Author: Sergei K. Lando
Publisher: American Mathematical Soc.
Release Date: 2003-10-21
In combinatorics, one often considers the process of enumerating objects of a certain nature, which results in a sequence of positive integers. With each such sequence, one can associate a generating function, whose properties tell us a lot about the nature of the objects being enumerated. Nowadays, the language of generating functions is the main language of enumerative combinatorics. This book is based on the course given by the author at the College of Mathematics of the Independent University of Moscow. It starts with definitions, simple properties, and numerous examples of generating functions. It then discusses various topics, such as formal grammars, generating functions in several variables, partitions and decompositions, and the exclusion-inclusion principle. In the final chapter, the author describes applications of generating functions to enumeration of trees, plane graphs, and graphs embedded in two-dimensional surfaces. Throughout the book, the reader is motivated by interesting examples rather than by general theories. It also contains a lot of exercises to help the reader master the material. Little beyond the standard calculus course is necessary to understand the book. It can serve as a text for a one-semester undergraduate course in combinatorics.
Author: Fred Roberts
Publisher: CRC Press
Release Date: 2009-06-03
Now with solutions to selected problems, Applied Combinatorics, Second Edition presents the tools of combinatorics from an applied point of view. This bestselling textbook offers numerous references to the literature of combinatorics and its applications that enable readers to delve more deeply into the topics. After introducing fundamental counting rules and the tools of graph theory and relations, the authors focus on three basic problems of combinatorics: counting, existence, and optimization problems. They discuss advanced tools for dealing with the counting problem, including generating functions, recurrences, inclusion/exclusion, and Pólya theory. The text then covers combinatorial design, coding theory, and special problems in graph theory. It also illustrates the basic ideas of combinatorial optimization through a study of graphs and networks.
This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas. The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions. Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions. Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4. The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya’s enumeration theorem using symmetric functions. Chapters 7 and 8 are more specialized than the preceding ones, covering consecutive pattern matches in permutations, words, cycles, and alternating permutations and introducing the reciprocity method as a way to define ring homomorphisms with desirable properties. Counting with Symmetric Functions will appeal to graduate students and researchers in mathematics or related subjects who are interested in counting methods, generating functions, or symmetric functions. The unique approach taken and results and exercises explored by the authors make it an important contribution to the mathematical literature.
Author: Lasse Rempe-Gillen
Publisher: American Mathematical Soc.
Release Date: 2013-12-11
How can you tell whether a number is prime? What if the number has hundreds or thousands of digits? This question may seem abstract or irrelevant, but in fact, primality tests are performed every time we make a secure online transaction. In 2002, Agrawal, Kayal, and Saxena answered a long-standing open question in this context by presenting a deterministic test (the AKS algorithm) with polynomial running time that checks whether a number is prime or not. What is more, their methods are essentially elementary, providing us with a unique opportunity to give a complete explanation of a current mathematical breakthrough to a wide audience. Rempe-Gillen and Waldecker introduce the aspects of number theory, algorithm theory, and cryptography that are relevant for the AKS algorithm and explain in detail why and how this test works. This book is specifically designed to make the reader familiar with the background that is necessary to appreciate the AKS algorithm and begins at a level that is suitable for secondary school students, teachers, and interested amateurs. Throughout the book, the reader becomes involved in the topic by means of numerous exercises.
Author: Philippe Flajolet
Publisher: Cambridge University Press
Release Date: 2009-01-15
Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.
This practically-oriented textbook presents an accessible introduction to discrete mathematics through a substantial collection of classroom-tested exercises. Each chapter opens with concise coverage of the theory underlying the topic, reviewing the basic concepts and establishing the terminology, as well as providing the key formulae and instructions on their use. This is then followed by a detailed account of the most common problems in the area, before the reader is invited to practice solving such problems for themselves through a varied series of questions and assignments. Topics and features: provides an extensive set of exercises and examples of varying levels of complexity, suitable for both laboratory practical training and self-study; offers detailed solutions to many problems, applying commonly-used methods and computational schemes; introduces the fundamentals of mathematical logic, the theory of algorithms, Boolean algebra, graph theory, sets, relations, functions, and combinatorics; presents more advanced material on the design and analysis of algorithms, including asymptotic analysis, and parallel algorithms; includes reference lists of trigonometric and finite summation formulae in an appendix, together with basic rules for differential and integral calculus. This hands-on study guide is designed to address the core needs of undergraduate students training in computer science, informatics, and electronic engineering, emphasizing the skills required to develop and implement an algorithm in a specific programming language.
Author: Ida Kantor
Publisher: American Mathematical Soc.
Release Date: 2015-08-27
Mathematics++ is a concise introduction to six selected areas of 20th century mathematics providing numerous modern mathematical tools used in contemporary research in computer science, engineering, and other fields. The areas are: measure theory, high-dimensional geometry, Fourier analysis, representations of groups, multivariate polynomials, and topology. For each of the areas, the authors introduce basic notions, examples, and results. The presentation is clear and accessible, stressing intuitive understanding, and it includes carefully selected exercises as an integral part. Theory is complemented by applications--some quite surprising--in theoretical computer science and discrete mathematics. The chapters are independent of one another and can be studied in any order. It is assumed that the reader has gone through the basic mathematics courses. Although the book was conceived while the authors were teaching Ph.D. students in theoretical computer science and discrete mathematics, it will be useful for a much wider audience, such as mathematicians specializing in other areas, mathematics students deciding what specialization to pursue, or experts in engineering or other fields.
Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.
Author: Ronald L. Graham
Publisher: Addison-Wesley Professional
Release Date: 1994-01-01
This book, updated and improved, introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills--the skills needed to solve complex problems, to evaluate horrendous-looking sums, to solve complex recurrence relations, and to discover subtle patterns in data. It is an indispensable text and reference, not only for computer scientists but for all technical professionals in virtually every discipline.
Author: V. K . Balakrishnan
Publisher: Courier Corporation
Release Date: 2012-04-30
This concise, undergraduate-level text focuses on combinatorics, graph theory with applications to some standard network optimization problems, and algorithms. More than 200 exercises, many with complete solutions. 1991 edition.
Richard Stanley's work in combinatorics revolutionized and reshaped the subject. His lectures, papers, and books inspired a generation of researchers. In this volume, these researchers explain how Stanley's vision and insights influenced and guided their own perspectives on the subject. As a valuable bonus, this book contains a collection of Stanley's short comments on each of his papers. This book may serve as an introduction to several different threads of ongoing research in combinatorics as well as giving historical perspective.
Author: V. V. I͡Ashchenko
Publisher: American Mathematical Soc.
Release Date: 2002
Learning about cryptography requires examining fundamental issues about information security. Questions abound, ranging from 'From whom are we protecting ourselves?' and 'How can we measure levels of security?' to 'What are our opponent's capabilities?' and 'What are their goals?' Answering these questions requires an understanding of basic cryptography. This book, written by Russian cryptographers, explains those basics.Chapters are independent and can be read in any order. The introduction gives a general description of all the main notions of modern cryptography: a cipher, a key, security, an electronic digital signature, a cryptographic protocol, etc. Other chapters delve more deeply into this material. The final chapter presents problems and selected solutions from ""Cryptography Olympiads for (Russian) High School Students"". This is an English translation of a Russian textbook. It is suitable for advanced high school students and undergraduates studying information security. It is also appropriate for a general mathematical audience interested in cryptography. Also on cryptography and available from the AMS is ""Codebreakers: Ame Beurling and the Swedish Crypto Program during World War II"".
"Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks."--Front flap.