Survey and Review
Authors: Nick Trefethen
Venue: SIAM Review (1999)

In planning the "new" SIAM Review, we looked back over old volumes. To convey some of the flavor of what we found, here is an entirely unscientific sample of one title from each of our first 40 years. I must emphasize that this is just a personal list, compiled arbitrarily, with no claim whatever to represent the "greatest hits" of our past. It omits hundreds of other equally remarkable articles and authors, from Ablowitz to Zelmer, from Ahlfors to Zaremba. 1959: Tjalling C. Koopmans and Augustus F. Bausch, "Selected topics in economics involving mathematical reasoning" 1960: Frank Harary, "Some historical and intuitive aspects of graph theory" 1961: S. Parter, "The use of linear graphs in Gauss elimination" 1962: Mark Kac, "Probability theory: Its role and its impact" 1963: Saul Gorn, "The computer and information sciences: A new basic discipline" 1964: S. Ulam, "Combinatorial analysis in infinite sets and some physical theories" 1965: Solomon Lefschetz, "Recent advances in the stability of nonlinear controls" 1966: Jürgen Moser, "On the theory of quasiperiodic motions" 1967: Eugene P. Wigner, "Random matrices in physics" 1968: Benoit B. Mandelbrot and John W. Van Ness, "Fractional Brownian motions, fractional noises and applications" 1969: C. C. Lin, "Dynamics of self-gravitating systems: Structure of galaxies" 1970: M. J. D. Powell, "A survey of numerical methods for unconstrained optimization" 1971: W. L. Miranker, "A survey of parallelism in numerical analysis" 1972: W. B. Gragg, "The Pade table and its relation to certain algorithms of numerical analysis" 1973: Ulf Grenander, "Computational probability and statistics" 1974: W. J. Cody, "The construction of numerical subroutine libraries" 1975: Donald Ludwig, "Geometrical theory for surface waves" 1976: Peter D. Lax, "Almost periodic solutions of the KdV equation" 1977: J. E. Dennis, Jr. and Jorge J. Moré, "Quasi-Newton methods, motivation and theory" 1978: Robert Endre Tarjan, "Complexity of combinatorial algorithms" 1979: Peter Henrici, "Fast Fourier methods in computational complex analysis" 1980: K. O. Friedrichs, "Von Neumann's Hilbert space theory and partial differential equations" 1981: C. W. Gear, "Numerical solution of ordinary differential equations: Is there anything left to do?" 1982: David S. Watkins, "Understanding the QR algorithm" 1983: Garrett Birkhoff, "Numerical fluid dynamics" 1984: J. M. Borwein and P. B. Borwein, "The arithmetic-geometric mean and fast computation of elementary functions" 1985: Joseph B. Keller, "Semiclassical mechanics" 1986: J. T. Stuart, "Taylor-vortex flow: A dynamical system" 1987: John W. Tukey, "Configural polysampling" 1988: J. L. Lions, "Exact controllability, stabilization and perturbations for dynamical systems" 1989: Gene H. Golub and Dianne P. O'Leary, "Some history of the conjugate gradient and Lanczos algorithms: 1948--1976" 1990: S. Smale, "Some remarks on the foundations of numerical analysis" 1991: Andrew J. Majda, "Vorticity, turbulence, and acoustics in fluid flow" 1992: Clovis C. Gonzaga, "Path-following methods for linear programming" 1993: Andrea Bertozzi and James McKenna, "Multidimensional residues, generating functions, and their applications to queueing networks" 1994: Ivo Babuska and Manil Suri, "The P and H-P versions of the finite element method, basic principles and properties" 1995: Thomas Kailath and Ali H. Sayed, "Displacement structure: Theory and applications" 1996: Lieven Vandenberghe and Stephen Boyd, "Semidefinite programming" 1997: Carla Savage, "A survey of combinatorial Gray codes" 1998: G. I. Barenblatt and A. J. Chorin, "New perspectives in turbulence: Scaling laws, asymptotics, and intermittency" What a grand tradition to build upon! And that, of course, is our goal with the new Survey and Review section: to continue to publish first-rate surveys of important topics from all across applied mathematics, scientific computing, and the many other areas of interest to the SIAM community. If journals still exist in 2039, we want readers looking back then to consider our second 40 years even more exciting than our first. Many Survey and Review articles will be invited by the board of Editors, but not all. Proposals for articles are welcome at any time on any topic; just contact the Section Editor, preferably electronically, with a few pages of information indicating the kind of article you would like to contribute. We will do our best to respond quickly with a decision as to whether we consider the proposed article of wide enough potential interest to request a full manuscript for review. The best survey articles combine a broad and even-handed view of a field with a strong organizing principle personal to the authors. In this issue we offer two examples from areas of applied mathematics that differ in every respect except their importance and topicality. Perelson and Nelson write about the nonlinear dynamics of AIDS, a topic where simple equations have extraordinary implications. Their review summarizes and extends some high-profile work by themselves and others in the last few years in which mathematical models have been developed in direct collaboration with clinical experiments. Readers will quickly be persuaded that AIDS is, among other things, a dynamical system; we must all hope that this kind of analysis will be one of the scientific steps that helps bring us closer to better treatment and eventual cure. Diaconis and Freedman synthesize a wide collection of seemingly disparate mathematical topics under the unifying theme of "iterated random functions." Their subject, in the general area of Markov chains, touches applications as diverse as condensed matter physics, queueing theory, random geometrical tilings, and image compression, including the surprising recent algorithms by Propp and Wilson for exactly random sampling from very large state spaces. The dozens of examples they describe are unified by a general theorem on convergence of these processes to a unique stationary distribution, and the brief supplementary article by the Editors illustrates some of the visual side of this topic. All the SIAM Review editors thank Drs. Perelson, Nelson, Diaconis, and Freedman for helping to inaugurate our second 40 years in style. Nick Trefethen Section Editor LNT@comlab.ox.ac.uk