Tung, K.K. 2007. Topics in Mathematical Modeling. Princeton, NJ: Princeton University Press; 336 pp, $45. ISBN 978–0–691–116426.
The world today is awash in textbooks on modeling. These range in level from texts for students with very little mathematical background to texts for graduate students and professionals. Many of these books use modeling as a foil to promote a particular agenda: dynamical systems, or nonlinear differential equations, or perhaps finite mathematics. These books are less interested in teaching and more interested in planting a flag.
The book under review is a refreshing departure from the sorts of polemics just described. Tung’s preface shows that he is a dyed-in-the-wool teacher of considerable talent whose only mission is to show the student how to take raw empirical data and turn it into a mathematical paradigm that can be analyzed. His prerequisites are solid but minimal: calculus and a smat-tering of ordinary differential equations (ODEs). He is wise to provide an appendix with a quick treatment of ODEs for those whose background is deficient. Tung also describes in the preface a clear path for those who wish to avoid the differential equations altogether.
Tung covers some of the usual modeling topics but also many others that are surprising and refreshing. Among the former are
• Fibonacci numbers,
• compound interest,
• radiocarbon dating,
• Kepler’s laws,
• nonlinear population models, and
• predator-prey problems.
Among the latter are
• global warming,
• marriage and divorce,
• analysis of the El Ni ˜no effect,
• the age of the Earth,
The UMAP Journal 29 (2) (2008) 175–184. c!Copyright 2008 by COMAP, Inc. All rights reserved.
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• the Broughton and Tacoma Narrows bridges,
• climate models,
• HIV modeling, and
• mapping the World Wide Web.
Tung uses a variety of techniques to analyze these different problems.
Among these are differential equations, dynamical systems, linearization, phase-plane analysis, and many others. One important feature of the book is that an entire chapter is devoted to each problem and its related ideas. In a calculus class, the student typically sees examples and problems that can be solved in a few lines. Here the student sees the substantive development of mathematical ideas over the course of a prolonged discussion. The book does not contain any proofs per se, but it has discussions that have the gravitas of proofs.
The writing in this book is delightful and elegant—almost literary in its beauty and precision. The presentation is thoughtful and readable. The organization is exemplary. As an instance, each chapter begins with a few words telling the reader exactly what mathematics will be needed for the discussion. Every chapter has a useful introduction. There are many interesting references of a philosophical or cultural nature.
The book contains plenty of entertaining graphics, photographs of math-ematicians, and other illustrative figures. The sections and their titles are chosen to give the reader a keen sense of the flow of ideas. The layout of the book is open and friendly. This is certainly an inviting text for students.
One of the truly critical components of a successful textbook is the ex-ercise sets. This text contains exex-ercises that are quite thought-provoking.
Each is a word problem that could be used for class or group discussion, and many of these problems could be developed into research projects or term papers. One might wonder whether it would have been propitious to include some elementary exercises as well. If I were to teach from this text—and I would certainly enjoy doing so—I would probably find myself hunting around in other texts for routine and drill exercises to give the stu-dents. (One may well puzzle over what these elementary exercises might consist of. But something must be provided to help bring students up to speed.) That would be too bad, for it is the author’s job to provide that sort of material. But I must stress that the exercises that are provided are the product of much research and thoughtful editing. They are quite valuable and instructive.
Another small criticism—or at least a comment—is that the book con-tains few if any displayed examples. One usually expects a textbook to have the format
• Introductory patter, then
• enunciation of idea, followed by
• illustrative example
for each key topic. Tung’s book deviates from that paradigm because in fact each chapter is a topic. Each chapter is an example. It actually does not make a great deal of sense—for what Tung is trying to achieve—to have displayed examples. Such items would be too trite.
But the point of the discussion in the last two paragraphs is worth not-ing: For many if not most students, this course, and this book, will be a first exposure to serious mathematical discourse. Here, for the first time, the student will see protracted mathematical reasoning directed toward a sophisticated and well-defined goal. This is not the place for cute little problems with three-line solutions. This is instead the venue for rather re-condite reasoning. Elementary exercises and elementary examples do not really have a place here. The student in a course like this will need to exert some effort in order to get something of value out of it. But the effort will be well rewarded.
Another important point is that this text illustrates, unlike any text in a previous or more elementary course, the symbiosis of mathematics with other parts of science and technology. Mathematics is not a cottage industry that caters primarily to its own whims. Rather, mathematics is the key to understanding much of the world around us. Surely Tung’s book illustrates this important point clearly and decisively. That is an incisive message for any student to see and understand, and it takes a good textbook to get the message across.
Steven G. Krantz, Professor of Mathematics, Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130;sk@math.wustl.edu.
Hunt, Earl. 2007. The Mathematics of Behavior. New York: Cambridge Uni-versity Press, 2007; x+346 pp, $80, $34.99 (P). ISBN 978–9–521–85012–4, 978–0–521–61522–8.
For too many mathematics professors, if they have any idea of industry, it is the government research laboratories. That is, they have no idea of the sort of jobs that most people working in industry with mathematics degrees experience. But I am going to give one piece of advice for the student facing industry, and then I will extrapolate from it back to academia.
Suppose that a student with a recent degree in mathematics receives two job offers that seem utterly equivalent in pay, conditions, security, and so on.
But there is one significant difference. In Corporation A, the worker will be working with many mathematical scientists in an environment where much of the work is inherently mathematical (and scientific). In Corporation B, the worker will be the house mathematician: the go-to person for mathematical questions.
Now, though the second job might sound like a nice opportunity, the worker is almost certainly better off to take the job with Corporation A. In Corporation A, you have mathematical workers who make work for one another. With any luck, they will support one another and collaborate on papers and generate more mathematical work. But in Corporation B, the worker is very likely to be lonely—the corporate pariah. The mathemat-ical advice that the worker is solicited to provide is likely to be met with incomprehension and suspicion.
How does this apply to academia? Suppose that a student has a de-gree in mathematics but wants to pursue a Ph.D. in cultural anthropology.
My advice is to pursue a degree in physical anthropology instead. Physi-cal anthropology is an area that is much more quantitative and where the mathematically-trained student has a much greater chance of fitting in. Af-ter receiving the Ph.D., there is little to keep the physical anthropologist from venturing into cultural anthropology at least occasionally. There is an unmistakable trend towards greater use of mathematics in each of the social sciences. I have even heard of departments in the social sciences recruiting mathematics majors on the grounds that it is easier to convert them to the discipline at hand than trying to convert social science students to mathematical methods. (This sort of thing seems to occur frequently in different areas and is one reason to major in mathematics.)
The book reviewed here is a survey of experimental psychology, a field that has a long tradition of mathematical modeling and in particular of first-rate statistical research. Psychology, as we are told on Day 1 in Psychology 100, is the study of behavior. Earl Hunt has practiced mathematical psy-chology for 50 years. The book is a fairly comprehensive survey of experi-mental psychology; it does not touch upon any area of clinical psychology, even when the controversies there may have mathematical content (largely through the use of statistics). I will not enumerate the topics covered in this book. Many of them will be familiar to anyone with a knowledge of applied mathematics. Some of the material will be familiar only to experimental psychologists—for example, the chapter on the physics of perception. That subject does not interest me much, but I found it valuable because of Hunt’s historical approach.
The book begins with Eratosthenes’ work on estimation of the Earth’s circumference. This is because Prof. Hunt first discusses the philosophy of mathematical modeling. His second chapter explores the foundations of probability. I think that he is a little too theoretical here, given the subject of the book; no one is going to learn probability here. He says (p. ix) that the book should be accessible to anyone with a “basic understanding of calculus, and most of the book will not even require that.” I would require some calculus for mathematical maturity. I would also expect the reader to have some knowledge of probability and of statistics.
Prof. Hunt, as an experimental psychologist, is quite good on statistical issues. But the reader should have some knowledge of it coming in. To
put it succinctly: A freshman with one semester of calculus can learn a lot from this book and should acquire the gist of many topics; a senior with many mathematics courses can learn a lot more and can get a remarkably clear idea of experimental psychology; but knowledge of statistics would be most useful to the reader. There is serious attention to Bayes’ Theo-rem, to covariance, to regression, and to factor analysis (a technique used by experimental psychologists probably more than by any other group).
This book will give the mature student the information necessary to make an informed decision about pursuing experimental psychology in a grad-uate program. One topic in experimental psychology is among the most controversial and dangerous topics in all of academe. I am referring to intelligence testing, which has been the subject of a remarkable amount of political posturing and uninformed commentary, in some cases by math-ematicians. Prof. Hunt’s discussion of that area is far more detailed and sophisticated than what readers see in the popular press (and by “popular press” I include venues such as Scientific American). However, it cannot be considered a comprehensive survey; it is instead a good introduction to a contentious subject at an appropriate level. The wars over intelligence test-ing are far more intense than those in mathematics and the hard sciences.
That controversy has had far more public exposure than most.
A similar war in clinical psychology, over “repressed memory,” has involved serious contributions by experimental psychology. Prof. Hunt shows the experimental approach to memory. He barely mentions the con-troversy and equivocates to some extent; other experimental psychologists have a great deal to say on this topic and there is no hint here of the intensity of the conflict. In fact, a survey of the battles over repressed memory is suf-ficient to show that conflicts in the social sciences tend to be more intense than those in math and the hard sciences and that social scientists do combat at an entirely different level. A corollary to this is that Ph.D. programs in these areas can be even more hazardous than those in the “hard” sciences.
Keeping this in mind, mathematics students should consider the social sci-ences for graduate study. (This is certainly true of economics. Economics in the U.S. and the U.K. is so mathematical that I think that mathematics majors have an advantage.)
The book reviewed here is a good survey of much of experimental psy-chology. It is also of interest to anyone pursuing the mathematics of soci-ology and to a lesser extent political science (for the treatment of Arrow’s theorem).
James M. Cargal, Mathematics Department, Troy University—Montgomery Cam-pus, 231 Montgomery St., Montgomery, AL 36104; jmcargal@sprintmail.
com.
Nahin, Paul J. Digital Dice: Computational Solutions to Practical Probability Problems. Princeton, NJ: Princeton University Press; xi + 262 pp, $27.95.
ISBN 978–0–691–12698–2.
The evolution of computer languages is a fascinating but complex topic.
But it is probably safe to say that prior to about 1985 most languages were quite useful for numerical computation (although there are obvious excep-tions such as COBOL, the dominant business language). However, in the late 1980s, C and then C++ became dominant in computer science depart-ments. The problem with C was twofold:
• it is a cryptic language; and
• it is designed to give the user access to the guts of the machine—a capa-bility largely irrelevant to numerical computation.
C (C++ actually) was replaced as the dominant language by Java. Java was designed for Web development; again, a purpose largely irrelevant to computation.
The result of all of this is that it is easy to encounter students who have passed courses in programming but who somehow do not know anything about simple control structures. Many of them cannot program a spread-sheet. (Spreadsheets can be remarkably efficient for a variety of tasks, es-pecially in discrete mathematics but also in areas such as differential equa-tions.) It appears that mathematicians are responding to this problem by doing their numerical programming in mathematical environments such as Mathematica, Maple, and Matlab. Also, I suspect that mathematics de-partments are teaching their majors programming themselves rather than sending them to the CS departments.
Nahin’s book reviewed here could be a useful for a course in mathe-matical programming. Paul J. Nahin is is professor emeritus of electrical engineering at the University of New Hampshire. In recent years, he has published a number of books. His An Imaginary Tale: The Story ofi[1998]
is I think the best introduction to complex arithmetic and analysis for the undergraduate in mathematics and the sciences. The sequel, Dr. Euler’s Fabulous Formula [2006], is something of a tour de force.
In 2000 Nahin published a book of probability problems, Dueling Idiots and Other Probability Puzzlers [2000], in which there is sometimes the sug-gestion of analyzing the problem by simulation. The book reviewed here is explicitly dedicated to that technique. What makes the book useful is that there is little formal probability. Density functions and probability dis-tributions do not come up; so the teacher using this book as a source can concentrate on programming and logic.
The book is divided into four parts: introduction, problems, solutions, and appendices. The solutions usually contain analysis as well as numerical results. Programming is in Matlab, which closely resembles structured Basic and functions effectively as readable pseudocode. The introduction
is an informative essay, and the nine appendices are themselves likely to be of interest to the reader.
Another current educational issue is that students in the mathematical sciences often do not seem to understand the power of computation. This is because the curriculum tends to lag behind technology—by decades.
For example, statistical methods, to a remarkable extent, reflect the tech-nology that was available to Ronald Fisher in the 1920s. Many statistical tests can be replaced by remarkably quick computational methods; more-over, these methods often are nonparametric and as such (I would say) esthetically superior to classical methods. The very first problem in Digital Dice is essentially a statistical test: Five dishwashers work in a restaurant.
In a one-week period, five dishes are broken and four of those are broken by one individual. The individual claims that this is merely bad luck. The problem is to calculate the probability that one individual would break four out of the five dishes under the assumption (null hypothesis) that each dishwasher is equally likely to break a dish. Here, though, I believe that author Nahin is in error. He calculates the probability that the one partic-ular worker would break four or more dishes under the null hypothesis.
The correct approach, I am sure, is to calculate the probability that some one employee would break four or more of the five dishes under the null hypothesis. In any case, that very question is a good one for the students to contemplate.
Much of the impetus of Digital Dice is the analysis of counterintuitive problems, with the idea that these problems might better motivate the stu-dents. There is a lot of truth to that; nothing is more motivating than results that are unexpected and surprising. These problems often motivate both approaches to the solution: analysis to understand why things work the way they do, and simulation to verify both the way things work and the analysis.
In my review of books on investment [Cargal 2006, 87–89], I discussed a problem from Morton Davis and I recapitulated his analysis, which leads to a highly counterintuitive result about a reasonable-seeming investment strategy. When I first encountered the problem in 1989, I was working in industry. To simulate the problem, I would wait until after work and run it on as many as 10 personal computers simultaneously (this was when a 25 MHz Intel-386 processor was considered fast). I didn’t doubt the analysis of the problem—it is at the precalculus level—but I had to see it with my own eyes.
The fact is, though, that counterintuitive problems are not necessary.
Students at this level usually are not acquainted with the gambler’s ruin problem, for example, and this book offers a great set of exercises. An-other writer, Julian Havil, has produced a remarkably similar set of books.
His Non-Plussed. . . [2007] is devoted to counterintuitive problems; and its sequel—which I believe is of even greater interest to mathematics majors—
Impossible?: Surprising Solutions to Counterintuitive Conundrums [2008] has
just appeared, with a prepublication blurb by Nahin. (Havil’s Gamma [2003]
would roughly correspond to Nahin’s Dr. Euler’s Fabulous Formula.) How-ever, the problems in Digital Dice are selected for computation exercises and are not necessarily as counterintuitive as those in the prior book Dueling Idiots. The most counterintuitive problem in Digital Dice is problem 14: Par-rondo’s Paradox, probably the most challenging probability paradox I have seen; it has received a fair amount of attention lately. As usual, Nahin both discusses computational simulation and provides an analysis. However, Havil provides probably a better analysis in Non-Plussed [2007].
All of the books by Nahin and Havil are worth having, including others not listed here. I particularly recommend Digital Dice for the task of teach-ing undergraduates in mathematics the fundamentals of computation and simulation.
References
Cargal, J.M. 2006. Review of books on investment. The UMAP Journal 27 (1): 81–90.
Havil, Julian. 2003. Gamma. Princeton, NJ: Princeton University Press.
. 2007. Non-Plussed! Mathematical Proof of Implausible Ideas. Prince-ton, NJ: Princeton University Press.
. 2008. Impossible?: Surprising Solutions to Counterintuitive Conun-drums. Princeton, NJ: Princeton University Press.
Nahin, Paul J. 2000. Dueling Idiots and Other Probability Puzzlers. Princeton, NJ: Princeton University Press.
. 1998. An Imaginary Tale: The Story ofi. Princeton, NJ: Princeton University Press.
. 2006. Dr. Euler’s Fabulous Formula. Princeton, NJ: Princeton University Press.
James M. Cargal, Mathematics Department, Troy University—Montgomery Cam-pus, 231 Montgomery St., Montgomery, AL 36104; jmcargal@sprintmail.
com.
Shiflet, Angela B., and George B. Shiflet. Introduction to Computational Sci-ence: Modeling and Simulation for the Sciences. Princeton, NJ: Princeton University Press, 2006; xxiv + 554 pp, $69.50. ISBN 0–691–12565–1.
Applied mathematics is almost synonymous with mathematical model-ing—which is why there is an annual issue of this journal devoted to a modeling competition. It might also explain why the editor constantly