This book is intended as a textbook for a first-year Ph. D. course in mathematics for economists and as a reference for graduate students in economics. It provides a self-contained, rigorous treatment of most of the concepts and techniques required to follow the standard first-year theory sequence in micro and macroeconomics. The topics covered include an introduction to analysis in metric spaces, differential calculus, comparative statics, convexity, static optimization, dynamical systems and dynamic optimization. The book includes a large number of applications to standard economic models and over two hundred fully worked-out problems.

In the two volumes that comprise this work Roger Penrose and Wolfgang Rindler introduce the calculus of 2-spinors and the theory of twistors, and discuss in detail how these powerful and elegant methods may be used to elucidate the structure and properties of space-time. In volume 1, Two-spinor calculus and relativistic fields, the calculus of 2-spinors is introduced and developed. Volume 2, Spinor and twistor methods in space-time geometry, introduces the theory of twistors, and studies in detail how the theory of twistors and 2-spinors can be applied to the study of space-time. This work will be of great value to all those studying relativity, differential geometry, particle physics and quantum field theory from beginning graduate students to experts in these fields.

Discrete optimization problems are everywhere, from traditional operations research planning (scheduling, facility location and network design); to computer science databases; to advertising issues in viral marketing. Yet most such problems are NP-hard; unless P = NP, there are no efficient algorithms to find optimal solutions. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first section is devoted to a single algorithmic technique applied to several different problems, with more sophisticated treatment in the second section. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithm courses, it will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

Available for the first time in paperback, The Quantum Theory of Fields is a self-contained, comprehensive, and up-to-date introduction to quantum field theory from Nobel Laureate Steven Weinberg. Volume II gives an up-to-date and self-contained account of the methods of quantum field theory, and how they have led to an understanding of the weak, strong, and electromagnetic interactions of the elementary particles. The presentation of modern mathematical methods is throughout interwoven with accounts of the problems of elementary particle physics and condensed matter physics to which they have been applied. Many topics are included that are not usually found in books on quantum field theory. The book contains much original material, and is peppered with examples and insights from the author's experience as a leader of elementary particle physics. Exercises are included at the end of each chapter.

The rapid endangerment and death of many minority languages across the world is a matter of widespread concern, not only among linguists and anthropologists but among all concerned with issues of cultural identity in an increasingly globalized culture. By some counts, only 600 of the 6,000 or so languages in the world are 'safe' from the threat of extinction. A leading commentator and popular writer on language issues, David Crystal asks the fundamental question, 'Why is language death so important?', reviews the reasons for the current crisis, and investigates what is being done to reduce its impact. This 2002 book contains not only intelligent argument, but moving descriptions of the decline and demise of particular languages, and practical advice for anyone interested in pursuing the subject further.

Ronald Dworkin occupies a distinctive place in both public life and philosophy. In public life, he is a regular contributor to The New York Review of Books and other widely read journals. In philosophy, he has written important and influential works on many of the most prominent issues in legal and political philosophy. In both cases, his interventions have in part shaped the debates he joined. His opposition to Robert Bork's nomination for the United States Supreme Court gave new centrality to debates about the public role of judges and the role of original intent in constitutional interpretation. His writings in legal philosophy have reoriented the modern debate about legal positivism and natural law. In political philosophy, he has shaped the ways in which people debate the nature of equality and has reframed debates about the sanctity of life.

The aim of this graduate-level text is to equip the reader with the basic tools and techniques needed for research in various areas of geometric analysis. Throughout, the main theme is to present the interaction of partial differential equations and differential geometry. More specifically, emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the underlying manifold and vice versa. For efficiency the author mainly restricts himself to the linear theory and only a rudimentary background in Riemannian geometry and partial differential equations is assumed. Originating from the author's own lectures, this book is an ideal introduction for graduate students, as well as a useful reference for experts in the field.

Advanced Grammar in Use Second edition is a fully updated version of the highly successful grammar title. This new revised edition focuses specifically on the complexities of grammatical choices that advanced students need to appreciate. It also includes a comprehensive Basic Grammar Reference so students can easily review their understanding of language areas they have previously studied. This edition, with answers and CD-ROM, is ideal for self-study. The CD-ROM includes hundreds of interactive exercises to reinforce the language learned in the book.

For many years scholars have sought to explain why the European states which emerged in the period before the French Revolution developed along such different lines. Why did some become absolutist and others constitutionalist? What enabled some to develop bureaucratic administrative systems, while others remained dependent upon patrimonial practices? This book presents a new theory of state-building in medieval and early modern Europe. Ertman argues that two factors - the organisation of local government at the time of state formation and the timing of sustained geo-military competition - can explain most of the variation in political regimes and in state infrastructures found across the continent during the second half of the eighteenth century. Drawing on insights developed in historical sociology, comparative politics, and economic history, this book makes a compelling case for the value of interdisciplinary approaches to the study of political development.

This classic account shows how the fall of Constantinople in May 1453, after a siege of several weeks, came as a bitter shock to Western Christendom. The city’s plight had been neglected, and negligible help was sent in this crisis. To the Turks, victory not only brought a new imperial capital, but guaranteed that their empire would last. To the Greeks, the conquest meant the end of the civilisation of Byzantium, and led to the exodus of scholars stimulating the tremendous expansion of Greek studies in the European Renaissance.

Learn Latin from the Romans is the only introductory Latin textbook to feature texts written by ancient Romans for Latin learners. These texts, the 'colloquia', consist of dialogues and narratives about daily life similar to those found in modern-language textbooks today, introducing learners to Roman culture as well as to Latin in an engaging, accessible, and enjoyable way. Students and instructors will find everything they need in one complete volume, including clear explanations of grammatical concepts and how Latin works, both British and American orders for all noun and adjective paradigms, 5,000 easy practice sentences, and over 150 longer passages (from the colloquia and a diverse range of other sources including inscriptions, graffiti, and Christian texts as well as Catullus, Cicero, and Virgil). Written by a leading Latin linguist with decades of language teaching experience, this textbook is suitable for introductory Latin courses worldwide.
Passages are taken from leading literary writers like Cicero, Catullus and Virgil, and from inscriptions and graffiti, as well as the 'colloquia'
Grammatical terminology is introduced from the start, allowing students to gain confidence immediately
All noun and adjective paradigms are given in both the British order and the American order
Includes over 5,000 short practice sentences and over 150 longer passages for continuous reading practice

David Crystal's classic English as a Global Language considers the history, present status and future of the English language, focusing on its role as the leading international language. English has been deemed the most 'successful' language ever, with 1500 million speakers internationally, presenting a difficult task to those who wish to investigate it in its entirety. However, Crystal explores the subject in a measured but engaging way, always backing up observations with facts and figures. Written in a detailed and fascinating manner, this is a book written by an expert both for specialists in the subject and for general readers interested in the English language.

This is the first comprehensive introduction to Support Vector Machines (SVMs), a new generation learning system based on recent advances in statistical learning theory. SVMs deliver state-of-the-art performance in real-world applications such as text categorisation, hand-written character recognition, image classification, biosequences analysis, etc., and are now established as one of the standard tools for machine learning and data mining. Students will find the book both stimulating and accessible, while practitioners will be guided smoothly through the material required for a good grasp of the theory and its applications. The concepts are introduced gradually in accessible and self-contained stages, while the presentation is rigorous and thorough. Pointers to relevant literature and web sites containing software ensure that it forms an ideal starting point for further study. Equally, the book and its associated web site will guide practitioners to updated literature, new applications, and on-line software.

String theory is one of the most exciting and challenging areas of modern theoretical physics. This book guides the reader from the basics of string theory to recent developments. It introduces the basics of perturbative string theory, world-sheet supersymmetry, space-time supersymmetry, conformal field theory and the heterotic string, before describing modern developments, including D-branes, string dualities and M-theory. It then covers string geometry and flux compactifications, applications to cosmology and particle physics, black holes in string theory and M-theory, and the microscopic origin of black-hole entropy. It concludes with Matrix theory, the AdS/CFT duality and its generalizations. This book is ideal for graduate students and researchers in modern string theory, and will make an excellent textbook for a one-year course on string theory. It contains over 120 exercises with solutions, and over 200 homework problems with solutions available on a password protected website for lecturers at www.cambridge.org/9780521860697.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.