Diskret matematik

GrundbokDenna bok är avsedd som litteratur för en första kurs i diskret matematik. Boken syftar särskilt till att ge förståelse för den diskreta matematikens roll vid modellering. Fenomenen som modelleras finner man exempelvis i områden som datateknik, elektronik och politik.FördjupningDenna bok är avsedd för fortsättningskurser i diskret matematik och passar särskilt väl som påbyggnad till Diskret matematik och diskreta modeller av samma författare. Boken innehåller även en engelsk-svensk ordlista och förslag till projektuppgifter.

Den här boken är främst avsedd för grundläggande kurser i diskret matematik vid universitet och högskolor. Framför allt riktar den sig till första- och andraårsstudenter på data-, matematik-, civilingenjörs- och högskoleingenjörsprogrammen.

Boken behandlar logik, mängdlära, funktioner, induktion, rekursion, kombinatorik, talteori, relationer, grafteori och boolesk algebra. Teoriavsnitten, som avslutas med testfrågor, innehåller ett stort antal exempel och många tillämpningar. Dessutom ingår cirka 450 övningar med svar varav drygt 100 också har ledningar eller lösningar.

Om författarnaArmen Asratian och Anders Björn är professorer och Bengt Ove Turesson är universitetslektor i matematik vid Linköpings universitet. De har alla mångårig erfarenhet av undervisning, speciellt i diskret matematik. De är också aktiva forskare. Armen Asratians specialitet är diskret matematik.

Biggs' Discrete Mathematics has been a best-selling textbook since the first and revised editions were published in 1986 and 1990, respectively. This second edition has been developed in response to undergraduate course changes and changes in students' needs. New to this edition are chapters on statements and proof, logical framework, and natural numbers and the integers, in addition to updated chapters from the previous edition. The new chapters are presented at a level suitable for mathematics and computer science students seeking a first approach to this broad and highly relevant topic. Each chapter contains newly developed tailored exercises, and miscellaneous exercises are presented throughout, providing the student with over 1000 individual tailored exercises. This edition is accompanied by a website www.oup.com/mathematics/discretemath containing hints and solutions to all exercises presented in the text, providing an invaluable resource for students and lecturers alike. The book is carefully structured, coherent and comprehensive, and is the ideal text for students seeking a clear introduction to discrete mathematics, graph theory, combinatorics, number theory, coding theory and abstract algebra.

A PRACTICAL GUIDE TO OPTIMIZATION PROBLEMS WITH DISCRETE OR INTEGER VARIABLES, REVISED AND UPDATEDThe revised second edition of Integer Programming explains in clear and simple terms how to construct custom-made algorithms or use existing commercial software to obtain optimal or near-optimal solutions for a variety of real-world problems. The second edition also includes information on the remarkable progress in the development of mixed integer programming solvers in the 22 years since the first edition of the book appeared. The updated text includes information on the most recent developments in the field such as the much improved preprocessing/presolving and the many new ideas for primal heuristics included in the solvers. The result has been a speed-up of several orders of magnitude. The other major change reflected in the text is the widespread use of decomposition algorithms, in particular column generation (branch-(cut)-and-price) and Benders’ decomposition. The revised second edition: Contains new developments on column generationOffers a new chapter on Benders’ algorithmIncludes expanded information on preprocessing, heuristics, and branch-and-cutPresents several basic and extended formulations, for example for fixed costnetwork flowsAlso touches on and briefly introduces topics such as non-bipartite matching, the complexity of extended formulations or a good linear program for the implementation of lift-and-project Written for students of integer/mathematical programming in operations research, mathematics, engineering, or computer science, Integer Programming offers an updated edition of the basic text that reflects the most recent developments in the field.

Enumerative combinatorics, in its algebraic and analytic forms, is vital to many areas of mathematics, from model theory to statistical mechanics. This book, which stems from many years' experience of teaching, invites students into the subject and prepares them for more advanced texts. It is suitable as a class text or for individual study. The author provides proofs for many of the theorems to show the range of techniques available, and uses examples to link enumerative combinatorics to other areas of study. The main section of the book introduces the key tools of the subject (generating functions and recurrence relations), which are then used to study the most important combinatorial objects, namely subsets, partitions, and permutations of a set. Later chapters deal with more specialised topics, including permanents, SDRs, group actions and the Redfield-Pólya theory of cycle indices, Möbius inversion, the Tutte polynomial, and species.

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdos's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

I Wish They'd Taught Me That: Overlooked and Omitted Topics in Mathematics concerns the topics which every undergraduate mathematics student "should" know but has probably never encountered. These topics are not the ones which dominate every syllabus, but those magnificent secrets that are beautiful, useful and accessible but which are inexplicably hidden away from the mainstream curriculum.Each chapter of this book concerns a different topic which students will almost certainly be unfamiliar with. Written in a lively, conversational style, by the end of each section the reader should feel equipped with the knowledge to explore the area more fully elsewhere.FeaturesTopics from a variety of areas of mathematics, including geometry, logic, analysis, algebra, numerical analysis, and topologyNumerous examples, diagrams, and exercisesCollections of resources where an interested reader can learn more about each topicNontechnical introductions to each chapter.For a full list of errata, maintained by the authors, please visit: https://juliangould.github.io/pages/IWTTMT_corrections.html

Over the past few decades, graph theory has developed into one of the central areas of modern mathematics, with close (and growing) connections to areas of pure mathematics such as number theory, probability theory, algebra and geometry, as well as to applied areas such as the theory of networks, machine learning, statistical physics, and biology. It is a young and vibrant area, with several major breakthroughs having occurred in just the past few years. This book offers the reader a gentle introduction to the fundamental concepts and techniques of graph theory, covering classical topics such as matchings, colourings and connectivity, alongside the modern and vibrant areas of extremal graph theory, Ramsey theory, and random graphs. The focus throughout is on beautiful questions, ideas and proofs, and on illustrating simple but powerful techniques, such as the probabilistic method, that should be part of every young mathematician's toolkit.

The Joy of Finite Mathematics: The Language and Art of Math teaches students basic finite mathematics through a foundational understanding of the underlying symbolic language and its many dialects, including logic, set theory, combinatorics (counting), probability, statistics, geometry, algebra, and finance.

Through detailed explanations of the concepts, step-by-step procedures, and clearly defined formulae, readers learn to apply math to subjects ranging from reason (logic) to finance (personal budget), making this interactive and engaging book appropriate for non-science, undergraduate students in the liberal arts, social sciences, finance, economics, and other humanities areas.

The authors utilize important historical facts, pose interesting and relevant questions, and reference real-world events to challenge, inspire, and motivate students to learn the subject of mathematical thinking and its relevance.

The book is based on the authors' experience teaching Liberal Arts Math and other courses to students of various backgrounds and majors, and is also appropriate for preparing students for Florida's CLAST exam or similar core requirements.

Highlighted definitions, rules, methods, and procedures, and abundant tables, diagrams, and graphs, clearly illustrate important concepts and methods Provides end-of-chapter vocabulary and concept reviews, as well as robust review exercises and a practice test Contains information relevant to a wide range of topics, including symbolic language, contemporary math, liberal arts math, social sciences math, basic math for finance, math for humanities, probability, and the C.L.A.S.T. exam Optional advanced sections and challenging problems are included for use at the discretion of the instructor Online resources include PowerPoint Presentations for instructors and a useful student manual

Adaptive Mobile Computing: Advances in Processing Mobile Data Sets explores the latest advancements in producing, processing and securing mobile data sets. The book provides the elements needed to deepen understanding of this trend which, over the last decade, has seen exponential growth in the number and capabilities of mobile devices. The pervasiveness, sensing capabilities and computational power of mobile devices have turned them into a fundamental instrument in everyday life for a large part of the human population. This fact makes mobile devices an incredibly rich source of data about the dynamics of human behavior, a pervasive wireless sensors network with substantial computational power and an extremely appealing target for a new generation of threats.

Offers a coherent and realistic image of today's architectures, techniques, protocols, components, orchestration, choreography and development related to mobile computing Explains state-of-the-art technological solutions for the main issues hindering the development of next-generation pervasive systems including: supporting components for collecting data intelligently, handling resource and data management, accounting for fault tolerance, security, monitoring and control, addressing the relation with the Internet of Things and Big Data and depicting applications for pervasive context-aware processing Presents the benefits of mobile computing and the development process of scientific and commercial applications and platforms to support them Familiarizes readers with the concepts and technologies that are successfully used in the implementation of pervasive/ubiquitous systems

A Relaxation Based Approach to Optimal Control of Hybrid and Switched Systems proposes a unified approach to effective and numerically tractable relaxation schemes for optimal control problems of hybrid and switched systems. The book gives an overview of the existing (conventional and newly developed) relaxation techniques associated with the conventional systems described by ordinary differential equations. Next, it constructs a self-contained relaxation theory for optimal control processes governed by various types (sub-classes) of general hybrid and switched systems. It contains all mathematical tools necessary for an adequate understanding and using of the sophisticated relaxation techniques.

In addition, readers will find many practically oriented optimal control problems related to the new class of dynamic systems. All in all, the book follows engineering and numerical concepts. However, it can also be considered as a mathematical compendium that contains the necessary formal results and important algorithms related to the modern relaxation theory.

Illustrates the use of the relaxation approaches in engineering optimization Presents application of the relaxation methods in computational schemes for a numerical treatment of the sophisticated hybrid/switched optimal control problems Offers a rigorous and self-contained mathematical tool for an adequate understanding and practical use of the relaxation techniques Presents an extension of the relaxation methodology to the new class of applied dynamic systems, namely, to hybrid and switched control systems

Stability, Control and Application of Time-Delay Systems gives a systematic description of these systems. It includes adequate designs of integrated modeling and control and frequency characterizations. Common themes revolve around creating certain synergies of modeling, analysis, control, computing and applications of time delay systems that achieve robust stability while retaining desired performance quality. The book provides innovative insights into the state-of-the-art of time-delay systems in both theory and practical aspects. It has been edited with an emphasis on presenting constructive theoretical and practical methodological approaches and techniques.

Unifies existing and emerging concepts concerning time delay dynamical systems Provides a series of the latest results in large-delay analysis and multi-agent and thermal systems with delays Gives in each chapter numerical and simulation results in order to reflect the engineering practice

Advances in Mathematics for Industry 4.0 examines key tools, techniques, strategies, and methods in engineering applications. By covering the latest knowledge in technology for engineering design and manufacture, chapters provide systematic and comprehensive coverage of key drivers in rapid economic development. Written by leading industry experts, chapter authors explore managing big data in processing information and helping in decision-making, including mathematical and optimization techniques for dealing with large amounts of data in short periods. Focuses on recent research in mathematics applications for Industry 4.0 Provides insights on international and transnational scales Identifies mathematics knowledge gaps for Industry 4.0 Describes fruitful areas for further research in industrial mathematics, including forthcoming international studies and research

Discrete Mathematics: Essentials and Applications offers a comprehensive survey of the area, particularly concentrating on the basic principles and applications of Discrete Mathematics. This up-to-date text provides proofs of significance, keeping the focus on numerous relevant examples and many pertinent applications. Written in a simple and clear tone, the title features insightful descriptions and intuitive explanations of all complex concepts and ensures a thorough understanding of the subject matter. Offers easy-to-understand coverage of the subject matter with a class-tested pedagogical approach Covers all topics in Discrete Math in a comprehensive yet not overwhelming way Includes numerous meaningful examples on all topics to bring insight, and relevant applications for all major topics

Differential Equations with Mathematica, Fifth Edition uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists.

Mathematica's diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica's built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, Mathematica can be used to perform the calculations encountered when solving a differential equation.

Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica's outstanding graphics capabilities.

Demonstrates how to take advantage of the advanced features of Mathematica Introduces the fundamental theory of ordinary and partial differential equations using Mathematica to solve typical problems of interest to students, instructors, scientists, and practitioners in many fields Showcases practical applications and case studies drawn from biology, physics, and engineering

In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go on to be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, now called the Markoff theory. Markoff's theory characterized certain quadratic forms and real numbers by extremal inequalities. Both classes are constructed using certain natural numbers, known as Markoff numbers, and they form part of a solution to the Markoff Diophantine equation. More basically, they are constructed using certain words, essentially Christoffel words.The link between Christoffel words and Markoff's theory was noted by Ferdinand Frobenius in 1913, but has been neglected in recent times. Motivated by this overlooked connection, this book looks to expand on the relationship between these two areas. The first part of the book focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words. This new edition includes many additional exercises and solutions, as well as expanded sections on quadratic forms and quadratic numbers, two new chapters on standard words and the commutator subgroup, and revised results in combinatorics on words.

This book provides a unique and unusual introduction to graph theory by one of the founding fathers, and will be of interest to all researchers in the subject. It is not intended as a comprehensive treatise, but rather as an account of those parts of the theory that have been of special interest to the author. Professor Tutte details his experience in the area, and provides a fascinating insight into how he was led to his theorems and the proofs he used. As well as being of historical interest it provides a useful starting point for research, with references to further suggested books as well as the original papers.The book starts by detailing the first problems worked on by Professor Tutte and his colleagues during his days as an undergraduate member of the Trinity Mathematical Society in Cambridge. It covers subjects such as comnbinatorial problems in chess, the algebraicization of graph theory, reconstruction of graphs, and the chromatic eigenvalues. In each case fascinating historical and biographical information about the author's research is provided.

The new edition of this text includes additional chapters on statements and proof, logical framework, and natural numbers and the integers. The updated chapters offer over 1000 tailored exercises, and an accompanying website contains hints and solutions to all exercises. The text is designed explicitly for mathematicians and computer scientists seeking a first approach to this important topic.

How many possible sudoku puzzles are there? In the lottery, what is the chance that two winning balls have consecutive numbers? Who invented Pascal's triangle? (it was not Pascal)Combinatorics, the branch of mathematics concerned with selecting, arranging, and listing or counting collections of objects, works to answer all these questions. Dating back some 3000 years, and initially consisting mainly of the study of permutations and combinations, its scope has broadened to include topics such as graph theory, partitions of numbers, block designs, design of codes, and latin squares. In this Very Short Introduction Robin Wilson gives an overview of the field and its applications in mathematics and computer theory, considering problems from the shortest routes covering certain stops to the minimum number of colours needed to colour a map with different colours for neighbouring countries.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Who first presented Pascal's triangle? (It was not Pascal.)Who first presented Hamiltonian graphs? (It was not Hamilton.)Who first presented Steiner triple systems? (It was not Steiner.) The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: 1) it constitutes the first book-length survey of the history of combinatorics; and 2) it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction by Donald E. Knuth to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler's contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections by the distinguished combinatorialist, Peter J. Cameron. This book is not expected to be read from cover to cover, although it can be. Rather, it aims to serve as a valuable resource to a variety of audiences. Combinatorialists with little or no knowledge about the development of their subject will find the historical treatment stimulating. A historian of mathematics will view its assorted surveys as an encouragement for further research in combinatorics. The more general reader will discover an introduction to a fascinating and too little known subject that continues to stimulate and inspire the work of scholars today.

Concentration inequalities for functions of independent random variables is an area of probability theory that has witnessed a great revolution in the last few decades, and has applications in a wide variety of areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry. Roughly speaking, if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value. This book offers a host of inequalities to illustrate this rich theory in an accessible way by covering the key developments and applications in the field. The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented. A self-contained introduction to concentration inequalities, it includes a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes. Written by leading experts in the field and containing extensive exercise sections this book will be an invaluable resource for researchers and graduate students in mathematics, theoretical computer science, and engineering.

This book deals with the analysis of the structure of complex networks by combining results from graph theory, physics, and pattern recognition. The book is divided into two parts. 11 chapters are dedicated to the development of theoretical tools for the structural analysis of networks, and 7 chapters are illustrating, in a critical way, applications of these tools to real-world scenarios. The first chapters provide detailed coverage of adjacency and metric and topological properties of networks, followed by chapters devoted to the analysis of individual fragments and fragment-based global invariants in complex networks. Chapters that analyse the concepts of communicability, centrality, bipartivity, expansibility and communities in networks follow. The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.

In 1875, Elwin Bruno Christoffel introduced a special class of words on a binary alphabet linked to continued fractions which would go onto be known as Christoffel words. Some years later, Andrey Markoff published his famous theory, the now called Markoff theory. It characterized certain quadratic forms and certain real numbers by extremal inequalities. Both classes are constructed using certain natural numbers — known as Markoff numbers — and they are characterized by a certain Diophantine equality. More basically, they are constructed using certain words — essentially the Christoffel words. The link between Christoffel words and the theory of Markoff was noted by Ferdinand Frobenius in 1913, but has been neglected in recent times. Motivated by this overlooked connection, this book looks to expand on the relationship between these two areas. Part 1 focuses on the classical theory of Markoff, while Part II explores the more advanced and recent results of the theory of Christoffel words.

The Nature of Complex Networks provides a systematic introduction to the statistical mechanics of complex networks and the different theoretical achievements in the field that are now finding strands in common.The book presents a wide range of networks and the processes taking place on them, including recently developed directions, methods, and techniques. It assumes a statistical mechanics view of random networks based on the concept of statistical ensembles but also features the approaches and methods of modern random graph theory and their overlaps with statistical physics.This book will appeal to graduate students and researchers in the fields of statistical physics, complex systems, graph theory, applied mathematics, and theoretical epidemiology.

Concentration inequalities for functions of independent random variables is an area of probability theory that has witnessed a great revolution in the last few decades, and has applications in a wide variety of areas such as machine learning, statistics, discrete mathematics, and high-dimensional geometry. Roughly speaking, if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value. This book offers a host of inequalities to illustrate this rich theory in an accessible way by covering the key developments and applications in the field. The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities to transportation arguments to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are also presented. A self-contained introduction to concentration inequalities, it includes a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes.Written by leading experts in the field and containing extensive exercise sections this book will be an invaluable resource for researchers and graduate students in mathematics, theoretical computer science, and engineering.

This book deals with the analysis of the structure of complex networks by combining results from graph theory, physics, and pattern recognition. The book is divided into two parts. 11 chapters are dedicated to the development of theoretical tools for the structural analysis of networks, and 7 chapters are illustrating, in a critical way, applications of these tools to real-world scenarios. The first chapters provide detailed coverage of adjacency and metric and topological properties of networks, followed by chapters devoted to the analysis of individual fragments and fragment-based global invariants in complex networks. Chapters that analyse the concepts of communicability, centrality, bipartivity, expansibility and communities in networks follow. The second part of this book is devoted to the analysis of genetic, protein residue, protein-protein interaction, intercellular, ecological and socio-economic networks, including important breakthroughs as well as examples of the misuse of structural concepts.