This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham's theorem on simplicial complexes. In addition, Sullivan's results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on:

- admissible or permissible inference rules

- the derivability of the admissible inference rules

- the structural completeness of logics

- the bases for admissible and valid inference rules.

There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered.

The book is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included.

The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook.

It is well known that "fuzziness"-informationgranulesand fuzzy sets as one of its formal manifestations- is one of important characteristics of human cognitionandcomprehensionofreality. Fuzzy phenomena existinnature and are encountered quite vividly within human society. The notion of a fuzzy set has been introduced by L. A. , Zadeh in 1965 in order to formalize human concepts, in connection with the representation of human natural language and computing with words. Fuzzy sets and fuzzy logic are used for mod- ing imprecise modes of reasoning that play a pivotal role in the remarkable human abilities to make rational decisions in an environment a?ected by - certainty and imprecision. A growing number of applications of fuzzy sets originated from the "empirical-semantic" approach. From this perspective, we were focused on some practical interpretations of fuzzy sets rather than being oriented towards investigations of the underlying mathematical str- tures of fuzzy sets themselves. For instance, in the context of control theory where fuzzy sets have played an interesting and practically relevant function, the practical facet of fuzzy sets has been stressed quite signi?cantly. However, fuzzy sets can be sought as an abstract concept with all formal underpinnings stemming from this more formal perspective. In the context of applications, it is worth underlying that membership functions do not convey the same meaning at the operational level when being cast in various contexts.

For one- or two-semester junior orsenior level courses in Advanced Calculus, Analysis I, or Real Analysis. This title is part of the Pearson Modern Classicsseries. This text prepares students for future coursesthat use analytic ideas, such as real and complex analysis, partial andordinary differential equations, numerical analysis, fluid mechanics, anddifferential geometry. This book is designed to challenge advanced studentswhile encouraging and helping weaker students. Offering readability,practicality and flexibility, Wade presents fundamental theorems and ideas froma practical viewpoint, showing students the motivation behind the mathematicsand enabling them to construct their own proofs.

Physical oncology has the potential to revolutionize cancer research and treatment. The fundamental rationale behind this approach is that physical processes, such as transport mechanisms for drug molecules within tissue and forces exchanged by cancer cells with tissue, may play an equally important role as biological processes in influencing progression and treatment outcome. This book introduces the emerging field of physical oncology to a general audience, with a focus on recent breakthroughs that help in the design and discovery of more effective cancer treatments. It describes how novel mathematical models of physical transport processes incorporate patient tissue and imaging data routinely produced in the clinic to predict the efficacy of many cancer treatment approaches, including chemotherapy and radiation therapy. By helping to identify which therapies would be most beneficial for an individual patient, and quantifying their effects prior to actual implementation in the clinic, physical oncology allows doctors to design treatment regimens customized to each patient's clinical needs, significantly altering the current clinical approach to cancer treatment and improving the outcomes for patients.

* Fosters a new consensus by articulating a balanced and integrative view to show the relevance of the empirical study of intentional interaction * Connects expression with moral psychology to give a detailed and perspicuous presentation of the second person interaction * A unique new resource for academics and students of social cognition, social and cognitive neuroscience, the cognitive sciences, philosophy, and metaphysics

* Fosters a new consensus by articulating a balanced and integrative view to show the relevance of the empirical study of intentional interaction * Connects expression with moral psychology to give a detailed and perspicuous presentation of the second person interaction * A unique new resource for academics and students of social cognition, social and cognitive neuroscience, the cognitive sciences, philosophy, and metaphysics

The Yang-Mills theory of gauge interactions is a prime example of interdisciplinary mathematics and advanced physics. Its historical development is a fascinating window into the ongoing struggle of mankind to understand nature. The discovery of gauge fields and their properties is the most formidable landmark of modern physics. The expression of the gauge field strength as the curvature associated to a given connection, places quantum field theory in the same geometrical footing as the gravitational field of general relativity which is naturally written in geometrical terms. The understanding of such geometrical property may help one day to write a unified field theory starting from symmetry principles.

Of course, there are remarkable differences between the standard gauge fields and the gravitational field, which must be understood by mathematicians and physicists before attempting such unification. In particular, it is important to understand why gravitation is not a standard gauge field.

This book presents an account of the geometrical properties of gauge field theory, while trying to keep the equilibrium between mathematics and physics. At the end we will introduce a similar approach to the gravitational field.

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback) This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincare's problem. Namely, there are given the upper estimations of the number of Poincare-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

The Center and Focus Problem: Algebraic Solutions and Hypotheses, M. N. Popa and V.V. Pricop, ISBN: 978-1-032-01725-9 (Hardback) This book focuses on an old problem of the qualitative theory of differential equations, called the Center and Focus Problem. It is intended for mathematicians, researchers, professors and Ph.D. students working in the field of differential equations, as well as other specialists who are interested in the theory of Lie algebras, commutative graded algebras, the theory of generating functions and Hilbert series. The book reflects the results obtained by the authors in the last decades. A rather essential result is obtained in solving Poincare's problem. Namely, there are given the upper estimations of the number of Poincare-Lyapunov quantities, which are algebraically independent and participate in solving the Center and Focus Problem that have not been known so far. These estimations are equal to Krull dimensions of Sibirsky graded algebras of comitants and invariants of systems of differential equations.

* Includes interdisciplinary contributions and brings together research on a range of extreme behaviors in one volume, by making theoretical links between different contexts * Explores the brain, hormones, and behaviour to offer insights into the mechanisms and processes that enable extremism to explain their occurrence and the conditions under which they may be likely to emerge * Ideal reading for high-level students taking courses on extremism, academics, and professionals dealing with extreme behavior

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

* Includes interdisciplinary contributions and brings together research on a range of extreme behaviors in one volume, by making theoretical links between different contexts * Explores the brain, hormones, and behaviour to offer insights into the mechanisms and processes that enable extremism to explain their occurrence and the conditions under which they may be likely to emerge * Ideal reading for high-level students taking courses on extremism, academics, and professionals dealing with extreme behavior

* Solid research basis, drawing on findings from a 4-year research project with in-depth interviews with judges, attorneys, and seasoned forensic neuropsychologists and psychologists as well as further interviews with professionals in other fields such as engineering, physics and economics. * Provides focused attention on how experts interact with judges, attorneys, and juries * Challenges experts to avoid the traps of professional jargon and traditional manners of presenting information/knowledge/opinions. * Provides a step-by-step approach to orienting the new academic to expert witnessing

Introduces a new web-based optimizer for Geometric algebra algorithms; Supports many programming languages as well as hardware; Covers the advantages of High-dimensional algebras; Includes geometrically intuitive support of quantum computing

Features: Suitable for PhD candidates and researchers. Requires prerequisites in set theory, general topology and abstract algebra, but is otherwise self-contained.

The Event of Psychopoetics overviews and investigates the notion of psychopoetics, a sociopsychological event that involves re-creative slips and that emerges under certain cultural conditions and power relations in the context of everyday interaction and through certain modes of dialoguing and conversing. This transdisciplinary text takes the reader through the thought processes of Deleuze, Guattari, Agamben, Maffesoli, Foucault, Butler, Haraway, and Braidotti, among others, addressing debates that are integral to the critique of psychology and its devices of subjectivization and normalization. Garcia takes a unique approach by reflecting on how psychopoetics contrasts institutionalized dialogues, while constantly emphasizing the generative and transformative potency of social worlds effectuated in the impetuous play of poetics. The book combines the rigor of academic research with the creative display of ideas that open diverse, suggestive lines of reflection on everyday interlocution and its possibilities of reinvention, modes of social existence, and the relation between subjectivity and the designs of power. A truly unique reading experience, this book is ideal for students, instructors, and researchers in the fields of philosophy, social psychology and sociological thought, discourse studies, literary theory, and cultural analysis.

The Event of Psychopoetics overviews and investigates the notion of psychopoetics, a sociopsychological event that involves re-creative slips and that emerges under certain cultural conditions and power relations in the context of everyday interaction and through certain modes of dialoguing and conversing. This transdisciplinary text takes the reader through the thought processes of Deleuze, Guattari, Agamben, Maffesoli, Foucault, Butler, Haraway, and Braidotti, among others, addressing debates that are integral to the critique of psychology and its devices of subjectivization and normalization. Garcia takes a unique approach by reflecting on how psychopoetics contrasts institutionalized dialogues, while constantly emphasizing the generative and transformative potency of social worlds effectuated in the impetuous play of poetics. The book combines the rigor of academic research with the creative display of ideas that open diverse, suggestive lines of reflection on everyday interlocution and its possibilities of reinvention, modes of social existence, and the relation between subjectivity and the designs of power. A truly unique reading experience, this book is ideal for students, instructors, and researchers in the fields of philosophy, social psychology and sociological thought, discourse studies, literary theory, and cultural analysis.

--Core textbook featuring accessible but advanced coverage of theory, research, and applications in nonverbal communication from renowned scholars --Usable for undergraduate and graduate courses in communication and psychology departments --Includes a new chapter on identity and impression management, as well as fully updated research coverage throughout --Online resources include an extensive instructor's manual and test bank

There has been a resurgence of interest in classical invariant theory driven by several factors: new theoretical developments; a revival of computational methods coupled with powerful new computer algebra packages; and a wealth of new applications, ranging from number theory to geometry, physics to computer vision. This book provides readers with a self-contained introduction to the classical theory as well as modern developments and applications. The text concentrates on the study of binary forms (polynomials) in characteristic zero, and uses analytical as well as algebraic tools to study and classify invariants, symmetry, equivalence and canonical forms. A variety of innovations make this text of interest even to veterans of the subject; these include the use of differential operators and the transform approach to the symbolic method, extension of results to arbitrary functions, graphical methods for computing identities and Hilbert bases, complete systems of rationally and functionally independent covariants, introduction of Lie group and Lie algebra methods, as well as a new geometrical theory of moving frames and applications. Aimed at advanced undergraduate and graduate students the book includes many exercises and historical details, complete proofs of the fundamental theorems, and a lively and provocative exposition.

Features Filled with instructive examples and exercises to help build understanding Suitable for researchers, professionals and students, both in mathematics and computer science Every chapter consists of exercises with solution provided online at www.Routledge.com/9780367720292

This work is at the crossroads of a number of mathematical areas, including algebraic geometry, several complex variables, differential geometry, and representation theory. It is the first book to cover complex tori, among the simplest of complex manifolds, which are important to research in the above areas. The book gives a systematic approach to the theory, presents new results, and includes an up-to-date bibliography.

Spatializing Social Media charts the theoretical and methodological challenges in analyzing and visualizing social media data mapped to geographic areas. It introduces the reader to concepts, theories, and methods that sit at the crossroads between spatial and social network analysis to unpack the conceptual differences between online and face-to-face social networks and the nonlinear effects triggered by social activity that overlaps online and offline. The book is divided into four sections, with the first accounting for the differences between space (the geometrical arrangements that structure and enable forms of interaction) and place (the mechanisms through which social meanings are attached to physical locations). The second section covers the rationale of social network analysis and the ontological differences, stating that relationships, more than individual and independent attributes, are key to understanding of social behavior. The third section covers a range of case studies that successfully mapped social media activity to geographically situated areas and considers the inflection of homophilous dependencies across online and offline social networks. The fourth and last section of the book explores a range of networks and discusses methods for and approaches to plotting a social network graph onto a map, including the purpose-built R package Spatial Social Media. The book takes a non-mathematical approach to social networks and spatial statistics suitable for postgraduate students in sociology, psychology and the social sciences.

This book intends to provide material for a graduate course on computational commutative algebra and algebraic geometry, highlighting potential applications in cryptography. Also, the topics in this book could form the basis of a graduate course that acts as a segue between an introductory algebra course and the more technical topics of commutative algebra and algebraic geometry.This book contains a total of 124 exercises with detailed solutions as well as an important number of examples that illustrate definitions, theorems, and methods. This is very important for students or researchers who are not familiar with the topics discussed. Experience has shown that beginners who want to take their first steps in algebraic geometry are usually discouraged by the difficulty of the proposed exercises and the absence of detailed answers. Therefore, exercises (and their solutions) as well as examples occupy a prominent place in this course.This book is not designed as a comprehensive reference work, but rather as a selective textbook. The many exercises with detailed answers make it suitable for use in both a math or computer science course.