The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules.
The volume presents the invited lectures of a conference devoted to "Infinite Length Modules," held at Bielefeld in September 1998, which brought together experts from quite different schools in order to survey surprising relations between algebra, topology and geometry. Some additional reports have been included in order to establish a unified picture. The collection of articles, written by well-known experts from all parts of the world, is conceived as a sort of handbook which provides an easy access to the present state of knowledge and its aim is to stimulate further development.

This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrange's theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.

Tensor Analysis and Nonlinear Tensor Functions embraces the basic fields of tensor calculus: tensor algebra, tensor analysis, tensor description of curves and surfaces, tensor integral calculus, the basis of tensor calculus in Riemannian spaces and affinely connected spaces, - which are used in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc.

The book suggests a new approach to definition of a tensor in space R3, which allows us to show a geometric representation of a tensor and operations on tensors. Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time.

It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua.

The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science.

Whereas many partial solutions and sketches for the odd-numbered exercises appear in the book, the Student Solutions Manual, written by the author, has comprehensive solutions for all odd-numbered exercises and large number of even-numbered exercises. This Manual also offers many alternative solutions to those appearing in the text. These will provide the student with a better understanding of the material. This is the only available student solutions manual prepared by the author of Contemporary Abstract Algebra, Tenth Edition and is designed to supplement that text. Table of Contents Integers and Equivalence Relations 0. Preliminaries Groups1. Introduction to Groups 2. Groups 3. Finite Groups; Subgroups 4. Cyclic Groups 5. Permutation Groups 6. Isomorphisms 7. Cosets and Lagrange's Theorem 8. External Direct Products 9. Normal Subgroups and Factor Groups 10. Group Homomorphisms 11. Fundamental Theorem of Finite Abelian Groups Rings12. Introduction to Rings 13. Integral Domains 14. Ideals and Factor Rings 15. Ring Homomorphisms 16. Polynomial Rings 17. Factorization of Polynomials 18. Divisibility in Integral Domains Fields Fields19. Extension Fields 20. Algebraic Extensions 21. Finite Fields 22. Geometric Constructions Special Topics23. Sylow Theorems 24. Finite Simple Groups 25. Generators and Relations 26. Symmetry Groups 27. Symmetry and Counting 28. Cayley Digraphs of Groups 29. Introduction to Algebraic Coding Theory 30. An Introduction to Galois Theory 31. Cyclotomic Extensions Biography Joseph A. Gallian earned his PhD from Notre Dame. In addition to receiving numerous national awards for his teaching and exposition, he has served terms as the Second Vice President, and the President of the MAA. He has served on 40 national committees, chairing ten of them. He has published over 100 articles and authored six books. Numerous articles about his work have appeared in the national news outlets, including the New York Times, the Washington Post, the Boston Globe, and Newsweek, among many others.

Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.

This book is dedicated to V.A. Yankov's seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov's results and their applications in algebraic logic, the theory of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov's revolutionary approach to constructive proof theory. The editors also include Yankov's contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics.

Originating from graduate topics courses given by the first author, this book functions as a unique text-monograph hybrid that bridges a traditional graduate course to research level representation theory. The exposition includes an introduction to the subject, some highlights of the theory and recent results in the field, and is therefore appropriate for advanced graduate students entering the field as well as research mathematicians wishing to expand their knowledge. The mathematical background required varies from chapter to chapter, but a standard course on Lie algebras and their representations, along with some knowledge of homological algebra, is necessary. Basic algebraic geometry and sheaf cohomology are needed for Chapter 10. Exercises of various levels of difficulty are interlaced throughout the text to add depth to topical comprehension. The unifying theme of this book is the structure and representation theory of infinite-dimensional locally reductive Lie algebras and superalgebras. Chapters 1-6 are foundational; each of the last 4 chapters presents a self-contained study of a specialized topic within the larger field. Lie superalgebras and flag supermanifolds are discussed in Chapters 3, 7, and 10, and may be skipped by the reader.

This book, the fourth book in the four-volume series in algebra, discusses Lie algebra and representation theory in detail. It covers topics such as semisimple Lie algebras, root systems, representation theory of Lie algebra, Chevalley groups and representation theory of Chevalley groups. Numerous motivating illustrations have been presented along with exercises, enabling readers to acquire a good understanding of topics which they can then use to find the exact or most realistic solutions to their problems.

This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Groebner Cover, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the Automatic Deduction of Geometric Theorems, Loci Computation and Envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Groebner Systems and Bases. It begins with Weispfenning's introduction of Comprehensive Groebner Systems (CGS) in 1992, and provides a complete description of the Groebner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed by Michael Wibmer and the author. In turn, the application part selects three problems for which the Groebner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hypersurfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described here have also been included in the software library "grobcov.lib" implemented in Singular by the author, and serve as a User Manual for it.

Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. This text traces the development of this theory from Quillen's fundamental calculation. It presents the stability theorems and low-dimensional results of A. Suslin, W. van der Kallen and others are presented. Coverage also examines the Friedlander-Milnor-conjecture concerning the homology of algebraic groups made discrete.

To our wives, Masha and Marian Interest in the so-called completely integrable systems with infinite num ber of degrees of freedom was aroused immediately after publication of the famous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky [75, 77, 96, 18, 66, 19J (see also [76]) on striking properties of the Korteweg-de Vries (KdV) equation. It soon became clear that systems of such a kind possess a number of characteristic properties, such as infinite series of symmetries and/or conservation laws, inverse scattering problem formulation, L - A pair representation, existence of prolongation structures, etc. And though no satisfactory definition of complete integrability was yet invented, a need of testing a particular system for these properties appeared. Probably one of the most efficient tests of this kind was first proposed by Lenard [19]' who constructed a recursion operator for symmetries of the KdV equation. It was a strange operator, in a sense: being formally integro-differential, its action on the first classical symmetry (x-translation) was well-defined and produced the entire series of higher KdV equations; but applied to the scaling symmetry, it gave expressions containing terms of the type J u dx which had no adequate interpretation in the framework of the existing theories. It is not surprising that P. Olver wrote "The de duction of the form of the recursion operator (if it exists) requires a certain amount of inspired guesswork. . . " [80, p.

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not' grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory arid the struc ture of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems," "chaos, synergetics and large-5cale order," which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathematics and Its Applications, is devoted to such (new) interrelations as exampla gratia: - a central concept which plays an important role in several different mathe matical and/or scientific specialized areas; - new applications of the results and ideas from one area of scientific en deavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another."

Thomas Harriot's "Artis analyticae praxis" is an essential work in the history of algebra. To some extent it is a development work of Viete, who was among the first to use literal symbols to stand for known and unknown quantities. But it was Harriot who took the crucial step of creating an entirely symbolic algebra, so that reasoning could be reduced to a quasi-mechanical manipulation of symbols. Although his algebra was still limited in scope (he insisted. for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern. Although Harriot's book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriot's achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond his comprehension, such as negative and complex roots of equations. The commentary included with the translation attempts to restore the Praxis to the state of Harriot's draft. Basing their work on manuscripts in the British Library, Pentworth House, and Lambeth Palace, the commentary contains some of Harriot's most novel and advanced mathematics, very little of which has been published in the past. It will provide the basis for a reassessment of the development of algebra.

The present work is the first ever English translation of the original text of Thomas Harriota (TM)s Artis Analyticae Praxis, first published in 1631 in Latin. Thomas Harriota (TM)s Praxis is an essential work in the history of algebra. Even though Harriota (TM)s contemporary, Viete, was among the first to use literal symbols to stand for known and unknown quantities, it was Harriott who took the crucial step of creating an entirely symbolic algebra. This allowed reasoning to be reduced to a quasi-mechanical manipulation of symbols. Although Harriota (TM)s algebra was still limited in scope (he insisted, for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern.

While Harriota (TM)s book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriota (TM)s achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond comprehension, such as negative and complex roots of equations.

The commentary included with this translation relates the contents of the Praxis to the corresponding pages in his manuscript papers, which enables much of Harriot's most novel and advanced mathematics to be explored. This publication will become an important contribution to the history of mathematics, and it will provide the basis for a reassessment of the development of algebra.

This book presents results about certain summability methods, such as the Abel method, the Norlund method, the Weighted mean method, the Euler method and the Natarajan method, which have not appeared in many standard books. It proves a few results on the Cauchy multiplication of certain summable series and some product theorems. It also proves a number of Steinhaus type theorems. In addition, it introduces a new definition of convergence of a double sequence and double series and proves the Silverman-Toeplitz theorem for four-dimensional infinite matrices, as well as Schur's and Steinhaus theorems for four-dimensional infinite matrices. The Norlund method, the Weighted mean method and the Natarajan method for double sequences are also discussed in the context of the new definition. Divided into six chapters, the book supplements the material already discussed in G.H.Hardy's Divergent Series. It appeals to young researchers and experienced mathematicians who wish to explore new areas in Summability Theory..

This book is a translation from Russian of Part I of the book Mathematics Through Problems: From Olympiads and Math Circles to Profession. The other two parts, Geometry and Combinatorics, will be published soon. The main goal of this book is to develop important parts of mathematics through problems. The author tries to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover and recreate much of elementary mathematics and start edging into the sophisticated world of topics such as group theory, Galois theory, and so on, thus building a bridge (by showing that there is no gap) between standard high school exercises and more intricate and abstract concepts in mathematics. Definitions and/or references for material that is not standard in the school curriculum are included. However, many topics in the book are difficult when you start learning them from scratch. To help with this, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions. The book is based on classes taught by the author at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.

This innovative monograph explores a new mathematical formalism in higher-order temporal logic for proving properties about the behavior of systems. Developed by the authors, the goal of this novel approach is to explain what occurs when multiple, distinct system components interact by using a category-theoretic description of behavior types based on sheaves. The authors demonstrate how to analyze the behaviors of elements in continuous and discrete dynamical systems so that each can be translated and compared to one another. Their temporal logic is also flexible enough that it can serve as a framework for other logics that work with similar models. The book begins with a discussion of behavior types, interval domains, and translation invariance, which serves as the groundwork for temporal type theory. From there, the authors lay out the logical preliminaries they need for their temporal modalities and explain the soundness of those logical semantics. These results are then applied to hybrid dynamical systems, differential equations, and labeled transition systems. A case study involving aircraft separation within the National Airspace System is provided to illustrate temporal type theory in action. Researchers in computer science, logic, and mathematics interested in topos-theoretic and category-theory-friendly approaches to system behavior will find this monograph to be an important resource. It can also serve as a supplemental text for a specialized graduate topics course.

Towards Inclusive Societies: Psychological and Sociological Perspectives focuses on the importance of building inclusive societies and communities for global human welfare within psychological, social, political, and cultural realms. It discusses the engagement of psychology and other social science disciplines on the need for building both cultural sensitivity and interdisciplinary dialogue. The volume presents the issues and consequences of globalization and diversity in the social and psychological domains and their role in shaping the physical and mental health of people. It systematically examines the various parameters of inclusivity such as equality, equity, social identity, social stigma, and coexistence of differences in socio-cultural behaviour. The volume focuses on the developments towards building inclusive societies in the South Asian countries including, India, Bangladesh, and Nepal. It also highlights the challenges and possibilities in making social-psychological discourses more inclusive. This book will be of interest to students, teachers, and scholars of psychology, cultural psychology, gender psychology, social psychology, sociology, and political science and social work. It will also be useful for psychologists, sociologists, social scientists, social workers, political scientists, and Gandhian philosophers.

Evolution and the Human-Animal Drive to Conflict examines how fundamental, universal animal drives, such as dominance/prevalence, survival, kinship, and "profit" (greed, advantage, whether of material or social nature), provide the basis for the evolutionary trap that promotes the unstable, conflictive, dominant-prone individual and group human behaviours. Examining this behavioural tension, this book argues that while these innate features set up behaviours that lean towards aggression influenced by social inequalities, the means implemented to defuse them resort to emotional and intellectual strategies that sponsor fanaticism and often reproduce the very same behaviours they intend to defuse. In addressing these concerns, the book argues that we should enhance our resources to promote solidarity, accept cultural differences, deter expansionist and uncontrolled profit drives, and achieve collective access towards knowledge and progress in living conditions. This entails promoting the redistribution of resources and creative labour access and avoiding policies that generate a fragmented world with collective and individual development disparities that invite and encourage dominance behaviours. This resource redistribution asserts that it is necessary to reformulate the global set of human priorities towards increased access to better living conditions, cognitive enhancement, a more amiable interaction with the ecosystem and non-aggressive cultural differences, promote universal access to knowledge, and enhance creativity and cultural convivence. These behavioural changes entail partial derangement of our ancestral animal drives camouflaged under different cultural profiles until the species succeeds in replacing the dominance of basic animal drives with prosocial, collective ones. Though it entails a formidable task of confronting financial, military, and religious powers and cultural inertias – human history is also a challenging, continuous experience in these domains – for the sake of our own self-identity and self-evaluation, we should reject any suggestion of not continuing embracing slowly constructing collective utopias channelled towards improving individual and collective freedom and creativeness. This book will interest academics and students in social, cognitive, and evolutionary psychology, the neurosciences, palaeoanthropology, philosophy, and anthropology.

This is the Proceedings of the ICM 2010 Satellite Conference on "Buildings, Finite Geometries and Groups" organized at the Indian Statistical Institute, Bangalore, during August 29 - 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups.

The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

This volume has its origins in the Barcelona Conference in Group Theory (July 2005) and the conference "Asymptotic and Probabilistic Methods in Geometric Group Theory" held in Geneva (June 2005). Twelve peer-reviewed research articles written by experts in the field present the most recent results in abstract and geometric group theory. In particular there are two articles by A. Juhasz.

This book gives the complete classification of Moufang polygons, starting from first principles. In particular, it may serve as an introduction to the various important algebraic concepts which arise in this classification including alternative division rings, quadratic Jordan division algebras of degree three, pseudo-quadratic forms, BN-pairs and norm splittings of quadratic forms. This book also contains a new proof of the classification of irreducible spherical buildings of rank at least three based on the observation that all the irreducible rank two residues of such a building are Moufang polygons. In an appendix, the connection between spherical buildings and algebraic groups is recalled and used to describe an alternative existence proof for certain Moufang polygons.

This book is an outgrowth of the Workshop on "Regulators in Analysis, Geom etry and Number Theory" held at the Edmund Landau Center for Research in Mathematical Analysis of The Hebrew University of Jerusalem in 1996. During the preparation and the holding of the workshop we were greatly helped by the director of the Landau Center: Lior Tsafriri during the time of the planning of the conference, and Hershel Farkas during the meeting itself. Organizing and running this workshop was a true pleasure, thanks to the expert technical help provided by the Landau Center in general, and by its secretary Simcha Kojman in particular. We would like to express our hearty thanks to all of them. However, the articles assembled in the present volume do not represent the proceedings of this workshop; neither could all contributors to the book make it to the meeting, nor do the contributions herein necessarily reflect talks given in Jerusalem. In the introduction, we outline our view of the theory to which this volume intends to contribute. The crucial objective of the present volume is to bring together concepts, methods, and results from analysis, differential as well as algebraic geometry, and number theory in order to work towards a deeper and more comprehensive understanding of regulators and secondary invariants. Our thanks go to all the participants of the workshop and authors of this volume. May the readers of this book enjoy and profit from the combination of mathematical ideas here documented.

Together with "Theory of Operator Algebras I, III" (EMS 124 and 127), this book, written by one of the most prominent researchers in the field of operator algebras, presents the theory of von Neumann algebras and non-commutative integration focusing on the group of automorphisms and the structure analysis.  It is part of the recently developed part of the "Encyclopaedia of Mathematical Sciences" on operator algebras and non-commutative geometry (see http://www.springer.de/math/ems/index.html). The book provides essential and comprehensive information for graduate students and researchers in mathematics and mathematical physics.