Loosely speaking, a family of k-regular graphs is
called a family of expanders if the size of the vertex sets goes to
infinity, while the graphs maintain good connectedness. Such graphs are
important in engineering applications such as network designs, complexity
theory, derandomization, coding theory and cryptography. Elementary
Number Theory, Group Theory, and Ramanujan Graphs is devoted to
constructing the Ramanujan graphs which are a family of expanders.
Moreover, these graphs provide an explicit example of an infinite family of
graphs with large girth and large chromatic number. The large girth and
chromatic number problem was originally solved by Erdös using the
probabilistic method, but this does not provide a construction of such
graphs. The book covers a considerable amount of mathematical ground in order to construct and prove the results about the Ramanujan graphs: linear algebra (eigenvalues and spectral gaps), number theory (sums of two and four squares and quadratic reciprocity), and group theory (general linear groups and representation theory of finite groups). Along the way the reader will also see operators between L2 spaces, Chebyshev polynomials, the ring of quaternions, metabelian groups, and Cayley graphs. The fact that all these topics are used to prove graph theory results is what makes this book so interesting. The book is broken up into four chapters covering graph theory, number theory, group theory, and the Ramanujan graphs. In fact, the first three chapters can be read independently and each one is interesting.
The preface of the book claims that this book could be used for an undergraduate course. Based on the topics above I will let you decide if it is appropriate for an undergraduate course at your institution. I think that it would be difficult to use at most undergraduate colleges even as a senior capstone type course. On the other hand, any of the first three chapters could be used for an independent study course with undergraduates. The book would make a nice elective course for graduate students since it pulls so many topics together. If you are looking for a book for a faculty seminar that isn't too discipline-specific, this would be a good choice.
Overall, the book is a well written and stimulating book. My only complaint is that the book doesn't actually give any examples of the applications. It does give references for the applications, but a couple of pages devoted to enlightening the reader about the application would have been worthwhile. [Thomas J. Pfaff]
Time is a big subject and The Little Book of
Time is a very little book. The book will probably be interesting to
enthusiasts, but might be a bit easy for others to put down. Possibly,
enthusiasts already know more than is in this book. The flavor is
philosophical, which is not surprising since Mainzer is a philosopher. In
any case, the book is a nice summary.
The first chapter relates the ancient view of time. The next four chapters follow the evolution of thinking about time based upon physics, beginning with classical physics, followed by relativity, quantum theory, and finally thermodynamics. An important test of a physical theory is that it is time reversible so that the directionality of time enters into physics only in very special places, like thermodynamics. Mainzer's treatment of this point is a bit sparse, even though he spends space on a bifurcation diagram without relating them strongly to his discussion of time. The last three chapters cover biological, psychological, and cultural time. Given the depth of these last three topics, the treatment is extremely brief.
Because the approach is philosophical and intellectual, topics are separated from their cultural and social connections even though history teaches us they are inextricable. For example, even in scientists' minds there were enormous conflicts about religion and time. Newton interpreted Genesis literally. Brydone related an observation by a Roman Catholic priest that some lava flows on Mt. Etna had to predate the creation. For writing this Brydone was castigated by Dr. Johnson, and the publisher inserted an apology into a later edition Brydone's book. Lord Kelvin showed the Earth had to be younger than the age required by geologic time. Darwin felt enormous internal conflict over the Biblical story of creation and the time necessary for selection and adaptation to work. The cultural and philosophical developments are inseparable.
Because all aspects of history interact, I would have included references such as History of the Hour, by Gerhard Dohrn-Van Rossum, The Quest for Longitude, ed. by William J. H. Andrewes, or even a popular book like Time's Pendulum, by Jo Ellen Barnett.[Thomas R. Berger]
Teachers who use history to motivate mathematics
are usually well aware of its power to transform the subject from alien
symbol-pushing to an exciting human adventure. Judging by the popularity of
"history in the classroom" sessions at mathematics meetings and the number
of accessible books written on the subject in recent years, we are now
witnessing a "historic" pedagogical shift. But while the sources are
plentiful, they are not balanced — and material for non-Western
cultures is often the most difficult to find. One popular history of
mathematics textbook, in fact, relegates all of Islamic and Chinese
mathematics to one section at the end of the chapter "Twilight of Greek
Mathematics: Diophantus"!
This state of affairs is especially lamentable because much of the mathematics we teach has roots in and connections with these cultures. In the case of Islam there is extra cause for dismay, since Islam falls naturally between Greece and the Renaissance as part of our Western tradition. One admirable effort to make Islamic mathematics accessible to educators has been J. L. Berggren's Episodes in the Mathematics of Medieval Islam. (Full disclosure: the author was the reviewer's doctoral supervisor.) Written in 1986 and inspired by Asger Aaboe's classic Episodes in the Early History of Mathematics, this book contains a wealth of classroom-ready examples of much of the mathematics one finds in high school and early college: arithmetic, geometry, algebra, and trigonometry.
Why review an 18-year-old book? For one, Springer has just released a paperback edition, identical to the original (other than the plain cover) at about half the price. For another, Episodes remains the only book-length source of Arabic mathematics accessible to teachers, and the material has not been outdated. Springer has taken the right step by issuing a paperback edition to get the book into the hands of a more general readership.
Some episodes fare better than others in today's classroom; for instance, few modern students (alas) will have the opportunity and patience to gain the background required to understand spherical astronomy. However, most examples in this book, properly motivated, can enliven an otherwise pedestrian lecture. For instance, Jamshid al-Kashi's method of determining 5th roots, made obsolete by the likes of Texas Instruments, motivates the study of the binomial theorem by allowing the student to see the theorem work itself out in a practical numeric context. The uses of conic sections through sundials and applications of geometry to Islamic façades demonstrate that the world of geometry, so often portrayed as pure and unsullied by outside influences, interacted with culture in significant and diverse ways. In algebra, the descriptions of various computational and geometric approaches to the solutions of quadratic and cubic equations emphasize the validity of casting problems in different ways to achieve different insights. The arithmetization of algebra by al-Karaji and others is one of many ways in which Islam transformed our Greek mathematical heritage into something much more familiar to our students. The close historical partnership between trigonometry and astronomy is underscored well. In some of my course evaluations, students recall with amazement our attempt to reproduce al-Biruni's method for finding the circumference of the earth using nothing more than a yardstick and a large piece of cardboard.
The re-issue of this gem is significant and welcomed. It will enrich your classes and deepen your perspective on mathematics and culture.[Glen van Brummelen]
Elementary Number Theory, Group Theory and Ramanujan Graphs, by Giuliana Davidoff, Peter Sarnak. and Alain Valette. London Mathematical Society Student Texts vol. 55. Cambridge University Press, 2003. Paperback, 154 pp., $25.00. ISBN 0521531438.
The Little Book of Time, by Klaus Mainzer. Springer-Verlag, 2002. Hardcover, 175 pp., $20.00. ISBN 0-387-95288-8.
Episodes in the Mathematics of Medieval Islam, by J. L. Berggren. Springer-Verlag, 1986 (hardcover), 2003 (softcover reprint). Paperback, 197 pp., $39.95. ISBN 0-387-40605-0.
Thomas J. Pfaff (tpfaff@ithaca.edu) teaches at Ithaca College.
Thomas R. Berger (trberger@colby.edu) is Carter Professor of Mathematics and Computer Science at Colby College.
Glen van Brummelen teaches at Bennington College.
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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.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Mon Apr 12 14:26:22 Eastern Daylight Time 2004