Kombinatorik och grafteori

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.

This thoroughly revised second edition provides an updated treatment of numerical linear algebra techniques for solving problems in data mining and pattern recognition. Adopting an application-oriented approach, the author introduces matrix theory and decompositions, describes how modern matrix methods can be applied in real life scenarios, and provides a set of tools that students can modify for a particular application.Building on material from the first edition, the author discusses basic graph concepts and their matrix counterparts. He introduces the graph Laplacian and properties of its eigenvectors needed in spectral partitioning and describes spectral graph partitioning applied to social networks and text classification. Examples are included to help readers visualize the results. This new edition also presents matrix-based methods that underlie many of the algorithms used for big data.The book provides a solid foundation to further explore related topics and presents applications such as classification of handwritten digits, text mining, text summarization, PageRank computations related to the Google search engine, and facial recognition. Exercises and computer assignments are available on a Web page that supplements the book.Matrix Methods in Data Mining and Pattern Recognition, Second Edition is primarily for undergraduate students who have previously taken an introductory scientific computing/numerical analysis course and graduate students in data mining and pattern recognition areas who need an introduction to linear algebra techniques.

This standard textbook on modern graph theory combines the authority of a classic with the engaging freshness of style that is the hallmark of active mathematics. It covers the core material of the subject, with concise yet complete proofs, while offering glimpses of more advanced methods in each field via one or two deeper results.This is a major new edition. Among many other improvements, it offers additional tools for applying the regularity lemma, brings the tangle theory of graph minors up to the cutting edge of current research, and addresses new topics such as chi-boundedness in perfect graph theory.The book can be used as a reliable text for an introductory graduate course and is also suitable for self-study.From the reviews:“Deep, clear, wonderful. This is a serious book about the heart of graph theory. It has depth and integrity.” Persi Diaconis & Ron Graham, SIAM Review“The book has received a very enthusiastic reception, which it amply deserves. A masterly elucidation of modern graph theory.” Bulletin of the Institute of Combinatorics and its Applications“Succeeds dramatically ... a hell of a good book.” MAA Reviews“ ... like listening to someone explain mathematics.” Bulletin of the AMS

Denna bok innehåller del II av den större studien och artikelsamlingen "Den första matematiken", som utgör en genomgång av några grundläggande begrepp och samband ur talteori och diskret matematik, med fördjupande eller reflekterande kommentarer. Artiklarna använder den speciella framställningsform eller layout som författaren valt att kalla för den holistiska. Boken riktar sig till studenter och lärare i matematik, data/IT, logik eller filosofi, men även till en allmän matematiskt eller filosofiskt intresserad publik.

The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. The general definitions were given in subsequent work of Drinfeld and Olshansky, and these algebras have since found numerous applications in and connections with mathematical physics, geometry, representation theory, and combinatorics. The book is an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras. A special algebraic technique, the $R$-matrix formalism, is developed and used as the main instrument for describing the structure of Yangians. A detailed exposition of the highest weight theory and the classification theorems for finite-dimensional irreducible representations of these algebras is given. The Yangian perspective provides a unifying picture of several families of Casimir elements for the classical Lie algebras and relations between these families.The Yangian symmetries play a key role in explicit constructions of all finite-dimensional irreducible representations of the orthogonal and symplectic Lie algebras via weight bases of Gelfand-Tsetlin type.

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation and differential posets.

Equivariant cohomology has become an indispensable tool in algebraic geometry and in related areas including representation theory, combinatorial and enumerative geometry, and algebraic combinatorics. This text introduces the main ideas of the subject for first- or second-year graduate students in mathematics, as well as researchers working in algebraic geometry or combinatorics. The first six chapters cover the basics: definitions via finite-dimensional approximation spaces, computations in projective space, and the localization theorem. The rest of the text focuses on examples – toric varieties, Grassmannians, and homogeneous spaces – along with applications to Schubert calculus and degeneracy loci. Prerequisites are kept to a minimum, so that one-semester graduate-level courses in algebraic geometry and topology should be sufficient preparation. Featuring numerous exercises, examples, and material that has not previously appeared in textbook form, this book will be a must-have reference and resource for both students and researchers for years to come.

Percolation theory is the study of an idealized random medium in two or more dimensions. The mathematical theory is mature, and continues to give rise to problems of special beauty and difficulty. Percolation is pivotal for studying more complex physical systems exhibiting phase transitions. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed. Much new material appears in this second edition, including: dynamic and static renormalization, strict inequalities between critical points, a sketch of the lace expansion, and several essays on related fields and applications.

This undergraduate text is designed for a single-semester introductory course in discrete mathematics. It is aimed at students of mathematics and computer science, as well as problem-solving enthusiasts or anyone with some familiarity with proofs seeking a concise introduction to the subject. The primary goal is to acquaint readers with key structures and ideas in combinatorial and discrete mathematics that will support further study in mathematics and its applications.Requiring minimal prerequisites, the book seeks to build appreciation while offering a clear foundation for developing combinatorial reasoning. Special attention is given to the basics of enumeration and generating functions. Background concepts are presented accessibly, and examples appear early and frequently. When suitable, multiple proofs are provided for the same result, offering insight into mathematical thinking and the interconnectedness of ideas. Each chapter concludes with a selection of exercises spanning a wide range of difficulty, and a collection of hints and solutions is included at the end of the book.

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.

Är det verkligen möjligt att reducera 5 miljoner spelrader, 170 miljoner, ja HELA Spelplanen med 99,9 % med mycket bra resultat?! Ja, bevisligen när författaren i denna bok introducerar Guds Matematik och skapar magi!! Matematiken besegrar slumpen i denna bok!! Hur skicklig är Du i att kunna tippa rätt vinstrad i spel som berör sporter med avdelningar, lopp och race som exempelvis inom trav- och galoppsport? Hur långt är det i praktiken möjligt att nå i reduceringen av ett spelsystem för att ändå ha kvar 1 -felsreducering, det vill säga 1 fel från alla rätt och därigenom mycket enklare kunna få alla rätt, det vill säga högsta vinsten? I Matematikens seger över slumpen: Jolly Trot & Gallop System - De Optimala Systemen 1 (Trav/galoppsystem), besvarar författaren bland annat dessa två frågor ingående genom att visa hur man räknar fram sin skicklighet samt bevisar med tydliga instruktioner, formler, tabeller, bilder och färdiga system hur det med bibehållen 1-felsreducering verkligen går att reducera spelsystem från gängse uppfattning inom matematiken, att en sådan reduceringsgrad ligger på drygt 91 %, till hela ofattbara 98,6 % i praktiken enligt kombinationsmatematiken för små system!! Författaren har på ett fantastiskt unikt sätt med delvis hjälp av framtagen matematik från den schweiziske sjuttonhundratals matematikern Leonhard Euler, bevisligen lyckats utveckla optimalt reducerade spelsystem och visar i formel att det är möjligt att fortsätta reducera ett sådant system mer än 99,999% ! (Men med Guds Matematik för stora system blir det förbi 99,9 %!!) Dessutom ger författaren mycket bra instruktioner genom tydliga illustrationer, intressanta tips och råd för den som i praktiken vill använda systemen för spel samt delar med sig av hans egna erfarenheter kring travspel och annat. Med andra ord är denna bok ett MÅSTE för alla matematiskt intresserade inom kombinatorik och spelsystem och givetvis för alla systemspelare såsom exempelvis trav- och galoppspelare!!

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.

Crossing Numbers of Graphs is the first book devoted to the crossing number, an increasingly popular object of study with surprising connections. The field has matured into a large body of work, which includes identifiable core results and techniques. The book presents a wide variety of ideas and techniques in topological graph theory, discrete geometry, and computer science. The first part of the text deals with traditional crossing number, crossing number values, crossing lemma, related parameters, computational complexity, and algorithms. The second part includes the rich history of alternative crossing numbers, the rectilinear crossing number, the pair crossing number, and the independent odd crossing number.It also includes applications of the crossing number outside topological graph theory.Aimed at graduate students and professionals in both mathematics and computer scienceThe first book of its kind devoted to the topicAuthored by a noted authority in crossing numbers

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Excellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required.Bibliography. 1979 edition.

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.

Updated to reflect current research and extended to cover more advanced topics as well as the basics, Algebraic Number Theory and Fermat’s Last Theorem, Fifth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers, initially from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work. New to the Fifth EditionPell's Equation x^2-dy^2=1: all solutions can be obtained from a single `fundamental' solution, which can be found using continued fractions.Galois theory of number field extensions, relating the field structure to that of the group of automorphisms.More material on cyclotomic fields, and some results on cubic fields.Advanced properties of prime ideals, including the valuation of a fractional ideal relative to a prime ideal, localisation at a prime ideal, and discrete valuation rings.Ramification theory, which discusses how a prime ideal factorises when the number field is extended to a larger one.A short proof of the Quadratic Reciprocity Law based on properties of cyclotomic fields. ThisValuations and p-adic numbers. Topology of the p-adic integers.Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

This volume presents papers from the 2nd Scandinavian Workshop on Algorithm Theory. The contributions describe original research on algorithms and data structures, in all areas, including combinatorics, computational geometry, parallel computing, and graph theory. The majority of the papers focus on the design and complexity analysis of: data structures, text algorithms, and sequential and parallel algorithms for graph problems and for geometric problems. Examples of tech- niques presented include: - efficient ways to find approximation algorithms for the maximum independent set problem and for graph coloring; - exact estimation of the expected search cost for skip lists; - construction of canonical representations of partial 2-trees and partial 3-trees in linear time; - efficient triangulation of planar point sets and convex polygons.

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.

Through three editions, Cryptography: Theory and Practice, has been embraced by instructors and students alike. It offers a comprehensive primer for the subject’s fundamentals while presenting the most current advances in cryptography.The authors offer comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the seemingly infinite and increasing amount of information circulating around the world.Key Features of the Fourth Edition: New chapter on the exciting, emerging new area of post-quantum cryptography (Chapter 9). New high-level, nontechnical overview of the goals and tools of cryptography (Chapter 1). New mathematical appendix that summarizes definitions and main results on number theory and algebra (Appendix A). An expanded treatment of stream ciphers, including common design techniques along with coverage of Trivium. Interesting attacks on cryptosystems, including: padding oracle attack correlation attacks and algebraic attacks on stream ciphers attack on the DUAL-EC random bit generator that makes use of a trapdoor. A treatment of the sponge construction for hash functions and its use in the new SHA-3 hash standard. Methods of key distribution in sensor networks. The basics of visual cryptography, allowing a secure method to split a secret visual message into pieces (shares) that can later be combined to reconstruct the secret. The fundamental techniques cryptocurrencies, as used in Bitcoin and blockchain. The basics of the new methods employed in messaging protocols such as Signal, including deniability and Diffie-Hellman key ratcheting.

Clearly written graduate-level text considers the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NP-complete problems, more. "Mathematicians wishing a self-contained introduction need look no further." — American Mathematical Monthly. 1982 edition.

An effective blend of carefully explained theory and practical applications, this text imparts the fundamentals of both information theory and data compression. Although the two topics are related, this unique text allows either topic to be presented independently, and it was specifically designed so that the data compression section requires no prior knowledge of information theory.The treatment of information theory, while theoretical and abstract, is quite elementary, making this text less daunting than many others. After presenting the fundamental definitions and results of the theory, the authors then apply the theory to memoryless, discrete channels with zeroth-order, one-state sources. The chapters on data compression acquaint students with a myriad of lossless compression methods and then introduce two lossy compression methods. Students emerge from this study competent in a wide range of techniques. The authors' presentation is highly practical but includes some important proofs, either in the text or in the exercises, so instructors can, if they choose, place more emphasis on the mathematics.Introduction to Information Theory and Data Compression, Second Edition is ideally suited for an upper-level or graduate course for students in mathematics, engineering, and computer science.Features:Expanded discussion of the historical and theoretical basis of information theory that builds a firm, intuitive grasp of the subject Reorganization of theoretical results along with new exercises, ranging from the routine to the more difficult, that reinforce students' ability to apply the definitions and results in specific situations. Simplified treatment of the algorithm(s) of Gallager and Knuth Discussion of the information rate of a code and the trade-off between error correction and information rate Treatment of probabilistic finite state source automata, including basic resul

This is a textbook for an introductory combinatorics course lasting one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course.Just as with the first three editions, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible to the talented and hardworking undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings, Eulerian and Hamiltonian cycles, and planar graphs.New to this edition are the Quick Check exercises at the end of each section. In all, the new edition contains about 240 new exercises. Extra examples were added to some sections where readers asked for them.The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, the theory of designs, enumeration under group action, generating functions of labeled and unlabeled structures and algorithms and complexity.The book encourages students to learn more combinatorics, provides them with a not only useful but also enjoyable and engaging reading.The Solution Manual is available upon request for all instructors who adopt this book as a course text. Please send your request to .The previous edition of this textbook has been adopted at various schools including UCLA, MIT, University of Michigan, and Swarthmore College. It was also translated into Korean.

This book can be seen as a continuation of Equations and Inequalities: El­ ementary Problems and Theorems in Algebra and Number Theory by the same authors, and published as the first volume in this book series. How­ ever, it can be independently read or used as a textbook in its own right. This book is intended as a text for a problem-solving course at the first­ or second-year university level, as a text for enrichment classes for talented high-school students, or for mathematics competition training. It can also be used as a source of supplementary material for any course dealing with combinatorics, graph theory, number theory, or geometry, or for any of the discrete mathematics courses that are offered at most American and Canadian universities. The underlying "philosophy" of this book is the same as that of Equations and Inequalities. The following paragraphs are therefore taken from the preface of that book.

This book shows how our lives are shaped not only by the choices we make, but by the choices we have.From dating, school and university applications to the job market, understand the most important decisions you’ll ever make with insights from a Nobel Prize-winner.Who Gets What and Why is a piquantly written, mind-expanding exploration of the markets that matter most to many of us. If you’ve ever sought a job or hired someone, applied to university or guided your child into a good school, asked someone out on a date or been asked out, you have participated in a matching market. They are everywhere around us and account for some of the biggest technological successes of the decade, like Uber and Airbnb. Matching markets can even be the gatekeeper of life itself, guiding how desperately ill patients receive scarce organs for transplants.Alvin E. Roth shared the 2012 Nobel Prize in economics for his pioneering research into market design – the principles that govern all kinds of markets where money isn’t the only factor in determining who gets what. His book reveals what factors make these markets work well – or badly – and shows us all how to recognise a good match and make smarter, more confident decisions.

Praise for the First Edition"Luck, Logic, and White Lies teaches readers of all backgrounds about the insight mathematical knowledge can bring and is highly recommended reading among avid game players, both to better understand the game itself and to improve one’s skills."– Midwest Book Review"The best book I've found for someone new to game math is Luck, Logic and White Lies by Jörg Bewersdorff. It introduces the reader to a vast mathematical literature, and does so in an enormously clear manner. . ."– Alfred Wallace, Musings, Ramblings, and Things Left Unsaid"The aim is to introduce the mathematics that will allow analysis of the problem or game. This is done in gentle stages, from chapter to chapter, so as to reach as broad an audience as possible . . . Anyone who likes games and has a taste for analytical thinking will enjoy this book."– Peter Fillmore, CMS NotesLuck, Logic, and White Lies: The Mathematics of Games, Second Edition considers a specific problem—generally a game or game fragment and introduces the related mathematical methods. It contains a section on the historical development of the theories of games of chance, and combinatorial and strategic games.This new edition features new and much refreshed chapters, including an all-new Part IV on the problem of how to measure skill in games. Readers are also introduced to new references and techniques developed since the previous edition.Features Provides a uniquely historical perspective on the mathematical underpinnings of a comprehensive list of games Suitable for a broad audience of differing mathematical levels. Anyone with a passion for games, game theory, and mathematics will enjoy this book, whether they be students, academics, or game enthusiasts Covers a wide selection of topics at a level that can be appreciated on a historical, recreational, and mathematical level.Jörg Bewersdorff (1958) studied mathematics from 1975 to 1982 at the University of Bonn and earned his PhD in 1985. In the same year, he started his career as game developer and mathematician. He served as the general manager of the subsidiaries of Gauselmann AG for more than two decades where he developed electronic gaming machines, automatic payment machines, and coin-operated Internet terminals.Dr. Bewersdorff has authored several books on Galois theory (translated in English and Korean), mathematical statistics, and object-oriented programming with JavaScript.*Here is the list of Errata for the second edition of Luck, Logic, and White Lies: The Mathematics of Games: http://bewersdorff-online.de/LLWL-errata.pdf

The idea behind this book is to provide the mathematical foundations for assessing modern developments in the Information Age. It deepens and complements the basic concepts, but it also considers instructive and more advanced topics. The treatise starts with a general chapter on algebraic structures; this part provides all the necessary knowledge for the rest of the book. The next chapter gives a concise overview of cryptography. Chapter 3 on number theoretic algorithms is important for developping cryptosystems, Chapter 4 presents the deterministic primality test of Agrawal, Kayal, and Saxena. The account to elliptic curves again focuses on cryptographic applications and algorithms. With combinatorics on words and automata theory, the reader is introduced to two areas of theoretical computer science where semigroups play a fundamental role.The last chapter is devoted to combinatorial group theory and its connections to automata.

Contents:Algebraic structuresCryptographyNumber theoretic algorithmsPolynomial time primality testElliptic curvesCombinatorics on wordsAutomataDiscrete infinite groups