This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper undergraduate or beginning graduate level.It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates with a treatment of Dedekind rings, including rings of algebraic integers.In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear, and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate the theory while others are general results that develop the theory further.

Descriptor linear systems theory is an important part in the general field of control systems theory, and has attracted much attention in the last two decades. In spite of the fact that descriptor linear systems theory has been a topic very rich in content, there have been only a few books on this topic. This book provides a systematic introduction to the theory of continuous-time descriptor linear systems and aims to provide a relatively systematic introduction to the basic results in descriptor linear systems theory. The clear representation of materials and a large number of examples make this book easy to understand by a large audience. General readers will find in this book a comprehensive introduction to the theory of descriptive linear systems. Researchers will find a comprehensive description of the most recent results in this theory and students will find a good introduction to some important problems in linear systems theory.

This monograph provides a modern introduction to the theory of quantales. First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research. This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic.

Based on extensive analysis of real-time, authentic crisis encounters collected in the UK and US, Crisis Talk: Negotiating with Individuals in Crisis sheds light on the relatively hidden world of communication between people in crisis and the professionals whose job it is to help them. The crisis situations explored in this book involve police hostage and crisis negotiators and emergency dispatchers interacting with individuals in crisis who threaten suicide or self-harm. The practitioners face various communicative challenges in these encounters, including managing strong emotions, resistance, hostility, and unresponsiveness. Using conversation analysis, Crisis Talk presents evidence on how practitioners deal with the interactional challenge of negotiating with people in crisis and how what they say shapes outcomes. Each chapter includes recommendations based on the detailed analysis of numerous cases of actual negotiation. Crisis Talk shows readers how every turn taken by negotiators can exacerbate or solve the communicative challenges created by crisis situations, making it a unique and invaluable text for academics in psychology, sociology, linguistic sciences, and related fields, as well as for practitioners engaging in crisis negotiation training or fieldwork.

This book is mainly intended for first-year University students who undertake a basic abstract algebra course, as well as instructors. It contains the basic notions of abstract algebra through solved exercises as well as a 'True or False' section in each chapter. Each chapter also contains an essential background section, which makes the book easier to use.

This edited volume brings together the latest research in understanding the nature, origins, and evolution of human sociability, one of the most intriguing aspects of human psychology. Sociability-our sophisticated ability to interact with others, imagine, plan, and execute interdependent behaviours-lies at the heart of our evolutionary success, and is the most important prerequisite for the development of increasingly elaborate civilizations. With contributions from internationally renowned researchers in areas of social psychology as well as anthropology and evolutionary psychology, this book demonstrates the role of social psychology in explaining how human sociability evolved, how it shapes our mental and emotional lives, and how it influences both large-scale civilizational practices and intimate interpersonal relations. Chapters cover the core psychological characteristics that shape human sociability, including such phenomena as the role of information exchange, affective processes, social norms, power relations, personal relationships, attachment patterns, personality characteristics, and evolutionary pressures. Featuring a wide variety of empirical and theoretical backgrounds, the book will be of interest to students and researchers in all areas of the social sciences, as well as practitioners and applied professionals who deal with issues related to sociability in their daily lives.

This edited volume brings together the latest research in understanding the nature, origins, and evolution of human sociability, one of the most intriguing aspects of human psychology. Sociability-our sophisticated ability to interact with others, imagine, plan, and execute interdependent behaviours-lies at the heart of our evolutionary success, and is the most important prerequisite for the development of increasingly elaborate civilizations. With contributions from internationally renowned researchers in areas of social psychology as well as anthropology and evolutionary psychology, this book demonstrates the role of social psychology in explaining how human sociability evolved, how it shapes our mental and emotional lives, and how it influences both large-scale civilizational practices and intimate interpersonal relations. Chapters cover the core psychological characteristics that shape human sociability, including such phenomena as the role of information exchange, affective processes, social norms, power relations, personal relationships, attachment patterns, personality characteristics, and evolutionary pressures. Featuring a wide variety of empirical and theoretical backgrounds, the book will be of interest to students and researchers in all areas of the social sciences, as well as practitioners and applied professionals who deal with issues related to sociability in their daily lives.

This book explores welfare politics, unemployment, and interventions in relation to the labour market from a critical psychological perspective. Using critical fieldwork and theory, the author explores the administration of the unemployed, and the drive to increase labour market participation through strategies of activation. There is a strong and coherent conceptual and theoretical framing for this work, with a critical perspective (essentially, question everything) taking centre stage. It will give an overall coherence in addressing the topic. The theoretical framing is cogent and, in combination with the critical perspective, works well for integrating the material and delivering a fresh approach to this topic. Psychology, Punitive Activation and Welfare will appeal to students engaging with critical psychology, unemployment or policy, by providing a distinct application of theoretical and methodological tools to think differently about the relationship between labour market non/participation, human misery, psychology, and frontline enactment of policy and research.

Psychology of Prejudice and Discrimination provides a comprehensive and compelling overview of what psychological theory and research have to say about the nature, causes, and reduction of prejudice and discrimination. It balances a detailed discussion of theories and selected research with applied examples that ensure the material is relevant to students. This edition has been thoroughly revised and updated and addresses several interlocking themes. It first looks at the nature of prejudice and discrimination, followed by a discussion of research methods. Next come the psychological underpinnings of prejudice: the nature of stereotypes, the conditions under which stereotypes influence responses to other people, contemporary theories of prejudice, and how individuals' values and belief systems are related to prejudice. Explored next are the development of prejudice in children and the social context of prejudice. The theme of discrimination is developed via discussions of the nature of discrimination, the experience of discrimination, and specific forms of discrimination, including gender, gender identity, sexual orientation, age, ability, and appearance. The concluding theme is the reduction of prejudice. The book is accompanied by a comprehensive website featuring an Instructor Manual that contains activities and tools to help with teaching a prejudice and discrimination course; PowerPoint slides for every chapter; and a Test Bank with short answer and multiple-choice exam questions for every chapter. This book is an essential companion for all students of prejudice and discrimination, including those in psychology, education, social work, business, communication studies, ethnic studies, and other disciplines. In addition to courses on prejudice and discrimination, this book will also appeal to those studying racism and diversity.

This book offers a timely exploration of our patterns of engagement with politics, news, and information in current high-choice information environments It analyzes the issue plaguing our society today - The spread of misinformation and its impact on the public sphere, our politics and our everyday lives The book offers insights into the processes that influence the supply of misinformation and factors influencing how and why people expose themselves to and process information that may support or contradict their beliefs and attitudes A team of authors from across a range of disciplines address the phenomena of knowledge resistance and its causes and consequences at the macro- as well as the micro-level The chapters take a philosophical look at the notion of knowledge resistance, before moving on to discuss issues such as misinformation and fake news, psychological mechanisms such as motivated reasoning in processes of selective exposure and attention, how people respond to evidence and fact-checking, the role of political partisanship, political polarization over factual beliefs, and how knowledge resistance might be counteracted This book will have a broad appeal to scholars and students interested in knowledge resistance, primarily within philosophy, psychology, media and communication, and political science, as well as journalists and policymakers

In several proofs from the theory of finite-dimensional Lie algebras, an essential contribution comes from the Jordan canonical structure of linear maps acting on finite-dimensional vector spaces. On the other hand, there exist classical results concerning Lie algebras which advise us to use infinite-dimensional vector spaces as well. For example, the classical Lie Theorem asserts that all finite-dimensional irreducible representations of solvable Lie algebras are one-dimensional. Hence, from this point of view, the solvable Lie algebras cannot be distinguished from one another, that is, they cannot be classified. Even this example alone urges the infinite-dimensional vector spaces to appear on the stage. But the structure of linear maps on such a space is too little understood; for these linear maps one cannot speak about something like the Jordan canonical structure of matrices. Fortunately there exists a large class of linear maps on vector spaces of arbi trary dimension, having some common features with the matrices. We mean the bounded linear operators on a complex Banach space. Certain types of bounded operators (such as the Dunford spectral, Foia decomposable, scalar generalized or Colojoara spectral generalized operators) actually even enjoy a kind of Jordan decomposition theorem. One of the aims of the present book is to expound the most important results obtained until now by using bounded operators in the study of Lie algebras."

The book presents an updated study of hypergroups, being structured on 12 chapters in starting with the presentation of the basic notions in the domain: semihypergroups, hypergroups, classes of subhypergroups, types of homomorphisms, but also key notions: canonical hypergroups, join spaces and complete hypergroups. A detailed study is dedicated to the connections between hypergroups and binary relations, starting from connections established by Rosenberg and Corsini. Various types of binary relations are highlighted, in particular equivalence relations and the corresponding quotient structures, which enjoy certain properties: commutativity, cyclicity, solvability.A special attention is paid to the fundamental beta relationship, which leads to a group quotient structure. In the finite case, the number of non-isomorphic Rosenberg hypergroups of small orders is mentioned. Also, the study of hypergroups associated with relations is extended to the case of hypergroups associated to n-ary relations. Then follows an applied excursion of hypergroups in important chapters in mathematics: lattices, Pawlak approximation, hypergraphs, topology, with various properties, characterizations, varied and interesting examples. The bibliography presented is an updated one in the field, followed by an index of the notions presented in the book, useful in its study.

Key problems and conjectures have played an important role in promoting the development of Ramsey theory, a field where great progress has been made during the past two decades, with some old problems solved and many new problems proposed. The present book will be helpful to readers who wish to learn about interesting problems in Ramsey theory, to see how they are interconnected, and then to study them in depth. This book is the first problem book of such scope in Ramsey theory. Many unsolved problems, conjectures and related partial results in Ramsey theory are presented, in areas such as extremal graph theory, additive number theory, discrete geometry, functional analysis, algorithm design, and in other areas. Most presented problems are easy to understand, but they may be difficult to solve. They can be appreciated on many levels and by a wide readership, ranging from undergraduate students majoring in mathematics to research mathematicians. This collection is an essential reference for mathematicians working in combinatorics and number theory, as well as for computer scientists studying algorithms. Contents Some definitions and notations Ramsey theory Bi-color diagonal classical Ramsey numbers Paley graphs and lower bounds for R(k, k) Bi-color off-diagonal classical Ramsey numbers Multicolor classical Ramsey numbers Generalized Ramsey numbers Folkman numbers The Erdos-Hajnal conjecture Other Ramsey-type problems in graph theory On van der Waerden numbers and Szemeredi's theorem More problems of Ramsey type in additive number theory Sidon-Ramsey numbers Games in Ramsey theory Local Ramsey theory Set-coloring Ramsey theory Other problems and conjectures

This book is intended as a textbook for a one-term senior undergraduate (or graduate) course in Ring and Field Theory, or Galois theory. The book is ready for an instructor to pick up to teach without making any preparations.The book is written in a way that is easy to understand, simple and concise with simple historic remarks to show the beauty of algebraic results and algebraic methods. The book contains 240 carefully selected exercise questions of varying difficulty which will allow students to practice their own computational and proof-writing skills. Sample solutions to some exercise questions are provided, from which students can learn to approach and write their own solutions and proofs. Besides standard ones, some of the exercises are new and very interesting. The book contains several simple-to-use irreducibility criteria for rational polynomials which are not in any such textbook.This book can also serve as a reference for professional mathematicians. In particular, it will be a nice book for PhD students to prepare their qualification exams.

Everyday Applications of Psychological Science explores several core areas of psychology, showing readers how to apply these principles to everyday situations in order to better their understanding of human behavior and improve their quality of life. The authors of this book, who are award-winning educators of psychology, have culled and collated the best practical research-based advice that psychological science can offer in an easy-to-read and digestible format. Lively and peppered with anecdotes, this book explores topical areas normally found in introductory psychology books but do so in a way that makes psychological science practical, accessible, and relevant to our readers. In Everyday Applications of Psychological Science, the best science that psychology has to offer is translated into life hacks that are applicable to improving readers' physical health, mental health, psychological wealth, relationships, and happiness. Everyday Applications of Psychological Science is vital reading for those interested in learning more about the field of psychology more generally and how aspects of it can be applied to daily life. Our approach may be of particular interest to current and prospective undergraduate students of psychology and those interested in learning more about mental health issues.

This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier Stokes equations. Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers. Contents Introduction to multigrid Robust multigrid technique Parallel multigrid methods Applications of multigrid methods in computational fluid dynamics

This book is dedicated to the work of Alasdair Urquhart. The book starts out with an introduction to and an overview of Urquhart's work, and an autobiographical essay by Urquhart. This introductory section is followed by papers on algebraic logic and lattice theory, papers on the complexity of proofs, and papers on philosophical logic and history of logic. The final section of the book contains a response to the papers by Urquhart. Alasdair Urquhart has made extremely important contributions to a variety of fields in logic. He produced some of the earliest work on the semantics of relevant logic. He provided the undecidability of the logics R (of relevant implication) and E (of relevant entailment), as well as some of their close neighbors. He proved that interpolation fails in some of those systems. Urquhart has done very important work in complexity theory, both about the complexity of proofs in classical and some nonclassical logics. In pure algebra, he has produced a representation theorem for lattices and some rather beautiful duality theorems. In addition, he has done important work in the history of logic, especially on Bertrand Russell, including editing Volume four of Russell's Collected Papers.

First comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables

Of interest to grad students and researchers in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering

Contains many open problems to spur future research

Rich and updated bibliography

This volume contains invited articles by top-notch experts who focus on such topics as: modular representations of algebraic groups, representations of quantum groups and crystal bases, representations of affine Lie algebras, representations of affine Hecke algebras, modular or ordinary representations of finite reductive groups, and representations of complex reflection groups and associated Hecke algebras.
"Representation Theory of Algebraic Groups and Quantum Groups" is intended for graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics.
Contributors:
H. H. Andersen, S. Ariki, C. Bonnaf, J. Chuang, J. Du, M. Finkelberg, Q. Fu, M. Geck, V. Ginzburg, A. Hida, L. Iancu, N. Jacon, T. Lam, G.I. Lehrer, G. Lusztig, H. Miyachi, S. Naito, H. Nakajima, T. Nakashima, D. Sagaki, Y. Saito, M. Shiota, J. Xiao, F. Xu, R. B. Zhang

This book, Algebraic Computability and Enumeration Models: Recursion Theory and Descriptive Complexity, presents new techniques with functorial models to address important areas on pure mathematics and computability theory from the algebraic viewpoint. The reader is first introduced to categories and functorial models, with Kleene algebra examples for languages. Functorial models for Peano arithmetic are described toward important computational complexity areas on a Hilbert program, leading to computability with initial models. Infinite language categories are also introduced to explain descriptive complexity with recursive computability with admissible sets and urelements. Algebraic and categorical realizability is staged on several levels, addressing new computability questions with omitting types realizably. Further applications to computing with ultrafilters on sets and Turing degree computability are examined. Functorial models computability is presented with algebraic trees realizing intuitionistic types of models. New homotopy techniques are applied to Marin Lof types of computations with model categories. Functorial computability, induction, and recursion are examined in view of the above, presenting new computability techniques with monad transformations and projective sets. This informative volume will give readers a complete new feel for models, computability, recursion sets, complexity, and realizability. This book pulls together functorial thoughts, models, computability, sets, recursion, arithmetic hierarchy, filters, with real tree computing areas, presented in a very intuitive manner for university teaching, with exercises for every chapter. The book will also prove valuable for faculty in computer science and mathematics.

Preventing Harmful Behaviour in Online Communities explores the ethics and logistics of censoring problematic communications online that might encourage a person to engage in harmful behaviour. Using an approach based on theories of digital rhetoric and close primary source analysis, Zoe Alderton draws on group dynamics research in relation to the way in which some online communities foster negative and destructive ideas, encouraging community members to engage in practices including self-harm, disordered eating, and suicide. This book offers insight into the dangerous gap between the clinical community and caregivers versus the pro-anorexia and pro-self-harm communities - allowing caregivers or medical professionals to understand hidden online communities young people in their care may be part of. It delves into the often-unanticipated needs of those who band together to resist the healthcare community, suggesting practical ways to address their concerns and encourage healing. Chapters investigate the alarming ease with which ideas of self-harm can infect people through personal contact, community unease, or even fiction and song and the potential of the internet to transmit self-harmful ideas across countries and even periods of time. The book also outlines the real nature of harm-based communities online, examining both their appeal and dangers, while also examining self-censorship and intervention methods for dealing with harmful content online. Rather than pointing to punishment or censorship as best practice, the book offers constructive guidelines that outline a more holistic approach based on the validity of expressing negative mood and the creation of safe peer support networks, making it ideal reading for professionals protecting vulnerable people, as well as students and academics in psychology, mental health, and social care.

This book examines ultrametric Banach algebras in general. It begins with algebras of continuous functions, and looks for maximal and prime ideals in connections with ultrafilters on the set of definition. The multiplicative spectrum has shown to be indispensable in ultrametric analysis and is described in the general context and then, in various cases of Banach algebras.Applications are made to various kind of functions: uniformly continuous functions, Lipschitz functions, strictly differentiable functions, defined in a metric space. Analytic elements in an algebraically closed complete field (due to M Krasner) are recalled with most of their properties linked to T-filters and applications to their Banach algebras, and to the ultrametric holomorphic functional calculus, with applications to spectral properties. The multiplicative semi-norms of Krasner algebras are characterized by circular filters with a metric and an order that are examined.The definition of the theory of affinoid algebras due to J Tate is recalled with all the main algebraic properties (including Krasner-Tate algebras). The existence of idempotents associated to connected components of the multiplicative spectrum is described.

Leibniz Algebras: Structure and Classification is designed to introduce the reader to the theory of Leibniz algebras. Leibniz algebra is the generalization of Lie algebras. These algebras preserve a unique property of Lie algebras that the right multiplication operators are derivations. They first appeared in papers of A.M Blokh in the 1960s, under the name D-algebras, emphasizing their close relationship with derivations. The theory of D-algebras did not get as thorough an examination as it deserved immediately after its introduction. Later, the same algebras were introduced in 1993 by Jean-Louis Loday , who called them Leibniz algebras due to the identity they satisfy. The main motivation for the introduction of Leibniz algebras was to study the periodicity phenomena in algebraic K-theory. Nowadays, the theory of Leibniz algebras is one of the more actively developing areas of modern algebra. Along with (co)homological, structural and classification results on Leibniz algebras, some papers with various applications of the Leibniz algebras also appear now. However, the focus of this book is mainly on the classification problems of Leibniz algebras. Particularly, the authors propose a method of classification of a subclass of Leibniz algebras based on algebraic invariants. The method is applicable in the Lie algebras case as well. Features: Provides a systematic exposition of the theory of Leibniz algebras and recent results on Leibniz algebras Suitable for final year bachelor's students, master's students and PhD students going into research in the structural theory of finite-dimensional algebras, particularly, Lie and Leibniz algebras Covers important and more general parts of the structural theory of Leibniz algebras that are not addressed in other texts

This timely interdisciplinary book brings together a wide spectrum of theoretical concepts and their empirical applications in relation to the COVID-19 pandemic, informing our understanding of the social and psychological bases of a global crisis. Written by an author team of psychologists and sociologists, the volume provides comprehensive coverage of phenomena such as fear, risk, judgement and decision making, threat and uncertainty, group identity and cohesion, social and institutional trust, and communication in the context of an international health emergency.The topics have been grouped into four main chapters, focusing on the individual, group, social, and communication perspectives of the issues affecting or being affected by the pandemic, based on over 740 classic and current references of peer-reviewed research and contextualized with an epidemiological perspective discussed in the introduction. The volume finishes with two special sections, with a chapter on cultural specificity of the social impact of pandemics, focusing specifically on both Islam and Hinduism, and a chapter on the cross-national differences in policy responses to the current health crisis. Providing not just a reference for academic research, but also short-term and long-term policy solutions based on successful strategies to combat adverse social, cognitive, and emotional consequences, this is the ideal resource for academics and policymakers interested in social and psychological determinants of individual reactions to pandemics, as well as in fields such as economics, management, politics, and medical care.

This timely interdisciplinary book brings together a wide spectrum of theoretical concepts and their empirical applications in relation to the COVID-19 pandemic, informing our understanding of the social and psychological bases of a global crisis. Written by an author team of psychologists and sociologists, the volume provides comprehensive coverage of phenomena such as fear, risk, judgement and decision making, threat and uncertainty, group identity and cohesion, social and institutional trust, and communication in the context of an international health emergency.The topics have been grouped into four main chapters, focusing on the individual, group, social, and communication perspectives of the issues affecting or being affected by the pandemic, based on over 740 classic and current references of peer-reviewed research and contextualized with an epidemiological perspective discussed in the introduction. The volume finishes with two special sections, with a chapter on cultural specificity of the social impact of pandemics, focusing specifically on both Islam and Hinduism, and a chapter on the cross-national differences in policy responses to the current health crisis. Providing not just a reference for academic research, but also short-term and long-term policy solutions based on successful strategies to combat adverse social, cognitive, and emotional consequences, this is the ideal resource for academics and policymakers interested in social and psychological determinants of individual reactions to pandemics, as well as in fields such as economics, management, politics, and medical care.