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This volume is easily accessible to young people and mathematicians unfamiliar with logic. It gives a terse historical picture of Model Theory and introduces the latest developments in the area. It further provides 'hands-on' proofs of elimination of quantifiers, elimination of imaginaries and other relevant matters. The book is for trainees and professional model theorists, and mathematicians working in Algebra and Geometry.
The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.
This is the first handbook focussing on classical social theory. It offers extensive discussions of debates, arguments, and discussions in classical theory and how they have informed contemporary sociological theory. The book pushes against the conventional classical theory pedagogy, which often focused on single theorists and their contributions, and looks at isolating themes capturing the essence of the interest of classical theorists that seem to have relevance to modern research questions and theoretical traditions. This book presents new approaches to thinking about theory in relationship to sociological methods.
This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.
Classical and Modern Social Theory is comprehensive introduction to the field, covering a wide historical range of thinkers, from the classical to the postmodernist, as well as key themes in social theory and a guide to the major debates. Designed for students with little or no background in social theory, this single volume covering both classic and contemporary theory introduces the basic concepts at the center of social theory in accessible language and provides readers with a useful reference source to the field.
The notion of complexity is an important contribution of logic to theoretical computer science and mathematics. This volume attempts to approach complexity in a holistic way, investigating mathematical properties of complexity hierarchies at the same time as discussing algorithms and computational properties. A main focus of the volume is on some of the new paradigms of computation, among them Quantum Computing and Infinitary Computation. The papers in the volume are tied together by an introductory article describing abstract properties of complexity hierarchies. This volume will be of great interest to both mathematical logicians and theoretical computer scientists, providing them with new insights into the various views of complexity and thus shedding new light on their own research.
The sheer complexity of computer systems has meant that automated reasoning, i.e. the ability of computers to perform logical inference, has become a vital component of program construction and of programming language design. This book meets the demand for a self-contained and broad-based account of the concepts, the machinery and the use of automated reasoning. The mathematical logic foundations are described in conjunction with practical application, all with the minimum of prerequisites. The approach is constructive, concrete and algorithmic: a key feature is that methods are described with reference to actual implementations (for which code is supplied) that readers can use, modify and experiment with. This book is ideally suited for those seeking a one-stop source for the general area of automated reasoning. It can be used as a reference, or as a place to learn the fundamentals, either in conjunction with advanced courses or for self study.
Theory is the bedrock of many sciences, providing a rigorous method to advance knowledge, through testing and falsifying hypotheses about observable phenomena. To begin with, the nascent field of HCI followed the scientific method borrowing theories from cognitive science to test theories about user performance at the interface. But HCI has emerged as an eclectic interdiscipline rather than a well-defined science. It now covers all aspects of human life, from birth to bereavement, through all manner of computing, from device ecologies to nano-technology. It comes as no surprise that the role of theory in HCI has also greatly expanded from the early days of scientific testing to include other functions such as describing, explaining, critiquing, and as the basis for generating new designs. The book charts the theoretical developments in HCI, both past and present, reflecting on how they have shaped the field. It explores both the rhetoric and the reality: how theories have been conceptualized, what was promised, how they have been used and which has made the most impact in the field -- and the reasons for this. Finally, it looks to the future and asks whether theory will continue to have a role, and, if so, what this might be. Table of Contents: Introduction / The Backdrop to HCI Theory / The Role and Contribution of Theory in HCI / Classical Theories / Modern Theories / Contemporary Theory / Discussion / Summary