Graduate Texts in Mathematics

"This book thoroughly examines the distribution of prime numbers in arithmetic progressions. It covers many classical results, including the Dirichlet theorem on the existence of prime numbers in arithmetic progressions, the theorem of Siegel, and functional equations of the L-functions and their consequences for the distribution of prime numbers. In addition, a simplified, improved version of the large sieve method is presented. The third edition includes a large number of revisions and corrections as well as a new section with references to more recent work in the field."--Jacket.

"This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; and applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations."--BOOK JACKET.

The purpose of this book is to use problems and their solutions to communicate to the reader many of the central techniques and ideas of Hilbert spaces. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks. Most of the problems are statements to be proved, but some are questions, and some are challenges. The second part, a very short one, consists of hints. The third part consists of solutions: proofs, answers, or constructions, depending on the nature of the problem. The solution may also introduce standard nomenclature, discuss the history of the subject, and mention pertinent references.--

This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique.

The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course.

The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. No background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasizing estimates of the efficiency of the techniques that arise from the theory. A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references.

This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included. New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable.