Mathematical Foundation for Computer Science

Discrete Mathematics is a textbook designed for the students of computer science engineering, information technology, and computer applications to help them develop the foundation of theoretical compu

?For undergraduate students in Computer Science and Computer Programming courses or beginning programmers A solid foundation in the basics of C++ programming will allow readers to create efficien

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

For business professionals and advanced undergraduates and graduate students in computer science and related fields, Akerkar (Technomathematics Research Foundation and Western Norway Research Institut

This book leads readers from a basic foundation to an advanced level understanding of algebra, logic and combinatorics. Perfect for graduate or PhD mathematical-science students looking for help in un

This book leads readers from a basic foundation to an advanced level understanding of algebra, logic and combinatorics. Perfect for graduate or PhD mathematical-science students looking for help in un

This book leads readers from a basic foundation to an advanced-level understanding of fluid and solid mechanics. Perfect for graduate or PhD mathematical-science students looking for help in understan

This book leads readers from a basic foundation to an advanced-level understanding of fluid and solid mechanics. Perfect for graduate or PhD mathematical-science students looking for help in understan

DISCRETE MATHEMATICS WITH APPLICATIONS, 5th Edition, explains complex, abstract concepts with clarity and precision and provides a strong foundation for computer science and upper-level mathematics co

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

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 short book, geared towards undergraduate students of computer science and mathematics, is specifically designed for a first course in mathematical logic.A proof of Godel's completeness theorem an

Introduction to Mathcad 15, 3/e is ideal for Freshman or Introductory courses in Engineering and Computer Science. Introduces Mathcad's basic mathematical and data analysis functions (e.g., trigonom

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

Intended for researchers and graduate students in theoretical computer science and mathematical logic, this volume contains accessible surveys by leading researchers from areas of current work in logical aspects of computer science, where both finite and infinite model-theoretic methods play an important role. Notably, the articles in this collection emphasize points of contact and connections between finite and infinite model theory in computer science that may suggest new directions for interaction. Among the topics discussed are: algorithmic model theory, descriptive complexity theory, finite model theory, finite variable logic, model checking, model theory for restricted classes of finite structures, and spatial databases. The chapters all include extensive bibliographies facilitating deeper exploration of the literature and further research.

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.

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.

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.

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.

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.

Mathematical modelling of physical and chemical systems is used extensively throughout science, engineering, and applied mathematics. To use mathematical models, one needs solutions to the model equations; this generally requires numerical methods. This book presents numerical methods and associated computer code in Matlab for the solution of a spectrum of models expressed as partial differential equations (PDEs). The authors focus on the method of lines (MOL), a well-established procedure for all major classes of PDEs, where the boundary value partial derivatives are approximated algebraically by finite differences. This reduces the PDEs to ordinary differential equations (ODEs) and makes the computer code easy to understand, implement, and modify. Also, the ODEs (via MOL) can be combined with any other ODEs that are part of the model (so that MOL naturally accommodates ODE/PDE models). This book uniquely includes a detailed line-by-line discussion of computer code related to the ass

Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, there is much coverage of techniques which could only be found in specialist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering or computer science background.

Originally published in 1988, this book presents an introduction to lambda-calculus and combinators without getting lost in the details of mathematical aspects of their theory. Lambda-calculus is treated here as a functional language and its relevance to computer science is clearly demonstrated. The main purpose of the book is to provide computer science students and researchers with a firm background in lambda-calculus and combinators and show the applicabillity of these theories to functional programming. The presentation of the material is self-contained. It can be used as a primary text for a course on functional programming. It can also be used as a supplementary text for courses on the structure and implementation of programming languages, theory of computing, or semantics of programming languages.

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.

Mathematics is not, and never will be, an empirical science, but mathematicians are finding that the use of computers and specialized software allows the generation of mathematical insight in the form of conjectures and examples, which pave the way for theorems and their proofs. In this way, the experimental approach to pure mathematics is revolutionizing the way research mathematicians work. As the first of its kind, this book provides material for a one-semester course in experimental mathematics that will give students the tools and training needed to systematically investigate and develop mathematical theory using computer programs written in Maple. Accessible to readers without prior programming experience, and using examples of concrete mathematical problems to illustrate a wide range of techniques, the book gives a thorough introduction to the field of experimental mathematics, which will prepare students for the challenge posed by open mathematical problems.

Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling of molecular structures, such as the Nobel prize winning fullerenes, occurring in chemistry and crystallography. The chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules. Here the authors give access to new results in the theory of polycycles and two-faced maps together with the relevant background material and mathematical tools for their study. Organised so that, after reading the introductory chapter, each chapter can be read independently from the others, the book should be accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography. Many of the results in the subject require the use of computer enumeration; the corresponding programs are available from the a

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.

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.

Distributed systems are fast becoming the norm in computer science. Formal mathematical models and theories of distributed behaviour are needed in order to understand them. This book proposes a distributed pi-calculus called Dpi, for describing the behaviour of mobile agents in a distributed world. It is based on an existing formal language, the pi-calculus, to which it adds a network layer and a primitive migration construct. A mathematical theory of the behaviour of these distributed systems is developed, in which the presence of types plays a major role. It is also shown how in principle this theory can be used to develop verification techniques for guaranteeing the behavior of distributed agents. The text is accessible to computer scientists with a minimal background in discrete mathematics. It contains an elementary account of the pi-calculus, and the associated theory of bisimulations. It also develops the type theory required by Dpi from first principles.

The modern view of causation can be traced back to the mechanistic science of Descartes, whose rejection of Aristotelian physics, with its concept of substantial forms, in favor of mechanical explanations was a turning-point in the history of philosophy. However the reasoning which led Descartes and other early moderns in this direction is not well understood. This book traces Descartes' groundbreaking theory of scientific explanation back to the mathematical demonstrations of Aristotelian mechanics and interprets these advances in light of the available arguments for and against substantial forms. It also examines how Descartes' new theory led him to develop a metaphysical foundation for his science that could avoid skeptical objections. It will appeal to a wide range of readers interested in the philosophy and science of the early modern period.

The modern view of causation can be traced back to the mechanistic science of Descartes, whose rejection of Aristotelian physics, with its concept of substantial forms, in favor of mechanical explanations was a turning-point in the history of philosophy. However the reasoning which led Descartes and other early moderns in this direction is not well understood. This book traces Descartes' groundbreaking theory of scientific explanation back to the mathematical demonstrations of Aristotelian mechanics and interprets these advances in light of the available arguments for and against substantial forms. It also examines how Descartes' new theory led him to develop a metaphysical foundation for his science that could avoid skeptical objections. It will appeal to a wide range of readers interested in the philosophy and science of the early modern period.

Ordinary Differential Equations introduces key concepts and techniques in the field and shows how they are used in current mathematical research and modelling. It deals specifically with initial value problems, which play a fundamental role in a wide range of scientific disciplines, including mathematics, physics, computer science, statistics and biology. This practical book is ideal for students and beginning researchers working in any of these fields who need to understand the area of ordinary differential equations in a short time.

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.

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.

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.

Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.

Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.

Today's pervasive computing and communications networks have created an intense need for secure and reliable cryptographic systems. Bringing together a fascinating mixture of topics in engineering, mathematics, computer science, and informatics, this book presents the timeless mathematical theory underpinning cryptosystems both old and new. Major branches of classical and modern cryptography are discussed in detail, from basic block and stream cyphers through to systems based on elliptic and hyperelliptic curves, accompanied by concise summaries of the necessary mathematical background. Practical aspects such as implementation, authentication and protocol-sharing are also covered, as are the possible pitfalls surrounding various cryptographic methods. Written specifically with engineers in mind, and providing a solid grounding in the relevant algorithms, protocols and techniques, this insightful introduction to the foundations of modern cryptography is ideal for graduate students and r