Series: Universitext
1st Edition., 2011, X, 467 p. 220 illus., 20 in color., Softcover
ISBN: 978-1-4419-8046-5
Due: January 29, 2011
Summarizing the rules of mathematical reasoning and how to construct proofs
Presents examples of formal and informal proofs
Includes examples of proofs by induction
Discusses public key cryptography, with a complete proof of the correctness of RSA Explicit, detailed algorithms for modular arithmetic
Explores graph flows and the max-flow min-cut theorem
Covers planar graphs
This book gives an introduction to discrete mathematics for beginning undergraduates and starts with a chapter on the rules of mathematical reasoning.
This book begins with a presentation of the rules of logic as used in mathematics where many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book deals with functions and relations, directed and undirected graphs and an introduction to combinatorics, partial orders and complete induction. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory where Eulerian and Hamiltonian cycles are discussed. This book also includes network flows, matchings, covering, bipartite graphs, planar graphs and state the graph minor theorem of Seymour and Robertson.
The book is highly illustrated and each chapter ends with a list of problems of varying difficulty. Undergraduates in mathematics and computer science will find this book useful.
Mathematical Reasoning, Proof Principles and Logic.- Relations, Functions, Partial Functions.- Graphs, Part I: Basic Notions.- Some Counting Problems; Multinomial Coefficients.- Partial Orders, GCD's, RSA, Lattices.- Graphs, Part II: More Advanced Notions.- Answers to Selected Problems.
Series: Monographs in Theoretical Computer Science. An EATCS Series
1st Edition., 2011, X, 480 p., Hardcover
ISBN: 978-3-642-17335-6
Due: March 23, 2011
This book provides foundations for software specification and formal software development from the perspective of work on algebraic specification, concentrating on developing basic concepts and studying their fundamental properties. These foundations are built on a solid mathematical basis, using elements of universal algebra, category theory and logic, and this mathematical toolbox provides a convenient language for precisely formulating the concepts involved in software specification and development. Once formally defined, these notions become subject to mathematical investigation, and this interplay between mathematics and software engineering yields results that are mathematically interesting, conceptually revealing, and practically useful.
The theory presented by the authors has its origins in work on algebraic specifications that started in the early 1970s, and their treatment is comprehensive. This book contains five kinds of material: the requisite mathematical foundations; traditional algebraic specifications; elements of the theory of institutions; formal specification and development; and proof methods.
While the book is self-contained, mathematical maturity and familiarity with the problems of software engineering is required; and in the examples that directly relate to programming, the authors assume acquaintance with the concepts of functional programming. The book will be of value to researchers and advanced graduate students in the areas of programming and theoretical computer science.
Introduction.- Universal Algebra.- Simple Equational Specifications.- Category Theory.- Working Within an Arbitrary Logical System.- Structured Specifications.- Parameterisation.- Formal Program Development.- Behavioural Specifications.- Proofs for Specifications.- Working with Multiple Logical Systems.- References.
Series: Springer Series in Statistics
1st Edition., 2011, XVIII, 488 p. 2 illus., Hardcover
ISBN: 978-1-4419-8341-1
Due: March 12, 2011
This book provides a theoretical foundation for the analysis of discrete data such as count and binary data in the longitudinal setup. Unlike the existing books, this book uses a class of auto-correlation structures to model the longitudinal correlations for the repeated discrete data that accommodates all possible Gaussian type auto-correlation models as special cases including the equi-correlation models. This new dynamic modelling approach is utilized to develop theoretically sound inference techniques such as the generalized quasi-likelihood (GQL) technique for consistent and efficient estimation of the underlying regression effects involved in the model, whereas the existing eworkingf correlations based GEE (generalized estimating equations) approach has serious theoretical limitations both for consistent and efficient estimation, and the existing random effects based correlations approach is not suitable to model the longitudinal correlations.
The book has exploited the random effects carefully only to model the correlations of the familial data. Subsequently, this book has modelled the correlations of the longitudinal data collected from the members of a large number of independent families by using the class of auto-correlation structures conditional on the random effects. The book also provides models and inferences for discrete longitudinal data in the adaptive clinical trial set up.
The book is mathematically rigorous and provides details for the development of estimation approaches under selected familial and longitudinal models. Further, while the book provides special cares for mathematics behind the correlation models, it also presents the illustrations of the statistical analysis of various real life data.
This book will be of interest to the researchers including graduate students in biostatistics and econometrics, among other applied statistics research areas.
Introduction. - An overview of linear models with correlated errors. - An overview of linear mixed models with correlated errors. - Generalized linear mixed models for familial count data. - Generalized linear mixed models for familial binary data. - Generalized linear models for longitudinal count data. - Generalized linear models for longitudinal binary data. - Generalized linear models for longitudinal count data with overdispersion. - Generalized linear models for longitudina binary data with overdispersion. - Generalized linear longitudinal mixed models. - Inferences for familiar data subject to non-response. - Inferences for longitudinal data subject to non-response. - References.
1st Edition., 2011, Approx. 350 p., Hardcover
ISBN: 978-3-7091-0447-7
Due: June 30, 2011
This work deals with several aspects of the Coulomb problem as well as with a large number of related multivariable hypergeometric functions. Intended for atomic physicists and mathematicians, it builds a bridge of understanding between both the Physical and Mathematical sciences. After a basic introduction on the Coulomb problem, it is divided into two parts. First, an original study of the two- and three-body Coulomb problem is presented with emphasis on related mathematical issues. Most of the aspects of this part have the common point that they involve a significant number of multivariable hypergeometric functions. These are studied in the second, mathematical, part of the book where emphasis is put on the relevant expressions for physical applications. This work will be an important tool in for atomic physicists dealing with two- and three-body Coulomb problems and includes aspects not addressed elsewhere. A significant number of hypergeometric functions in two, three and more variables related to physical problems are presented and studied. The aspect of considering both mathematical and physical aspects in connection with hypergeometric functions makes the book unique.