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2000

Volume 42, Issue 4, pp. 553-768


Survey and Review

Nick Trefethen, Section Editor

SIAM Rev. 42, pp. 553-553 (1 page)

Online Publication Date: August 04, 2006

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Science, and applied mathematics, proceeds by recognizing quantities that can be viewed as infinite or infinitesimal. We treat an iron bar as a continuum, ignoring the fact that it is made of molecules. We treat the planets as point masses, ignoring the fact that they are imperfect spheres. And so on.
The article by Chapman in this issue represents an exceptionally highly developed example of this art. His problem is the behavior of type-II superconductors, a booming technological topic since the discovery of high-temperature superconductivity in 1986. In a magnetic field, a type-II superconducting material may divide between superconducting regions and nonsuperconducting ``vortices.' As a certain Ginzburg--Landau parameter increases, the number of vortices approaches infinity---should we model this as a continuum of vortices? But that is just the beginning. Further parameters describe the distance between impurities in the material, the rate of change of the applied magnetic field, and the magnetic penetration depth, any or all of which may approach zero. A whole hierarchy of mathematical models is appropriate to handle the different combinations that may arise, and all of them have potential relevance to technology. Chapman attempts to shed light on each member of this hierarchy. It will probably be decades before we see which of his equations prove to be important.
Our second article, by Hethcote, also describes a hierarchy of mathematical models---now ODEs as well as PDEs. The scientific application could not be more different, or more urgent: Hethcote's subject is infectious diseases such as smallpox, polio, measles, and AIDS. He begins with the continuum approximation that populations are infinite. The hierarchy comes from many other effects that are important for some diseases but near enough to infinitesimal for others, such as transference of temporary passive immunity from mothers to newborns, latency between exposure and infectivity, dependence on age, spatial heterogeneity, and variation among subpopulations. Since smallpox has been eradicated worldwide, why is there little hope for eradication of measles? Why don't China and India vaccinate against rubella? Hethcote gives quantitative answers to such questions, focusing his discussion around parameter \textit{R}$_{\mbox{\sffamily\scriptsize 0}}$, the "basic reproduction number." His epidemiological models with age dependence lead to computer simulations for comparing vaccination programs for measles in Africa and whooping cough in the United States. Pick up this fascinating article and start reading! It will enrich your view of our world and your appreciation of applied mathematics.

A Hierarchy of Models for Type-II Superconductors

S. J. Chapman

SIAM Rev. 42, pp. 555-598 (44 pages) | Cited 16 times

Online Publication Date: August 04, 2006

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A hierarchy of models for type-II superconductors is presented. Through appropriate asymptotic limits we pass from the mesoscopic Ginzburg--Landau model to the London model with isolated superconducting vortices as line singularities, to vortex-density models, and finally to macroscopic critical-state models.

The Mathematics of Infectious Diseases

Herbert W. Hethcote

SIAM Rev. 42, pp. 599-653 (55 pages) | Cited 347 times

Online Publication Date: August 04, 2006

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Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

Problems and Techniques

Joe Flaherty, Section Editor

SIAM Rev. 42, pp. 655-655 (1 page)

Online Publication Date: August 04, 2006

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This installment of Problems and Techniques features articles on periodic billiard ball trajectories and the reduction of nearly quadratic polynomial Hamiltonians.
Lorenz Halbeisen and Norbert Hungerbühler treat us, in our first article, to a study of periodic billiard ball trajectories on triangular tables. The problem of determining whether or not a billiard ball has a periodic path in a planar domain having a boundary composed of smooth curves has a long history with roots in the eighteenth century. G. D. Birkhoff, for example, established the existence of periodic trajectories on smooth convex domains in 1927. However, the case with nonsmooth boundaries is more difficult, and many situations on polygonal domains are still unresolved. The existence of periodic motion on acute triangles is perhaps the simplest case with motion following the orthoptic triangle with "bounces" at the bases of the three altitudes of the triangle. The existence of periodic trajectories on obtuse triangles is still undetermined and our authors describe cases where periodic motion can be established. They also examine the stability of such motion relative to small perturbations in the domain shape. I don't know if this analysis will improve your game, but it is indeed interesting mathematics. I hope that you enjoy it.
In our second article, Jesús Palacián and Patricia Yanguas describe a procedure to reduce the number of degrees of freedom of two-dimensional Hamiltonians that are polynomial functions of the coordinates and momenta with a dominant part that is a homogeneous quadratic function. The popular Fermi--Pasta--Ulam and truncated Toda lattice systems are three-particle systems that may be expressed in this form after a suitable transformation of variables. The Hénon and Heiles family of systems also have the appropriate form. In these circumstances, our authors show how to reduce the number of degrees of freedom of the Hamilitonian system by one. The reduced Hamiltonians are then integrable and the reduced phase space is two-dimensional rather than four-dimensional. Thus, the phase-space diagrams are easily visualized, and this simplifies understanding. The authors characterize the reduced Hamiltonian systems in terms of a collection of normal forms and present conclusions regarding the perturbed Hamiltonian that result from properties of the reduced system. Lots of interesting views of the phase spaces for several reduced Hamiltonians are shown.

On Periodic Billiard Trajectories in Obtuse Triangles

Lorenz Halbeisen and Norbert Hungerbühler

SIAM Rev. 42, pp. 657-670 (14 pages) | Cited 1 time

Online Publication Date: August 04, 2006

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In 1775, J. F. de Tuschis a Fagnano observed that in every acute triangle, the orthoptic triangle represents a periodic billiard trajectory, but to the present day it is not known whether or not in every obtuse triangle a periodic billiard trajectory exists. The limiting case of right triangles was settled in 1993 by F. Holt, who proved that all right triangles possess periodic trajectories. The same result had appeared independently in the Russian literature in 1991, namely in the work of G. A. Gal'perin, A. M. Stepin, and Y. B. Vorobets. The latter authors discovered in 1992 a class of obtuse triangles which contain particular periodic billiard paths. In this article, we review the above-mentioned results and some of the techniques used in the proofs and at the same time show for an extended class of obtuse triangles that they contain periodic billiard trajectories.

Reduction of Polynomial Planar Hamiltonians with Quadratic Unperturbed Part

Jesús Palacián and Patricia Yanguas

SIAM Rev. 42, pp. 671-691 (21 pages) | Cited 8 times

Online Publication Date: August 04, 2006

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We classify the possible normal forms of quadratic Hamiltonians in 2 dimensions. Then we give a method to reduce by one the number of degrees of freedom of an arbitrary polynomial Hamiltonian whose principal part is quadratic in positions and moments. The reduction procedure is based on the extension of an integral of the unperturbed part to the whole system, up to a certain order. The corresponding reduced phase spaces have dimension 2 and are described by means of the set of invariants associated to the reduction.

SIGEST

SIAM Rev. 42, pp. 693-693 (1 page)

Online Publication Date: August 04, 2006

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Readers of SIAM Review are always happy to learn about difficult and important real-world problems that can be modeled mathematically and solved numerically. And we are even happier when serious mathematical issues arise during the modeling, when the latest numerical methods are needed to solve the problem, and when the modeling and solution techniques lead to new research directions in mathematics, algorithms, and the associated applications areas. All of these properties are present in this issue's SIGEST paper, "Free Material Design via Semidefinite Programming: The Multiload Case with Contact Conditions," by A. Ben-Tal, M. Kocvara, A. Nemirovski, and J. Zowe, which originally appeared in SIAM Journal on Optimization, volume 9, September 1999.
This paper is informative on several levels. The authors begin with a highly readable overview of free material design problems in structural engineering and then discuss how to formulate the very difficult multiload problem, which turns out to require new mathematical and computational approaches. A crucial element is that calculation of the optimal elasticity matrix can be reduced to a semidefinite programming problem---optimization of an affine function of an (unknown) symmetric matrix subject to linear and positive-semidefiniteness constraints. Finally, the paper describes the success of semidefinite programming in solving three structural design problems.
The appearance of semidefinite programming (SDP) in this paper is striking. Although the mathematical roots of SDP go back to the sixties, it has dramatically come to the fore only within the last few years---for example, it was the single most popular topic at both the 1999 SIAM Conference on Optimization and the 2000 International Symposium on Mathematical Programming. The analysis in this SIGEST paper illustrates one of the earliest applications of SDP, to structural optimization; SDP is also the foundation of much recent work on approximation algorithms for NP-hard combinatorial optimization problems. Every indication is that SDP will continue to grow as both theoretical paradigm and solution technique.

Free Material Design via Semidefinite Programming: The Multiload Case with Contact Conditions

A. Ben-Tal, M. Kovara, A. Nemirovski, and J. Zowe

SIAM Rev. 42, pp. 695-715 (21 pages) | Cited 2 times

Online Publication Date: August 04, 2006

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Free material design deals with the question of finding the stiffest structure with respect to one or more given loads which can be made when both the distribution of material and the material itself can freely vary. The case of one single load has been discussed in several recent papers, and an efficient numerical approach was presented in [M. Kovara, M. Zibulevsky, and J. Zowe, RAIRO Modél. Math. Anal. Numér., 32 (1998), pp. 255--281]. We attack here the multiload situation (understood in the worst-case sense), which is of much more interest for applications but also significantly more challenging from both the theoretical and the numerical points of view. After a series of transformation steps we reach a problem formulation for which we can prove existence of a solution; a suitable discretization leads to a semidefinite programming problem for which modern polynomial time algorithms of interior point type are available. A number of numerical examples demonstrate the efficiency of our approach.

Education

Bobby Schnabel, Section Editor

SIAM Rev. 42, pp. 717-717 (1 page)

Online Publication Date: August 04, 2006

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Both papers in this issue's Education Section use mathematical modeling to help explain very commonplace and interesting phenomena: the design of light reflectors such as fluorescent lights, and the operation of water rockets. In doing so they show the power of bringing calculus and differential equations to bear upon interesting, accessible phenomena. And one of them may allow students to go play with water rockets as part of their homework assignment!
The reflective properties of given curves, such as conic sections, are often studied in elementary calculus as a means of understanding the geometric properties of the curves. More interesting and challenging is the inverse problem: given a desired level of illumination over a given region from a point light source, find a reflector curve that achieves this. The paper "Exploring Reflection: Designing Light Reflectors for Uniform Illumination," by Gary De Young, applies a combination of concepts and techniques from calculus, geometry, and differential equations to the design of reflectors that achieve desired illumination properties. The approach taken results in a system of differential equations whose solution describes the desired curve. A specific example derived in detail is a reflector curve that provides uniform illumination over a straight line segment at some given distance from the source, which is applicable, for example, to the design of a suitable reflector for a flat diffusion panel for a fluorescent light tube. The material should be suitable either for third-semester calculus courses or in differential equations. As the author says, the paper should help to "lead students away from finding solutions to equations, toward finding equations that describe solutions."
"Hydrodynamics of a Water Rocket," by Joseph Prusa, presents the application of mass, momentum, and energy balance laws to the determination of the trajectory of a simple water rocket. Faculty will appreciate that the context permits the use of a variety of aspects of applied mathematics, including scale analysis, model development and approximation, and numerical methods for initial-value problems. Additionally, physical experiments may be conducted and comparisons made of model predictions with real experimental data. Students should find this aspect particularly appealing. The material is intended to be suitable for courses or topics in fluid dynamics at either the advanced undergraduate or the graduate level. The modeling exercise will provide a great opportunity for students to bring together skills in physical reasoning, analysis, and numerical methods. Besides, how many exercises in applied mathematics carry the caution "be careful not to get too wet"?
FREE

Hydrodynamics of a Water Rocket

Joseph M. Prusa

SIAM Rev. 42, pp. 719-726 (8 pages) | Cited 2 times

Online Publication Date: August 04, 2006

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Applications of mass, momentum, and energy balances are central to the teaching of fluid dynamics. In this study, a model to determine the trajectory of a water rocket is given that is far simpler than the system of coupled partial differential equations that typically results in modern hydrodynamic problems of interest. This makes the problem an excellent choice for a student project---it can reasonably be completed with a day or two of effort. In addition to the fundamental mathematics, this problem offers opportunities in scale analysis, numerical methods for IVPs, balance principles in accelerated frames of reference, and the collection and assessment of flight test data.
FREE

Exploring Reflection: Designing Light Reflectors for Uniform Illumination

Gary W. De Young

SIAM Rev. 42, pp. 727-735 (9 pages) | Cited 1 time

Online Publication Date: August 04, 2006

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Many students first learn of the law of reflection when they are told about the reflective properties of conics. Typically students are exposed to a curve or surface, and then its reflective properties are derived or verified. In this paper we explore reflection by starting from a desired property of an unknown reflector and proceeding to find a reflector with those properties. This is done in the context of finding a reflector that produces uniform illumination of nearby objects. The reflectors considered are those whose analysis can be reduced to studying curves in two dimensions.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 42, pp. 737-768 (32 pages)

Online Publication Date: August 04, 2006

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This issue's featured review by Iserles of Batterson's biography on Stephen Smale makes us reflect on the unfortunate short supply of biographies and autobiographies of twentieth-century mathematicians. We know how creative some of our colleagues are and something about their fascinating range of activities. Wouldn't mathematicians and the general public like to learn more about the lives of, for example, Mary Cartwright, Martin Kruskal, Peter Lax, Norman Levinson, James Lighthill, or Gian-Carlo Rota?
The steady publication of less-than-mediocre texts and monographs, often featuring poor editing and high prices, likewise misrepresents the value and vitality of current work in applied mathematics. I regret to report that potential reviewers often state that they have little to praise in some new books. Shouldn't we be encouraging publishers to emphasize quality and discourage involvement with those producing junk?
Fortunately, the reviews that follow describe lots of good books we should be reading. Many thanks to their authors for sharing their informed opinions with us.
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