Books

A Radical Approach to Lebesgue's Theory of Integration This is a sequel to A Radical Approach to Real Analysis (ARATRA). That book ended with Riemann's definition of the integral. That is where this text begins. All of the topics that one might expect to find in an undergraduate analysis book that were not in ARATRA are contained here, including the topology of the real number line, fundamentals of set theory, transfinite cardinals, the Bolzano–Weierstrass theorem, and the Heine–Borel theorem. I did not include them in the first volume because I felt I could not do them justice there and because, historically, they are quite sophisticated insights that did not arise until the second half of the 19th century.
This book owes a tremendous debt to Thomas Hawkins' Lebesgue's Theory of Integration: Its Origins and Development. Like ARATRA, my book is not intended to be read as a history of the development of analysis. Rather, it is a textbook informed by history, attempting to communicate the motivations, uncertainties, and difficulties surrounding the key concepts. Click here for a list of corrections.

A Course in Computational Number Theory, co-authored with Stan Wagon,published by Springer-Verlag under the Key College Publishing label. We have a Mathematica file of the strong pseduoprimes: StrongPseudoprimeData.m and a list of corrections. The file CNT.m is consistent with Mathematica versions 6.0and 7.0.
This is an introduction to number theory couched in an exploratory, computation rich setting that makes extensive use of Mathematica. Topics include the Euclidean Algorithm, modular arithmetic, linear congruences, Chinese Remainder Theorem, Fermat's Little Theroem and pseudoprimes, Euler's phi, perfect numbers, primitive roots and orders, distribution of primes, prime testing and certification, RSA, check digits, factoring algorithms, quadratic residues and reciprocity, Pepin's test, continued fractions, Pell's equation, CFRAC, Lucas sequences for prime certification and factorization and the Lucas-Lehmer algorithm, representations as sums of squares, Gaussian primes.

Proofs and Confirmations: the Story of the Alternating Sign Matrix Conjecture, published jointly by the Mathematical Association of America (Spectrum Series) and Cambridge University Press (or Cambridge University Press, NY) 1999. This is the story of the proof of the alternating sign matrix conjecture written at a level accessible to anyone who has had a course in linear algebra. It describes recent research in algebraic combinatorics, using this example to illustrate the surprising twists and turns of actual mathematical research. It is also an opportunity to explore some of the related fields that fed into the ultimate solution. These include partition theory, plane partitions, symmetric functions, hypergeometric and basic hypergeometric series, lattice path counting problems, and the Yang-Baxter equations of statistical mechanics. A notebook of the Mathematica commands is available as well as corrections. Solutions and hints for selected exercises in chapters 1-4 are available as either a PostScript or a PDF file. Note that for some reason I do not understand, the latter is upside down which is not a hindrance if you want to print it, but does make it difficult to read it from a screen. Kim-Ee Yeoh at Wisconsin has posted JAVA programs for finding and counting alternating sign matrices.

A Radical Approach to Real Analysis 2nd edition, Mathematical Association of America, 2006. This is an introduction to real analysis that begins with the problems the led to the development of this subject. It starts with Fourier series and the difficulties it presented for mathematicians of the early 1800s. It presents both successes and failures and explains how and why the fundamental definitions and theorems of real analysis came to be.

Click here to access the Web Resources for the second edition of A Radical Approach to Real Analysis

Second Year Calculus: from Celestial Mechanics to Special Relativity, Springer-Verlag, 1991. This is a vector calculus textbook that empahsizes the language of differential forms and the physical motivation for the topics encountered. The first and third chapters describe celestial mechanics and the latter chapters deal with electricity and magnetism and show how the symmetries of Maxwell's equations lead to special relativity. The book concludes with a proof that E=mc^2. There are two pdf files of to the 4th printing: calc_corrections-1.pdf and calc_corrections-2.pdf.

Factorization and Primality Testing, Springer-Verlag, 1989. This is really an introduction to Number Theory that is built around around the twin problems of how to determine whether a large integer is prime and, if it is not, how to factor it into its prime factors. It includes descriptions of the RSA public-key cryptosystem, the Multiple Polynomial Quadratic Sieve, and the Elliptic Curve methods for factorization and primality testing. Available ftp files include corrections and a pdf file of the corrections.

The Rademacher Legacy to Mathematics, edited with George Andrews and L. Alayne Parson, The Centenary Conference in Honor of Hans Radmacher, July 21-25, 1992, The Pennsylvania State University, #166 in Contemporary Mathematics. American Mathematical Society, Providence, Rhode Island. 1994.

Analytic and Combinatorial Generalizations of the Rogers-Ramanujan Identities, Memoirs of the American Mathematical Society #227, March, 1980

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