and engineering 2 1.3 Control system examples 7 1.4 Connections to other modeling frameworks 15 1.5 Notes 22 Chapter 2 The solution concept 25 2.1 Data of a hybrid system 25 2.2 Hybrid time domains and hybrid arcs 26 2.3 Solutions and their basic properties 29 2.4 Generators for classes of switching signals 35 2.5 Notes 41 Chapter 3 Uniform asymptotic stability, an initial treatment 43 3.1 Uniform global pre-asymptotic stability 43 3.2 Lyapunov functions 50 3.3 Relaxed Lyapunov conditions 60 3.4 Stability from containment 64 3.5 Equivalent characterizations 68 3.6 Notes 71 Chapter 4 Perturbations and generalized solutions 73 4.1 Differential and difference equations 73 4.2 Systems with state perturbations 76 4.3 Generalized solutions 79 4.4 Measurement noise in feedback control 84 4.5 Krasovskii solutions are Hermes solutions 88 4.6 Notes 94 Chapter 5 Preliminaries from set-valued analysis 97 5.1 Set convergence 97 5.2 Set-valued mappings 101 5.3 Graphical convergence of hybrid arcs 107 5.4 Differential inclusions 111 5.5 Notes 115 Chapter 6 Well-posed hybrid systems and their properties 117 6.1 Nominally well-posed hybrid systems 117 6.2 Basic assumptions on the data 120 6.3 Consequences of nominal well-posedness 125 6.4 Well-posed hybrid systems 132 6.5 Consequences of well-posedness 134 6.6 Notes 137 Chapter 7 Asymptotic stability, an in-depth treatment 139 7.1 Pre-asymptotic stability for nominally well-posed systems 141 7.2 Robustness concepts 148 7.3 Well-posed systems 151 7.4 Robustness corollaries 153 7.5 Smooth Lyapunov functions 156 7.6 Proof of robustness implies smooth Lyapunov functions 161 7.7 Notes 167 Chapter 8 Invariance principles 169 8.1 Invariance and omega-limits 169 8.2 Invariance principles involving Lyapunov-like functions 170 8.3 Stability analysis using invariance principles 176 8.4 Meagre-limsup invariance principles 178 8.5 Invariance principles for switching systems 181 8.6 Notes 184 Chapter 9 Conical approximation and asymptotic stability 185 9.1 Homogeneous hybrid systems 185 9.2 Homogeneity and perturbations 189 9.3 Conical approximation and stability 192 9.4 Notes 196 Appendix: List of Symbols 199 Bibliography 201 Index 211

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Book SynopsisThis book develops a new theory of multi-parameter singular integrals associated with Carnot-Caratheodory balls. Brian Street first details the classical theory of Calderon-Zygmund singular integrals and applications to linear partial differential equations. He then outlines the theory of multi-parameter Carnot-Caratheodory geometry, where the mainTable of Contents*FrontMatter, pg. i*Contents, pg. v*Preface, pg. ix*1. The Calderon-Zygmund Theory I: Ellipticity, pg. 1*2. The Calderon-Zygmund Theory II: Maximal Hypoellipticity, pg. 39*3. Multi-parameter Carnot-Caratheodory Geometry, pg. 198*4. Multi-parameter Singular Integrals I: Examples, pg. 223*5. Multi-parameter Singular Integrals II: General Theory, pg. 268*Appendix A. Functional Analysis, pg. 363*Appendix B. Three Results from Calculus, pg. 376*Appendix C. Notation, pg. 380*Bibliography, pg. 383*Index, pg. 393

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Book SynopsisThis book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineeTrade Review"Written by an expert in the area, the book presents in an accessible manner a lot of important results from the realm of positive matrices and of their applications... The book can be used for graduate courses in linear algebra, or as supplementary material for courses in operator theory, and as a reference book by engineers and researchers working in the applied field of quantum information."--S. Cobzas, Studia Universitatis Babes-Bolyai, Mathematica "There is no obvious competitor for Bhatia's book, due in part to its focus, but also because it contains some very recent material drawn from research articles. Beautifully written and intelligently organised, Positive Definite Matrices is a welcome addition to the literature. Readers who admired his Matrix Analysis will no doubt appreciate this latest book of Rajendra Bhatia."--Douglas Farenick, Image "This is an outstanding book. Its exposition is both concise and leisurely at the same time."--Jaspal Singh Aujla, Zentralblatt MATHTable of ContentsPreface vii Chapter 1: Positive Matrices 1 1.1 Characterizations 1 1.2 Some Basic Theorems 5 1.3 Block Matrices 12 1.4 Norm of the Schur Product 16 1.5 Monotonicity and Convexity 18 1.6 Supplementary Results and Exercises 23 1.7 Notes and References 29 Chapter 2: Positive Linear Maps 35 2.1 Representations 35 2.2 Positive Maps 36 2.3 Some Basic Properties of Positive Maps 38 2.4 Some Applications 43 2.5 Three Questions 46 2.6 Positive Maps on Operator Systems 49 2.7 Supplementary Results and Exercises 52 2.8 Notes and References 62 Chapter 3: Completely Positive Maps 65 3.1 Some Basic Theorems 66 3.2 Exercises 72 3.3 Schwarz Inequalities 73 3.4 Positive Completions and Schur Products 76 3.5 The Numerical Radius 81 3.6 Supplementary Results and Exercises 85 3.7 Notes and References 94 Chapter 4: Matrix Means 101 4.1 The Harmonic Mean and the Geometric Mean 103 4.2 Some Monotonicity and Convexity Theorems 111 4.3 Some Inequalities for Quantum Entropy 114 4.4 Furuta's Inequality 125 4.5 Supplementary Results and Exercises 129 4.6 Notes and References 136 Chapter 5: Positive Definite Functions 141 5.1 Basic Properties 141 5.2 Examples 144 5.3 Loewner Matrices 153 5.4 Norm Inequalities for Means 160 5.5 Theorems of Herglotz and Bochner 165 5.6 Supplementary Results and Exercises 175 5.7 Notes and References 191 Chapter 6: Geometry of Positive Matrices 201 6.1 The Riemannian Metric 201 6.2 The Metric Space Pn 210 6.3 Center of Mass and Geometric Mean 215 6.4 Related Inequalities 222 6.5 Supplementary Results and Exercises 225 6.6 Notes and References 232 Bibliography 237 Index 247 Notation 253

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Book SynopsisTrade Review"The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts."--Gerd Faltings, Zentralblatt MATHTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Foreword, pg. ix*Chapter I. Representations of the fundamental group and the torsor of deformations. An overview, pg. 1*Chapter II. Representations of the fundamental group and the torsor of deformations. Local study, pg. 27*Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects, pg. 179*Chapter IV. Cohomology of Higgs isocrystals, pg. 307*Chapter V. Almost etale coverings, pg. 449*Chapter VI. Covanishing topos and generalizations, pg. 485*Facsimile : A p-adic Simpson correspondence, pg. 577*Bibliography, pg. 595*Indexes, pg. 599

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Book SynopsisTrade Review"This well-written book prepares readers to take a real analysis course by carefully defining and proving all concepts one needs for this type of course. . . . Throughout the book, the style is incredibly reader friendly, and the author's enthusiasm for the subject is very clear." * Choice *"I can imagine this book proving useful to a motivated student who is finding the transition into analysis challenging through traditional textbooks."---Dominic Yeo, Mathematical GazetteTable of ContentsPreliminaries 1 1 Introduction 3 2 Basic Math and Logic* 6 3 Set Theory* 14 Real Numbers 25 4 Least Upper Bounds* 27 5 The Real Field* 35 6 Complex Numbers and Euclidean Spaces 46 Topology 59 7 Bijections 61 8 Countability 68 9 Topological Definitions* 79 10 Closed and Open Sets* 90 11 Compact Sets* 98 12 The Heine-Borel Theorem* 108 13 Perfect and Connected Sets 117 Sequences 127 14 Convergence* 129 15 Limits and Subsequences* 138 16 Cauchy and Monotonic Sequences* 148 17 Subsequential Limits 157 18 Special Sequences 166 19 Series* 174 20 Conclusion 183 Acknowledgments 187 Bibliography 189 Index 191

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Book SynopsisTrade ReviewOne of Amazon.com's 2013 Best Science Books One of Choice's Outstanding Academic Titles for 2013 Honorable Mention for the 2013 PROSE Award in Popular Science & Mathematics, Association of American Publishers "As Fortnow describes... P versus NP is 'one of the great open problems in all of mathematics' not only because it is extremely difficult to solve but because it has such obvious practical applications. It is the dream of total ease, of the confidence that there is an efficient way to calculate nearly everything, 'from cures to deadly diseases to the nature of the universe,' even 'an algorithmic process to recognize greatness.'... To postulate that P ? NP, as Fortnow does, is to allow for a world of mystery, difficulty, and frustration--but also of discovery and inquiry, of pleasures pleasingly delayed."--Alexander Nazaryan, New Yorker "Fortnow effectively initiates readers into the seductive mystery and importance of P and NP problems."--Publishers Weekly "Fortnow's book is just the ticket for bringing one of the major theoretical problems of our time to the level of the average citizen--and yes, that includes elected officials."--Veit Elser, Science "Without bringing formulas or computer code into the narrative, Fortnow sketches the history of this class of questions, convincingly demonstrates their surprising equivalence, and reveals some of the most far-reaching implications that a proof of P = NP would bring about. These might include tremendous advances in biotechnology (for instance, more cures for cancer), information technology, and even the arts. Verdict: Through story and analogy, this relatively slim volume manages to provide a thorough, accessible explanation of a deep mathematical question and its myriad consequences. An engaging, informative read for a broad audience."--J.J.S. Boyce, Library Journal "A provocative reminder of the real-world consequences of a theoretical enigma."--Booklist "The definition of this problem is tricky and technical, but in The Golden Ticket, Lance Fortnow cleverly sidesteps the issue with a boiled-down version. P is the collection of problems we can solve quickly, NP is the collection of problems we would like to solve. If P = NP, computers can answer all the questions we pose and our world is changed forever. It is an oversimplification, but Fortnow, a computer scientist at Georgia Institute of Technology, Atlanta, knows his stuff and aptly illustrates why NP problems are so important."--Jacob Aron, New Scientist "Fortnow's book does a fine job of showing why the tantalizing question is an important one, with implications far beyond just computer science."--Rob Hardy, Commercial Dispatch "A great book... [Lance Fortnow] has written precisely the book about P vs. NP that the interested layperson or IT professional wants and needs."--Scott Aaronson, Shtetl-Optimized blog "[The Golden Ticket] is a book on a technical subject aimed at a general audience... Lance's mix of technical accuracy with evocative story telling works."--Michael Trick, Michael Trick's Operations Research Blog "Thoroughly researched and reviewed. Anyone from a smart high school student to a computer scientist is sure to get a lot of this book. The presentation is beautiful. There are few formulas but lots of facts."--Daniel Lemire's Blog "An entertaining discussion of the P versus NP problem."--Andrew Binstock, Dr. Dobb's "The Golden Ticketis an extremely accessible and enjoyable treatment of the most important question of theoretical computer science, namely whether P is equal to NP."--Choice "The book is accessible and useful for practically anyone from smart high school students to specialists... [P]erhaps the interest sparked by this book will be the 'Golden Ticket' for further accessible work in this area. And perhaps P=NP will start to become as famous as E=mc2."--Michael Trick, INFORMS Journal of Computing "In any case, it is excellent to have a nontechnical book about the P versus NP question. The Golden Ticket offers an inspiring introduction for nontechnical readers to what is surely the most important open problem in computer science."--Leslie Ann Goldberg, LMS Newsletter "The Golden Ticket does a good job of explaining a complex concept in terms that a secondary-school student will understand--a hard problem in its own right, even if not quite NP."--Physics World "[The Golden Ticket] is fun to read and can be fully appreciated without any knowledge in (theoretical) computer science. Fortnow's efforts to make the difficult material accessible to non-experts should be commended."--Andreas Maletti, Zentralblatt MATH "This is a fabulous book for both educators and students at the secondary school level and above. It does not require any particular mathematical knowledge but, rather, the ability to think. Enjoy the world of abstract ideas as you experience an intriguing journey through mathematical thinking."--Gail Kaplan, Mathematics Teacher "Fortnow's book provides much of the background and personal information on the main characters involved in this problem--notably Steven Cook, with a cameo appearance by Kurt Godel--that one does not get in the more technical treatments. There is a lot of information in this book, and the serious computer science student is sure to learn from it."--James M. Cargal, UMAP JournalTable of ContentsPreface ix Chapter 1 The Golden Ticket 1 Chapter 2 The Beautiful World 11 Chapter 3 P and NP 29 Chapter 4 The Hardest Problems in NP 51 Chapter 5 The Prehistory of P versus NP 71 Chapter 6 Dealing with Hardness 89 Chapter 7 Proving P <> NP 109 Chapter 8 Secrets 123 Chapter 9 Quantum 143 Chapter 10 The Future 155 Acknowledgments 163 Chapter Notes and Sources 165 Index 171

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Book SynopsisTrade Review"This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way."---Jonathan Shock, Mathemafrica"Reading Nahin is like reading through a select library of ancient Babylonian mathematical clay tablets. Surprises abound. . . . Nahin weaves much colorful history into his narrative."---Andrew Simoson, Mathematical Intelligencer"Engaging. . . . The book contains a wealth of original problems. . . . An enjoyable read."---Antonín Slavík, Zentralblatt MATH"This reviewer found himself being drawn to a variety of unfamiliar settings with much interest and even fascination." * Choice *"I certainly enjoyed [the book]!"---Alan Stevens, Mathematics Today"The potential audience for this book should be fairly large and go from highly talented high school students up through professionals in any STEM field."---Geoffrey Dietz, MAA Reviews

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Book SynopsisTrade Review"This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way."---Jonathan Shock, Mathemafrica"Reading Nahin is like reading through a select library of ancient Babylonian mathematical clay tablets. Surprises abound. . . . Nahin weaves much colorful history into his narrative."---Andrew Simoson, Mathematical Intelligencer"Engaging. . . . The book contains a wealth of original problems. . . . An enjoyable read."---Antonín Slavík, Zentralblatt MATH"This reviewer found himself being drawn to a variety of unfamiliar settings with much interest and even fascination." * Choice *"I certainly enjoyed [the book]!"---Alan Stevens, Mathematics Today"The potential audience for this book should be fairly large and go from highly talented high school students up through professionals in any STEM field."---Geoffrey Dietz, MAA Reviews

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Book SynopsisThe present volume reflects both the diversity of Bochner's pursuits in pure mathematics and the influence his example and thought have had upon contemporary researchers. Originally published in 1971. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguisheTable of Contents*Frontmatter, pg. i*Foreword, pg. vii*Contents, pg. ix*On the Group of Automorphisms of a Symplectic Manifold, pg. 1*On the Minimal Immersions of the Two-sphere in a Space of Constant Curvature, pg. 27*Intersections of Cantor Sets and Transversality of Semigroups, pg. 41*Kahlersche Mannigfaltigkeiten mit hyper-q-konvexem Rand, pg. 61*Iteration of Analytic Functions of Several Variables, pg. 81*A Class of Positive-Difinite Functions, pg. 93*Local Noncommutative Analysis, pg. 111*Linearization of the Product omicronf Orthogonal Polynomials, pg. 131*Eisenstein Series on Tube Domains, pg. 139*Laplace-Fourier Transformation, the Foundation for Quantum Information Theory and Linear Physics, pg. 157*An Integral Equation Related to the Schroedinger Equation with an Application to Integration in Function Space, pg. 175*A Lower Bound for the Smallest Eigenvalue of the Laplacian, pg. 195*The Integral Equation Method in Scattering Theory, pg. 201*Group Algebra Bundles, pg. 229*Quadratic Periods of Hjperelliptic Abelian Integrals, pg. 239*The Existence of Complementary Series, pg. 249*Some Recent Developments in the Theory of Singular Perturbations, pg. 261*Sequential Convergence in Lattice Groups, pg. 273*A Group-theoretic Lattice-point Problem, pg. 291*The Riemann Surface of Klein with 168 Automorphisms, pg. 297*Envelopes of Holomorphy of Domains in Complex Lie Groups, pg. 309*Automorphisms of Commutative Banach Algebras, pg. 319*Historical Notes on Analyticity as a Concept in Functional Analysis, pg. 325*A -Almost Automorphic Functions, pg. 345

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Book SynopsisNonlinear dynamics has been successful in explaining complicated phenomena in well-defined low-dimensional systems. Now it is time to focus on real-life problems that are high-dimensional or ill-defined, for example, due to delay, spatial extent, stochasticity, or the limited nature of available data. How can one understand the dynamics of such systems? Written by international experts, Nonlinear Dynamics and Chaos: Where Do We Go from Here? assesses what the future holds for dynamics and chaos. The chapters address one or more of the broad and interconnected main themes: neural and biological systems, spatially extended systems, and experimentation in the physical sciences. The contributors offer suggestions as to what they see as the way forward, often in the form of open questions for future research.Trade Review"This handsome volume is the proceedings of a conference held in Bristol in 2001, which had the aim of charting new directions for the exploration of nonlinear dynamical systems. The editors must be commended for their work: the individual chapters have been given a clean, uniform style that reflects a serious effort to present the volume as a unified book rather than a recollection of articles, with several cross-references between the chapters. The book is also remarkably free of typographical errors. I heartily recommend this collection to students looking for some direction (as long as they don't think this is all of nonlinear dynamics!)." -UK Nonlinear News, May 2003 "This timely and important book is a record of papers presented at a conference in Bristol and is very well edited, and produced … The very richness of this book, in both theory and real-world applications, makes it difficult to summarize and even more difficult to put down." -Nonlinear Dynamics, Psychology and Life Sciences "The book is written by authors who are champions of their field. All researchers in nonlinear dynamics should have access to this book. It is a valuable resource of references and it contains a lot of ideas and open problems in various fields. One might think of it as a catalogue of problems in nonlinear dynamics. The introduction of the book is a 'must-read.' It presents the nature and the philosophy of the book (and the symposium). Reading the introduction, the editors clearly have done a great job of managing each of the invited lecturers to translate the philosophy of the symposium into their lectures … my impression is that all authors did a good job presenting the excitement of their research and addressing the interesting questions. This book in general is a valuable addition to the literature of the theory and practice of nonlinear dynamics and chaos." -Theo Tuwankotta, Institute of Technology,ITB, Bandung, IndonesiaTable of ContentsPreface. Bifurcation and Degenerate Decomposition in Multiple Time Scale Dynamical Systems. Many-body quantum mechanics. Unfolding Complexity: Hereditory Dynamical Systems-New Bifurcation Schemes and High Dimensional Chaos. Creating stability out of instability.Signal or Noise? A nonlinear dynamics approach to spatiotemporal communication. Outstanding problems in the theory of pattern formation. Is Chaos relevant to Fluid Mechanics?. Time-Reversed Acoustics and Chaos.Reduction methods applied to nonlocally coupled oscillator systems. A prime number of prime questions about vortex dynamcis in nonlinear media. Spontaneous pattern formation in primary visual cortex. Models for Pattern Formation in Development. Spatiotemporal nonlinear dynamics: a new beginning. Author index.

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Book Synopsis1. Two-sided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variables.- 1. Newton's polygon of a polynomial in two variables.- 2. Polynomials admitting of two-sided estimates.- 3. N Quasi-elliptic polynomials in two variables.- 4. N Quasi-elliptic differential operators.- Appendix to 4.- 2. Parabolic operators associated with Newton's polygon.- 1. Polynomials correct in Petrovski?'s sense.- 2. Two-sided estimates for polynomials in two variables satisfying Petrovski?'s condition. N-parabolic polynomials.- 3. Cauchy's problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable.- 4. Stable-correct and parabolic polynomials in several variables.- 5. Cauchy's problem for stable-correct differential operators with variable coefficients.- 3. Dominantly correct operators.- 1. Strictly hyperbolic operators.- 2. Dominantly correct polynomials in two variables.- 3. Dominantly correct differential operators with variable coefficients (the case of two variables).- 4. Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables).- 4. Operators of principal type associated with Newton's polygon.- 1. Introduction. Operators of principal and quasi-principal type.- 2. Polynomials of N-principal type.- 3. The main L2 estimate for operators of N-principal type.- Appendix to 3.- 4. Local solvability of differential operators of N-principal type.- Appendix to 4.- 5. Two-sided estimates in several variables relating to Newton's polyhedra.- 1. Estimates for polynomials in ?n relating to Newton's polyhedra.- 2. Two-sided estimates insome regions in ?n relating to Newton's polyhedron. Special classes of polynomials and differential operators in several variables.- 6. Operators of principal type associated with Newton's polyhedron.- 1. Polynomials of N-principal type.- 2. Estimates for polynomials of N-principal type in regions of special form.- 3. The covering of ?n by special regions associated with Newton's polyhedron.- 4. Differential operators of ?n-principal type with variable coefficients.- Appendix to 4.- 7. The method of energy estimates in Cauchy's problem 1. Introduction. The functional scheme of the proof of the solvability of Cauchy's problem.- 2. Sufficient conditions for the existence of energy estimates.- 3. An analysis of conditions for the existence of energy estimates.- 4. Cauchy's problem for dominantly correct differential operators.- References.Table of Contents1. Two-sided estimates for polynomials related to Newton’s polygon and their application to studying local properties of partial differential operators in two variables.- §1. Newton’s polygon of a polynomial in two variables.- §2. Polynomials admitting of two-sided estimates.- §3. N Quasi-elliptic polynomials in two variables.- §4. N Quasi-elliptic differential operators.- Appendix to §4.- 2. Parabolic operators associated with Newton’s polygon.- §1. Polynomials correct in Petrovski?’s sense.- §2. Two-sided estimates for polynomials in two variables satisfying Petrovski?’s condition. N-parabolic polynomials.- §3. Cauchy’s problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable.- §4. Stable-correct and parabolic polynomials in several variables.- §5. Cauchy’s problem for stable-correct differential operators with variable coefficients.- 3. Dominantly correct operators.- §1. Strictly hyperbolic operators.- §2. Dominantly correct polynomials in two variables.- §3. Dominantly correct differential operators with variable coefficients (the case of two variables).- §4. Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables).- 4. Operators of principal type associated with Newton’s polygon.- §1. Introduction. Operators of principal and quasi-principal type.- §2. Polynomials of N-principal type.- §3. The main L2 estimate for operators of N-principal type.- Appendix to §3.- §4. Local solvability of differential operators of N-principal type.- Appendix to §4.- 5. Two-sided estimates in several variables relating to Newton’s polyhedra.- §1. Estimates for polynomials in ?n relating to Newton’s polyhedra.- §2. Two-sided estimates in some regions in ?n relating to Newton’s polyhedron. Special classes of polynomials and differential operators in several variables.- 6. Operators of principal type associated with Newton’s polyhedron.- §1. Polynomials of N-principal type.- §2. Estimates for polynomials of N-principal type in regions of special form.- §3. The covering of ?n by special regions associated with Newton’s polyhedron.- §4. Differential operators of ?n-principal type with variable coefficients.- Appendix to §4.- 7. The method of energy estimates in Cauchy’s problem §1. Introduction. The functional scheme of the proof of the solvability of Cauchy’s problem.- §2. Sufficient conditions for the existence of energy estimates.- §3. An analysis of conditions for the existence of energy estimates.- §4. Cauchy’s problem for dominantly correct differential operators.- References.

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Book Synopsis1 Introduction to the Locally Trivial Bundles Theory.- 2 Homotopy Invariants of Vector Bundles.- 3 Geometric Constructions of Bundles.- 4 Calculation Methods in K-Theory.- 5 Elliptic Operators on Smooth Manifolds and K-Theory.- 6 Some Applications of Vector Bundle Theory.- References.Table of ContentsPreface. 1. Introduction to the Locally Trivial Bundles Theory. 2. Homotopy Invariants of Vector Bundles. 3. Geometric Constructions of Bundles. 4. Calculation Methods in K-Theory. 5. Elliptic Operators on Smooth Manifolds and K-Theory. 6. Some Applications of Vector Bundle Theory. Index. References.

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Book SynopsisPreface to the Third Edition.- Preface to the Second Edition.- Preface to the First Edition.- The Dirac Delta Function and Delta Sequences.- The Schwartz-Sobolev Theory of Distributions.- Additional Properties of Distributions.- Distributions Defined by Divergent Integrals.- Distributional Derivatives of Functions with Jump Discontinuities.- Tempered Distributions and the Fourier Transforms.- Direct Products and Convolutions of Distributions.- The Laplace Transform.- Applications to Ordinary Differential Equations.- Applications to Partial Differential Equations.- Applications to Boundary Value Problems.- Applications to Wave Propagation.- Interplay between Generalized Functions and the Theory of Moments.- Linear Systems.- Miscellaneous Topics.- References.- Index.Trade Review"This book on generalized functions is suitable for physicists, engineers and applied mathematicians. The author presents the notion of generalized functions, their properties and their applications for solving ordinary differential equations and partial differential equations. ... The author demonstrates through various examples that familiarity with generalized functions is very helpful for students in mathematics, physical sciences and technology. The proposed exercises are very good for better understanding of notions and properties presented in the chapters. The book contains new topics and important features." —Mathematica "The advantage of this text is in carefully gathered examples explaining how to use corresponding properties.... Even the standard material connecting with partial and ordinary differential equations is rewritten in modern terminology." —Zentralblatt (Review of a previous edition) "The author has done an excellent job in presenting examples and in displaying the calculational techniques associated with distributions and the applications. Throughout the book there are a wealth of examples concerning the distributional topics and caluclations introduced and concering the applications, and the examples are presented in detail." ---Zentralblatt (Review of the 1st edition) "The collaboration of physicists or engineers and mathematics, which is more and more popular and necessary in modern investigations, requires…a common language. The book under review provides this language…. [It] is a well written book, most of the material is accessible to senior undergraduate and graduate students in mathematical, physical and engineering sciences…. [The] book will [also] be useful…for specialists in ODEs, PDEs, functional analysis, [and] physicists, engineers, and lecturers." —Acta. Sci. Math. (Review of a previous edition) "An exceptionally clear exposition... The exercises at the end of each chapter are well-chosen." —The American Mathematical Monthly (Review of a previous edition) "This fully revised edition of well-received book expands the treatment of fundamental concepts and theoretical background material delineates connections to a variety of applications in mathematical physics, elasticity, wave propagation, magnetohydrodynamics, linear systems, probability and statistics, optical control problems in economics, and more. It has many new topics and [features] driven by additional examples and exercises. . . It presents a wealth of applications that connot be found in any other single source. the book will be important reading for graduate students in physics and engineering." --- Educational Book ReviewTable of ContentsPreface to the Third Edition * Preface to the Second Edition * Preface to the First Edition * Chapter 1. The Dirac Delta Function and Delta Sequences * 1.1 The Heaviside Function * 1.2 The Dirac Delta Function * 1.3 The Delta Sequences * 1.4 A Unit Dipole * 1.5 The Heaviside Sequences * Exercises * Chapter 2. The Schwartz-Sobolev Theory of Distributions * 2.1 Some Introductory Definitions * 2.2 Test Functions * 2.3 Linear Functionals and the Schwartz–Sobolev Theory of Distributions * 2.4 Examples * 2.5 Algebraic Operations on Distributions * 2.6 Analytic Operations on Distributions * 2.7 Examples * 2.8 The Support and Singular Support of a Distribution Exercises * Chapter 3. Additional Properties of Distributions * 3.1 Transformation Properties of the Delta Distributions * 3.2 Convergence of Distributions * 3.3 Delta Sequences with Parametric Dependence * 3.4 Fourier Series * 3.5 Examples * 3.6 The Delta Function as a Stieltjes Integral Exercises * Chapter 4. Distributions Defined by Divergent Integrals * 4.1 Introduction * 4.2 The Pseudofunction H(x)/x n , n = 1, 2,3, * 4.3 Functions with Algebraic Singularity of Order m * 4.4 Examples * Exercises * Chapter 5. Distributional Derivatives of Functions with Jump Discontinuities * 5.1 Distributional Derivatives in R 1 * 5.2 Moving Surfaces of Discontinuity in R n , n 2 * 5.3 Surface Distributions * 5.4 Various Other Representations * 5.5 First-Order Distributional Derivatives * 5.6 Second Order Distributional Derivatives * 5.7 Higher-Order Distributional Derivatives * 5.8 The Two-Dimensional Case * 5.9 Examples * 5.10 The Function Pf ( l/r ) and its Derivatives * Chapter 6. Tempered Distributions and the Fourier Transforms * 6.1 Preliminary Concepts * 6.2 Distributions of Slow Growth (Tempered Distributions) * 6.3 The Fourier Transform * 6.4 Examples * Exercises * Chapter 7. Direct Products and Convolutions of Distributions * 7.1 Definition of the Direct Product * 7.2 The Direct Product of Tempered Distributions * 7.3 The Fourier Transform of the Direct Product of Tempered Distributions * 7.4 The Convolution * 7.5 The Role of Convolution in the Regularization of the Distributions * 7.6 The Dual Spaces E and E' * 7.7 Examples * 7.8 The Fourier Transform of the Convolution * 7.9 Distributional Solutions of Integral Equations * Exercises * Chapter 8. The Laplace Transform * 8.1 A Brief Discussion of the Classical Results * 8.2 The Laplace Transform of the Distributions * 8.3 The Laplace Transform of the Distributional Derivatives and Vice Versa * 8.4 Examples * Exercises * Chapter 9. Applications to Ordinary Differential Equations * 9.1 Ordinary Differential Operators * 9.2 Homogeneous Differential Equations * 9.3 Inhomogeneous Differentational Equations: The Integral of a Distribution * 9.4 Examples * 9.5 Fundamental Solutions and Green's Functions * 9.6 Second Order Differential Equations with Constant Coefficients * 9.7 Eigenvalue Problems * 9.8 Second Order Differential Equations with Variable Coefficients * 9.9 Fourth Order Differential Equations * 9.10 Differential Equations of n th Order * 9.11 Ordinary Differential Equations with Singular Coefficients * Exercises * Chapter 10. Applications to Partial Differential Equations * 10.1 Introduction * 10.2 Classical and Generalized Solutions * 10.3 Fundamental Solutions * 10.4 The Cauchy–Riemann Operator * 10.5 The Transport Operator * 10.6 The Laplace Operator * 10.7 The Heat Operator * 10.8 The Schroedinger Operator * 10.9 The Helmholtz Operator * 10.10 The Wave Operator * 10.11 The Inhomogeneous Wave Equation * 10.12 The Klein–Gordon Operator * Exercises * Chapter 11. Applications to Boundary Value Problems * 11.1 Poisson's Equation * 11.2 Dumbbell-Shaped Bodies * 11.3 Uniform Axial Distributions * 11.4 Linear Axial Distributions * 11.5 Parabolic Axial Distributions * 11.6 The Four-Order Polynomial Distribution, n = 7; Spheroidal Cavities * 11.7 The Polarization Tensor for a Spheroid * 11.8 The Virtual Mass Tensor for a Spheroid * 11.9 The Electric and Magnetic Polarizability Tensors * 11.10 The Distributional Approach to Scattering Theory * 11.11 Stokes Flow * 11.12 Displacement-Type Boundary Value Problems in Elastostatics * 11.13 The Extension to Elastodynamics * 11.14 Distributions on Arbitrary Lines * 11.15 Distributions on Plane Curves * 11.16 Distributions on a Circular Disk * Chapter 12. Applications to Wave Propagation * 12.1 Introduction * 12.2 The Wave Equation * 12.3 First-Order Hyperbolic Systems * 12.4 Aerodynamic Sound Generation * 12.5 The Rankine–Hugoniot Conditions * 12.6 Wave Fronts That Carry Infinite Singularities * 12.7 Kinematics of Wave Fronts * 12.8 Derivation of the Transport Theorems for Wave Fronts * 12.9 Propagation of Wave Fronts Carrying Multilayer Densities * 12.10 Generalized Functions with Support on the Light Cone * 12.11 Examples * Chapter 13. Interplay Between Generalized Functions and the Theory of Moments * 13.1 The Theory of Moments * 13.2 Asymptotic Approximation of Integrals * 13.3 Applications to the Singular Perturbation Theory * 13.4 Applications to Number Theory * 13.5 Distributional Weight Functions for Orthogonal Polynomials * 13.6 Convolution Type Integral Equations Revisited * 13.7 Further Applications * Chapter 14. Linear Systems * 14.1 Operators * 14.2 The Step Response * 14.3 The Impulse Response * 14.4 The Response to an Arbitrary Input * 14.5 Generalized Functions as Impulse Response Functions * 14.6 The Transfer Function * 14.7 Discrete-Time Systems * 14.8 The Sampling Theorem * Chapter 15. Miscellaneous Topics * 15.1 Applications to Probability and Statistics * 15.2 Applications to Mathematical Economics * 15.3 Periodic Generalized Functions * 15.4 Microlocal Theory * References * Index

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Book SynopsisFirst-Order, Quasi-Linear Equations and Method of Characteristics.- Mathematical Models.- Classification of Second-Order Linear Equations.- The Cauchy Problem and Wave Equations.- Fourier Series and Integrals with Applications.- Method of Separation of Variables.- Eigenvalue Problems and Special Functions.- Boundary-Value Problems and Applications.- Higher-Dimensional Boundary-Value Problems.- Green's Functions and Boundary-Value Problems.- Integral Transform Methods with Applications.- Nonlinear Partial Differential Equations with Applications.- Numerical and Approximation Methods.- Tables of Integral Transforms.Table of ContentsPreface to the Fourth Edition Preface to the Third Edition Introduction First-Order, Quasi-Linear Equations and Method of Characteristics Mathematical Models Classification of Second-Order Linear Equations The Cauchy Problem and Wave Equations Fourier Series and Integrals with Applications Method of Separation of Variables Eigenvalue Problems and Special Functions Boundary-Value Problems and Applications Higher-Dimensional Boundary-Value Problems Green's Functions and Boundary-Value Problems Integral Transform Methods with Applications Nonlinear Partial Differential Equations with Applications Numerical and Approximation Methods Tables of Integral Transforms Answers and Hints to Selected Exercises Appendix: Some Special Functions and Their Properties Bibliography Index

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Book SynopsisPresents an introduction to functional analysis and the initial fundamentals of $C^*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide an account of the introductory portions of this important and technically difficult subject.Table of ContentsLinear spaces Basics of Hilbert space and linear operators Banach algebras Elementary $C^*$-algebra theory Elementary von Neumann algebra theory Bibliography Index of notation Index.

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Book SynopsisThere are incredibly rich connections between classical analysis and number theory. This title uncovers interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. It is suitable for a graduate course or for independent reading.Table of ContentsThe modular group and elliptic function theory Theta functions with characteristics Function theory for the modular group $\Gamma$ and its subgroups Theta constant identities Partition theory: Ramanujan congruences and generalizations Identities related to partition functions Combinatorial and number theoretic applications Bibliography Bibliographical notes Index.

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Book SynopsisIn addition to the standard topics, this volume includes topics not often found in an algebra book, such as inequalities, and the elements of substitution theory. Especially extensive is Chrystal's treatment of the infinite series, infinite products, and (finite and infinite) continued fractions.

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Book SynopsisContains lectures from the CBMS Regional Conference held at Harvey Mudd College, June 1977. This monograph consists of applications to nonlinear differential equations of the author's coincidental degree. It includes an bibliography covering many aspects of the modern theory of nonlinear differential equations and the theory of nonlinear analysis.Table of ContentsIntroduction Suggestions for the readerSuggestions for the reader Fredholm mappings of index zero and linear boundary value problems Degree theory for some classes of mappings Duality theorems for several fixed point operators associated to periodic problems for ordinary differential equations Existence theorems for equations in normed spaces Boundary value problems for second order nonlinear vector differential equations Periodic solutions of ordinary differential equations with one-sided growth restrictions Bound sets for functional differential equations The index of isolated zeros of some mappings Bifurcation theory Periodic solutions of autonomous ordinary differential equations around an equilibrium References.

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Book SynopsisHelps in the understanding of continuous and differentiable functions. This book emphasises on real functions of a single variable. It contains topics that include: continuous functions, the intermediate value property, uniform continuity, mean value theorems, Taylors formula, convex functions, and sequences and series of functions.Trade ReviewA very stimulating problem book ... a useful supplement to any graduate textbok in mathematical analysis MAA OnlineTable of ContentsProblems: Limits and continuity Differentiation Sequences and series of functions Solutions: Limits and continuity Differentiation Sequences and series of functions Bibliography Index.

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Book SynopsisNature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations. This book employs ingredients from each of these subjects to tell the mathematical story of soap films.Trade Review... a book like Oprea's has been sorely needed MAA OnlineTable of ContentsSurface tension A quick trip through differential geometry and complex variables The mathematics of soap films The calculus of variations and shape Maple, soap films and minimal surfaces Bibliography Index.

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Book SynopsisGives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory).Table of ContentsIntroduction Braided tensor categories Ribbon categories Modular tensor categories 3-dimensional topological quantum field theory Modular functor Moduli spaces and complex modular functor Wess-Zumino-Witten model Bibliography Index Index of notation.

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Book SynopsisIntroduces the notion of topological degree and develops its basic properties. This book uses these properties in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz's global bifurcation theorem. It is suitable as a graduate level textbook and a supplementary course text.Trade Reviewextremely stimulating Zentralblatt fur MathematikTable of ContentsTopological approach: Finite dimensions Topological degree in Banach space Bifurcation theory Further topological methods Monotone operators and the min-max theorem Generalized implicit function theorems Bibliography.

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Book SynopsisEmphasizes the conceptual and historical continuity of analytic function theory. This work covers topics including elliptic functions, entire and meromorphic functions, as well as conformal mapping. It features chapters on majorization and on functions holomorphic in a half-plane.Table of ContentsVolume II:; 10. Analytic continuation:; 10.1 Introduction; 10.2 Rearrangements of power series; 10.3 Analytic functions; 10.4 Singularities; 10.5 Borel monogenic functions; 10.6 Multivalued functions and Riemann surfaces; 10.7 Law of permanence of functional equations; 11. Singularities and representation of analytic functions:; 11.1 Holomorphy-preserving transformations: I. Integral operators; 11.2 Holomorphy-preserving transformations: II. Differential operators; 11.3 Power series with analytic coefficients; 11.4 Analytic continuation in a star; 11.5 Polynomial series; 11.6 Composition theorems; 11.7 Gap theorems and noncontinuable power series; 12. Algebraic functions:; 12.1 Local properties; 12.2 Critical points; 12.3 Newton's diagram; 12.4 Riemann surfaces; some concepts of algebraic geometry; 12.5 Rational functions on the surface and Abelian integrals; 13. Elliptic functions:; 13.1 Doubly-periodic functions; 13.2 The functions of Weierstrass; 13.3 Some further properties of elliptic functions; 13.4 On the functions of Jacobi; 13.5 The theta functions; 13.6 Modular functions; 14. Entire and meromorphic functions:; 14.1 Order relations for entire functions; 14.2 Entire functions of finite order; 14.3 Functions with real zeros; 14.4 Characteristic functions; 14.5 Picard's and Landau's theorems; 14.6 The second fundamental theorem; 14.7 Defect relations; 15. Normal families:; 15.1 Schwarz's lemma and hyperbolic measure; 15.2 Normal families; 15.3 Induced convergence; 15.4 Applications; 16. Lemniscates:; 16.1 Chebichev polynomials; 16.2 The transfinite diameter; 16.3 Additive set functions; Radon-Stieltjes integrals; 16.4 Logarithmic capacity; 16.5 Green's function; Hilbert's theorem; 16.6 Runge's theorem; 16.7 Overconvergence; 17. Conformal mapping:; 17.1 Riemann's mapping theorem; 17.2 The kernel function; 17.3 Fekete polynomials and the exterior mapping problem; 17.4 Univalent functions; 17.5 The boundary problem; 17.6 Special mappings; 17.7 The theorem of Bloch; 18. Majorization:; 18.1 The Phragmen-Lindelof principle; 18.2 Dirichlet's problem; Lindelof's principle; 18.3 Harmonic measure; 18.4 The Nevanlinna-Ahlfors-Heins theorems; 18.5 Subordination; 19. Functions holomorphic in a half-plane:; 19.1 The Hardy-Lebesgue classes; 19.2 Bounded functions; 19.3 Growth-measuring functions; 19.4 Remarks on Laplace-Stieltjes integrals Bibliography Index.

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