John J. Watkins - Böcker

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.

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.

Topics in Commutative Ring Theory is a textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra. Commutative ring theory arose more than a century ago to address questions in geometry and number theory. A commutative ring is a set-such as the integers, complex numbers, or polynomials with real coefficients--with two operations, addition and multiplication. Starting from this simple definition, John Watkins guides readers from basic concepts to Noetherian rings-one of the most important classes of commutative rings--and beyond to the frontiers of current research in the field. Each chapter includes problems that encourage active reading--routine exercises as well as problems that build technical skills and reinforce new concepts. The final chapter is devoted to new computational techniques now available through computers. Careful to avoid intimidating theorems and proofs whenever possible, Watkins emphasizes the historical roots of the subject, like the role of commutative rings in Fermat's last theorem.He leads readers into unexpected territory with discussions on rings of continuous functions and the set-theoretic foundations of mathematics. Written by an award-winning teacher, this is the first introductory textbook to require no prior knowledge of ring theory to get started. Refreshingly informal without ever sacrificing mathematical rigor, Topics in Commutative Ring Theory is an ideal resource for anyone seeking entry into this stimulating field of study

Across the Board is the definitive work on chessboard problems. It is not simply about chess but the chessboard itself--that simple grid of squares so common to games around the world. And, more importantly, the fascinating mathematics behind it. From the Knight's Tour Problem and Queens Domination to their many variations, John Watkins surveys all the well-known problems in this surprisingly fertile area of recreational mathematics. Can a knight follow a path that covers every square once, ending on the starting square? How many queens are needed so that every square is targeted or occupied by one of the queens? Each main topic is treated in depth from its historical conception through to its status today. Many beautiful solutions have emerged for basic chessboard problems since mathematicians first began working on them in earnest over three centuries ago, but such problems, including those involving polyominoes, have now been extended to three-dimensional chessboards and even chessboards on unusual surfaces such as toruses (the equivalent of playing chess on a doughnut) and cylinders.Using the highly visual language of graph theory, Watkins gently guides the reader to the forefront of current research in mathematics. By solving some of the many exercises sprinkled throughout, the reader can share fully in the excitement of discovery. Showing that chess puzzles are the starting point for important mathematical ideas that have resonated for centuries, Across the Board will captivate students and instructors, mathematicians, chess enthusiasts, and puzzle devotees.

The natural numbers have been studied for thousands of years, yet most undergraduate textbooks present number theory as a long list of theorems with little mention of how these results were discovered or why they are important. This book emphasizes the historical development of number theory, describing methods, theorems, and proofs in the contexts in which they originated, and providing an accessible introduction to one of the most fascinating subjects in mathematics. Written in an informal style by an award-winning teacher, Number Theory covers prime numbers, Fibonacci numbers, and a host of other essential topics in number theory, while also telling the stories of the great mathematicians behind these developments, including Euclid, Carl Friedrich Gauss, and Sophie Germain. This one-of-a-kind introductory textbook features an extensive set of problems that enable students to actively reinforce and extend their understanding of the material, as well as fully worked solutions for many of these problems. It also includes helpful hints for when students are unsure of how to get started on a given problem.* Uses a unique historical approach to teaching number theory * Features numerous problems, helpful hints, and fully worked solutions * Discusses fun topics like Pythagorean tuning in music, Sudoku puzzles, and arithmetic progressions of primes * Includes an introduction to Sage, an easy-to-learn yet powerful open-source mathematics software package * Ideal for undergraduate mathematics majors as well as non-math majors * Digital solutions manual (available only to professors)