Mathematics of Discrete Structures for Computer Science

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local or non-local boundary conditions. The theory is introduced at a level suitable for graduate students. Numerous examples are then given to illustrate the peculiar properties of the Dirac operator, and the role of boundary conditions in heat-kernel asymptotics and quantum field theory. Topics covered include the introduction of spin-structures in Riemannian and Lorentzian manifolds; applications of index theory; heat-kernel asymptotics for operators of Laplace type; quark boundary conditions; one-loop quantum cosmology; conformally covariant operators; and the role of the Dirac operator in some recent investigations of four-manifolds. This volume provides graduate students with a rigorous introduction and researchers with a valuable refer

This international and interdisciplinary collection presents and discusses the many issues and educational practices that are touched on by constructivism. Drawing on perspectives from a range of different fields (ethics, mathematics education, philosophy, social psychology, science education, social studies), this book invites us to reposition ourselves in relation to the major currents that have influenced education in this century, namely pragmatism, genetic epistemology, and social interactionism. The essays call for new reflection on the questions that are central to the project of education and that, in particular, involve the validity of knowledge and types of knowledge, the compartmentalization of school subjects, the mediating role of teachers, and, above all, the ends of education. In so doing, this book relaunches the discussion on constructivism's potential for the social empowerment of groups and individuals.

This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics. With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory.

This book is a comprehensive introduction to the mechanical properties of ceramics, and is designed primarily as a textbook for undergraduate and graduate students in materials science and engineering. Over the past 25 years ceramics have become key materials in the development of many new technologies as scientists have been able to design these materials with new structures and properties. An understanding of the factors that influence their mechanical behavior and reliability is essential. This book will introduce the reader to current concepts in the field. It contains problems and exercises to help readers develop their skills. Although designed principally as a textbook for advanced undergraduates and graduate students, this book will also be of value as a supplementary text for more general courses and to industrial scientists and engineers involved in the development of ceramic-based products, materials selection and mechanical design.

An in-depth account of graph theory, written for serious students of mathematics and computer science. It reflects the current state of the subject and emphasises connections with other branches of pu

How does our Universe evolve? And how did structures like stars and galaxies form? In recent years, scientists' understanding of these profound questions has developed enormously. This book presents a clear and detailed picture of contemporary cosmology for the general reader. Unlike existing popular books on cosmology, After the First Three Minutes does not gloss over details, nor shy away from explaining the underlying concepts. Instead, with a lucid and informal style, the author introduces all the relevant background and then carefully pieces together an engaging story of the evolution of our Universe. We are left with a state-of-the-art picture of scientists' current understanding in cosmology, and a keen taste of the excitement of this fast-moving science. Throughout, no mathematics is used; and all technical jargon is clearly introduced and reinforced in a handy glossary at the end of the book. For general readers who want to get to grips with what we really do and don't know a

The third edition of this text has been completely rewritten and revised. It is intended for first- and second-year undergraduates in chemistry taking physical chemistry courses, and for undergraduates in other science and engineering subjects that require an understanding of chemistry. The author gives more attention to the solid and liquid states than is found in other texts on this subject, and introduces topics such as computer simulation and quasicrystals. Each chapter concludes with a set of problems, to which there are solution notes, designed to lead the reader to familiarity with the subject and its application in new situations. Computer programs designed to assist the reader are downloadable from the World Wide Web, from the time of publication. Detailed solutions to the problems will also be available via the World Wide Web. See http://www.cup.cam.ac.uk/stm/laddsolutions.htm. This modern text on physical chemistry will be of interest to undergraduate students in chemistry a

Self-organized criticality (SOC) is based upon the idea that complex behavior can develop spontaneously in certain multi-body systems whose dynamics vary abruptly. This book is a clear and concise introduction to the field of self-organized criticality, and contains an overview of the main research results. The author begins with an examination of what is meant by SOC, and the systems in which it can occur. He then presents and analyzes computer models to describe a number of systems, and he explains the different mathematical formalisms developed to understand SOC. The final chapter assesses the impact of this field of study, and highlights some key areas of new research. The author assumes no previous knowledge of the field, and the book contains several exercises. It will be ideal as a textbook for graduate students taking physics, engineering, or mathematical biology courses in nonlinear science or complexity.

Since the early 1990s, genetic programming (GP) a discipline whose goal is to enable the automatic generation of computer programs has emerged as one of the most promising paradigms for fast, producti

This text for advanced undergraduates and graduates reading applied mathematics, electrical, mechanical, or control engineering, employs block diagram notation to highlight comparable features of line

Michael Powell is one of the world's foremost figures in numerical analysis. This volume, first published in 1997, is derived from invited talks given at a meeting celebrating his 60th birthday and, reflecting Powell's own achievements, focuses on innovative work in optimisation and in approximation theory. The individual papers have been written by leading authorities in their subjects and are a mix of expository articles and surveys. They have all been reviewed and edited to form a coherent volume for this important discipline within mathematics, with highly relevant applications throughout science and engineering.

This fascinating book explains why materials behave as they do. In a completely non-technical style, using only basic arithmetic, the author explains how the properties of materials result from the way they are composed of atoms and why they have the properties they do: for example, why copper and rubies are coloured, why metals conduct heat better than glass, why magnets attract iron nails but not brass pins, and how superconductors can conduct electricity without resistance. The book is intended for general readers, and uses mainly words, pictures and analogies, with only a minimum of very simple mathematics. The author explains how it is possible to understand the basic properties of matter, and translates the technical jargon of physics into a language that can be understood by anyone with an interest in science who wants to know why the world around us behaves in the way that it does.

Robert Axelrod is widely known for his groundbreaking work in game theory and complexity theory. He is a leader in applying computer modeling to social science problems. His book The Evolution of Coo

What is the timescale for the settlement and cracking of an old stone building? How do the elegant flying buttresses of a Gothic cathedral safely transfer thrust to the foundations? And what is the effect of wind on a stone spire, or bell-ringing on a church tower? These and many other questions pertinent to the upkeep of old stone structures are answered in this clear and authoritative guide. With a firm scientific basis, but without the use of complex mathematics, the author provides a thorough and intuitive understanding of masonry structures. The basis of masonry analysis is introduced in the first two chapters, after which individual structures - including piers, pinnacles, towers, vaults and domes - are considered in more detail. This lucid and informative text will be of particular interest to structural engineers, practising architects and others involved in the renovation and care of old stone buildings.

Information is a central topic in computer science, cognitive science and philosophy. In spite of its importance in the 'information age', there is no consensus on what information is, what makes it possible, and what it means for one medium to carry information about another. Drawing on ideas from mathematics, computer science and philosophy, this book addresses the definition and place of information in society. The authors, observing that information flow is possible only within a connected distribution system, provide a mathematically rigorous, philosophically sound foundation for a science of information. They illustrate their theory by applying it to a wide range of phenomena, from file transfer to DNA, from quantum mechanics to speech act theory.

Dr. Ecco is a mathematical detective and puzzle solver. In this book readers are invited to join him in solving nearly 40 puzzles inspired by methods in computer science and mathematics, including The

Various problems in computer science are 'hard', that is NP-complete, and so not realistically computable; thus in order to solve them they have to be approximated. This book is a survey of the basic techniques for approximating combinatorial problems using parallel algorithms. Its core is a collection of techniques that can be used to provide parallel approximations for a wide range of problems (for example, flows, coverings, matchings, travelling salesman problems, graphs), but in order to make the book reasonably self-contained, the authors provide an introductory chapter containing the basic definitions and results. A final chapter deals with problems that cannot be approximated, and the book is ended by an appendix that gives a convenient summary of the problems described in the book. This is an up-to-date reference for research workers in the area of algorithms, but it can also be used for graduate courses in the subject.

The aim of this book is to provide the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications.

The aim of this book is to provide the reader with a basic understanding of Fourier series, Fourier transforms and Laplace transforms. The book is an expanded and polished version of the authors' notes for a one semester course, for students of mathematics, electrical engineering, physics and computer science. Prerequisites for readers of this book are a basic course in both calculus and linear algebra. Otherwise the material is self-contained with numerous exercises and various examples of applications.

This new and exciting interactive resource centers around fourteen high quality computer simulations covering essential topics in solid state physics. The simulations cover x-ray diffraction, phonons, electron states and dynamics, semiconductors, magnetism, and dislocations. Users can vary different characteristics and immediately see the results in animations and graphical displays. The companion book is essential for effective use of the simulations. It guides the user through hundreds of exercises and examples, illustrates fundamental physical principles, and contains notes on the relevant physics. This paperback (without CD-ROM) is intended for students who have access to the simulations on a local area network. This package provides an interactive resource for those studying solid state physics at advanced undergraduate or graduate level. It will also be of interest and value to researchers in physics, materials science, electrical engineering, chemistry and chemical engineerin

This book treats the very special and fundamental mathematical properties that hold for a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, other parts of mathematics, statistics and theoretical physics. The emphasis throughout this book is on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.

A textbook for a semester course in any discipline that requires knowledge of complex systems. Assumes a previous course in discrete mathematics as generally taught in computer science and systems sci

Combinatorics on words, or finite sequences, is a field which grew simultaneously within disparate branches of mathematics such as group theory and probability. It has grown into an independent theory finding substantial applications in computer science automata theory and linguistics. This volume is the first to attempt to present a thorough treatment of this theory. All of the main results and techniques are covered. The presentation is accessible to undergraduate and graduate level students in mathematics and computer science as well as to specialists in all branches of applied mathematics. Topics discussed include: Thue's square free words, Van der Waerden's theorem, Ramsey's theorem; factorization of free monoids including application to free Lie algebras; equations in words, etc.

This book gives an introduction to theories of computability from a mathematically sophisticated point of view. It treats not only 'the' theory of computability (created by Alan Turing and others in the 1930s), but also a variety of other theories (of Boolean functions, automata and formal languages). These are addressed from the classical perspective of their generation by grammars and from the modern perspective as rational cones. The treatment of the classical theory of computable functions and relations takes the form of a tour through basic recursive function theory, starting with an axiomatic foundation and developing the essential methods in order to survey the most memorable results of the field. This authoritative account by one of the leading lights of the subject will prove exceptionally useful reading for graduate students, and researchers in theoretical computer science and mathematics.

This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading.

This series is for people—adults and teenagers—who are interested in computer programming because it's fun. The three volumes use the Logo programming language as the vehicle for an exploration of com

Joyce, Chaos, and Complexity studies the manifold relations among twentieth-century mathematics and Science, James Joyce's fiction, and the critical reception of Joyce's work. Calling for profound rea

Computational learning theory is a subject which has been advancing rapidly in the last few years. The authors concentrate on the probably approximately correct model of learning, and gradually develop the ideas of efficiency considerations. Finally, applications of the theory to artificial neural networks are considered. Many exercises are included throughout, and the list of references is extensive. This volume is relatively self contained as the necessary background material from logic, probability and complexity theory is included. It will therefore form an introduction to the theory of computational learning, suitable for a broad spectrum of graduate students from theoretical computer science and mathematics.

Ideal for use in a microprocessor course in electrical engineering or computer science,Software and Hardware Engineering: Motorola M68HC11 provides an introduction to the architecture and design of ha

Mathematicians from Leibniz to Hilbert have sought to mechanise the verification of mathematical proofs. Developments arising out of Gödel's proof of his incompleteness theorem showed that no computer program could automatically prove true all the theorems of mathematics. In practice, however, there are a number of sophisticated automated reasoning programs that are quite effective at checking mathematical proofs. Now in paperback, this book describes the use of a computer program to check the proofs of several celebrated theorems in metamathematics including Gödel's incompleteness theorem and the Church–Rosser theorem. The computer verification using the Boyer–Moore theorem prover yields precise and rigorous proofs of these difficult theorems. It also demonstrates the range and power of automated proof checking technology. The mechanisation of metamathematics itself has important implications for automated reasoning since metatheorems can be applied by labour-saving devices to simpli

A textbook for the advanced undergraduate or graduate student beginning a serious study of X-ray crystallography. It will be of interest both to those intending to become professional crystallographers and to those physicists, chemists, biologists, geologists, metallurgists and others who will use it as a tool in their research. All major aspects of crystallography are covered - the geometry of crystals and their symmetry, theoretical and practical aspects of diffracting X-rays by crystals and how the data may be analysed to find the symmetry of the crystal and its structure. Recent advances are fully covered, including the synchrotron as a source of X-rays, methods of solving structures from power data and the full range of techniques for solving structures from single-crystal data. A suite of computer programs is provided for carrying out many operations of data-processing and solving crystal structures - including by direct methods. While these are limited to two dimensions they ful

This is a unique book on the mathematics of signals written for hearing-science researchers. Designed to follow an introductory text on psychoacoustic, Signals, Sound, and Sensation takes the reader

An introduction to basic water science for use in a one-quarter course in elementary hydrology for undergraduate nonscience majors without extensive background in mathematics, discussing the physical

Finite fields are algebraic structures in which there is much research interest and they have been shown to have a wide range of applications. These proceedings give a state-of-the-art account of the area of finite fields and their applications in communications (coding theory, cryptology), combinatorics, design theory, quasirandom points, algorithms and their complexity. Typically, theory and application are tightly interwoven in the survey articles and original research papers included here. These also demonstrate inter-connections with other branches of pure mathematics such as number theory, group theory and algebraic geometry. This volume is an invaluable resource for any researcher in finite fields or related areas.

Eric Temple Bell (1883–1960) was a distinguished mathematician and a best selling popularizer of mathematics. Under the name of John Taine, he also published science fiction novels. Constance Reid gi

Bayesian probability theory has emerged not only as a powerful tool for building computational theories of vision, but also as a general paradigm for studying human visual perception. This 1996 book provides an introduction to and critical analysis of the Bayesian paradigm. Leading researchers in computer vision and experimental vision science describe general theoretical frameworks for modelling vision, detailed applications to specific problems and implications for experimental studies of human perception. The book provides a dialogue between different perspectives both within chapters, which draw on insights from experimental and computational work, and between chapters, through commentaries written by the contributors on each others' work. Students and researchers in cognitive and visual science will find much to interest them in this thought-provoking collection.

Now in paperback, Topology via Logic is an advanced textbook on topology for computer scientists. Based on a course given by the author to postgraduate students of computer science at Imperial College, it has three unusual features. First, the introduction is from the locale viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the methods of locale theory are freely exploited. Third, there is substantial discussion of some computer science applications. Although books on topology aimed at mathematics exist, no book has been written specifically for computer scientists. As computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap.

The rewards and dangers of speculating in the modern financial markets have come to the fore in recent times with the collapse of banks and bankruptcies of public corporations as a direct result of ill-judged investment. At the same time, individuals are paid huge sums to use their mathematical skills to make well-judged investment decisions. Here now is the first rigorous and accessible account of the mathematics behind the pricing, construction and hedging of derivative securities. Key concepts such as martingales, change of measure, and the Heath-Jarrow-Morton model are described with mathematical precision in a style tailored for market practitioners. Starting from discrete-time hedging on binary trees, continuous-time stock models (including Black-Scholes) are developed. Practicalities are stressed, including examples from stock, currency and interest rate markets, all accompanied by graphical illustrations with realistic data. A full glossary of probabilistic and financial terms

This book is about abduction, 'the logic of Sherlock Holmes', and about how some kinds of abductive reasoning can be programmed in a computer. The work brings together Artificial Intelligence and philosophy of science and is rich with implications for other areas such as, psychology, medical informatics, and linguistics. It also has subtle implications for evidence evaluation in areas such as accident investigation, confirmation of scientific theories, law, diagnosis, and financial auditing. The book is about certainty and the logico-computational foundations of knowledge; it is about inference in perception, reasoning strategies, and building expert systems.

Recent and ongoing developments in science and technology - such as the prevention and treatment of disease through genetics and the development of increasingly sophisticated computer systems with wide-ranging applications - hold out the promise of vastly improving the quality of human life, but they can also raise serious ethical, legal, and public policy questions. The thirteen essays in this volume address these questions and related issues from a variety of perspectives. Written by prominent philosophers, economists, and legal theorists, these essays offer valuable insights into the nature of scientific innovation and its implications for our social policies and practices.