It is general consensus that Combinatorics has developed into a full-fledged mathematical discipline whose beginnings as a charming pastime have long since been left behind and whose great signifi cance for other branches of both pure and applied mathematics is only beginning to be realized. The last ten years have witnessed a tremendous outburst of activity both in relatively new fields such as Coding Theory and the Theory of Matroids as well as in' more time honored endeavors such as Generating Functions and the Inver sion Calculus. Although the number of text books on these subjects is slowly increasing, there is also a great need for up-to-date surveys of the main lines of research designed to aid the beginner and serve as a reference for the expert. It was the aim of the Advanced Study Institute "Higher Combinatorics" in Berlin, 1976, to help fulfill this need. There were five sections: I. Counting Theory, II. Combinatorial Set Theory and Order Theory, III. Matroids, IV. Designs and V. Groups and Coding Theory, with three principal lecturers in each section. Expanded versions of most lectures form the contents of this book. The Institute was designed to offer, especially to young researchers, a comprehen sive picture of the most interesting developments currently under way. It is hoped that these proceedings will serve the same purpose for a wider audience."

This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices, and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions. The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important applications in cryptography, quantum information theory, communications, and networking.

This book shows how the Bayesian Approach (BA) improves well known heuristics by randomizing and optimizing their parameters. That is the Bayesian Heuristic Approach (BHA). The ten in-depth examples are designed to teach Operations Research using Internet. Each example is a simple representation of some impor tant family of real-life problems. The accompanying software can be run by remote Internet users. The supporting web-sites include software for Java, C++, and other lan guages. A theoretical setting is described in which one can discuss a Bayesian adaptive choice of heuristics for discrete and global optimization prob lems. The techniques are evaluated in the spirit of the average rather than the worst case analysis. In this context, "heuristics" are understood to be an expert opinion defining how to solve a family of problems of dis crete or global optimization. The term "Bayesian Heuristic Approach" means that one defines a set of heuristics and fixes some prior distribu tion on the results obtained. By applying BHA one is looking for the heuristic that reduces the average deviation from the global optimum. The theoretical discussions serve as an introduction to examples that are the main part of the book. All the examples are interconnected. Dif ferent examples illustrate different points of the general subject. How ever, one can consider each example separately, too."

Graph theory is a specific concept that has numerous applications throughout many industries. Despite the advancement of this technique, graph theory can still yield ambiguous and imprecise results. In order to cut down on these indeterminate factors, neutrosophic logic has emerged as an applicable solution that is gaining significant attention in solving many real-life decision-making problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistency, and indeterminacy. However, empirical research on this specific graph set is lacking. Neutrosophic Graph Theory and Algorithms is a collection of innovative research on the methods and applications of neutrosophic sets and logic within various fields including systems analysis, economics, and transportation. While highlighting topics including linear programming, decision-making methods, and homomorphism, this book is ideally designed for programmers, researchers, data scientists, mathematicians, designers, educators, researchers, academicians, and students seeking current research on the various methods and applications of graph theory.

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."

This book presents methods of solving problems in three areas of elementary combinatorial mathematics: classical combinatorics, combinatorial arithmetic, and combinatorial geometry. In each topic, brief theoretical discussions are immediately followed by carefully worked-out examples of increasing degrees of difficulty, and by exercises that range from routine to rather challenging. While this book emphasizes some methods that are not usually covered in beginning university courses, it nevertheless teaches techniques and skills that are useful not only in the specific topics covered here. There are approximately 310 examples and 650 exercises. Jiri Herman is the headmaster of a prestigious secondary school (Gymnazium) in Brno, Radan Kucera is Associate Professor of Mathematics at Masaryk University in Brno, and Jaromir Simsa is a researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic. The translator, Karl Dilcher, is Professor of Mathematics at Dalhousie University in Canada. This book can be seen as a continuation of the previous book by the same authors and also translated by Karl Dilcher, Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory (Springer-Verlag 2000).

The study of random graphs was begun by Paul Erdos and Alfred Renyi in the 1960s and now has a comprehensive literature. A compelling element has been the threshold function, a short range in which events rapidly move from almost certainly false to almost certainly true. This book now joins the study of random graphs (and other random discrete objects) with mathematical logic. The possible threshold phenomena are studied for all statements expressible in a given language. Often there is a zero-one law, that every statement holds with probability near zero or near one. The methodologies involve probability, discrete structures and logic, with an emphasis on discrete structures.The book will be of interest to graduate students and researchers in discrete mathematics.

A unique approach illustrating discrete distribution theory through combinatorial methods

This book provides a unique approach by presenting combinatorial methods in tandem with discrete distribution theory. This method, particular to discreteness, allows readers to gain a deeper understanding of theory by using applications to solve problems. The author makes extensive use of the reduction approach to conditional distributions of independent random occupancy numbers, and provides excellent studies of occupancy and sequential occupancy distributions, convolutions of truncated discrete distributions, and compound and mixture distributions.

Combinatorial Methods in Discrete Distributions begins with a brief presentation of set theory followed by basic counting principles. Fundamental principles of combinatorics, finite differences, and discrete probability are included to give readers the necessary foundation to the topics presented in the text.

A thorough examination of the field is provided and features: Stirling numbers and generalized factorial coefficientsOccupancy and sequential occupancy distributionsn-fold convolutions of truncated distributionsCompound and mixture distributions

Thoroughly worked examples aid readers in understanding complex theory and discovering how theory can be applied to solve practical problems. An appendix with hints and answers to the exercises helps readers work through the more complex sections. Reference notes are provided at the end of each chapter, and an extensive bibliography offers readers a resource for additional information on specialized topics.

The switching net.work is an important. classic research area in t.ele- communication and comput.er net.works. It.s import.ancc st.ems from both theory and practice. In fact, some open problems, such as Benes conjec- ture on shuffle-exchange networks and Chung-Rmis conjecture on multi- rate rearrangeability, still attract many researchers and the further de- velopment in optical networks requires advanced technology in optical switching networks. In 1997, we had a workshop in switching networks held in NSF Sci- ence and Technology Center in Discrete Mathematics and Theoretical Computer Science (DIMACS), at Princeton University. This workshop was very successful. Many participants wished to have a similar activity every two or three years. This book is a result of such a wish. We are putting together SOllle important developments in this area during last. several years, including articles ill fault-tolerance, rearrang{~ability. non- blocking, optical networks. random permutation generat.ioll. and layout complexity. SOlllC of thos(~ art ides are research papers alld SOIllC an' sur- veys. All articles were reviewed. We would like to uWlItioll two special problems studied in those articles.

A useful introduction to this topic for both students and researchers, with an emphasis on applications and practicalities rather than on a formal development. It is based on the popular software package for graphical modelling, MIM, freely available for downloading from the Internet. Following a description of some of the basic ideas of graphical modelling, subsequent chapters describe particular families of models, including log-linear models, Gaussian models, and models for mixed discrete and continuous variables. Further chapters cover hypothesis testing and model selection. Chapters 7 and 8 are new to this second edition and describe the use of directed, chain, and other graphs, complete with a summary of recent work on causal inference.

Boundary value problems which have variational expressions in form of inequal ities can be divided into two main classes. The class of boundary value prob lems (BVPs) leading to variational inequalities and the class of BVPs leading to hemivariational inequalities. The first class is related to convex energy functions and has being studied over the last forty years and the second class is related to nonconvex energy functions and has a shorter research "life" beginning with the works of the second author of the present book in the year 1981. Nevertheless a variety of important results have been produced within the framework of the theory of hemivariational inequalities and their numerical treatment, both in Mathematics and in Applied Sciences, especially in Engineering. It is worth noting that inequality problems, i. e. BVPs leading to variational or to hemivariational inequalities, have within a very short time had a remarkable and precipitate development in both Pure and Applied Mathematics, as well as in Mechanics and the Engineering Sciences, largely because of the possibility of applying and further developing new and efficient mathematical methods in this field, taken generally from convex and/or nonconvex Nonsmooth Analy sis. The evolution of these areas of Mathematics has facilitated the solution of many open questions in Applied Sciences generally, and also allowed the formu lation and the definitive mathematical and numerical study of new classes of interesting problems."

The Guinness Book made records immensely popular. This book is devoted, at first glance, to present records concerning prime numbers. But it is much more. It explores the interface between computations and the theory of prime numbers. The book contains an up-to-date historical presentation of the main problems about prime numbers, as well as many fascinating topics, including primality testing. It is written in a language without secrets, and thoroughly accessible to everyone. The new edition has been significantly improved due to a smoother presentation, many new topics and updated records.

This book contains the extended abstracts presented at the 12th International Conference on Power Series and Algebraic Combinatorics (FPSAC '00) that took place at Moscow State University, June 26-30, 2000. These proceedings cover the most recent trends in algebraic and bijective combinatorics, including classical combinatorics, combinatorial computer algebra, combinatorial identities, combinatorics of classical groups, Lie algebra and quantum groups, enumeration, symmetric functions, young tableaux etc...

Combinatorial Engineering of Decomposable Systems presents a morphological approach to the combinatorial design/synthesis of decomposable systems. Applications involve the following: design (e.g., information systems; user's interfaces; educational courses); planning (e.g., problem-solving strategies; product life cycles; investment); metaheuristics for combinatorial optimization; information retrieval; etc.

In the spectrum of mathematics, graph theory which studies a mathe matical structure on a set of elements with a binary relation, as a recognized discipline, is a relative newcomer. In recent three decades the exciting and rapidly growing area of the subject abounds with new mathematical devel opments and significant applications to real-world problems. More and more colleges and universities have made it a required course for the senior or the beginning postgraduate students who are majoring in mathematics, computer science, electronics, scientific management and others. This book provides an introduction to graph theory for these students. The richness of theory and the wideness of applications make it impossi ble to include all topics in graph theory in a textbook for one semester. All materials presented in this book, however, I believe, are the most classical, fundamental, interesting and important. The method we deal with the mate rials is to particularly lay stress on digraphs, regarding undirected graphs as their special cases. My own experience from teaching out of the subject more than ten years at University of Science and Technology of China (USTC) shows that this treatment makes hardly the course di: fficult, but much more accords with the essence and the development trend of the subject."

This book provides clear explanatory text, illustrative mathematics and algorithms, demonstrations of the iterative process, pseudocode, and well-developed examples for applications of the branch-and-bound paradigm to important problems in combinatorial data analysis.

Supplementary material, such as computer programs, are provided on the world wide web.

Dr. Brusco is an editorial board member for the Journal of Classification, and a member of the Board of Directors for the Classification Society of North America.

Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.

This book is a revised edition of the monograph which appeared under the same title in the series Research Notes in Theoretical Computer Science, Pit man, in 1986. In addition to a general effort to improve typography, English, and presentation, the main novelty of this second edition is the integration of some new material. Part of it is mine (mostly jointly with coauthors). Here is brief guide to these additions. I have augmented the account of categorical combinatory logic with a description of the confluence properties of rewriting systems of categor ical combinators (Hardin, Yokouchi), and of the newly developed cal culi of explicit substitutions (Abadi, Cardelli, Curien, Hardin, Levy, and Rios), which are similar in spirit to the categorical combinatory logic, but are closer to the syntax of A-calculus (Section 1.2). The study of the full abstraction problem for PCF and extensions of it has been enriched with a new full abstraction result: the model of sequential algorithms is fully abstract with respect to an extension of PCF with a control operator (Cartwright, Felleisen, Curien). An order extensional model of error-sensitive sequential algorithms is also fully abstract for a corresponding extension of PCF with a control operator and errors (Sections 2.6 and 4.1). I suggest that sequential algorithms lend themselves to a decomposition of the function spaces that leads to models of linear logic (Lamarche, Curien), and that connects sequentiality with games (Joyal, Blass, Abramsky) (Sections 2.1 and 2.6)."

This monograph provides a theoretical treatment of the problems related to the embeddability of graphs. Among these problems are the planarity and planar embeddings of a graph, the Gaussian crossing problem, the isomorphisms of polyhedra, surface embeddability, problems concerning graphic and cographic matroids and the knot problem from topology to combinatorics are discussed. Rectilinear embeddability, and the net-embeddability of a graph, which appears from the VSLI circuit design and has been much improved by the author recently, is also illustrated. Furthermore, some optimization problems related to planar and rectilinear embeddings of graphs, including those of finding the shortest convex embedding with a boundary condition and the shortest triangulation for given points on the plane, the bend and the area minimizations of rectilinear embeddings, and several kinds of graph decompositions are specially described for conditions efficiently solvable. At the end of each chapter, the Notes Section sets out the progress of related problems, the background in theory and practice, and some historical remarks. Some open problems with suggestions for their solutions are mentioned for further research.

Graph models are extremely useful for a large number of applications as they play an important role as structuring tools. They allow to model net structures - like roads, computers, telephones, social networks - instances of abstract data structures - like lists, stacks, trees - and functional or object oriented programming. The focus of this highly self-contained book is on homomorphisms and endomorphisms, matrices and eigenvalues.

Theory of Association Schemes is the first concept-oriented treatment of the structure theory of association schemes. It contains several recent results which appear for the first time in book form. The generalization of Sylow 's group theoretic theorems to scheme theory arises as a consequence of arithmetical considerations about quotient schemes. The theory of Coxeter schemes (equivalent to the theory of buildings) emerges naturally and yields a purely algebraic proof of Tits main theorem on buildings of spherical type. Also a scheme-theoretic characterization of Glauberman 's Z*-involutions is included. The text is self-contained and accessible for advanced undergraduate students.

The analysis of orthogonal polynomials associated with general weights has been a major theme in classical analysis this century. The use of potential theory since the early 1980¿s had a dramatic influence on the development of orthogonal polynomials associated with weights on the real line. For many applications of orthogonal polynomials, for example in approximation theory and numerical analysis, it is not asymptotics but certain bounds that are most important. In this monograph, the authors define and discuss their classes of weights, state several of their results on Christoffel functions, Bernstein inequalities, restricted range inequalities, and record their bounds on the orthogonal polynomials as well as their asymptotic results. This book will be of interest to researchers in approximation theory and potential theory, as well as in some branches of engineering.

Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from basic notions of combinatorial enumeration to a variety of topics, ranging from algebra to statistical physics. The book is organized in three parts: Basics, Methods, and Topics. The aim is to introduce readers to a fascinating field, and to offer a sophisticated source of information for professional mathematicians desiring to learn more. There are 666 exercises, and every chapter ends with a highlight section, discussing in detail a particularly beautiful or famous result.

The study of combinatorial block designs is a vibrant area of combinatorial mathematics with connections to finite geometries, graph theory, coding theory and statistics. The practice of ordering combinatorial objects can trace its roots to bell ringing which originated in 17th century England, but only emerged as a significant modern research area with the work of F. Gray and N. de Bruijn. These two fascinating areas of mathematics are brought together for the first time in this book. It presents new terminology and concepts which unify existing and recent results from a wide variety of sources. In order to provide a complete introduction and survey, the book begins with background material on combinatorial block designs and combinatorial orderings, including Gray codes -- the most common and well-studied combinatorial ordering concept -- and universal cycles. The central chapter discusses how ordering concepts can be applied to block designs, with definitions from existing (configuration orderings) and new (Gray codes and universal cycles for designs) research. Two chapters are devoted to a survey of results in the field, including illustrative proofs and examples. The book concludes with a discussion of connections to a broad range of applications in computer science, engineering and statistics. This book will appeal to both graduate students and researchers. Each chapter contains worked examples and proofs, complete reference lists, exercises and a list of conjectures and open problems. Practitioners will also find the book appealing for its accessible, self-contained introduction to the mathematics behind the applications.