Series: Dolciani Mathematical Expositions No. 32
Hardback (ISBN-13: 9780883853382)
This quick and easy-to-use guide provides a solid grounding in the fundamental area of complex variables. Copious figures and examples are used to illustrate the principal ideas, and the exposition is lively and inviting. In addition to important ideas from the Cauchy theory, the author also includes the Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping, and dozens of other central topics. An undergraduate taking a first look at the subject, or a graduate student preparing for their qualifying exams, will find this book to be both a valuable resource and a useful companion to more exhaustive texts in the field. For mathematicians and non-mathematicians alike.
* A quick and friendly guide, illustrated with copious figures and examples * Suitable for both mathematicians and non-mathematicians * A highly useful companion to more exhaustive texts in the field
Preface; 1. The complex plane; 2. Complex line integrals; 3. Applications of the Cauchy theory; 4. Isolated singularities and Laurent series; 5. The argument principle; 6. The geometric theory of holomorphic functions; 7. Harmonic functions; 8. Infinite series and products; 9. Analytic continuation.
Klaus Rothfs pioneering research in the field of number theory has led to important and substantial breakthroughs in many areas, including sieve theory, diophantine approximation, and irregularities of distribution. His work on the Thue-Siegel-Roth Theorem earned him a Fields Medal in 1958 - the first British mathematician to receive the honour. Analytic Number Theory: Essays in Honour of Klaus Roth comprises 32 essays from close colleagues and leading experts in those fields in which he has worked, and provides a great insight into the historical development of the subject matter and the importance of Rothfs contributions to number theory and beyond. His influence is also discussed in relation to more recent mathematical advances. Extensive lists of references make this a valuable source for research mathematicians in many areas, an introductory overview of the subject for beginning research students, and a fitting long-awaited tribute to a great mathematician.
* Contains 32 contributions from leading figures in the field of number theory and many related areas * Explores the impact of Klaus Rothfs research across a number of mathematical fields, including sieve theory and diophantine approximations * An extensive list of references makes this a valuable source for all research mathematicians
Contents
Preface; Acknowledgments; Klaus Roth at 80; Numbers with a large prime factor II Roger Baker and Glyn Harman; Character sums with Beatty sequences on Burgess-type intervals William D. Banks and Igor E. Shparlinski; The Hales-Jewett number is exponential: game-theoretic consequences Jozsef Beck, Wesley Pegden and Sujith Vijay; Classical metric diophantine approximation revisited Victor Beresnevich, Vasily Bernik, Maurice Dodson and Sanju Velani; The sum-product phenomenon and some of its applications J. Bourgain; Integral points on cubic hypersurfaces T. D. Browning and D. R. Heath-Brown; Binary additive problems and the circle method, multiplicative sequences and convergent sieves Jorg Brudern; On the convergents to algebraic numbers Yann Bugeaud; Complexity bounds via Rothfs method of orthogonal functions Bernard Chazelle; Some of Rothfs ideas in discrepancy theory William Chen and Giancarlo Travaglini; Congruences and ideals Harold G. Diamond and H. Halberstam; Elementary geometry of Hilbert spaces applied to abelian groups P. D. T. A. Elliott; New bounds for Szemeredifs theorem II: a new bound for r4(N) Ben Green and Terence Tao; One-sided discrepancy of linear hyperplanes in finite vector spaces Nils Hebbinghaus, Tomasz Schoen and Anand Srivastav; How small must ill-distributed sets be? H. A. Helfgott and A. Venkatesh; On the power-free values of polynomials in two variables C. Hooley; On a question of Browning and Heath-Brown Nicholas M. Katz; Good distribution of values of sparse polynomials modulo a prime Sergei Konyagin; Diophantine approximation and continued fractions in power series fields A. Lasjaunias; On transfer inequalities in diophantine approximation Michel Laurent; On exponential sums with multiplicative coefficients Helmut Maier; Multiplicative dependence of values of algebraic functions David Masser; Linear forms in logarithms, and simultaneous diophantine approximation Bernard de Mathan; The Caccetta-Haggkvist conjecture and additive number theory Melvyn B. Nathanson; L2 discrepancy and multivariate integration Erich Novak and Henryk Wozniakowski; Irregularities of sequences relative to long arithmetic progressions A. Sarkozy and C. L. Stewart; The number of solutions of a linear homogeneous congruence II A. Schinzel, with an appendix by Jerzy Kaczorowski; The diophantine equation ¿1x1 . . . ¿1xn = f (x1,c,xn) Wolfgang M. Schmidt; Approximation exponents for function fields Dinesh S. Thakur; On generating functions in additive number theory I R. C. Vaughan; Words and transcendence Michel Waldschmidt; Rothfs theorem, integral points and certain ramified covers of ¯1 Umberto Zannier.
Paperback (ISBN-13: 9780521092975)
The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields.
1. G and other normal modal propositional logics; 2. Peano Arithmetic; 3. The box as Bew; 4. Some applications of G; 5. Semantics for G and other modal logics; 6. Canonical models; 7. The completeness and decidability of G; 8. Trees for G; 9. Calculating the truth-values of fixed points; 10. Rosser's theorem; 11. The fixed-point theorem; 12. Solovay's completeness theorems; 13. An S4-preserving proof-theoretical treatment of modality; 14. The Craig Interpolation Lemma for G.
EMS Textbooks in Mathematics
ISBN 978-3-03719-048-7
September 2008, 578 pages, hardcover, 16 x 23 cm.
This book is written as a textbook on algebraic topology which covers the material for introductory courses (homotopy and homology), background material (manifolds, cell complexes, fibre bundles), and more advanced applications of the basic tools and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). A special feature is the rich supply of nearly 500 exercises and problems at the end of each section. The book recommends to start an introductory course with homotopy theory. For this purpose, basic classical results are presented with new simplified elementary proofs. Alternatively, one could start more traditionally with singular homology. Later chapters include material which has not appeared before in textbooks as well as new simplified proofs for some more advanced results.
Prerequisites are basic point set topology (as recalled in the first chapter), some acquaintance with basic algebra (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (masters) students to the basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included.
EMS Tracts in Mathematics Vol. 6
ISBN 978-3-03719-026-5
September 2008, 395 pages, hardcover, 17 x 24 cm.
Multivariate problems occur in many applications. These problems are defined on spaces of d-variate functions and d can be huge ? in the hundreds or even in the thousands. Some high-dimensional problems can be solved efficiently to within Ã, i.e., the cost increases polynomially in Ã?1 and d. However, there are many multivariate problems for which even the minimal cost increases exponentially in d. This exponential dependence on d is called intractability or the curse of dimensionality.
This is the first of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to algorithms using linear information consisting of arbitrary linear functionals. The theory for multivariate problems is developed in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in Ã?1 and d. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in Ã?1 and d. The study of tractability was initiated about 15 years ago. This is the first research monograph on this subject.
Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. But many practically important problems are solved today for huge d in a reasonable time. One of the most intriguing challenges of theory is to understand why this is possible. Multivariate problems may become tractable if they are defined over weighted spaces with properly decaying weights. In this case, all variables and groups of variables are moderated by weights. The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability.
The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It may be also suitable for graduate courses and seminars. The text concludes with a list of thirty open problems that can be good candidates for future tractability research.
EMS Tracts in Mathematics Vol. 7
ISBN 978-3-03719-019-7
September 2008, 265 pages, hardcover, 17 x 24 cm.
Wavelets have emerged as an important tool in analyzing functions containing discontinuities and sharp spikes. They were developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, turbulence, human vision, radar, earthquake prediction, and pure mathematics applications such as solving partial differential equations.
This book develops a theory of wavelet bases and wavelet frames for function spaces on various types of domains. Starting with the usual spaces on Euclidean spaces and their periodic counterparts, the exposition moves on to so-called thick domains (including Lipschitz domains and snowflake domains). Especially, wavelet expansions and extensions to corresponding spaces on Euclidean n-spaces are developed. Finally, spaces on smooth and cellular domains and related manifolds are treated.
Although the presentation relies on the recent theory of function spaces, basic notation and classical results are repeated in order to make the text self-contained.
The book is addressed to two types of readers: researchers in the theory of function spaces who are interested in wavelets as new effective building blocks for functions, and scientists who wish to use wavelet bases in classical function spaces for various applications. Adapted to the second type of readers, the preface contains a guide to where one finds basic definitions and key assertions.