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The Nature of Mathematical Modeling

Neil A. Gershenfeld
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mark McKibben
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A plethora of new books concerning mathematical modeling have recently appeared on the market (e.g., A Course in Mathematical Modeling by D. Mooney and R. Swift, A Concrete Approach to Mathematical Modeling by M. Mesterton-Gibbons, and Mathematical Modeling in the Environment by C. Hadlock). Each author takes a stab at "defining" what the contents of a course on mathematical modeling should be, and each has a different flavor entirely. The disparity among such texts, all presumably addressing the same topic, gives rise to two different characterizations of the philosophies used to write them. The book is either written for a very broad audience and demands only a moderate course in pre-calculus as a prerequisite, or it is intended for a graduate level audience (in mathematics or hard sciences) and draws on a wide variety of results/techniques taught in courses throughout the undergraduate mathematics curriculum. Personally, it seems that the essence of a true modeling course is the latter and the former is more a course in elementary problem solving. Neil Gershenfeld, author of The Nature of Mathematical Modeling, adopts this higher level approach in this book.

The Nature of Mathematical Modeling is a 344-page (224 pages are devoted to content and the remainder various useful appendices) compendium of (over 50) topics/techniques that arise in many different realms of the world of applied mathematics. Now, you might ask "Why use the word compendium?", since this might suggest that the book is a magnum opus constituting 1000 pages or more and addressing literally hundreds of techniques. But while reading through the book my initial reaction was exactly that I was reading an encyclopedia (much like Eric Weinstein's The CRC Concise Encyclopedia of Mathematics). In fact I think this is at least partially intentional, for the author makes the following comment in the book's description:

... Each of these essential topics would be the worthy subject of a dedicated text, but such a narrow treatment obscures the connections among old and new approaches for modeling.

I wholeheartedly agree with this claim. But then he goes on to further say:

By covering so much material so compactly, this book helps bring it to a much broader audience.

This desperately needs to be put into context. It is indicated in the preface that the book is self-contained, other than a basic knowledge of calculus, linear algebra, and basic programming skills. This suggests that the book could be used in an undergraduate course, perhaps at the junior level. However, such an audience, unless exceptionally bright, will become quickly lost in the jungle of techniques. Why? Because their background does not enable them to truly understand or even appreciate the connections among these different worlds that the author is making. I would not use this text in an undergraduate course. The author feels that exposure to such a variety of applications and techniques might inspire students to initiate further investigation of those topics they found interesting. It is my feeling that this text would frighten them away from higher-level mathematics altogether. If, however, the intended audience consists of graduate students of mathematics, engineering, and physics (or experts in any of these fields), then I believe the book could have a positive impact and effectively illustrate the vital connections that exist among all of the techniques discussed.

The exposition of the text is fluent and engrossing, and the author often illustrates a concept with a very well planned example and/or diagram. Also, a thorough list of standard references that discuss the topics he addresses in more detail accumulates throughout the text. Each section is very short (say 3 - 4 pages) and ends with a couple of exercises for the reader to complete. Thorough solutions to these problems are collected as a rather large appendix at the end of the book. This is undoubtedly an important feature of the text, as it gives guidance to a novice reader. Other appendices concerning various graphing software and programming advice are also included.

As a mathematician I found many of the topics discussed in the text interesting and worthy of further investigation. I would certainly recommend the text to a colleague interested in understanding the connections that exist among the seemingly unrelated techniques of applied mathematics.

Mark McKibben ([email protected]) is assistant professor of mathematics at Goucher College in Baltimore, Maryland. His research areas are nonlinear analysis, abstract evolution equations, and integral equations. His most recent work deals with abstract nonlinear nonlocal Cauchy problems in abstract spaces and controllability of solutions to such problems. He is the co-author of the book Algebra (with Dave Keck and Shane Rosanbalm, Wiley, 1998, 2nd edition) and is currently writing texts in real analysis and differential equations.

Preface; 1. Introduction; Part I. Analytical Models: 2. Ordinary differential and difference equations; 3. Partial differential equations; 4. Variational principles; 5. Random systems; Part II. Numerical Models: 6. Finite differences: ordinary difference equations; 7. Finite differences: partial differential equations; 8. Finite elements; 9. Cellular automata and lattice gases; Part III. Observational Models: 10. Function fitting; 11. Transforms; 12. Architectures; 13. Optimization and search; 14. Clustering and density estimation; 15. Filtering and state estimation; 16. Linear and nonlinear time series; Appendix 1. Graphical and mathematical software; Appendix 2. Network programming; Appendix 3. Benchmarking; Appendix 4. Problem solutions; Bibliography.