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The gratifying response to Counterexamples in analysis (CEA) was followed, when the book went out of print, by expressions of dismay from those who were unable to acquire it. The connection of the present volume with CEA is clear, although the sights here are set higher. In the quarter-century since the appearance of CEA, mathematical education has taken some large steps reflected in both the undergraduate and graduate curricula. What was once taken as very new, remote, or arcane is now a well-established part of mathematical study and discourse. Consequently the approach here is designed to match the observed progress. The contents are intended to provide graduate and ad vanced undergraduate students as well as the general mathematical public with a modern treatment of some theorems and examples that constitute a rounding out and elaboration of the standard parts of algebra, analysis, geometry, logic, probability, set theory, and topology. The items included are presented in the spirit of a conversation among mathematicians who know the language but are interested in some of the ramifications of the subjects with which they routinely deal. Although such an approach might be construed as demanding, there is an extensive GLOSSARY jlNDEX where all but the most familiar notions are clearly defined and explained. The object ofthe body of the text is more to enhance what the reader already knows than to review definitions and notations that have become part of every mathematician's working context.
This book consists of short descriptions of 106 mathematical theorems, which belong to the great achievements of 21st century mathematics but require relatively little mathematical background to understand their formulation and appreciate their importance. The selected theorems of this volume, chosen from the famous Annals of Mathematics journal, cover a broad range of topics from across mathematics. Each theorem description is essentially self-contained, can be read independently of the others, and requires as little preliminary knowledge as possible. Although the sections often start with an informal discussion and toy examples, all the necessary definitions are included and each description culminates in the precise formulation of the corresponding theorem. Filling the gap between surveys written for mathematicians and popular mathematics, this book is intended for readers with a keen interest in contemporary mathematics.
Even the simplest mathematical abstraction of the phenomena of reality the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model. Analysis regards the line, and the functions on it, in the unity of the whole system of their algebraic and topological properties, with the fundamental deductions about them obtained by using the interplay between the algebraic and topological structures. The same picture is observed at higher stages of abstraction. Algebra studies linear spaces, groups, rings, modules, and so on. Topology studies structures of a different kind on arbitrary sets, structures that give mathe matical meaning to the concepts of a limit, continuity, a neighborhood, and so on. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Thus, the basic object of study in functional analysis consists of objects equipped with compatible algebraic and topological structures.
This text examines fundamental and general existence theorems, along with uniqueness theorems and Picard iterants, and applies them to properties of solutions and linear differential equations. 1954 edition.
This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.
Mathematics is indeed fun as this little book testifies. This book presents a unique collection of mathematical ideas, theories, theorems, conjectures, rules, facts, equations, formulas, paradoxes, fallacies and puzzles with short, simple and witty explanations that require no background in mathematics.
Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. Now William Dunham gives them the attention they deserve. Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics. A rare combination of the historical, biographical, and mathematical, Journey Through Genius is a fascinating introduction to a neglected field of human creativity. “It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash.” —Isaac Asimov
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.
Many boundary value problems are equivalent to Au=O (1) where A : X --+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional rand inf.
A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.