Read This!

The MAA Online book review column

Briefly Noted

July 2006

At this point, we are adding some 30 reviews a month to MAA Reviews. This column contains (sometimes abbreviated versions of) some of the reviews that have appeared there (fairly) recently.

There are many introductory textbooks on probability and statistics, for two reasons: first, there is a huge market for such books because many university courses require students to take one or more semesters of statistics; second, it is difficult to present this material well. If it were easy to strike a balance between theoretical rigor and practical application, after all, the perfect text would already have been written.

Oliver Ibe's Fundamentals of Applied Probability and Random Processes is informed by his experiences teaching the introductory probability and statistics course to junior and senior engineering students at the University of Massachusetts-Lowell, where he is a professor in the Department of Electrical and Computer Engineering. This book presents a straightforward exposition of the basics of probability and statistics, starting with basic definitions of sample space and events, and proceeding through fairly advanced topics not always included in a one-semester statistics course. Each chapter is broken down into small subunits, making this a useful reference book as well as a textbook. The material is presented clearly, and solved problems are included in the text. The text layout is particularly good, with lots of white space and logical use of headers which make it easy to locate a particular topic within a chapter. There are exercises at the end of each chapter, but no solutions provided. No reference is made to a web page or any other types of supporting materials.

Fundamentals of Applied Probability and Random Processes could be used as a probability text in many contexts, including beginning statistics classes at the graduate level. Its usefulness is not be limited to engineering departments: the examples used as illustrations are drawn from many fields. It could also serve as a self-teaching text, although the fact that apparently no solutions are available for the end-of-chapter problems makes it less useful for that purpose. However, this text assumes readers are comfortable with mathematical notation and competent in at least freshman calculus: students without good mathematical preparation (I'm thinking of many graduate student in the social sciences, for instance) may find their eyes glazing over mid-way through the second chapter.

[Sarah Boslaugh; posted to MAA Reviews 07/02/2006]

Geometry: A Self-Teaching Guide gives the reader a basic course in high school geometry. The text covers standard topics like the midpoint and distance formula, angles, polygons, triangles, circles, perimeters, area, volume, and conic sections. For the most part, it assumes no math background except arithmetic, though a few sections use algebraic concepts, like factoring and completing the square.

The text is most appropriate for high school students that need a study guide, homeschoolers that want a basic geometry handbook or adults that want a refresher course. As the title denotes, it is a self-teaching guide and as such features a pre-test in each chapter, lots of worked examples, and post-tests to diagnose your progress. The guide also fosters problem-solving skills as most chapters include a section on applications where students must process and assimilate all of the concepts learned in the chapter.

A flaw in the text is that often the authors present formulas and clearly explain how to use the formulas, but don't explain where they come from or why they work. I am not expecting rigorous proofs or derivations of formulas, but an informal explanation of where, for example, the distance formula comes from or a diagram showing why (n – 2) x 180 gives the sum of angles in a polygon (in degrees), may help students remember and understand the concepts better.

Slavin and Crisonino encourage you to "work your way through this book problem by problem." This is entirely possible as the prose is readable and the problems are carefully explained line by line. The guide offers a concise and structured way to learn basic geometry.

[Kara Shane Colley; posted to MAA Reviews 07/19/2006]

According to the author, Huygens, The Man Behind the Principle is the first complete biography of the famed physicist and mathematician that includes his development and background. This is a surprising claim considering the wide-reaching contributions to the sciences that were made by Huygens.

Huygens looks to be a very diligently referenced work. The book is a thoroughly footnoted and cross-referenced chronicle of the life and background of the physicist. The research behind the book seems thorough, with plentiful footnotes on each page. The Bibliography and "Further Reading" sections span almost thirty pages.

This diligence, however, limits the audience for the book; it is not an easy read. The tone is formal enough to make casual reading a chore. The mathematician or physicist looking for a discussion of Huygens' contributions will also be disappointed as the mathematical content is limited to just a few pages spread through the volume.

Overall, Huygens, The Man Behind the Principle will likely be an excellent reference for the historian. Anyone else without a specific interest in Huygens will have a hard time with it.

[Steven Frankel; posted to MAA Reviews 07/09/2006]

In the original meaning, a "handbook" was a small book that one could hold in one's hand. The idea was that such a book could be carried around and therefore serve as a convenient reference. As the word's meaning developed, it came to mean "a compendious book or treatise for guidance in any art, occupation, or study," as the OED puts it. This Handbook of Elliptic and Hyperelliptic Curve Cryptography definitely falls within the latter definition. It has more than 800 pages and weighs in at almost four pounds. No one would want to carry this monster around. It clearly aims for fairly complete coverage of the basics of public-key cryptography using elliptic and hyperelliptic curves.

The structure of the book is interesting. The first chapter gives an introduction to public-key cryptography at a fairly abstract level. By page 7, the RSA system has already been described, and by the end of the chapter we have a general framework in place for cryptographic systems based on various kinds of discrete logarithm (DL)problems. Thus, in the rest of the volume, all one needs to do is generate an appropriate DL problem and refer to the first chapter.

The following section sets up the necessary algebraic background, quickly running from groups, rings, and fields to the cohomology of algebraic varieties. Then comes a section on "elementary arithmetic" that focuses mostly on algorithms for arithmetic (exponentiation in a group, "infinite-precision" integer arithmetic, finite field arithmetic, and p-adic numbers). The third section takes up the arithmetic of algebraic curves, focusing mostly on the elliptic and hyperelliptic cases. The concluding sections home in on issues related to cryptography, discussing point counting, discrete logarithms, and the practical implementation of these systems.

This is not, of course, the place to go to begin to learn this material, but it should serve as a very useful reference. In particular, specialists in arithmetical algebraic geometry who would like to learn more about algorithmic issues will find it very useful. I suspect that specialists in cryptography who want to find out about elliptic curve cryptography will find it tough sledding (depending, of course, on their background in algebra and number theory). Most undergraduates will find it very hard, but they may find certain chapters useful, especially if they are interested in finding out about applications of abstract algebra. Libraries should make sure they have a copy.

[Fernando Q. Gouvêa; posted to MAA Reviews 07/05/2006]

Publication Data

Geometry: A Self-Teaching Guide, by Steve Slavin and Ginny Crisonino. John Wiley, 2005. Paperback, 276 pages, $17.95. ISBN 0471386340.

Huygens: The Man Behind the Principle, by C. D. Andriesse. Cambridge University Press, 2005. Hardcover, 440 pages, $95.00. ISBN 0521850908.

Handbook of Elliptic and Hyperelliptic Curve Cryptography, by Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, and Frederik Vercauteren. Chapman & Hall/CRC, 2005. Hardcover, 808 pages, $99.95. ISBN 1-58488-518-1.

Sarah Boslaugh is a Senior Statistical Data Analyst in the Department of Pediatrics at the Washington University School of Medicine in St. Louis, MO. She wrote An Intermediate Guide to SPSS Programming: Using Syntax for Data Management with Sage Publications in 2005 and is currently writing Secondary Data Sources for Public Health: A Practical Guide for Cambridge University Press. She is also Editor-in-Chief of The Encyclopedia of Epidemiology which will be published by Sage in 2007.

Kara Shane Colley studied physics at Dartmouth College and math education at Teachers College. She has taught math and physics to middle school, high school, and community college students in the U.S., the Marshall Islands, and England . Currently, She is volunteering aboard the Halfmoon, a replica of Henry Hudson’s 17th century ship, docked in Albany, NY. Contact her at karashanecolley@yahoo.com.

Steven Frankel is an undergraduate Engineering student at The Cooper Union in New York, NY. His primary interests lie on the line between Electrical Engineering and Mathematics, including Signal Processing and Control Systems. He can be contacted at franke2@cooper.edu.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME. He is the editor of FOCUS Online and MAA Reviews.

Back Issues:

• April 1999: Reprints of "classics", books on applied mathematics.
• May 1999: Books on math and physics, both popular and technical.
• July 1999: Various kinds of geometry, and classics in applied mathematics.
• August 1999: What's happening, Dirichlet, and Cryptonomicon.
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• February 2000: Fallacies, Flaws, and Flimflam, measuring the universe, and primary sources at USMA.
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• May 2000: Grassmann, Smarandache, and a really big prime.
• June 2000: A problem book, a history book, a collection of biographies, and a historical survey of reciprocity laws.
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• September 2000: excursions, history, and some high-level expository writing.
• November 2000: a festival of math videos and the Fourier-analytic proof of quadratic reciprocity.
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• December, 2005: lattices, irrational numbers, topology, and linear models.
• January, 2006: algebra, geometry, bubbles, jaws, Greek mathematics, and quotations.
• February, 2006: classics in statistics and in mathematics education, a math league, and transcendence.
• March, 2006: Halmos, Riemann, statistical models, and a contest problem book.
• April, 2006: Real analysis, maxima and minima, geometry, and the IMO.
• May, 2006: numbers, statistics, algebraic numbers, and problems from New Mexico.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.