This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses.An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer.Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work.The first three chapters develop the basic theory of elliptic curves.For this edition, the text has been completely revised and updated.
One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Violence Assessment and Intervention: The Practitioner’s Handbook, now in its third edition, provides a proven methodology, grounded in the current empirical research and the authors’ experience in successfully assessing and managing thousands of cases in a variety of contexts and environments, for analyzing concerning behaviors and potential threatening situations, and taking action in these challenging, dynamic environments before tragedy occurs. Threat and violence assessment and management is an essential process in reducing violence and its consequences. The ongoing challenge for those assessors, particularly in common workplace environments (e.g., educational settings, public agency settings, and business settings), is applying the applicable behavioral science research in a practical and effective manner to maximize safety. The book begins by demonstrating the threat and violence assessment process from the point of the initial call and proceeds through the steps that quantify the situation and determine the appropriate response. The next section covers information gathering, victimology, and formulas and tools for risk assessment. Finally, the book explores organizational influences, school violence, ethics, security and consultation issues; the formation and running of threat management teams, and relevant laws related to violence assessment. This book is a valuable reference for human resource professionals, security professionals, mental health practitioners, law enforcement personnel, and lawyers who are members of threat assessment teams, provide threat and violence assessment and management consultations, as well as expert witnesses in cases involving workplace violence, school violence, security negligence; or wrongful termination or disputed school disciplinary actions related to aggressive, threatening, or violent behavior.
Covers the whole civil aviation scene regarding aircraft registered in Britain, as well as those overseas airliners likely to be seen in British skies.
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