Read This!

The MAA Online book review column

Briefly Noted

February 2005

The last chapter demonstrates how PDEs are used in the life sciences. This inclusion is in step with a major initiative called Meeting the Challenges: Education Across the Biological, Mathematical and Computer Sciences, a joint project of the Mathematical Association of America (MAA), the National Science Foundation Division of Undergraduate Education (NSF DUE), the National Institute of General Medical Sciences (NIGSM), the American Association for the Advancement of Science (AAAS) and the American Society for Microbiology (ASM). The overwhelming consensus of these groups is that the formerly distinct disciplines of mathematics and biology needed to generate more interdisciplinary courses if future biologists are to be properly trained. The Mathematical Association of America recently published the book, Math & Bio 2010: Linking Undergraduate Disciplines which is a description of the initiative. Clearly, movements like this demonstrate the wisdom of including such a chapter.

An occasional solution in the symbolic mathematics package Maple is also included, showing the reader how the problem can be coded and what the solution will look like. There are a large number of exercises, but no solutions are given. I consider this to be a major weakness. My opinion is that if you are going to include exercises in a math book, then solutions to at least some of them should be included. As a student I always appreciated it, and as a math instructor who has not worked with PDEs for many years, I would welcome the ability to immediately verify the veracity of my recollections.

The emphasis is on the explanations of the form of the solution strategy and how to implement it. While most of the techniques are presented in the form of a theorem, I was hard pressed to find any that were justified with a proof. The coverage is appropriate and the explanations are understandable and thorough. Therefore, I would not hesitate to use it as a textbook. [Charles Ashbacher]

Graphs, Algorithms and Optimization, by William Kocay and Donald L. Kreher, was written as a textbook for computer science majors in an upper division or graduate level course on graph theory involving some programming. This book covers a wide variety of elementary graph theory topics, including bipartite graphs, trees, connectivity, Euler tours, Hamiltonian cycles, digraphs, graph coloring, and planarity. In addition there is a very interesting chapter about graphs on surfaces, and three chapters treating topics in linear programming.

The book is written in an easygoing style, and the proofs are concisely presented and easy to follow. The figures are also clearly drawn and are well utilized to illustrate key concepts and arguments in proofs. The exercises do a good job guiding students through examples to illustrate important definitions and theorems. I do think the exercises might be a bit challenging to a student with a weak background in either the basics graph theory or general mathematical proof techniques — perhaps an appendix with hints or solutions would be a nice addition.

This book seems to be well-suited to its intended purpose for upper division computer science majors (with a reasonable math background). The authors describe all of the algorithms discussed using such general programming constructs as if-then statements, for loops, and while loops, not tied to any specific programming language. While most of the "pseudocode" the authors use is self-explanatory, some of the more complex algorithms can look a little overwhelming written in this way. This is something most upper division computer science majors will be familiar with, and they will have little trouble understanding the algorithms.

As a final remark, I think Graphs, Algorithms and Optimization would serve as a fine textbook for an undergraduate graph theory course for math majors. I believe that even students with little or no programming background could learn a lot of mathematics from this book, especially if the instructor were to place minimal emphasis on the programming and pseudocode algorithms. All the positive comments made earlier about the book — that it is well-written, the proofs are easy to follow, the figures complement the text, and the exercises are helpful to student understanding — in my opinion make it a strong choice for this purpose as well. [Frederick M. Butler]

The first edition of Hirsch and Smale's Differential Equations, Dynamical Systems, and Linear Algebra has been a standard on mathematical bookshelves for three decades. Now middle-aged, this 30 year old text has gotten a facelift and a new convertible. Well, not really a new convertible, but it did drop the faithful-yet-staid "and Linear Algebra" from the title (but not from the contents), to replaced by the alluring young "and An Introduction to Chaos". Along the way, it picked up a third author, the prolific and entertaining Bob Devaney.

The first hint of the new approach appears subtly on page 1. The first example (dx/dt = ax) is word-for-word the same as the old text, with this exception: in 1974, the authors wrote, "Here a denotes a constant." The new text reads, "Also, a is a parameter; for each value of a we have a different differential equation." Readers familiar with the first edition will find much that is familiar, but with an increased emphasis on the behavior of families of solutions.

Of course the material in the book has been reorganized a bit. Not only has some content shifted during rewriting, but, as noted above, there is a consistent emphasis on understanding the geometry and behavior of solutions to differential equations. The exercises (which were earlier called "problems") have moved from the end of each section to the end of each chapter. In addition, there are three chapters with new material: The Lorenz System, Discrete Dynamical Systems, and Homoclinic Phenomena. Starting about a third of the way through the book, each chapter concludes with an "Exploration" that allows students to pursue an extended, open-ended project.

The "facelift" mentioned above is the addition of many new graphs and improvements in the appearance of the old ones. The book is nice to look at — even fairly easy to skim — because of these. Devaney's quirky humor pokes through occasionally. My favorite example is Figure 4.1 on page 63, which depicts the behavior of families of solutions to X' = AX depending on the trace and determinant of the matrix A. The caption reads, "The trace determinant plane. Any resemblance to any of the authors' faces is purely coincidental." Sure enough, when I glanced back at the figure, I saw a smiley face among the collected graphs. This visual mnemonic is cute enough that it's easy to remember; this picture alone is worth the $80 price tag, I think. [Annalisa Crannell]

Quite early in the development of the theory of algebraic numbers, mathematicians noticed that there were deep formal analogies between that theory and the theory of algebraic functions (with complex coefficients, at first) of one variable. Dedekind and Weber, for example, wrote a book dedicated to pursuing exactly this idea, recasting the theory of algebraic functions in entirely algebraic terms, highlighting the similarity with the theory of algebraic number fields.

Early in the 20th century, it became clear that this analogy is especially strong when one studies function fields in finite characteristic of transcendence degree one, that is, finite extensions of the field F_p(t) of rational functions with coefficients in the field with p elements. These are the "Function Fields" with which this book deals.

The analogy can be, and has been, read in several ways. In a famous letter to his sister, André Weil highlighted the idea that function fields might be easier than number fields. For example, a natural analogue of the Riemann Hypothesis can be proved in the function field case. Similarly, the cyclotomic fields that are obtained by adjoining roots of unity in the number field case can be thought of as analogous to the fields we get by enlarging the field of constants from F_p to F_p^r, since every element of a finite field is a root of unity. This analogy was the seed of "Iwasawa Theory."

Going the other way, however, can be even more interesting. We can try to study, in the context of function fields, structures that are analogous to certain interesting structures in the number field case. This was the motivation for the development of "Drinfeld Modules", which (with their generalizations) are the main topic of Thakur's book. Drinfeld Modules are classified in part by their rank. If the rank is one, they are a kind of characteristic p version of the multiplicative group, leading to a whole new notion, for example, of what a "root of unity" is. In the case of rank two, they are a little like elliptic curves. In higher rank they are stranger, with no obvious number field analogue. The result is a theory that is rich, complex, and fascinating.

Dinesh Thakur is one of the leaders in the study of Drinfeld Modules and related themes. He describes the book as "just a tour of some topics I enjoyed learning and working on", which makes it seem lighter and less thorough than it actually is, but does give a little bit of the flavor. It is dense with mathematics, but there is also motivation and discussion. The overall feeling is that of a very helpful survey of a very interesting field.

Unfortunately, the publisher has not provided the author with a good copy-editor. There is far too much fractured English, most often in way that could easily be fixed with a little time and a blue pencil. This is particularly the case in the Preface; as things get more technical, the very formal nature of the language helps minimize the problem. This is a real pity. The publisher has done the author a disservice that may limit his readership, or at least reduce the readers' enjoyment of a very nice monograph. [Fernando Q. Gouvêa]

Publication Data

Graphs, Algorithms and Optimization, by William Kocay and Donald L. Kreher. Chapman & Hall/CRC, 2004. Hardcover, 504 pp., $89.95. ISBN 1584883960.

Differential Equations, Dynamical Systems, and an Introduction to Chaos, by Morris Hirsch, Stephen Smale, and Robert L. Devaney. Elsevier, 2003. Hardcover, 425 pp., $79.95. ISBN 0123497035.

Function Field Arithmetic, by Dinesh Thakur. World Scientific, 2004. Hardcover, 404 pp., $92.00. ISBN 981-238-839-7.

Charles Ashbacher (cashbacher@yahoo.com) teaches at Mount Mercy College in Cedar Rapids, Iowa.

Frederick M. Butler is Assistant Professor of Mathematics at the Institute for Mathematics Learning, West Virginia University.

Annalisa Crannell's primary research is in topological dynamical systems, but she is also active in developing curricular materials for courses on "Mathematics and Art" as well as materials for writing across the curriculum.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

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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.