About 60 years ago, R. Brauer introduced "block theory"; his purpose was to study the group algebra kG of a finite group G over a field k of nonzero characteristic p: any indecomposable two-sided ideal that also is a direct summand of kG determines a G-block.
But the main discovery of Brauer is perhaps the existence of families of infinitely many nonisomorphic groups having a "common block"; i.e., blocks having mutually isomorphic "source algebras."
In this book, based on a course given by the author at Wuhan University in 1999, all the concepts mentioned are introduced, and all the proofs are developed completely. Its main purpose is the proof of the existence and the uniqueness of the "hyperfocal subalgebra" in the source algebra. This result seems fundamental in block theory; for instance, the structure of the source algebra of a nilpotent block, an important fact in block theory, can be obtained as a corollary.

The exceptional layout of this bilingual edition featuring 2 columns per page (one English, one Chinese) sharing the displayed mathematical formulas is the joint achievement of the author and A. Arabia.

Helmut Koch's classic is now available in English. Competently translated by Franz Lemmermeyer, it introduces the theory of pro-p groups and their cohomology. The book contains a postscript on the recent development of the field written by H. Koch and F. Lemmermeyer, along with many additional recent references.

The last decade witnessed an increasing interest of mathematicians in prob lems originated in mathematical physics. As a result of this effort, the scope of traditional mathematical physics changed considerably. New problems es pecially those connected with quantum physics make use of new ideas and methods. Together with classical and functional analysis, methods from dif ferential geometry and Lie algebras, the theory of group representation, and even topology and algebraic geometry became efficient tools of mathematical physics. On the other hand, the problems tackled in mathematical physics helped to formulate new, purely mathematical, theorems. This important development must obviously influence the contemporary mathematical literature, especially the review articles and monographs. A considerable number of books and articles appeared, reflecting to some extend this trend. In our view, however, an adequate language and appropriate methodology has not been developed yet. Nowadays, the current literature includes either mathematical monographs occasionally using physical terms, or books on theoretical physics focused on the mathematical apparatus. We hold the opinion that the traditional mathematical language of lem mas and theorems is not appropriate for the contemporary writing on mathe matical physics. In such literature, in contrast to the standard approaches of theoretical physics, the mathematical ideology must be utmost emphasized and the reference to physical ideas must be supported by appropriate mathe matical statements. Of special importance are the results and methods that have been developed in this way for the first time."

Lattice-valued Logic aims at establishing the logical foundation for uncertain information processing routinely performed by humans and artificial intelligence systems. In this textbook for the first time a general introduction on lattice-valued logic is given. It systematically summarizes research from the basic notions up to recent results on lattice implication algebras, lattice-valued logic systems based on lattice implication algebras, as well as the corresponding reasoning theories and methods. The book provides the suitable theoretical logical background of lattice-valued logic systems and supports newly designed intelligent uncertain-information-processing systems and a wide spectrum of intelligent learning tasks.

Every Abelian group can be related to an associative ring with an identity element, the ring of all its endomorphisms. Recently the theory of endomor phism rings of Abelian groups has become a rapidly developing area of algebra. On the one hand, it can be considered as a part of the theory of Abelian groups; on the other hand, the theory can be considered as a branch of the theory of endomorphism rings of modules and the representation theory of rings. There are several reasons for studying endomorphism rings of Abelian groups: first, it makes it possible to acquire additional information about Abelian groups themselves, to introduce new concepts and methods, and to find new interesting classes of groups; second, it stimulates further develop ment of the theory of modules and their endomorphism rings. The theory of endomorphism rings can also be useful for studies of the structure of additive groups of rings, E-modules, and homological properties of Abelian groups. The books of Baer [52] and Kaplansky [245] have played an important role in the early development of the theory of endomorphism rings of Abelian groups and modules. Endomorphism rings of Abelian groups are much stu died in monographs of Fuchs [170], [172], and [173]. Endomorphism rings are also studied in the works of Kurosh [287], Arnold [31], and Benabdallah [63].

Some Historical Background This book deals with the cohomology of groups, particularly finite ones. Historically, the subject has been one of significant interaction between algebra and topology and has directly led to the creation of such important areas of mathematics as homo logical algebra and algebraic K-theory. It arose primarily in the 1920's and 1930's independently in number theory and topology. In topology the main focus was on the work ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among others) made significant contributions. The main thrust of the early work here was to try to understand the meanings of the low dimensional homology groups of a space X. For example, if the universal cover of X was three connected, it was known that H2(X; A. ) depends only on the fundamental group of X. Group cohomology initially appeared to explain this dependence. In number theory, group cohomology arose as a natural device for describing the main theorems of class field theory and, in particular, for describing and analyzing the Brauer group of a field. It also arose naturally in the study of group extensions, N"

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. It includes differentiable manifolds, tensors and differentiable forms. Lie groups and homogenous spaces, integration on manifolds, and in addition provides a proof of the de Rham theorem via sheaf cohomology theory, and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem. Those interested in any of the diverse areas of mathematics requiring the notion of a differentiable manifold will find this beginning graduate-level text extremely useful.

Symmetry is a key ingredient in many mathematical, physical, and biological theories. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of Lie groups and Lie algebras, Symmetry, Representations, and Invariants is a significant reworking of an earlier highly-acclaimed work by the authors. The result is a comprehensive introduction to Lie theory, representation theory, invariant theory, and algebraic groups, in a new presentation that is more accessible to students and includes a broader range of applications.

The philosophy of the earlier book is retained, i.e., presenting the principal theorems of representation theory for the classical matrix groups as motivation for the general theory of reductive groups. The wealth of examples and discussion prepares the reader for the complete arguments now given in the general case.

Key Features of Symmetry, Representations, and Invariants (1) Early chapters suitable for honors undergraduate or beginning graduate courses, requiring only linear algebra, basic abstract algebra, and advanced calculus; (2) Applications to geometry (curvature tensors), topology (Jones polynomial via symmetry), and combinatorics (symmetric group and Young tableaux); (3) Self-contained chapters, appendices, comprehensive bibliography; (4) More than 350 exercises (most with detailed hints for solutions) further explore main concepts; (5) Serves as an excellent main text for a one-year course in Lie group theory; (6) Benefits physicists as well as mathematicians as a reference work.

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are "geo metrically indistinguishable," that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude."

From reviews of the German edition: "This is an exciting text and a refreshing contribution to an area in which challenges continue to flourish and to captivate the viewer. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions." Mathematical Reviews

Written in the spirit of Liboff's acclaimed text on Quantum Mechanics, this introduction to group theory offers an exceptionally clear presentation with a good sense of what to explain, which examples are most appropriate, and when to give a counter-example.

This book gathers concepts of information across diverse fields -physics, electrical engineering and computational science - surveying current theories, discussing underlying notions of symmetry, and showing how the capacity of a system to distinguish itself relates to information. The author develops a formal methodology using group theory, leading to the application of Burnside's Lemma to count distinguishable states. This provides a tool to quantify complexity and information capacity in any physical system.

This book treats Jacques Tit's beautiful theory of buildings, making that theory accessible to readers with minimal background. It covers all three approaches to buildings, so that the reader can choose to concentrate on one particular approach. Beginners can use parts of the new book as a friendly introduction to buildings, but the book also contains valuable material for the active researcher.

This book is suitable as a textbook, with many exercises, and it may also be used for self-study.

The new edition of this well received primer on rigorous aspects of symmetry breaking presents a more detailed and thorough discussion of the mechanism of symmetry breaking in classical field theory in relation with the Noether theorem. Moreover, the link between symmetry breaking without massless Goldstone bosons in Coulomb systems and in gauge theories is made more explicit. A subject index has been added and a number of misprints have been corrected.

The Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.

This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

The authors describe the important generalization of the original Weil conjectures, as given by P. Deligne in his fundamental paper "La conjecture de Weil II." The authors follow the important and beautiful methods of Laumon and Brylinski which lead to a simplification of Deligne's theory. Deligne's work is closely related to the sheaf theoretic theory of perverse sheaves. In this framework Deligne's results on global weights and his notion of purity of complexes obtain a satisfactory and final form. Therefore the authors include the complete theory of middle perverse sheaves. In this part, the l-adic Fourier transform is introduced as a technique providing natural and simple proofs. To round things off, there are three chapters with significant applications of these theories.

The aim of this book is to extend the understanding of the fundamental role of generalizations of Lie and related non-commutative and non-associative structures in Mathematics and Physics. This is a thematic volume devoted to the interplay between several rapidly exp- ding research ?elds in contemporary Mathematics and Physics, such as generali- tions of the main structures of Lie theory aimed at quantization and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, n- commutative geometry and applications in Physics and beyond. The speci?c ?elds covered by this volume include: * Applications of Lie, non-associative and non-commutative associative structures to generalizations of classical and quantum mechanics and non-linear integrable systems, operadic and group theoretical methods; * Generalizations and quasi-deformations of Lie algebras such as color and super Lie algebras, quasi-Lie algebras, Hom-Lie algebras, in?nite-dimensional Lie algebras of vector ?elds associated to Riemann surfaces, quasi-Lie algebras of Witt type and their central extensions and deformations important for in- grable systems, for conformal ? eld theory and for string theory; * Non-commutative deformation theory, moduli spaces and interplay with n- commutativegeometry,algebraicgeometryandcommutativealgebra,q-deformed differential calculi and extensions of homological methods and structures; * Crossed product algebras and actions of groups and semi-groups, graded rings and algebras, quantum algebras, twisted generalizations of coalgebras and Hopf algebra structures such as Hom-coalgebras, Hom-Hopf algebras, and super Hopf algebras and their applications to bosonisation, parastatistics, parabosonic and parafermionic algebras, orthoalgebas and root systems in quantum mechanics;

The mathematical theory of control became a ?eld of study half a century ago in attempts to clarify and organize some challenging practical problems and the methods used to solve them. It is known for the breadth of the mathematics it uses and its cross-disciplinary vigor. Its literature, which can befoundinSection93ofMathematicalReviews, wasatonetimedominatedby the theory of linear control systems, which mathematically are described by linear di?erential equations forced by additive control inputs. That theory led to well-regarded numerical and symbolic computational packages for control analysis and design. Nonlinear control problems are also important; in these either the - derlying dynamical system is nonlinear or the controls are applied in a n- additiveway.Thelastfourdecadeshaveseenthedevelopmentoftheoretical work on nonlinear control problems based on di?erential manifold theory, nonlinear analysis, and several other mathematical disciplines. Many of the problems that had been solved in linear control theory, plus others that are new and distinctly nonlinear, have been addressed; some resulting general de?nitions and theorems are adapted in this book to the bilinear case

Includes a rich variety of exercises to accompany the exposition of Coxeter groups

Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra. This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).

This book, designed for advanced graduate students and post-graduate researchers, introduces Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.

The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory.
The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment.
The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

This new Reader aims to guide students through some of the key readings on the subject of terrorism and political violence.

In an age when there is more written about terrorism than anyone can possibly read in a lifetime, it has become increasingly difficult for students and scholars to navigate the literature. At the same time, courses and modules on terrorism studies are developing at a rapid rate. To meet this challenge, this wide-ranging Reader seeks to equip the aspiring student, based anywhere in the world, with a comprehensive introduction to the study of terrorism. Containing many of the most influential and groundbreaking studies from the world s leading experts, drawn from several academic disciplines, this volume is the essential companion for any student of terrorism and political violence.

The Reader, which starts with a detailed Introduction by the editors, is divided into seven sections, each of which contains a short introduction as well as a guide to further reading and student discussion questions:

• Terrorism in Historical Context
• Definitions
• Understanding and Explaining Terrorism
• Terrorist Movements
• Terrorist Behaviour
• Counterterrorism
• Current and Future Trends in Terrorism.

This Reader will be essential reading for students of Terrorism and Political Violence, and highly recommended for students of Security Studies, War and Conflict Studies and Political Science in general, as well as for practitioners in the field of counter-terrorism and homeland security.

Contributors: David C. Rapoport, Isabelle Duyvesteyn, Jack Gibbs, Leonard Weinberg, Ami Pedahzur, Sivan Hirsch-Hoefler, Alex Schmid, Martha Crenshaw, Max Taylor, John Horgan, Magnus Ranstorp, C.J.M. Drake, Ehud Sprinzak, Jennifer S. Holmes, Sheila Amin Gutierrez de Pineres, Kevin M. Curtin, Xavier Raufer, Donatella della Porta, Robert Pape, Mia Bloom, Chris Dishman, Andrew Silke, Muhammad Hanif bin Hassan, Gary Ackerman, Bruce Hoffman, John Mueller, Mohammed Hafez, Karla J. Cunningham, Jonathan Tonge, Lorenzo Vidino and Michael Barkun.