This book gives a new foundation for the theory of links in 3-space modeled on the modern developmentby Jaco, Shalen, Johannson, Thurston et al. of the theory of 3-manifolds. The basic construction is a method of obtaining any link by "splicing" links of the simplest kinds, namely those whose exteriors are Seifert fibered or hyperbolic. This approach to link theory is particularly attractive since most invariants of links are additive under splicing. Specially distinguished from this viewpoint is the class of links, none of whose splice components is hyperbolic. It includes all links constructed by cabling and connected sums, in particular all links of singularities of complex plane curves. One of the main contributions of this monograph is the calculation of invariants of these classes of links, such as the Alexander polynomials, monodromy, and Seifert forms.

There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields. These papers are dedicated to the memory of Ralph H. Fox, one of the world's leading topologists, by colleagues, former students, and friends. In knot theory, papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten. Of these several are devoted to the study of branched covering spaces over knots and links, while others utilize the braid groups of Artin. Cossey and Smythe, Stallings, and Strasser address themselves to group theory. In his contribution Stallings describes the calculation of the groups In/In+1 where I is the augmentation ideal in a group ring RG. As a consequence, one has for each k an example of a k-generator l-relator group with no free homomorphs. In the third part, papers by Birman, Cappell, Milnor, Montesinos, Papakyriakopoulos, and Shalen comprise the treatment of 3-manifolds. Milnor gives, besides important new results, an exposition of certain aspects of our current knowledge regarding the 3- dimensional Brieskorn manifolds.

In the past three decades, local search has grown from a simple heuristic idea into a mature field of research in combinatorial optimization that is attracting ever-increasing attention. Local search is still the method of choice for NP-hard problems as it provides a robust approach for obtaining high-quality solutions to problems of a realistic size in reasonable time. "Local Search in Combinatorial Optimization" covers local search and its variants from both a theoretical and practical point of view, each topic discussed by a leading authority. This book is an important reference and invaluable source of inspiration for students and researchers in discrete mathematics, computer science, operations research, industrial engineering, and management science.

In addition to the editors, the contributors are Mihalis Yannakakis, Craig A. Tovey, Jan H. M. Korst, Peter J. M. van Laarhoven, Alain Hertz, Eric Taillard, Dominique de Werra, Heinz Muhlenbein, Carsten Peterson, Bo Soderberg, David S. Johnson, Lyle A. McGeoch, Michel Gendreau, Gilbert Laporte, Jean-Yves Potvin, Gerard A. P. Kindervater, Martin W. P. Savelsbergh, Edward J. Anderson, Celia A. Glass, Chris N. Potts, C. L. Liu, Peichen Pan, Iiro Honkala, and Patric R. J. Ostergard."

This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. It introduces readers to fundamental theories, such as Craine's work on fuzzy interval graphs, fuzzy analogs of Marczewski's theorem, and the Gilmore and Hoffman characterization. It also introduces them to the Fulkerson and Gross characterization and Menger's theorem, the applications of which will be discussed in a forthcoming book by the same authors. This book also discusses in detail important concepts such as connectivity, distance and saturation in fuzzy graphs. Thanks to the good balance between the basics of fuzzy graph theory and new findings obtained by the authors, the book offers an excellent reference guide for advanced undergraduate and graduate students in mathematics, engineering and computer science, and an inspiring read for all researchers interested in new developments in fuzzy logic and applied mathematics.

Dieses Lehrbuch vermittelt die Grundlagen und Konzepte der modernen Kombinatorik in anschaulicher Weise. Die verstandliche Darlegung richtet sich an Studierende der Mathematik, der Naturwissenschaften, der Informatik und der Wirtschaftswissenschaften und erlaubt einen einfachen und beispielorientierten Zugang zu den Methoden der Kombinatorik. Beginnend mit den Grundaufgaben der Kombinatorik wird der Leser Schritt fur Schritt mit weiterfuhrenden Themen wie erzeugende Funktionen, Rekurrenzgleichungen und der Moebiusinversion sowie Graphenpolynomen und endlichen Automaten vertraut gemacht. Eine Vielzahl von Beispielen und UEbungsaufgaben mit Loesungen erleichtern das Verstandnis und dienen der Vertiefung und praktischen Anwendung des Lehrstoffes. Die vorliegende dritte Auflage ist komplett durchgesehen und deutlich erweitert um das Thema Kombinatorische Klassen und weitere, auch fur die praktische Anwendung wichtige Graphenpolynome.

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.

This much-awaited new edition of Biggs' best-selling text includes new chapters on statements and proof, logical framework, and natural numbers and the integers, in addition to updated chapters, over 1000 tailored exercises and an accompanying website containing hints and solutions to all exercises. The text is designed explicitly for mathematicians and computer scientists seeking a first approach to this important topic.

This textbook, suitable for an early undergraduate up to a graduate course, provides an overview of many basic principles and techniques needed for modern data analysis. In particular, this book was designed and written as preparation for students planning to take rigorous Machine Learning and Data Mining courses. It introduces key conceptual tools necessary for data analysis, including concentration of measure and PAC bounds, cross validation, gradient descent, and principal component analysis. It also surveys basic techniques in supervised (regression and classification) and unsupervised learning (dimensionality reduction and clustering) through an accessible, simplified presentation. Students are recommended to have some background in calculus, probability, and linear algebra. Some familiarity with programming and algorithms is useful to understand advanced topics on computational techniques.

This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include: graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.

The importance of mathematics competitions has been widely recognized for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded.
Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in thefinal toolchest.
An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement.
As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum.

The importance of mathematics competitions has been widely recognized for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded.
Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in thefinal toolchest.
An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement.
As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum.

This textbook provides an accessible introduction to the rich and beautiful area of hyperplane arrangement theory, where discrete mathematics, in the form of combinatorics and arithmetic, meets continuous mathematics, in the form of the topology and Hodge theory of complex algebraic varieties. The topics discussed in this book range from elementary combinatorics and discrete geometry to more advanced material on mixed Hodge structures, logarithmic connections and Milnor fibrations. The author covers a lot of ground in a relatively short amount of space, with a focus on defining concepts carefully and giving proofs of theorems in detail where needed. Including a number of surprising results and tantalizing open problems, this timely book also serves to acquaint the reader with the rapidly expanding literature on the subject. Hyperplane Arrangements will be particularly useful to graduate students and researchers who are interested in algebraic geometry or algebraic topology. The book contains numerous exercises at the end of each chapter, making it suitable for courses as well as self-study.

Like Descartes and Pascal, Hans Hahn (1879-1934) was both an eminent mathematician and a highly influential philosopher. He founded the Vienna Circle and was the teacher of both Kurt Goedel and Karl Popper. His seminal contributions to functional analysis and general topology had a huge impact on the development of modern analysis. Hahn's passionate interest in the foundations of mathematics, vividly described in Sir Karl Popper's foreword (which became his last essay), had a decisive influence upon Goedel. Like Freud, Musil and Schoenberg, Hahn became a pivotal figure in the feverish intellectual climate of Vienna between the two wars. Volume 1: The first volume of Hahn's Collected Works contains his path-breaking contributions to functional analysis, the theory of curves, and ordered groups. These papers are commented on by Harro Heuser, Hans Sagan, and Laszlo Fuchs. Volume 2: The second volume deals with functional analysis, real analysis and hydrodynamics. The commentaries are written by Wilhelm Frank, Davis Preiss, and Alfred Kluwick. Volume 3: In the third volume, Hahn's writings on harmonic analysis, measure and integration, complex analysis and philosophy are collected and commented on by Jean-Pierre Kahane, Heinz Bauer, Ludger Kaup, and Christian Thiel. This volume also contains excerpts of Hahn's letters and accounts by his students and colleagues.

A textbook suitable for undergraduate courses. The materials are presented very explicitly so that students will find it very easy to read. A wide range of examples, about 500 combinatorial problems taken from various mathematical competitions and exercises are also included.

Dieses Kryptographiebuch ist geschrieben fur Studierende der Mathematik, Informatik, Physik, Elektrotechnik oder andere Leser mit mathematischer Grundbildung und wurde in vielen Vorlesungen erfolgreich eingesetzt. Es behandelt die aktuellen Techniken der modernen Kryptographie, zum Beispiel Verschlusselung und digitale Signaturen. Das Buch vermittelt auf elementare Weise alle mathematischen Grundlagen, die zu einem prazisen Verstandnis der Kryptographie noetig sind, mit vielen Beispielen und UEbungen. Die Leserinnen und Leser erhalten ein fundiertes Verstandnis der modernen Kryptographie und werden in die Lage versetzt Forschungsliteratur zur Kryptographie zu verstehen.

Introducing the reader to the mathematics beyond complex networked systems, these lecture notes investigate graph theory, graphical models, and methods from statistical physics. Complex networked systems play a fundamental role in our society, both in everyday life and in scientific research, with applications ranging from physics and biology to economics and finance. The book is self-contained, and requires only an undergraduate mathematical background.

Diskrete und kontinuierliche Methoden der mathematischen Optimierung werden in diesem Lehrbuch integriert behandelt. Nach einer Einfuhrung werden konvexe Mengen (mit einer Anwendung auf notwendige Optimalitatsbedingungen bei Ungleichungsrestriktionen) behandelt, gefolgt von einer genaueren Betrachtung des Spezialfalls von Polyedern und dessen Zusammenhang zum Linearen Programmieren. Eine ausfuhrliche Darstellung des Simplexverfahrens schliesst diesen Teil ab. Danach wird die Konvexitat von Funktionen (inklusive einiger Abschwachungen) untersucht und fur ein grundliches Studium von Optimalitatskriterien sowie der Lagrange-Dualitat verwendet. Schliesslich folgen noch ein Ausblick auf allgemeine Algorithmen sowie ein kurzer Anhang zur affinen Geometrie. In der Neuauflage ist Anordnung und Darstellung des behandelten Stoffs nochmals grundlich im Sinne der aktuellen BA-Studiengange Mathematik, Wirtschaftswissenschaften und Informatik uberarbeitet worden.

Graphentheorie: eine Theorie, oder einfach eine Methode, um Sachverhalte an- schaulich darzustellen, die auch in anderer Weise erfassbar waren? Beides trifft zu: Tatsachlich ist die Graphentheorie heute bereits ein recht weit entwickelter Zweig der diskreten Mathematik, mit dem sich Spezialisten befassen. Auf der anderen Seite bietet die Abstraktion von Problemstellungen aus der "rea- len" Welt auf die beiden urspriinglich der Geometrie entliehenen Elemente "Punkt" und "Kante" eine reizvolle Gelegenheit, auch fUr die Losung von Aufgaben eher kombinatorischer Natur an die bildliche Vorstellung zu appellieren. Dieses Buch ist aus dem Manuskript einer einsemestrigen Vorlesung entstan- den, die ich mehrere Male an der Eidgenossischen Technischen Hochschule Ziirich (ETH) fUr Studenten der Informatik und der Mathematik in mittleren Semestern gehalten habe. Der Text richtet sich - ausser an dies en Kreis - durchaus auch an Studenten von Hoheren Technischen Lehranstalten (bzw. Fachhochschulen), sowie an Gymnasiallehrer. Die letzteren werden zwar den Stoff kaum als geschlos- senen Lehrgang beniitzen, konnen jedoch eventuell einzelne herausgegriffene Teile im Unterricht der oberen Klassen einbringen. Das Buch solI zunachst eine erste EinfUhrung in die Graphentheorie vermitteln - wer nur diese sucht, kann im Extremfall die Kapitel 3 und 4, sowie in den folgenden Kapiteln alles, was sich auf Algorithmen bezieht, weglassen.

Succinct representation and fast access to large amounts of data are challenges of our time. This unique book suggests general approaches of 'complexity of descriptions'. It deals with a variety of concrete topics and bridges between them, while opening new perspectives and providing promising avenues for the 'complexity puzzle'.

The theory of lattices, initiated by Dedekind in the past centu- ry, and revived in the thirties by Garrett Birkhoff, F. Klein-Barmen, ore, and von Neumann, is only in our time coming into its own. The fledgling theory was handicapped by a contingent historical circumstance. The peculiarities of mathematical personality of the founders made lattice theory less welcome to the mathematical public of the time than it otherwise might have been. Thus Dedekind was wi- dely thought in his time to be far too abstract for his own good, and some of his peers, notably Kronecker, did not hesitate to state their loud and clear disapproval. Later on, the tempers of Garrett Birkhoff and John von Neumann clashed with those of some of the "mainstream"' mathematicians of their time. Norman Levinson once related to me the following anecdote about von Neumann. Invited to deliver the weekly mathematics colloquium at Harvard sometime in the thirties, he chose the subject of his current interest, namely, continuous geometries. At the end of the lecture, as the public was streaming out, G. H. Hardy, who was at the time visiting Cambridge, was overheard whispering to G. D. Birkhoff (Gar- rett's father): "He is quite clearly a very brilliant man, but why does he waste his time on this stuff?" I myself, when still an assistant professor, was once stopped in the hall of M. I. T.

The second volume of this work contains Parts 2 and 3 of the "Handbook of Coding Theory". Part 2, "Connections", is devoted to connections between coding theory and other branches of mathematics and computer science. Part 3, "Applications", deals with a variety of applications for coding.

Gli Automi sono modelli matematici di macchine digitali di grande interesse sia dal punto di vista teorico che applicativo. La teoria degli Automi Finiti costituisce una delle parti fondamentali dell Informatica Teorica. Questo volume fornisce, per la prima volta, nel panorama didattico italiano una trattazione matematicamente rigorosa della teoria degli Automi Finiti e delle macchine sequenziali generalizzate nell ambito della teoria algebrica dei semigruppi. Il volume, la cui lettura presuppone solamente conoscenze elementari di algebra, si rivolge agli studenti sia dei corsi di laurea magistrale e specialistica che di master e di dottorato in Informatica, in Matematica, ed in Ingegneria. Il libro e anche uno strumento utilissimo per gli studiosi di Informatica e, in particolare, di Informatica Teorica, ai quali fornisce una trattazione completa e rigorosa della teoria algebrica degli Automi. Ogni capitolo ha una sezione di esercizi ed una di note bibliografiche. La risoluzione della maggior parte degli esercizi e riportata alla fine del volume.