Mathematical Foundation for Computer Science

Cambridge University's Lucasian Professorship of Mathematics is one of the most celebrated academic positions in the world. Since its foundation in 1663, the chair has been held by seventeen men who represent some of the best and most influential minds in science and technology. Principally a social history of mathematics and physics, the story of these great natural philosophers and mathematical physicists is told here by some of the finest historians of science. The journey begins with the search for a benefactor able to establish a 'mathematicus professor honorarius', and travels through the life and work of the professors, exploring aspects from the heroic to the absurd. Covering both the great similarities and the extreme differences in mathematical physics over the last four centuries, this informative work offers interesting perspectives on world-famous scientists including Isaac Newton, Charles Babbage, G. G. Stokes, Paul Dirac and Stephen Hawking.

Digital Design: An Embedded Systems Approach Using VHDL provides a foundation in digital design for students in computer engineering, electrical engineering and computer science courses. It takes an u

Many aspects of the internal and external workings of computers can be viewed as a series of communication processes. Communication complexity is the mathematical theory of such communication processes. It is also often used as an abstract model of other aspects of computation. This book surveys this mathematical theory, concentrating on the question of how much communication is necessary for any particular process. The first part of the book is devoted to the simple two-party model introduced by Yao in 1979, which is still the most widely studied model. The second part treats newer models developed to deal with more complicated communication processes. Finally, applications of these models, including computer networks, VLSI circuits, and data structures, are treated in the third part of the book. This is an essential resource for graduate students and researchers in theoretical computer science, circuits, networks and information theory.

This is the 2005 second edition of a highly successful and well-respected textbook on the numerical techniques used to solve partial differential equations arising from mathematical models in science, engineering and other fields. The authors maintain an emphasis on finite difference methods for simple but representative examples of parabolic, hyperbolic and elliptic equations from the first edition. However this is augmented by new sections on finite volume methods, modified equation analysis, symplectic integration schemes, convection-diffusion problems, multigrid, and conjugate gradient methods; and several sections, including that on the energy method of analysis, have been extensively rewritten to reflect modern developments. Already an excellent choice for students and teachers in mathematics, engineering and computer science departments, the revised text includes more latest theoretical and industrial developments.

Providing a basic foundation for advanced graduate study and research in the mechanics of solids, this 2004 treatise contains a systematic development of the fundamentals of finite inelastic deformations of heterogeneous materials. The book combines the mathematical rigour of solid mechanics with the physics-based micro-structural understanding of the material science, to present a coherent picture of finite inelastic deformation of single and polycrystalline metals, over broad ranges of strain rates and temperatures. It also includes a similarly rigourous and experimentally based development of the quasi-static deformation of cohesionless granular materials that support the applied loads through contact friction. Every effort has been made to provide a thorough treatment of the subject, rendering the book accessible to students in solid mechanics and in the mechanics of materials. This book integrates rigourous mathematical description of finite deformations seamlessly with mechanisms

Alfred Tarski, one of the greatest logicians of all time, is widely thought of as 'the man who defined truth'. His work on the concepts of truth and logical consequence are cornerstones of modern logic, influencing developments in mathematics, philosophy, linguistics, and computer science. Tarski was a charismatic teacher and zealous promoter of his view of logic as the foundation of all rational thought, a bon vivant and a womanizer, who played the 'great man' to the hilt. A fortuitous trip to the United States at the outbreak of World War II saved his life and turned his career around, even while it separated him from his family for years. From the cafés of Warsaw and Vienna to the mountains and deserts of California, this first full-length biography places Tarski in the social, intellectual, and historical context of his times and presents a frank, vivid picture of a personally and professionally passionate man - interlaced with an account of his major scientific achievements.

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. The heart of this book is the language of generalized operads. This is as natural and transparent a language for higher category theory as the language of sheaves is for algebraic geometry, or vector spaces for linear algebra. It is introduced carefully, then used to give simple descriptions of a variety of higher categorical structures. In particular, one possible definition of n-category is discussed in detail, and some common aspects of other possible definitions are established. This is the first book on the subject and lays its foundations. It will appeal to both graduate students and established researchers who wish to become acquainted with this modern branch of mathematics.

This textbook is an introduction to algebra via examples. The book moves from properties of integers, through other examples, to the beginnings of group theory. Applications to public key codes and to error correcting codes are emphasised. These applications, together with sections on logic and finite state machines, make the text suitable for students of computer science as well as mathematics students. Attention is paid to historical development of the mathematical ideas. This second edition contains new material on mathematical reasoning skills and a new chapter on polynomials has been added. The book was developed from first-level courses taught in the UK and USA. These courses proved successful in developing not only a theoretical understanding but also algorithmic skills. This book can be used at a wide range of levels: it is suitable for first- or second-level university students, and could be used as enrichment material for upper-level school students.

A First Course in Combinatorial Optimization is a text for a one-semester introductory graduate-level course for students of operations research, mathematics, and computer science. It is a self-contained treatment of the subject, requiring only some mathematical maturity. Topics include: linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. Central to the exposition is the polyhedral viewpoint, which is the key principle underlying the successful integer-programming approach to combinatorial-optimization problems. Another key unifying topic is matroids. The author does not dwell on data structures and implementation details, preferring to focus on the key mathematical ideas that lead to useful models and algorithms. Problems and exercises are included throughout as well as references for further study.

A First Course in Combinatorial Optimization is a text for a one-semester introductory graduate-level course for students of operations research, mathematics, and computer science. It is a self-contained treatment of the subject, requiring only some mathematical maturity. Topics include: linear and integer programming, polytopes, matroids and matroid optimization, shortest paths, and network flows. Central to the exposition is the polyhedral viewpoint, which is the key principle underlying the successful integer-programming approach to combinatorial-optimization problems. Another key unifying topic is matroids. The author does not dwell on data structures and implementation details, preferring to focus on the key mathematical ideas that lead to useful models and algorithms. Problems and exercises are included throughout as well as references for further study.

Cambridge University's Lucasian Professorship of Mathematics is one of the most celebrated academic positions in the world. Since its foundation in 1663, the chair has been held by seventeen men who represent some of the best and most influential minds in science and technology. Principally a social history of mathematics and physics, the story of these great natural philosophers and mathematical physicists is told here by some of the finest historians of science. The journey begins with the search for a benefactor able to establish a 'mathematicus professor honorarius', and travels through the life and work of the professors, exploring aspects from the heroic to the absurd. Covering both the great similarities and the extreme differences in mathematical physics over the last four centuries, this informative work offers interesting perspectives on world-famous scientists including Isaac Newton, Charles Babbage, G. G. Stokes, Paul Dirac and Stephen Hawking.

Mobile systems, whose components communicate and change their structure, now pervade the informational world and the wider world of which it is a part. The science of mobile systems is as yet immature, however. This book presents the pi-calculus, a theory of mobile systems. The pi-calculus provides a conceptual framework for understanding mobility, and mathematical tools for expressing systems and reasoning about their behaviours. The book serves both as a reference for the theory and as an extended demonstration of how to use pi-calculus to describe systems and analyse their properties. It covers the basic theory of pi-calculus, typed pi-calculi, higher-order processes, the relationship between pi-calculus and lambda-calculus, and applications of pi-calculus to object-oriented design and programming. The book is written at the graduate level, assuming no prior acquaintance with the subject, and is intended for computer scientists interested in mobile systems.

Term rewriting systems developed out of mathematical logic and are an important part of theoretical computer science. They consist of sequences of discrete transformation steps where one term is replaced with another and have applications in many areas, from functional programming to automatic theorem proving and computer algebra. This 2003 book starts at an elementary level with the earlier chapters providing a foundation for the rest of the work. Much of the advanced material appeared here for the first time in book form. Subjects treated include orthogonality, termination, completion, lambda calculus, higher-order rewriting, infinitary rewriting and term graph rewriting. Many exercises are included with selected solutions provided on the web. A comprehensive bibliography makes this book ideal both for teaching and research. A chapter is included presenting applications of term rewriting systems, with many pointers to actual implementations.

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume II, on formal (ZFC) set theory, incorporates a self-contained 'chapter 0' on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques will provide the reader with a solid foundation in set theory and provides a context for the presentation of advanced topics such as absoluteness, relative consistency results, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing.

This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realised its worth. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. Ideal for anyone wanting to learn modern modal logic.

Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introd

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Basics of Holography is a general introduction to the subject written by a leading worker in the field. It begins with the theory of holographic imaging, the characteristics of the reconstructed image, and the various types of holograms. Practical aspects of holography are then described, including light sources, the characteristics of recording media and recording materials, as well as methods for producing different types of holograms and computer-generated holograms. Finally, important applications of holography are discussed, such as high-resolution imaging, holographic optical elements, information processing, and holographic interferometry. The book includes comprehensive reference sections and appendices summarizing some useful mathematical results. Numerical problems with their solutions are provided at the ends of chapters. This is an invaluable resource for advanced undergraduate and graduate students as well as researchers in science and engineering who would like to learn m

Basics of Holography is a general introduction to the subject written by a leading worker in the field. It begins with the theory of holographic imaging, the characteristics of the reconstructed image, and the various types of holograms. Practical aspects of holography are then described, including light sources, the characteristics of recording media and recording materials, as well as methods for producing different types of holograms and computer-generated holograms. Finally, important applications of holography are discussed, such as high-resolution imaging, holographic optical elements, information processing, and holographic interferometry. The book includes comprehensive reference sections and appendices summarizing some useful mathematical results. Numerical problems with their solutions are provided at the ends of chapters. This is an invaluable resource for advanced undergraduate and graduate students as well as researchers in science and engineering who would like to learn m

Provides an innovative hands-on introduction to techniques for specifying the behaviour of software components. It is primarily intended for use as a text book for a course in the 2nd or 3rd year of Computer Science and Computer Engineering programs, but it is also suitable for self-study. Using this book will help the reader improve programming skills and gain a sound foundation and motivation for subsequent courses in advanced algorithms and data structures, software design, formal methods, compilers, programming languages, and theory. The presentation is based on numerous examples and case studies appropriate to the level of programming expertise of the intended readership. The main topics covered are techniques for using programmer-friendly assertional notations to specify, develop, and verify small but non-trivial algorithms and data representations, and the use of state diagrams, grammars, and regular expressions to specify and develop recognizers for formal languages.

Provides an innovative hands-on introduction to techniques for specifying the behaviour of software components. It is primarily intended for use as a text book for a course in the 2nd or 3rd year of Computer Science and Computer Engineering programs, but it is also suitable for self-study. Using this book will help the reader improve programming skills and gain a sound foundation and motivation for subsequent courses in advanced algorithms and data structures, software design, formal methods, compilers, programming languages, and theory. The presentation is based on numerous examples and case studies appropriate to the level of programming expertise of the intended readership. The main topics covered are techniques for using programmer-friendly assertional notations to specify, develop, and verify small but non-trivial algorithms and data representations, and the use of state diagrams, grammars, and regular expressions to specify and develop recognizers for formal languages.

Structural proof theory is a branch of logic that studies the general structure and properties of logical and mathematical proofs. This book is both a concise introduction to the central results and methods of structural proof theory, and a work of research that will be of interest to specialists. The book is designed to be used by students of philosophy, mathematics and computer science. The book contains a wealth of results on proof-theoretical systems, including extensions of such systems from logic to mathematics, and on the connection between the two main forms of structural proof theory - natural deduction and sequent calculus. The authors emphasize the computational content of logical results. A special feature of the volume is a computerized system for developing proofs interactively, downloadable from the web and regularly updated.

Written for a broad audience of students in mathematics, computer science, operations research, statistics, and engineering, this textbook presents a short, lively survey of several fascinating non-calculus topics in modern applied mathematics. Coverage includes probability, mathematical finance, graphs, linear programming, statistics, computer science algorithms, and groups. A key feature is the abundance of interesting examples not normally found in standard finite mathematics courses, such as options pricing and arbitrage, tournaments, and counting formulas. The author assumes a level of mathematical sophistication at the beginning calculus level, that is, students should have had at least a course in pre-calculus, and the added sophistication attained from studying calculus would be useful.

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to sel

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Oriented matroids are a very natural mathematical concept which presents itself in many different guises and which has connections and applications to many different areas. These include discrete and computational geometry, combinatorics, convexity, topology, algebraic geometry, operations research, computer science and theoretical chemistry. This is the second edition of the first comprehensive, accessible account of the subject. It is intended for a diverse audience: graduate students who wish to learn the subject from scratch; researchers in the various fields of application who want to concentrate on certain aspects of the theory; specialists who need a thorough reference work; and others at academic points in between. A list of exercises and open problems ends each chapter. For the second edition, the authors have expanded the bibliography greatly to ensure that it remains comprehensive and up-to-date, and they have also added an appendix surveying research since the work was firs

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.

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.

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.

A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logicLogic of Mathematics combin

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 book covers elementary aspects of category theory and topos theory for graduate students in mathematics, computer science, and logic; it has few mathematical prerequisites, and uses categorical me

Atmospheric analysis is the method of transforming the enormous and scattered database of atmospheric observations into the powerful mathematical and physical tools required to meet the growing demand for atmospheric data. Atmospheric Data Analysis is intended to fill a void in the atmospheric science literature and curricula. The book is self contained, and includes topics important in several other fields outside atmospheric observation, including atmospheric dynamics and statistics. It outlines the physical and mathematical basis of all aspects of atmospheric analysis. The emphasis is on the theoretical foundation of the subject and most of the development is analytic, but many practical considerations and examples are introduced. There are numerous exercises at the end of each chapter to aid the student in comprehending the material.

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic (intuitionistic type theories) is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificia

In this book, which was originally published in 1985, Arto Salomaa gives an introduction to certain mathematical topics central to theoretical computer science: computability and recursive functions, formal languages and automata, computational complexity and cryptography. Without sacrificing readability, the presentation is essentially self-contained, with detailed proofs of all statements provided. Professor Salomaa is well known for his books in this area. The present work provides an insight into the basics, together with explanations of some of the more important developments in the field.

Recursion theory – now a well-established branch of pure mathematics, having grown rapidly over the last 35 years – deals with the general (abstract) theory of those operations which we conceive as being `computable' by idealized machines. The theory grew out of, and is usually still regarded, as a branch of mathematical logic. This book is a collection of advanced research/survey papers by eminent research workers in the field, based on their lectures given at the Leeds Logic Colloquium 1979. As such it provides an up-to-date view of current ideas and developments in the field of recursion theory as a whole. The individual contributions fit together naturally so as to provide an overview of all the main areas of research in the field. It will therefore be an important and invaluable source for advanced researchers and research students in mathematics and computer science (particularly in Europe, USA and USSR).

By the time students have done some programming in one or two languages and have learnt the common ways of representing information in a computer, they will want to embark upon further study of theoretical or applied topics in computer science. Most will encounter problems that require for their solution one or more of the techniques described in this book: for example problems depending upon the formation and solution of different equations; the task of making lists of possible alternatives and of answering questions about them; or the search for discrete optima. Written by the same authors as the highly successful Information Representation and Manipulation in a Computer, this book describes algorithms of mathematical methods and illustrates their application with examples. The mathematical background needed is elementary algebra and calculus. Numerous exercises are provided, with hints to their solutions.

This book provides a comprehensive foundation of machine learning. To answer the questions of what to learn, how to learn, what to get from learning, and how to evaluate, as well as what is meant by learning, the book focuses on the fundamental basics of machine learning, its methodology, theory, algorithms, and evaluations, together with some philosophical thinking on comparison between machine learning and human learning for machinery intelligence.The book is organized as follows: Introduction (Chapter 1), Evaluation (Chapter 2), Supervised learning (Chapters 3, 4, and 5), Unsupervised learning (Chapter 6), Representation learning (Chapter 7), Problem decomposition (Chapter 8), Ensemble learning (Chapter 9), Deep learning (Chapter 10), Application (Chapter 11), and Challenges (Chapter 12).The book can be used as a textbook for college, undergraduate, graduate and PhD students majored in computer science, automation, electronic engineering, communication, ect. It can also be used as

Rosen's Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors.

In the past several decades the classical Perron–Frobenius theory for nonnegative matrices has been extended to obtain remarkably precise and beautiful results for classes of nonlinear maps. This nonlinear Perron–Frobenius theory has found significant uses in computer science, mathematical biology, game theory and the study of dynamical systems. This is the first comprehensive and unified introduction to nonlinear Perron–Frobenius theory suitable for graduate students and researchers entering the field for the first time. It acquaints the reader with recent developments and provides a guide to challenging open problems. To enhance accessibility, the focus is on finite dimensional nonlinear Perron–Frobenius theory, but pointers are provided to infinite dimensional results. Prerequisites are little more than basic real analysis and topology.