Richard T. Cox

ALGEBRA OF PROBABLE INFERENCE

paperback | 0-8018-6982-X

December 2001 paperback, 127 pp.

"[This book] is, in my opinion one of the most important ever written on the foundations of probability theory, and the greatest advance in the conceptual, as opposed to the purely mathematical, formulation of the theory since Laplace." -- E. T. Jaynes, American Journal of Physics

In Algebra of Probable Inference, Richard T. Cox develops and demonstrates that probability theory is the only theory of inductive inference that abides by logical consistency. Cox does so through a functional derivation of probability theory as the unique extension of Boolean Algebra thereby establishing, for the first time, the legitimacy of probability theory as formalized by Laplace in the 18th century.

Perhaps the most significant consequence of Cox's work is that probability represents a subjective degree of plausible belief relative to a particular system but is a theory that applies universally and objectively across any system making inferences based on an incomplete state of knowledge. Cox goes well beyond this amazing conceptual advancement, however, and begins to formulate a theory of logical questions through his consideration of systems of assertions -- a theory that he more fully developed some years later. Although Cox's contributions to probability are acknowledged and have recently gained worldwide recognition, the significance of his work regarding logical questions is virtually unknown. The contributions of Richard Cox to logic and inductive reasoning may eventually be seen to be the most significant since Aristotle.

William T. Trotter

COMBINATORICS AND PARTIALLY ORDERED SETS
Dimension Theory

paperback | 0-8018-6977-3
Johns Hopkins Studies in the Mathematical Sciences
In Association with the Department of Mathematical Sciences, The Johns Hopkins University
January 2002 paperback, 328 pp.



"Eminently suitable as a self-study or supervised text in partially ordered sets and combinatorics." -- Siam Review

Primarily intended for research mathematicians and computer scientists, Combinatorics and Partially Ordered Sets: Dimension Theory also serves as a useful text for advanced students in either field. William Trotter concentrates on combinatorial topics for finite partially ordered sets, and with dimension theory serving as a unifying theme, research on partially ordered sets or posets is linked to more traditional topics in combinatorial mathematics -- including graph theory, Ramsey theory, probabilistic methods, hypergraphs, algorithms, and computational geometry. The book's most important contribution is to collect, organize, and explain the many theorems on partially ordered sets in a way that makes them available to the widest possible audience.

Chapters: Introduction to Dimension

Crowns, Splits, Stacks, Sums and Products
Characterization Problems for Posets, Lattices, Graphs, and Families of Sets
Hypergraph Coloring, Computational Complexity, and Irreducible Posets
Planar Posets and Trees
Planar Graphs, Planar Maps and Convex Polytopes
Probabilistic Methods in Dimension Theory
Interval and Geometric Containment Orders
Greedy Dimension, Back-Tracking, and Depth First Search
Products of Chains of Bounded Length
Large Minimal Realizers
Chapters: Introduction to Dimension

Crowns, Splits, Stacks, Sums and Products
Characterization Problems for Posets, Lattices, Graphs, and Families of Sets
Hypergraph Coloring, Computational Complexity and Irreducible Posets
Planar Maps, and Convex Polytopes
Probabilistic Methods in Dimension Theory
Interval and Geometric Containment Orders
Greedy Dimension, Back-Tracking and Depth First Search
Products of Chains of Bounded Length
Large Minimal Realizers

David Joyner

ADVENTURES IN GROUP THEORY
Rubik's Cube, Merlin's Machine, and Other Mathematical Toys

| hardcover | 0-8018-6945-5
| paperback | 0-8018-6947-1
April 2002, 264 pp.


"This is a book on group theory that lives outside the usual rather dry regime of typical mathematics texts. In setting the book squarely among these puzzles, the underlying mathematics comes alive in quite spectacular fashion. The author achieves this goal admirably here. The text is well organized and written in an interesting and very readable manner." -- Ian W. Knowles, University of Alabama , Birmingham

Group theory deals with symmetry, in the most abstract form possible. It is a core part of the undergraduate math curriculum, and forms part of the training of theoretical physicists and chemical crystallographers. Group theory has tended to be very dry--until now. David Joyner uses mathematical toys (primarily the Rubik's Cube and its more modern cousins, the Megaminx, the Pyraminx, and so on) as well as other mathematical examples (e.g., bell ringing) to breathe new life into a time-honored subject.

"Why," asks the author, "should two such different topics, mechanical puzzles and abstract group theory, be related? This book takes the reader on an intellectual trip to answer this curiosity." Adventures in Group Theory will not only appeal to all math enthusiasts and interested general readers but will also find use in the classroom as a wonderful supplementary text in any abstract algebra or group theory course.

A.W.F. Edwards

PASCAL'S ARITHMETICAL TRIANGLE
The Story of a Mathematical Idea

| paperback | 0-8018-6946-3
May 2002 paperback, 224 pp.

"An impressive culmination of meticulous research into original sources, this definitive study constitutes the first full-length history of the Arithmetic Triangle."--Mathematics of Computation

"A fascinating book... giving new insights into the early history of probability theory and combinatorics, and incidentally providing much stimulating material for teachers of mathematics."--G.A. Barnard, International Statistical Institute Review

"Scrupulously researched... Carries the reader along in a rewarding manner. It is a scientific who-dun-it, and one must admire the author for the scholarly yet unpedantic manner in which he disperses some of the mists of antiquity."--A.W. Kemp, Biometrics

"Recommended not only to historians and mathematicians, but also to students seeking to put some life into the dry treatment of these topics to which they have doubtless been subjected."--Ivor Grattan-Guinness, Annals of Science

Imagine having some marbles, pebbles, or other objects that you want to lay out in a neat triangular pattern. How many do you need to end up with a complete triangle? Three will do; so will 6, 10, 15, and... These numbers are called triangular numbers. Ask the same question for a triangular pyramid. Four will do; so will 10, 20, 35, and... the so-called pyramidal numbers. This book looks at the discovery of the multiplicity of properties and uses triangular numbers and their many extensions possess.

Although often displayed in a triangular array named after the seventeenth-century French philosopher Blaise Pascal, triangular numbers were known many centuries earlier. In this book A.W.F. Edwards traces the Arithmetical Triangle back to its roots in Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra, and gives an account of the progressive solution of combinatorial problems from the earliest recorded examples to the work of Renaissance and later mathematicians. He shows how Pascal's work -- so modern in style -- in establishing the properties of the numbers and their application in various fields led to Newton's discovery of the binomial theorem for fractional and negative indices and to Leibniz's discovery of calculus.

Bart K. Holland

WHAT ARE THE CHANCES?
Voodoo Deaths, Office Gossip, and Other Adventures in Probability

Using examples drawn from daily life and history, the author explains what probability is and how it works.

| hardcover | 0-8018-6941-2
May 2002, 192 pp., 7 line drawings

"Though there are many books on the market that deal with applications of the theory of probabilities and statistics, none contain the variety of examples taken from everyday life found in this book. Holland first arouses the curiosity of the reader, then satisfies it in a remarkable way." -- Florin Diacu, University of Victoria, BC, Pacific Institute for the Mathematical Sciences

Our lives are governed by chance. But what, exactly, is chance? In this book, accomplished statistician and storyteller Bart Holland takes us on a tour of the world of probability. Weaving together tales from real life -- from the spread of the bubonic plague in Medieval Europe or the number of Prussian cavalrymen kicked to death by their horses, through IQ test results and deaths by voodoo curse, to why you have to wait in line for rides at Disneyworld -- Holland captures the reader's imagination with surprising examples of probability in action, everyday events that can profoundly affect our lives but are controlled by just one number.

As Holland explains, even chance events are governed by the laws of probability and follow regular patterns called statistical laws. He shows how such laws are successfully applied, with great benefit, in fields as diverse as the insurance industry, the legal system, medical research, aerospace engineering, and climatology. Whether you have only a distant recollection of high school algebra or use differential equations every day, this book offers examples of the impact of chance that will amuse and astonish.

Don S. Lemons

AN INTRODUCTION TO STOCHASTIC PROCESSES
IN PHYSICS

Containing "On the Theory of Brownian Motion" by Paul Langevin translated by Anthony Gythiel

| hardcover | 0-8018-6866-1
| paperback | 0-8018-6867-X
May 2002, 160 pp., 27 line drawings

"This is a clear, well-written, and valuable book. It is both original and important because it ties together much disparate material scattered throughout the literature into a coherent and readable form." -- Gregory N. Derry, Loyola College

"This book will be much appreciated by those who wish to teach, without going into excessive and demanding mathematical details, a little more than can be covered by analysing a one-dimensional random walk on a lattice or solving the Langevin equation. The author covers a lot of ground in very few pages. The last chapter, entitled 'Fluctuations without Dissipation,' gives his admirably slim volume its own flavor. I will have no hesitation in recommending the book to my students." -- K. Razi Naqvi, Norwegian University of Science and Technology

"This is a lucid, masterfully written introduction to an often difficult subject and a text which belongs on the bookshelf of every student of statistical physics. I have every confidence that the accessibility of the presentation and the insight offered within will make it a classic reference in the field."--Dr. Brian J. Albright, Applied Physics Division, Los Alamos National Laboratory

"Professor Lemons's book has reclaimed the field of stochastic processes for physics. For too long it has been taught as a highly mathematical subject devoid of its roots in the physical sciences. Professor Lemons's book shows how the subject grew historically from early fundamental problems in physics, and how the greater minds, like Einstein, used its methods to solve problems that are still important today. The book is not only a good introduction for students, but an excellent guide for the professional." -- William Peter, Advance Power Technologies, Inc.

This book provides an accessible introduction to stochastic processes in physics and describes the basic mathematical tools of the trade: probability, random walks, and Wiener and Ornstein-Uhlenbeck processes. It includes end-of-chapter problems and emphasizes applications.

An Introduction to Stochastic Processes in Physics builds directly upon early-twentieth-century explanations of the "peculiar character in the motions of the particles of pollen in water" as described, in the early nineteenth century, by the biologist Robert Brown. Lemons has adopted Paul Langevin's 1908 approach of applying Newton's second law to a "Brownian particle on which the total force included a random component" to explain Brownian motion. This method builds on Newtonian dynamics and provides an accessible explanation to anyone approaching the subject for the first time. Students will find this book a useful aid to learning the unfamiliar mathematical aspects of stochastic processes while applying them to physical processes that he or she has already encountered.