David M. Bressoud
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.
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.
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.
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.