This volume of the "Mathematics and Culture" series is dedicated to Italian artist Armando Pizzicato. The work of Pollock is also discussed, thanks to the collaboration of the Venice Guggenheim Collection. Mathematics creates beauty in architecture, from topology to the projects of Gehry and Piano to the muqarnas of Islam. The fourth dimension is made visible in these pages.

This is a text that contains the latest in thinking and the best in practice. It provides a state-of-the-art statement on tertiary teaching from a multi-perspective standpoint. No previous book has attempted to take such a wide view of the topic. The book will be of special interest to academic mathematicians, mathematics educators, and educational researchers. It arose from the ICMI Study into the teaching and learning of mathematics at university level (initiated at the conference in Singapore, 1998).

This volume honours the eminent mathematicians Vera Sos and Andras Hajnal. The book includes survey articles reviewing classical theorems, as well as new, state-of-the-art results. Also presented are cutting edge expository research papers with new theorems and proofs in the area of the classical Hungarian subjects, like extremal combinatorics, colorings, combinatorial number theory, etc. The open problems and the latest results in the papers are sure to inspire further research.

Ian Stewart's up-to-the-minute guide to the cosmos moves from the formation of the Earth and its Moon to the planets and asteroids of the solar system and from there out into the galaxy and the universe. He describes the architecture of space and time, dark matter and dark energy, how galaxies form, why stars implode, how everything began, and how it will end. He considers parallel universes, what forms extra-terrestrial life might take, and the likelihood of Earth being hit by an asteroid. Mathematics, Professor Stewart shows, has been the driving force in astronomy and cosmology since the ancient Babylonians. He describes how Kepler's work on planetary orbits led Newton to formulate his theory of gravity, and how two centuries later irregularities in the motion of Mars inspired Einstein's theory of general relativity. In crystal-clear terms he explains the fundamentals of gravity, spacetime, relativity and quantum theory, and shows how they all relate to each other. Eighty years ago the discovery that the universe is expanding led to the Big Bang theory of its origins. This in turn led cosmologists to posit features such as dark matter and dark energy. But does dark matter exist? Could another scientific revolution be on the way to challenge current scientific orthodoxy? These are among the questions Ian Stewart raises in his quest through the realms of astronomy and cosmology.

This comprehensive, detailed reference to Mathematica provides the reader with both a working knowledge of Mathematica programming in general and a detailed knowledge of key aspects of Mathematica needed to create the fastest, shortest, and most elegant implementations possible to solve problems from the natural and physical sciences. The Guidebook gives the user a deeper understanding of Mathematica by instructive implementations, explanations, and examples from a range of disciplines at varying levels of complexity. "Programming" covers the structure of Mathematica expressions, after an overview of the syntax of Mathematica, its programming, graphic, numeric and symbolic capabilities in chapter 1. Chapter 2-6 cover hierarchical construction of all Mathematica objects out of symbolic expressions, the definition of functions, the recognition of patterns and their efficient application, program flows and program structuring, the manipulation of lists, and additional topics. An Appendix contains some general references on algorithms and applications of computer algebra, Mathematica itself and comparisons of various algebra systems. The multiplatform CD contains Mathematica 4.1 notebooks with detailed descriptions and explanations of the Mathematica commands needed in that chapter and used in applications, supplemented by a variety of mathematical, physical, and graphic examples and worked out solutions to all exercises. The Mathematica Guidebook is an indispensible resource for practitioners, researchers and professionals in mathematics, the sciences, and engineering. It will find a natural place on the bookshelf as an essential reference work.

From whatever domain they come, engineers are faced daily with optimization problems that requires conflicting objectives to be met. This monograph systematically presents several multiobjective optimization methods accompanied by many analytical examples. Each method or definition is clarified, when possible, by an illustration. Multiobjective Optimization treats not only engineering problems, e.g in mechanics, but also problems arising in operations research and management. It explains how to choose the most suitable method to solve a given problem and uses three primary application examples: optimization of the numerical simulation of an industrial process; sizing of a telecommunication network; and decision-aid tools for the sorting of bids. This book is intended for engineering students, and those in applied mathematics, algorithmics, economics (operational research), production management, and computer scientists.

The IMA Summer Program on Probability and Partial Differential Equations in Modern Applied Mathematics took place July 21-August 1, 2003. The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering.

A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in analysis, as well as offer new perspectives on the phenomena for modeling purposes. In addition, such approaches can be effective in sorting out multiple scale structure and in the development of both non-Monte Carlo as well as Monte Carlo type numerical methods.

There is also a growing recognition of a role in the inclusion of stochastic terms in the modeling of complex flows, and the addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.

This volume consists of original contributions by researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. The volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and sciences.

Contemporary Abstract Algebra, Eleventh Edition is intended for a course whose main purpose is to enable students to do computations and write proofs. This text stresses the importance of obtaining a solid introduction to the traditional topics, while at the same time presenting abstract algebra as a contemporary and very much active subject, which is currently being used by working physicists, chemists, and computer scientists.

For nearly four decades, this classic text has been widely appreciated by instructors and students alike. The book offers an enjoyable read and conveys and develops enthusiasm for the beauty of the topics presented. It is comprehensive, lively, and engaging.

Students will learn how to do computations and write proofs. A unique feature of the book are exercises that build the skill of generalizing, a skill that students should develop, but rarely do.

Examples elucidate the definitions, theorems, and proof techniques; exercises facilitate understanding, provide insight, and develop the ability to do proofs. The hallmark features of previous editions of the book are enhanced in this edition. These include:

A good mixture of approximately 1900 computational and theoretical exercises appearing in each chapter that synthesizes concepts from multiple chapters

Back-of-the-book skeleton solutions and hints to odd-numbered exercises

Over 300 worked-out examples ranging from routine computations to the more challenging

Links to interactive True/False questions with comments

Links to computer exercises that utilize interactive software available on the author's website, stressing guessing and making conjectures

Many applications from scientific and computing fields, as well as some from everyday life

Numerous historical notes and biographies that spotlight the people and events behind the mathematics

Motivational and humorous quotations

Hundreds of figures, photographs, and tables

Changes to the eleventh edition include new exercises, examples, biographies, and quotes, and an enrichment of the discussion portions. These changes accentuate and enhance the hallmark features that have made previous editions of the book a comprehensive, lively, and engaging introduction to the subject.

While many partial solutions and sketches for the odd-numbered exercises appear in the book, an Instructor’s Solutions Manual offers solutions for all the exercises. A Student's Solution Manual has comprehensive solutions for all odd-numbered exercises, many even-numbered exercises, and numerous alternative solutions as well.

Table of Contents

1 Introduction to Groups

2 Groups

3 Finite Groups; Subgroups

4 Cyclic Groups

5 Permutation Groups

6 Ismorphisms

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

12 Introduction to Rings

13 Integral Domains

14 Ideals and Factor Rings

15 Ring Homomorphisms

16 Polynomial Rings

17 Factorization of Polynomials

18 Divisibilty in Integral Domains

19 Extension Fields

20 Algebraic Extensions

21 Finite Fields

22 Geometric Constructions

23 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

This book collects survey papers in the fields of entropy, search and complexity, summarizing the latest developments in their respective areas. More than half of the papers belong to search theory which lies on the borderline of mathematics and computer science, information theory and combinatorics, respectively. The book will be useful to experienced researchers as well as young scientists and students both in mathematics and computer science.

What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing.

Process calculi are among the most successful models of concurrent systems. Various behavior equivalences between processes are central notions in CCS (calculus of communicating systems) and other process calculi. In the real applications, specification and implementation are described as two processes, and correctness of programs is treated as a certain behavior equivalence between them. The purpose of this book is to establish a theory of approximate correctness and infinite evolution of concurrent programs by employing some notions and tools from point-set topology. This book is restricted to CCS for simplicity, but the main idea also applies to some other process calculi. The concept of bisimulation limits, useful for the understanding and analysis of infinite evolution of processes, is introduced. In addition, the notions of near bisimulations and bisimulation indexes, suitable in describing approximate correctness of concurrent programs, are proposed. The book will be of particular interest to researchers in the fields of theoretical computer science, especially theory of concurrency and hybrid systems, and graduate students in related disciplines. It will also be valuable to practical system designers developing concurrent and/or real-time systems.

Algebraic Geometry provides an impressive theory targeting the understanding of geometric objects defined algebraically. Geometric Modeling uses every day, in order to solve practical and difficult problems, digital shapes based on algebraic models. In this book, we have collected articles bridging these two areas. The confrontation of the different points of view results in a better analysis of what the key challenges are and how they can be met. We focus on the following important classes of problems: implicitization, classification, and intersection. The combination of illustrative pictures, explicit computations and review articles will help the reader to handle these subjects.

Connecting Math Concepts: Comprehensive Edition works as a core program or as a Tier 3 intervention for at-risk students. Facts, procedures, conceptual understanding, applications, and problem solving skills are combined in this program to provide a comprehensive curriculum for students. The Student eBook and Textbook includes teacher and student-guided exercises online The Student Workbook contains mastery tests and twopractice tests.

In 1978, in the foreword to Weeding and Sowing: A Preface to a Science of Mathematics Education, Hans Freudenthal wrote that his book is a preface to a science that does not exist. Almost 20 years later, does his claim still hold true? The present book is the result of the reflection of many individuals in mathematics education on this and related questions. Is mathematics education a science? Is it a discipline? In what sense? What is its place within other domains of research and academic disciplines? What accounts for its specificity? In the book, the reader will find a range of possible answers to these questions, a variety of analyses of the actual directions of research in different countries, and a number of visions for the future of research in mathematics education. The book is a result of an ICMI Study, whose theme was formulated as: What is Research in Mathematics Education and What are Its Results?'. One important outcome of this study was the realization of the reasons for the difficulty of the questions that the study was posing, leading possibly to a set of other questions, better suited to the actual concerns and research practices of mathematics education researchers. The book addresses itself to researchers in mathematics education and all those working in their neighborhood who are concerned with the problems of the definition of this new scientific domain emerging at their borders.

(Very preliminary)A tribute to the vision and legacy of Israel Gelfand, the invited papers in this volume reflect the unity of mathematics as a whole, with particular emphasis on the many connections among the fields of geometry, physics, and representation theory. Written by leading mathematicians, the text is broadly divided into two sections: the first is devoted to developments at the intersection of geometry and physics, and the second to representation theory and algebraic geometry. Topics include conformal field theory, K-theory, noncommutative geometry, gauge theory, representations of infinite-dimensional Lie algebras, and various aspects of the Langlands program. Graduate students and researchers will benefit from and find inspiration in this broad and unique work, which brings together fundamental results in a number of disciplines and highlights the rewards of an interdisciplinary approach to mathematics and physics.Contributors: M. Atiyah, A. Beilinson, J. Bernstein, A. Connes, P. Deligne, R. Dijkgraaf, D. Gaitsgory, M. Gromov, F. Hirzebruch, M. Hopkins, D. Kazhdan, F. Kirwan, M. Kontsevich, B. Kostant, G. Lusztig, D. McDuff, H. Nakajima, S. Novikov, P. Sarnak, A.

This book presents the proceedings of Positivity VII, held from 22-26 July 2013, in Leiden, the Netherlands. Positivity is the mathematical field concerned with ordered structures and their applications in the broadest sense of the word. A biyearly series of conferences is devoted to presenting the latest developments in this lively and growing discipline. The lectures at the conference covered a broad spectrum of topics, ranging from order-theoretic approaches to stochastic processes, positive solutions of evolution equations and positive operators on vector lattices, to order structures in the context of algebras of operators on Hilbert spaces. The contributions in the book reflect this variety and appeal to university researchers in functional analysis, operator theory, measure and integration theory and operator algebras. Positivity VII was also the Zaanen Centennial Conference to mark the 100th birth year of Adriaan Cornelis Zaanen, who held the chair of Analysis in Leiden for more than 25 years and was one of the leaders in the field during his lifetime.

What is Time? Assuming no prior specialized knowledge by the reader, the book raises specific, hitherto overlooked questions about how time works, such as how and why anyone can be made to be, at the very same instant, simultaneous with events that are actually days apart. It examines abiding issues in the physics of time or at its periphery which still elude a full explanation such as delayed choice experiments, the brain's perception of time during saccadic masking, and more and suggests that these phenomena can only exist because they ultimately obey applicable mathematics, thereby agreeing with a modern view that the universe and everything within it, including the mind, are ultimately mathematical structures. It delves into how a number of conundrums, such as the weak Anthropic Principle, could be resolved, and how such resolutions could be tested experimentally. All its various threads converge towards a same new vision of the ultimate essence of time, seen as a side effect from a deeper reality.

This is the second volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. BjA, rstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. KollAr, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. MArz. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. MA1/4ller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. PAschel, L. Polterovich, L. Pyber, N. SimAnyi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua.

"Elsevier's Dictionary of Mathematics" contains 11,652 entries with more than 4,750 cross-references. Selection of the terms was based either on their significance or on their frequency of use according to authoritative encyclopedias, dictionaries and textbooks. Included are both modern developments and contemporary changes in terminology as well as recently established terms.

The terminology covers all the major branches from elementary to advanced subjects: arithmetic, algebra, geometry, set theory, discrete mathematics, logic, Boolean algebra, linear algebra, matrix algebra, calculus, differential equations, vector algebra, field theory, probability theory and statistics, optimization, numerical methods, mathematical programming, modern algebra, algebraic structures, computer algebra, category theory, applied mathematics, theory of automata and formal languages, theory of games, theory of graphs, as well as some commonly used entries in computer architecture, hardware, communications, system and application software, microprogramming, etc.

This work will provide readers, writers and translators with a guide of the most widely used terms and collections in the area, and will prove to be a useful tool for all professionals exploring the multilingual scientific terminology.

This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by: K. Behrend N. Bergeron S. K. Donaldson J. Dubedat B. Duplantier G. Faltings E. Getzler G. Kings R. Mazzeo J. Millson C. Moeglin W. Muller R. Rhodes D. Roessler S. Sheffield A. Teleman G. Tian K-I. Yoshikawa H. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

STP Caribbean Mathematics has been revised and updated to address the demands of mathematics syllabuses in the region and provide students with a firm foundation for success at CSEC (R). Workbook 1 in this series is for use alongside STP Caribbean Mathematics Book 1, and offers students opportunities to practise key mathematical skills and concepts. Its focus on practicing the core aspects of mathematics helps to reinforce students' knowledge and understanding. Workbook 1 also includes answers to the activities.

This book discusses mathematics learners in transition and their practices in different contexts; the institutional and socio-cultural framing of the transition processes involved; and the communication and negotiation of mathematical meanings during transition. Providing both empirical studies and significant theoretical reflections, it will appeal to researchers and postgraduate students in mathematics education, cultural psychology, multicultural education, immigrant and indigenous education.

In unserer technisierten Welt stossen wir uberall auf Mathematik. Mathematik ist eine Basiswissenschaft und der Schlussel fur bahnbrechende Innovationen. Sie macht viele Produkte und Dienstleistungen uberhaupt erst moglich und ist damit ein wichtiger Produktions- und Wettbewerbsfaktor.

Im vorliegenden Buch berichten 19 grosse internationale Unternehmen sowie die Bundesagentur fur Arbeit wie unverzichtbar Mathematik fur ihren Erfolg heute geworden ist. Ein spannender und lehrreicher Einblick in die Mathematik, der mit oft zitierten und negativen Vorurteilen grundlich aufraumt."