Optional sections designated as "excursion" and "exploration" present interesting sidelights of graph theory and touch upon topics that allow students the opportunity to experiment and use their imaginations. Three appendixes review important facts about sets and logic, equivalence relations and functions, and the methods of proof. The text concludes with solutions or hints for odd-numbered exercises, in addition to references, indexes, and a list of symbols.
Chapters 0–3 cover fundamental operations involving sets and the principle of mathematical induction, and standard combinatorial topics: basic counting principles, permutations, combinations, the inclusion-exclusion principle, generating functions, recurrence relations, and an introduction to the analysis of algorithms. Applications are emphasized wherever possible and more than 200 exercises at the ends of these chapters help students test their grasp of the material.
Chapters 4 and 5 survey graphs and digraphs, including their connectedness properties, applications of graph coloring, and more, with stress on applications to coding and other related problems. Two important problems in network optimization ― the minimal spanning tree problem and the shortest distance problem ― are covered in the last two chapters. A very brief nontechnical exposition of the theory of computational complexity and NP-completeness is outlined in the appendix.
Modern cryptology has been described as the science of the integrity of information, covering all aspects like confidentiality, authenticity and non-repudiation and also including the protocols required for achieving these aims. In both theory and practice it requires notions and constructions from three major disciplines: computer science, electronic engineering and mathematics. Within mathematics, group theory, the theory of finite fields, and elementary number theory as well as some topics not normally covered in courses in algebra, such as the theory of Boolean functions and Shannon theory, are involved.Although essentially self-contained, a degree of mathematical maturity on the part of the reader is assumed, corresponding to his or her background in computer science or engineering. Algebra for Cryptologists is a textbook for an introductory course in cryptography or an upper undergraduate course in algebra, or for self-study in preparation for postgraduate study in cryptology.
This highly versatile text provides mathematical background used in a wide variety of disciplines, including mathematics and mathematics education, computer science, biology, chemistry, engineering, communications, and business.
Some of the major features and strengths of this textbook
More than 1,600 exercises, ranging from elementary to challenging, are included with hints/answers to all odd-numbered exercises.
Descriptions of proof techniques are accessible and lively.
Students benefit from the historical discussions throughout the textbook.
"Seamless R and C++ integration with Rcpp" is simply a wonderful book. For anyone who uses C/C++ and R, it is an indispensable resource. The writing is outstanding. A huge bonus is the section on applications. This section covers the matrix packages Armadillo and Eigen and the GNU Scientific Library as well as RInside which enables you to use R inside C++. These applications are what most of us need to know to really do scientific programming with R and C++. I love this book. -- Robert McCulloch, University of Chicago Booth School of Business
Rcpp is now considered an essential package for anybody doing serious computational research using R. Dirk's book is an excellent companion and takes the reader from a gentle introduction to more advanced applications via numerous examples and efficiency enhancing gems. The book is packed with all you might have ever wanted to know about Rcpp, its cousins (RcppArmadillo, RcppEigen .etc.), modules, package development and sugar. Overall, this book is a must-have on your shelf. -- Sanjog Misra, UCLA Anderson School of Management
The Rcpp package represents a major leap forward for scientific computations with R. With very few lines of C++ code, one has R's data structures readily at hand for further computations in C++. Hence, high-level numerical programming can be made in C++ almost as easily as in R, but often with a substantial speed gain. Dirk is a crucial person in these developments, and his book takes the reader from the first fragile steps on to using the full Rcpp machinery. A very recommended book! -- Søren Højsgaard, Department of Mathematical Sciences, Aalborg University, Denmark
"Seamless R and C ++ Integration with Rcpp" provides the first comprehensive introduction to Rcpp. Rcpp has become the most widely-used language extension for R, and is deployed by over one-hundred different CRAN and BioConductor packages. Rcpp permits users to pass scalars, vectors, matrices, list or entire R objects back and forth between R and C++ with ease. This brings the depth of the R analysis framework together with the power, speed, and efficiency of C++.
Dirk Eddelbuettel has been a contributor to CRAN for over a decade and maintains around twenty packages. He is the Debian/Ubuntu maintainer for R and other quantitative software, edits the CRAN Task Views for Finance and High-Performance Computing, is a co-founder of the annual R/Finance conference, and an editor of the Journal of Statistical Software. He holds a Ph.D. in Mathematical Economics from EHESS (Paris), and works in Chicago as a Senior Quantitative Analyst.
The second edition adds a discussion of vector auto-regressive, structural vector auto-regressive, and structural vector error-correction models. To analyze the interactions between the investigated variables, further impulse response function and forecast error variance decompositions are introduced as well as forecasting. The author explains how these model types relate to each other.
The introductory chapter presents an overview of the foundations of the subject, consisting of topics such as logic, set theory, relations, functions, algebraic structures, and graphs. The subsequent chapters provide detailed coverage of each of these topics as well as major areas of discrete mathematics such as combinatorics, lattices and Boolean algebras. Major applications such as computer models and computation, coding theory, cryptography and databases are dealt with in the final chapters of the book. In addition to this, a new chapter on matrices is included in this edition of the book, which forms a part of MCA course curriculum.
The book is replete with features which enable the building of a firm foundation of the underlying principles of the subject and also provide adequate scope for testing the comprehension acquired by the students. Each chapter contains numerous worked-out examples within the main discussion as well as several chapter-end Supplementary Examples for revision. The Self-Test and Exercises at the end of each chapter provide large numbers of objective type questions and problems respectively. Answers to objective type questions and hints to exercises are also provided. All these pedagogic features, together with thorough coverage of the subject matter, make this book a readable text for beginners as well as advanced learners of the subject.
The lively and accessible text requires only minimal scientific background. Designed for senior college or beginning graduate-level students, it assumes only elementary calculus and basic probability theory for the first part, and ordinary differential equations and continuous probability for the second section. All problems require students to study and create models, encouraging their active participation rather than a mechanical approach.
Beyond the classroom, this volume will prove interesting and rewarding to anyone concerned with the development of mathematical models or the application of modeling to problem solving in a wide array of applications.
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This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade.
The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual.
Key Features:Suitable for a variety of courses for students in both Mathematics and Computer Science.Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematicsConcise, clear and uncluttered presentation with numerous examples.Covers some applications including cryptographic systems, discrete probability and network algorithms.
Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study.
The remaining six chapters are more advanced, covering graph theory algorithms and computer programs, graphs in switching and coding theory, electrical network analysis by graph theory, graph theory in operations research, and more. Instructors may combine these chapters with the preceding material for courses in a variety of fields, including electrical engineering, computer science, operations research, and applied mathematics.
Each chapter includes full-colour illustrations to clarify the mathematical descriptions, and in some cases, equations are also coloured to reveal vital algebraic patterns. The numerous worked examples will consolidate comprehension of abstract mathematical concepts.
Foundation Mathematics for Computer Science covers number systems, algebra, logic, trigonometry, coordinate systems, determinants, vectors, matrices, geometric matrix transforms, differential and integral calculus, and reveals the names of the mathematicians behind such inventions. During this journey, John Vince touches upon more esoteric topics such as quaternions, octonions, Grassmann algebra, Barycentric coordinates, transfinite sets and prime numbers. Whether you intend to pursue a career in programming, scientific visualisation, systems design, or real-time computing, you should find the author’s literary style refreshingly lucid and engaging, and prepare you for more advanced texts.
The first two chapters provide the foundations of graph theoretic notions and algorithmic techniques. The remaining chapters discuss the topics of planarity testing, embedding, drawing, vertex- or edge-coloring, maximum independence set, subgraph listing, planar separator theorem, Hamiltonian cycles, and single- or multicommodity flows.
Suitable for a course on algorithms, graph theory, or planar graphs, the volume will also be useful for computer scientists and graph theorists at the research level. An extensive reference section is included.
Mathematical modelling is a highly useful methodology designed to enable mathematicians, physicists and other scientists to formulate equations from a given nonmathematical situation. In this elegantly written volume, a distinguished theoretical chemist and engineer sets down helpful rules not only for setting up models but also for solving the mathematical problems they pose and for evaluating models.
The author begins with a discussion of the term "model," followed by clearly presented examples of the different types of models (finite, statistical, stochastic, etc.). He then goes on to discuss the formulation of a model and how to manipulate it into its most responsive form. Along the way Dr. Aris develops a delightful list of useful maxims for would-be modellers. In the final chapter he deals not only with the empirical validation of models but also with the comparison of models among themselves, as well as with the extension of a model beyond its original "domain of validity."
Filled with numerous examples, this book includes three appendices offering further examples treated in more detail. These concern longitudinal diffusion in a packed bed, the coated tube chromatograph with Taylor diffusion and the stirred tank reactor. Six journal articles, a useful list of references and subject and name indexes complete this indispensable, well-written guide.
"A most useful, readable-and stimulating-book, to be read both for pleasure and for enlightenment." — Bulletin of the Institute of Mathematics and Its Applications
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Answers are found to many questions that usually remain unanswered: Why is the truth table for logical implication so unintuitive? Why are there no recipes to design proofs? Where do these numerous mathematical rules come from? What issues in logic, mathematics, and computer science still remain unresolved? And the perennial question: In what ways are we going to use this material? Additionally, the selection of topics presented reflects many major accomplishments from the twentieth century and includes applications in game theory and Nash'sequilibrium, Gale and Shapley's match making algorithms, Arrow's Impossibility Theorem in voting, to name a few.
From the reviews of the first edition:
"...All the results are proved in full detail from first principles...remarkably, the arithmetic laws on the rational numbers are proved, step after step, starting from the very definitions!...This is a valuable reference text and a useful companion for anybody wondering how basic mathematical concepts can be rigorously developed within set theory."
"Rigorous and modern in its theoretical aspect, attractive as a detective novel in its applied aspects, this paper book deserves the attention of both beginners and advanced students in mathematics, logic and computer sciences as well as in social sciences."
Optimization modeling and algorithms are key components toproblem-solving across various fields of research, from operationsresearch and mathematics to computer science and engineering.Addressing the importance of the algorithm design process.Deterministic Operations Research focuses on the design ofsolution methods for both continuous and discrete linearoptimization problems. The result is a clear-cut resource forunderstanding three cornerstones of deterministic operationsresearch: modeling real-world problems as linear optimizationproblem; designing the necessary algorithms to solve theseproblems; and using mathematical theory to justify algorithmicdevelopment.
Treating real-world examples as mathematical problems, theauthor begins with an introduction to operations research andoptimization modeling that includes applications form sportsscheduling an the airline industry. Subsequent chapters discussalgorithm design for continuous linear optimization problems,covering topics such as convexity. Farkas’ Lemma, and thestudy of polyhedral before culminating in a discussion of theSimplex Method. The book also addresses linear programming dualitytheory and its use in algorithm design as well as the Dual SimplexMethod. Dantzig-Wolfe decomposition, and a primal-dual interiorpoint algorithm. The final chapters present network optimizationand integer programming problems, highlighting various specializedtopics including label-correcting algorithms for the shortest pathproblem, preprocessing and probing in integer programming, liftingof valid inequalities, and branch and cut algorithms.
Concepts and approaches are introduced by outlining examplesthat demonstrate and motivate theoretical concepts. The accessiblepresentation of advanced ideas makes core aspects easy tounderstand and encourages readers to understand how to think aboutthe problem, not just what to think. Relevant historical summariescan be found throughout the book, and each chapter is designed asthe continuation of the “story” of how to both modeland solve optimization problems by using the specificproblems-linear and integer programs-as guides. The book’svarious examples are accompanied by the appropriate models andcalculations, and a related Web site features these models alongwith Maple™ and MATLAB® content for the discussedcalculations.
Thoroughly class-tested to ensure a straightforward, hands-onapproach, Deterministic Operations Research is an excellentbook for operations research of linear optimization courses at theupper-undergraduate and graduate levels. It also serves as aninsightful reference for individuals working in the fields ofmathematics, engineering, computer science, and operations researchwho use and design algorithms to solve problem in their everydaywork.
Examples are drawn from fields as diverse as function approximation, spatial statistics, boundary value problems, machine learning, surrogate modeling and finance. Researchers from those and other fields can recreate the results within using the documented MATLAB code, also available through the online library. This combination of a strong theoretical foundation and accessible experimentation empowers readers to use positive definite kernels on their own problems of interest.
"...complete, up-to-date coverage of computational complexitytheory...the book promises to become the standard reference oncomputational complexity." -Zentralblatt MATH
A thorough revision based on advances in the field ofcomputational complexity and readers’ feedback, the SecondEdition of Theory of Computational Complexity presentsupdates to the principles and applications essential tounderstanding modern computational complexity theory. The newedition continues to serve as a comprehensive resource on the useof software and computational approaches for solving algorithmicproblems and the related difficulties that can be encountered.
Maintaining extensive and detailed coverage, Theory ofComputational Complexity, Second Edition, examines the theoryand methods behind complexity theory, such as computational models,decision tree complexity, circuit complexity, and probabilisticcomplexity. The Second Edition also features recentdevelopments on areas such as NP-completeness theory, as wellas:A new combinatorial proof of the PCP theorem based on thenotion of expander graphs, a research area in the field of computerscienceAdditional exercises at varying levels of difficulty to furthertest comprehension of the presented materialEnd-of-chapter literature reviews that summarize each topic andoffer additional sources for further study
Theory of Computational Complexity, Second Edition, is anexcellent textbook for courses on computational theory andcomplexity at the graduate level. The book is also a usefulreference for practitioners in the fields of computer science,engineering, and mathematics who utilize state-of-the-art softwareand computational methods to conduct research.Athorough revision based on advances in the field of computationalcomplexity and readers’feedback,the Second Edition of Theory of Computational Complexity presentsupdates to theprinciplesand applications essential to understanding modern computationalcomplexitytheory.The new edition continues to serve as a comprehensive resource onthe use of softwareandcomputational approaches for solving algorithmic problems and therelated difficulties thatcanbe encountered.Maintainingextensive and detailed coverage, Theory of ComputationalComplexity, SecondEdition,examines the theory and methods behind complexity theory, such ascomputationalmodels,decision tree complexity, circuit complexity, and probabilisticcomplexity. The SecondEditionalso features recent developments on areas such as NP-completenesstheory, as well as:•A new combinatorial proof of the PCP theorem based on the notion ofexpandergraphs,a research area in the field of computer science•Additional exercises at varying levels of difficulty to furthertest comprehension ofthepresented material•End-of-chapter literature reviews that summarize each topic andoffer additionalsourcesfor further studyTheoryof Computational Complexity, Second Edition, is an excellenttextbook for courses oncomputationaltheory and complexity at the graduate level. The book is also auseful referenceforpractitioners in the fields of computer science, engineering, andmathematics who utilizestate-of-the-artsoftware and computational methods to conduct research.
The logical systems presented are: propositional logic, first-order logic, resolution and its application to logic programming, Hoare logic for the verification of sequential programs, and linear temporal logic
for the verification of concurrent programs.
The third edition has been entirely rewritten and includes new chapters on central topics of modern computer science: SAT solvers and model checking.
Covering all the mathematical techniques required to resolve geometric problems and design computer programs for computer graphic applications, each chapter explores a specific mathematical topic prior to moving forward into the more advanced areas of matrix transforms, 3D curves and surface patches. Problem-solving techniques using vector analysis and geometric algebra are also discussed.
All the key areas are covered including: Numbers, Algebra, Trigonometry, Coordinate geometry, Transforms, Vectors, Curves and surfaces, Barycentric coordinates, Analytic geometry.
Plus – and unusually in a student textbook – a chapter on geometric algebra is included.
The notes are written in a rather colloquial style, presenting the subject matter in the same form as it can be explained in a classroom. For instructors, this will minimize the amount of effort required to prepare their blackboard presentations.
As prerequisites, the book only relies on standard calculus, an introductory course on matrices, and some basic computer programming skills. As a new feature, these notes are supplemented by two sets of videos from the author's Youtube channel. These videos contain a complete set of live lectures given in Spring 2015, together with a complete set of short tutorials, from 5 to 15 minutes each.
A set of homework problems is included at the end of each chapter. Homework projects cover a variety of applications, in connection with population dynamics, engineering, mechanics, image reconstruction, etc. A complete set of solutions is available for instructors, upon request.
New to the Fifth Edition
New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings New examples, figures, and applications to illustrate concepts and theorems Expanded historical discussions of well-known mathematicians and problems More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book Reorganization of sections into subsections to make the material easier to read Bolded definitions of terms, making them easier to locate
Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subject’s fascinating history and presents a host of interesting problems and diverse applications.
- the first category highlights those essential parts of the text that demonstrate the use of new commands in Mathematica whilst solving each problem presented;
- the second comprises problems that further demonstrate the use of commands previously introduced to tackle different situations; and
- the third presents more challenging problems for further study.
The intention is to enable the reader to learn from the codes, thus avoiding long and exhausting explanations.
While based on a computer algebra course taught to undergraduate students of mathematics, science, engineering and finance, the book also includes chapters on calculus and solving equations, and graphics, thus covering all the basic topics in Mathematica. With its strong focus upon programming and problem solving, and an emphasis on using numerical problems that do not need any particular background in mathematics, this book is also ideal for self-study and as an introduction to researchers who wish to use Mathematica as a computational tool.
Mathematica®: A Problem-Centered Approach comes with a free 30 day trial of the Wolfram Mathematica(R) software'
One of the youngest, most vital areas of applied mathematics,combinatorial optimization integrates techniques fromcombinatorics, linear programming, and the theory of algorithms.Because of its success in solving difficult problems in areas fromtelecommunications to VLSI, from product distribution to airlinecrew scheduling, the field has seen a ground swell of activity overthe past decade.
Combinatorial Optimization is an ideal introduction to thismathematical discipline for advanced undergraduates and graduatestudents of discrete mathematics, computer science, and operationsresearch. Written by a team of recognized experts, the text offersa thorough, highly accessible treatment of both classical conceptsand recent results. The topics include:
* Network flow problems
* Optimal matching
* Integrality of polyhedra
Featuring logical and consistent exposition, clear explanations ofbasic and advanced concepts, many real-world examples, and helpful,skill-building exercises, Combinatorial Optimization is certain tobecome the standard text in the field for many years to come.
This book will show you how to manipulate data in modern R structures and includes connecting R to data bases such as SQLite, PostgeSQL, and MongoDB. The book closes with a hands-on section to get R running in the cloud. Each chapter also includes a detailed bibliography with references to research articles and other resources that cover relevant conceptual and theoretical topics.
What You Will LearnWrite and document R functionsMake an R package and share it via GitHub or privatelyAdd tests to R code to insure it works as intendedBuild packages automatically with GitHubUse R to talk directly to databases and do complex data managementRun R in the Amazon cloudGenerate presentation-ready tables and reports using R
Who This Book Is For
Working professionals, researchers, or students who are familiar with R and basic statistical techniques such as linear regression and who want to learn how to take their R coding and programming to the next level.
The text consists of six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebra. It explains in great detail the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebra. In addition to examples from engineering and science applications, proofs of required results are provided without leaving out critical details. The Preface suggests ways in which the book can be used with or without an intensive study of proofs.
This book will be a useful reference for graduate or advanced undergraduate students in engineering, science, and mathematics. It will also appeal to professionals in engineering and science, such as practicing engineers who want to see how numerical linear algebra problems can be solved using a programming language such as MATLAB, MAPLE, or Mathematica.Six introductory chapters that thoroughly provide the required background for those who have not taken a course in applied or theoretical linear algebraDetailed explanations and examplesA through discussion of the algorithms necessary for the accurate computation of the solution to the most frequently occurring problems in numerical linear algebraExamples from engineering and science applications
Discrete Mathematics for Computing presents the essential mathematics needed for the study of computing and information systems. The subject is covered in a gentle and informal style, but without compromising the need for correct methodology. It is perfect for students with a limited background in mathematics.
This new edition includes:
• An expanded section on encryption
• Additional examples of the ways in which theory can be applied to problems in computing
• Many more exercises covering a range of levels, from the basic to the more advanced
This book is ideal for students taking a one-semester introductory course in discrete mathematics - particularly for first year undergraduates studying Computing and Information Systems.
PETER GROSSMAN has worked in both MA26 and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing mathematics courses for Computing students. He has also applied his skills in areas as diverse as calculator design, irrigation systems and underground mine layouts. He lives and works in Melbourne, Australia.
Since its inception in the 1960s, the theory of random graphs hasevolved into a dynamic branch of discrete mathematics. Yet despitethe lively activity and important applications, the lastcomprehensive volume on the subject is Bollobas's well-known 1985book. Poised to stimulate research for years to come, this new workcovers developments of the last decade, providing a much-needed,modern overview of this fast-growing area of combinatorics. Writtenby three highly respected members of the discrete mathematicscommunity, the book incorporates many disparate results from acrossthe literature, including results obtained by the authors and somecompletely new results. Current tools and techniques are alsothoroughly emphasized. Clear, easily accessible presentations makeRandom Graphs an ideal introduction for newcomers to the field andan excellent reference for scientists interested in discretemathematics and theoretical computer science. Special featuresinclude:
* A focus on the fundamental theory as well as basic models ofrandom graphs
* A detailed description of the phase transition phenomenon
* Easy-to-apply exponential inequalities for large deviationbounds
* An extensive study of the problem of containing smallsubgraphs
* Results by Bollobas and others on the chromatic number of randomgraphs
* The result by Robinson and Wormald on the existence of Hamiltoncycles in random regular graphs
* A gentle introduction to the zero-one laws
* Ample exercises, figures, and bibliographic references