The Princeton Companion to Applied Mathematics - Hardcover

Hardcover

ISBN 10: 0691150397 ISBN 13: 9780691150390

Publisher: Princeton University Press, 2015

Synopsis

The must-have compendium on applied mathematics

This is the most authoritative and accessible single-volume reference book on applied mathematics. Featuring numerous entries by leading experts and organized thematically, it introduces readers to applied mathematics and its uses; explains key concepts; describes important equations, laws, and functions; looks at exciting areas of research; covers modeling and simulation; explores areas of application; and more.

Modeled on the popular Princeton Companion to Mathematics, this volume is an indispensable resource for undergraduate and graduate students, researchers, and practitioners in other disciplines seeking a user-friendly reference book on applied mathematics.

• Features nearly 200 entries organized thematically and written by an international team of distinguished contributors
• Presents the major ideas and branches of applied mathematics in a clear and accessible way
• Explains important mathematical concepts, methods, equations, and applications
• Introduces the language of applied mathematics and the goals of applied mathematical research
• Gives a wide range of examples of mathematical modeling
• Covers continuum mechanics, dynamical systems, numerical analysis, discrete and combinatorial mathematics, mathematical physics, and much more
• Explores the connections between applied mathematics and other disciplines
• Includes suggestions for further reading, cross-references, and a comprehensive index

About this title

About the Author

Nicholas J. Higham is the Richardson Professor of Applied Mathematics at The University of Manchester. Mark R. Dennis is professor of theoretical physics at the University of Bristol. Paul Glendinning is professor of applied mathematics at The University of Manchester. Paul A. Martin is professor of applied mathematics at the Colorado School of Mines. Fadil Santosa is professor of mathematics at the University of Minnesota. Jared Tanner is professor of the mathematics of information at the University of Oxford.

From the Back Cover

"Monumental and comprehensive, The Princeton Companion to Applied Mathematics does a breathtaking job of conveying the richness, depth, and vitality of today's applied mathematics. Better still, it does so in a remarkably clear and friendly voice. An instant classic."--Steven Strogatz, Cornell University and author of The Joy of x

"The perfect companion to the earlier Companion. This book will be a landmark for decades ahead."--Nick Trefethen, University of Oxford

"The Princeton Companion to Applied Mathematics is a tour de force, an incredible collection that should be on the shelf of every mathematician, of every scientist, and indeed of anyone fascinated with mathematics. It has something for everyone from beginner to advanced researcher, and is written in a style that will encourage reading outside one's area. It is comprehensive, authoritative, and beautifully written and organized, and I know of nothing comparable."--Simon A. Levin, Princeton University and editor of The Princeton Guide to Ecology

"This remarkable volume takes on an impossible mission and completes it with style, precision, and scholarship. Collected here are definitive accounts of the major topics in modern applied mathematics by world leaders on the subject. The breadth of topics is enormous. Higham and his collaborators have compiled a special source that will be invaluable to researchers, students, and specialists alike."--J. Tinsley Oden, University of Texas at Austin

"A lot of modern science is applied math. Thus, The Princeton Companion to Applied Mathematics is a remarkable compendium of insight from across the sciences. Whether you want new insights into your own field or just want to learn something interesting about the world, this book is an ideal place to start."--John H. Miller, Carnegie Mellon University and the Santa Fe Institute

"The editors are distinguished scientists and, perhaps of even more significance for a reference book like this, excellent expositors of mathematics across a range of levels."--Jan Hesthaven, Swiss Federal Institute of Technology in Lausanne

"A welcome contribution to the field."--Andrea L. Bertozzi, University of California, Los Angeles

"There is nothing like The Princeton Companion to Applied Mathematics, which will be extremely useful to both researchers and the mathematically inclined public. We have needed a book like this."--Steven Strogatz, author of The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math

Excerpt. © Reprinted by permission. All rights reserved.

The Princeton Companion to Applied Mathematics

By Nicholas J. Higham

PRINCETON UNIVERSITY PRESS

Copyright © 2015 Princeton University Press
All rights reserved.

ISBN: 978-0-691-15039-0

Contents

Preface, ix, 
Contributors, xiii, 
Part I Introduction to Applied Mathematics, 
Part II Concepts, 
PART III Equations, Laws, and Functions of Applied Mathematics, 
Part IV Areas of Applied Mathematics, 
PART V Modeling, 
PART VI Example Problems, 
PART VII Application Areas, 
PART VIII Final Perspectives, 

CHAPTER 1

Part I

Introduction to Applied Mathematics

I.1 What Is Applied Mathematics?

Nicholas J. Higham

1 The Big Picture

Applied mathematics is a large subject that interfaces with many other fields. Trying to define it is problematic, as noted by William Prager and Richard Courant, who set up two of the first centers of applied mathematics in the United States in the first half of the twentieth century, at Brown University and New York University, respectively. They explained that:

Precisely to define applied mathematics is next to impossible. It cannot be done in terms of subject matter: the borderline between theory and application is highly subjective and shifts with time. Nor can it be done in terms of motivation: to study a mathematical problem for its own sake is surely not the exclusive privilege of pure mathematicians. Perhaps the best I can do within the framework of this talk is to describe applied mathematics as the bridge connecting pure mathematics with science and technology.

Prager (1972)

Applied mathematics is not a definable scientific field but a human attitude. The attitude of the applied scientist is directed towards finding clear cut answers which can stand the test of empirical observation. To obtain the answers to theoretically often insuperably difficult problems, he must be willing to make compromises regarding rigorous mathematical completeness; he must supplement theoretical reasoning by numerical work, plausibility considerations and so on.

Courant (1965)

Garrett Birkhoff offered the following view in 1977, with reference to the mathematician and physicist Lord Rayleigh (John William Strutt, 1842–1919):

[ILLUSTRATION OMITTED]

Essentially, mathematics becomes "applied" when it is used to solve real-world problems "neither seeking nor avoiding mathematical difficulties" (Rayleigh).

Rather than define what applied mathematics is, one can describe the methods used in it. Peter Lax stated of these methods, in 1989, that:

Some of them are organic parts of pure mathematics: rigorous proofs of precisely stated theorems. But for the greatest part the applied mathematician must rely on other weapons: special solutions, asymptotic description, simplified equations, experimentation both in the laboratory and on the computer.

Here, instead of attempting to give our own definition of applied mathematics we describe the various facets of the subject, as organized around solving a problem. The main steps are described in figure 1. Let us go through each of these steps in turn.

Modeling a problem. Modeling is about taking a physical problem and developing equations — differential, difference, integral, or algebraic — that capture the essential features of the problem and so can be used to obtain qualitative or quantitative understanding of its behavior. Here, "physical problem" might refer to a vibrating string, the spread of an infectious disease, or the influence of people participating in a social network. Modeling is necessarily imperfect and requires simplifying assumptions. One needs to retain enough aspects of the system being studied that the model reproduces the most important behavior but not so many that the model is too hard to analyze. Different types of models might be feasible (continuous, discrete, stochastic), and for a given type there can be many possibilities. Not all applied mathematicians carry out modeling; in fact, most join the process at the next step.

Analyzing the mathematical problem. The equations formulated in the previous step are now analyzed and, ideally, solved. In practice, an explicit, easily evaluated solution usually cannot be obtained, so approximations may have to be made, e.g., by discretizing a differential equation, producing a reduced problem. The techniques necessary for the analysis of the equations or reduced problem may not exist, so this step may involve developing appropriate new techniques. If analytic or perturbation methods have been used then the process may jump from here directly to validation of the model.

Developing algorithms. It may be possible to solve the reduced problem using an existing algorithm — a sequence of steps that can be followed mechanically without the need for ingenuity. Even if a suitable algorithm exists it may not be fast or accurate enough, may not exploit available structure or other problem features, or may not fully exploit the architecture of the computer on which it is to be run. It is therefore often necessary to develop new or improved algorithms.

Writing software. In order to use algorithms on a computer it is necessary to implement them in software. Writing reliable, efficient software is not easy, and depending on the computer environment being targeted it can be a highly specialized task. The necessary software may already be available, perhaps in a package or program library. If it is not, software is ideally developed and documented to a high standard and made available to others. In many cases the software stage consists simply of writing short programs, scripts, or notebooks that carry out the necessary computations and summarize the results, perhaps graphically.

Computational experiments. The software is now run on problem instances and solutions obtained. The computations could be numeric or symbolic, or a mixture of the two.

Validation of the model. The final step is to take the results from the experiments (or from the analysis, if the previous three steps were not needed), interpret them (which may be a nontrivial task), and see if they agree with the observed behavior of the original sys- tem. If the agreement is not sufficiently good then the model can be modified and the loop through the steps repeated. The validation step may be impossible, as the system in question may not yet have been built (e.g., a bridge or a building).

Other important tasks for some problems, which are not explicitly shown in our outline, are to calibrate parameters in a model, to quantify the uncertainty in these parameters, and to analyze the effect of that uncertainty on the solution of the problem. These steps fall under the heading of UNCERTAINTY QUANTIFICATION [II.34].

Once all the steps have been successfully completed the mathematical model can be used to make predictions, compare competing hypotheses, and so on. A key aim is that the mathematical analysis gives new insights into the physical problem, even though the mathematical model may be a simplification of it.

A particular applied mathematician is most likely to work on just some of the steps; indeed, except for relatively simple problems it is rare for one person to have the skills to carry out the whole process from modeling to computer solution and validation.

In some cases the original problem may have been communicated by a scientist in a different field. A significant effort can be required to understand what the mathematical problem is and, when it is eventually solved, to translate the findings back into the language of the relevant field. Being able to talk to people out- side mathematics is therefore a valuable skill for the applied mathematician.

It would be wrong to give the impression that all applied mathematics is done in the context of modeling. Frequently, a mathematical problem will be tack- led because of its inherent interest (see the quote from Prager above) with the hope or expectation that a relevant application will be found. Indeed some applied mathematicians spend their whole careers working in this way. There are many examples of mathematical results that provide the foundations for important practical applications but were developed without knowledge of those applications (sections 3.1 and 3.2 provide such examples).

Before the twentieth century, applied mathematics was driven by problems in astronomy and mechanics. In the twentieth century physics became the main driver, with other areas such as biology, chemistry, economics, engineering, and medicine also providing many challenging mathematical problems from the 1950s onward. With the massive and still-growing amounts of data available to us in today's digital society we can expect information, in its many guises, to be an increasingly important influence on applied mathematics in the twenty-first century.

For more on the definition and history of applied mathematics, including the development of the term "applied mathematics," see the article HISTORY OF APPLIED MATHEMATICS [I.6].

2 Applied Mathematics and Pure Mathematics

The question of how applied mathematics compares with pure mathematics is often raised and has been discussed by many authors, sometimes in controversial terms. We give a few highlights.

Paul Halmos wrote a 1981 paper provocatively titled "Applied mathematics is bad mathematics." However, much of what Halmos says would not be disputed by many applied mathematicians. For example:

Pure mathematics can be practically useful and applied mathematics can be artistically elegant. ... Just as pure mathematics can be useful, applied mathematics can be more beautifully useless than is sometimes recognized. ...

Applied mathematics is an intellectual discipline, not a part of industrial technology. ...

Not only, as is universally admitted, does the applied need the pure, but, in order to keep from becoming inbred, sterile, meaningless, and dead, the pure needs the revitalization and the contact with reality that only the applied can provide.

G. H. Hardy's book A Mathematician's Apology (1940) is well known as a defense of mathematics as a subject that can be pursued for its own sake and beauty. As such it contains some criticism of applied mathematics:

But is not the position of an ordinary applied mathematician in some ways a little pathetic? If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. "Imaginary" universes are so much more beautiful than this stupidly constructed "real" one; and most of the ?nest products of an applied mathematician's fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.

Halmos and Hardy were pure mathematicians. Applied mathematicians C. C. Lin and L. A. Segel offer some insights in the introductory chapter of their classic 1974 book Mathematics Applied to Deterministic Problems in the Natural Sciences:

The differences in motivation and objectives between pure and applied mathematics — and the consequent differences in emphasis and attitude — must be fully recognized. In pure mathematics, one is often dealing with such abstract concepts that logic remains the only tool permitting judgment of the correctness of a theory. In applied mathematics, empirical verification is a necessary and powerful judge. However ... in some cases (e.g., celestial mechanics), rigorous theorems can be proved that are also valuable for practical purposes. On the other hand, there are many instances in which new mathematical ideas and new mathematical theories are stimulated by applied mathematicians or theoretical scientists.

They also opine that:

Much second-rate pure mathematics is concealed beneath the trappings of applied mathematics (and vice versa). As always, knowledge and taste are needed if quality is to be assured.

The applied versus pure discussion is not always taken too seriously. Chandler Davis quotes the applied mathematician Joseph Keller as saying, "pure mathematics is a subfield of applied mathematics"!

The discussion can also focus on where in the spectrum a particular type of mathematics lies. An interesting story was told in 1988 by Clifford Truesdell of his cofounding in 1952 of the Journal of Rational Mechanics and Analysis (which later became Archive for Rational Mechanics and Analysis). He explained that

In those days papers on the foundation of continuum mechanics were rejected by journals of mathematics as being applied, by journals of "applied" mathematics as being physics or pure mathematics, by journals of physics as being mathematics, and by all of them as too long, too expensive to print, and of interest to no one.

3 Applied Mathematics in Everyday Life

We now give three examples of applied mathematics in use in everyday life. These examples were chosen because they can be described without delving into too many technicalities and because they illustrate different characteristics. Some of the terms used in the descriptions are explained in THE LANGUAGE OF APPLIED MATHEMATICS [I.2].

3.1 Searching Web Pages

In the early to mid-1990s — the early days of the World Wide Web — search engines would find Web pages that matched a user's search query and would order the results by a simple criterion such as the number of times that the search query appears on a page. This approach became unsatisfactory as the Web grew in size and spammers learned how to influence the search results. From the late 1990s onward, more sophisticated criteria were developed, based on analysis of the links between Web pages. One of these is Google's PageRank algorithm [VI.9]. Another is the hyperlink induced topic search (HITS) algorithm of Kleinberg.

The HITS algorithm is based on the idea of deter mining hubs and authorities. Authorities are Web pages with many links to them and for which the linking pages point to many authorities. For example, the New York Times home page or a Wikipedia article on a popular topic might be an authority. Hubs are pages that point to many authorities. An example might be a page on a programming language that provides links to useful pages about that language but that does not necessarily contain much content itself. The authorities are the pages that we would like to rank higher among pages that match a search term. However, the definition of hubs and authorities is circular, as each depends on the other.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2015
• Language: English
• ISBN 10: 0691150397
• ISBN 13: 9780691150390
• Binding: Hardcover
• Number of pages: 1032
• Editor: Higham Nicholas J.