This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.

Warning: After you read this book you will believe in alien life. The possible existence of extraterrestrial life is a phenomenon that has sparked more debate, controversy, fascination, and passion than any in recent history. Outpourings of UFO sightings, alien abductions, and covert government activity point to a truth which-as much as some people would like to-simply won't die. Now, in this provocative new book, bestselling author and UFOlogist Whitley Strieber boldly explores the vast territory of alien encounters, uncovering the most conclusive evidence of all: physical evidence that aliens may really be here. Marvel as Whitley Strieber tells his own compelling story-and those of countless others-while you discover: -Shocking new close encounters, many involving groups of people -Thousands of sightings worldwide, many captured on video -Shocking evidence of five mysterious implants surgically removed from human bodies -And much, much more! The most compelling question in the universe has remained unanswered for centuries. Now, finally, there is...Confirmation. With a gallery of thought-provoking photos!

According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes's original paper to contemporary formal learning theory. In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh.