Showing papers in "Siam Review in 1993"

Semi-infinite programming: theory, methods, and applications

Abstract: Starting from a number of motivating and abundant applications in §2, including control of robots, eigenvalue computations, mechanical stress of materials, and statistical design, the authors describe a class of optimization problems which are referred to as semi-infinite, because their constraints bound functions of a finite number of variables on a whole region. In §§3–5, first- and second-order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints. Another main effort for achieving simplification is through duality in §6. There, algebraic properties of finite linear programming are brought to bear on duality theory in semi-infinite programming. Section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent (SQP-type) methods. Taking this differentiable point of view, this paper can be considered to be complementar...

On the early history of the singular value decomposition

Abstract: This paper surveys the contributions of five mathematicians—Eugenio Beltrami (1835–1899), Camille Jordan (1838–1921), James Joseph Sylvester (1814–1897), Erhard Schmidt (1876–1959), and Hermann Weyl (1885–1955)—who were responsible for establishing the existence of the singular value decomposition and developing its theory.

Lagrange multipliers and optimality

Abstract: Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces su...

Qualitative theory of compartmental systems

Abstract: Dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations, many nonlinear, that are called compartmental systems. In this paper, the authors define compartmental systems, specify their relations to other nonnegative systems, and discuss examples of applications.The authors review the qualitative results on linear and nonlinear compartmental systems, including their relation to cooperative systems. They review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems, providing a framework for unifying them under a few general theorems. In the course of that they complete the solution of a problem posed by Bellman and show that closed nonlinear, autonomous, n-compartment systems can show the full gamut of possible behaviors of systems of ODES.Finally, to provide additional structure to this study, the authors show how to partiti...

Mathematical models for hysteresis

Abstract: The various existing classical models for hysteresis, Preisach, Ishlinskii, Duhem–Madelung, are surveyed, as well a more modern treatments by contemporary workers The emphasis is on a clear mathematical description of the formulation and properties of each model In addition the authors try to make the reader aware of the many open questions in the study of hysteresis

The Perron-Frobenius theorem and the ranking of football teams

Abstract: The author describes four different methods to rank teams in uneven paired competition and shows how each of these methods depends in some fundamental way on the Perron–Frobenius theorem.

Some perspectives on the eigenvalue problem

Abstract: This expository paper explores the relationships among a number of algorithms for solving eigenvalue problems, including the power method, subspace iteration, the $QR$ algorithm, and the Arnoldi and symmetric Lanczos algorithms. The symmetric Lanczos algorithm is shown to be identical to the three-term recursion (Stieltjes procedure) for computing orthogonal polynomials with respect to a measure on the real line. The connection between measures on the line and symmetric tridiagonal (Jacobi) matrices is investigated. If such a matrix is transformed by a step of the $QR$ algorithm, there is a corresponding transformation in the measure. The tridiagonal matrices are also exploited for the construction of Gaussian quadrature formulas for measures on the line. The developments on the real line are replicated with suitable modifications on the unit circle via Lanczos-like procedures for unitary operators. The best-known procedure of this type is the recursion of Szego for computing orthogonal polynomials on the...

The warranty problem: its statistical and game theoretic aspects

Abstract: Manufacturers of consumer products, such as automobiles, usually offer warranties guaranteeing the product or its parts, for example, for five years or 50,000 miles, whichever comes first. There are at least two issues of interest to applied mathematicians and statisticians that arise from warranty considerations. The first is the specification of an optimum price-warranty combination, and the second is the forecast of a reserve fund to meet warranty claims against the product. The former involves a consideration of the item’s reliability, its rate of usage, the consumer’s attitude toward a specific warranty, and the competitor’s actions, all of which lead towards a game theoretic formulation of the problem. The latter involves the analysis of a special kind of time series in two-dimensions, for which the use of dynamic linear models appears to be natural. This is an expository paper. Due to the timeliness of the topic, the focus here is on problem definition, its scope, and its formulation in a manner th...

Automatic Differentiation of Algorithms: Theory, Implementation, and Application (Andreas Griewank and George F. Corliss, eds.)

Abstract: Those ofyou whoknow Ken Meyerand Dick Hall may have already purchased this book. For those of you who don’t know these authors, I’d like to give you my impressions. The principal author, Meyer, has long been a leading researcher in dynamics. His writing style is careful, complete, accurate, and down to earth. He is a mathematician and this means that he is vitally concerned with giving careful definitions and establishing true statements. He avoids abstraction for its own sake, but he is comfortable using it when necessary. He tells you what is really going on. Hall is also a leading researcher in dynamical systems. His work has been primarily devoted to the study of twist maps, the subject of his chapter of this book. His instinct is to simplify, to go to the very heart of a problem, and to explain it in a way that conveys insight. His exposition is outstanding. In their preface the authors describe the book this way:

A review of regenerative processes

Abstract: The authors present an expository review of the theory of regenerative processes starting with the more traditional notions and then moving on to some of the more recent and modern developments. Fi...

Sinc Methods for Quadrature and Differential Equations (J. Lund and K. L. Bowers)

Abstract: In algebraic geometry many natural problems, such as those arising in invariant theory, in the solution of polynomial systems of interest to scientists and engineers, and in the computation of syzygies of ideals, lead to computations that prove too tedious to pursue by hand. The widespread availability of powerful, relatively easy to use computers has spurred a return to these basic questions, and a rethinking of what reasonable problems are. Algorithms based on the work of Buchberger and many other mathematicians have led to a wide array of computational algebraic tools that are readily available on many computers. The very readable, excellent book being reviewed is an introduction to algebraic geometry emphasizing the algorithmic, computational viewpoint. This book will be valuable to many different constituencies. The core book consists of four chapters covering the correspondence between results on affine varieties and basic commutative algebra, a solid introduction to Grtibner bases, resultants, and elimination theory. The book then goes on to sample a number of more advanced topics, including basic material on the kinematics of planar robots, some invariant theory of finite groups, projective varieties, and dimension. All the chapters contain many very good examples and exercises. It will be easy to use this textbook as the basis for an innovative algebraic geometry course for undergraduate mathematics majors or for graduate students in a wide variety of subjects such as computer science and mechanical engineering. Mathematicians and scientists who are not specialists in computer algebra, and graduate students learning algebraic geometry by one of the many other approaches to the subject will also find this book useful and pleasant reading.

The Global Dynamics of Cellular Automata (Andrew Wuensche and Mike Lesser)

Abstract: or noninteger dilation factors, multidimensional wavelet bases, wavelets on intervals, and wavelet packet constructions with increased frequency resolution. Only the outlines are discussed here, but at this point the reader can move on to the books [BF], [C2], [M], and [R], or to the literature for more details or for applications of wavelets to specific problems. Also, [C1] contains an introductory treatment of some material only briefly mentioned by Daubechies, including spline-wavelets and wavelet packets. In summary, this is a clearly written introduction to the mathematics of wavelets that provides solid background material on most of the major aspects of the current theory. Especially appealing is the way in which the relationships between wavelets and other areas are pointed out. This text would be well-suited for an introductory graduate-level course on wavelets; with some careful thought on the part of the instructor it could also be used at the advanced undergraduate level. feel certain that this will be the major introductory text on wavelets for some time to come. It will definitely be a welcome addition to the library of anyone interested in leaming the basics of wavelets.

Mathematical models of running

Abstract: A number of mathematical theories to describe the momentum and energy balances associated with running have been advocated since the pioneering work of the British biologist A.V. Hill in the 1920s. Since the various models lead to quite different conclusions and predictions about the balances associated with running, it is instructive to examine critically the assumptions underlying them. So, for example, depending on the model employed, it has been predicted that, for a sprinter, the relative expenditure of energy in overcoming air resistance lies somewhere in the range of 2 to $16\% $. This wide range of values derives mainly from the various estimates of the propulsive force exerted during sprinting, indicating just how disparate the predictions among the models are.In the early 1970s Kelley built on the ideas of Hill to develop a complete theoretical model for competitive running in which the goal of the athlete is to cover the race distance in a minimum amount of time (subject to certain constraints ...

Multidimensional residues, generating functions, and their application to queueing networks

Abstract: This paper has two goals: the first is to introduce applied mathematicians to a new technique involving several complex variable residue theory. This is a multidimensional extension of a well-known result for functions of one complex variable [P Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978] that yields useful asymptotic information when applied to generating functions. The authors begin by reviewing the residue theory of one complex variable, noting that there are useful sets of relationships between the residues of certain rational functions. They use the one-dimensional theory to rederive and extend a well-known result of Koenigsberg [P G. Harrison, Oper. Res., 33 (1985), pp. 464–468]. These relationships, to a degree, can be found for residues of the analogous rational functions of several complex variables. The second goal is to apply these tools to the study of product form queueing networks.It is well known that for generating functions of a single v...

An equilibrium theory of dislocation continua

Abstract: A homogenised model for elastic media containing large numbers of dislocations is described. First, discrete dislocations are discussed from the mathematical and crystallographical points of view, and their stress fields are calculated. These building blocks are averaged to construct a homogenised model in which the dislocation distribution is averaged to a number density tensor. Equilibrium configurations are then considered and it is proved that, in the absence of externally applied stresses, the only possible finite, simply connected distributions are ones in which all components of stress vanish everywhere. Some examples are given of these zero-stress everywhere (ZSE) distributions, and their geometrical interpretation is considered in terms of the plastic distortion tensor, which shows that they are equivalent to local rotations of the crystal lattice. Finally, some conjectures are made about the response of a cellular ZSE distribution to an applied stress, introducing the idea of “polarization” by a...

Mixed and Hybrid Finite Element Methods (Franco Brezzi and Michel Fortin)

Abstract: and concludes with an application of filtering theory. The book has few examples or concrete cases. Most such things are presented as exercises, which also serve to summarize various results from the literature. Notes and comments at the end of the book are instructive and point out further references. I enjoyed the book immensely. It has an unhurried, careful and elegant style. The treatment is well thought out, the conditions feel right and natural, and the arrangement of the material follows an inexorable logic. This is the best book on the foundations of stochastic flows.

$H^\infty $-Optimal Control and Related Minimax Design Problems (Tamer Basar and Pierre Bernhard)

Abstract: this important topic, introducing it as a topic in reliability in its own right. The material is developed from a historical perspective, and takes the reader from the inception of the theory of load sharing models to that which is current in this topic. One of the authors specializes in this topic, among other topics, and has contributed much to its development; this specialization is well reflected in the treatment given here. I heartily recommend Chapter 9 to anyone who wishes to learn more about load sharing and the strength of fiber bundles, for here is an example of a strong partnership between engineering, probability, and statistics. The review of any book should end with answers to at least the following two questions. The first is whether the reviewer has learned anything new from the book. The second is whether the reviewer would recommend that others read the book or, for example, adopt it as a textbook. In my case the answer to both is an emphatic \"Yes!\

Mathematical models and the design of public health policy: HIV and antiviral therapy

Abstract: Antiviral therapy acting to increase the incubation period of HIV-1, without significantly reducing infectiousness, will necessarily increase the prevalence of infection, unless accompanied by appropriate levels of behaviour change. However, the number of AIDS-related deaths may decline despite a large increase in numbers of seropositives, if the latter is balanced by an increase in the average life-span of an infected individual. In a homogeneous population, treatment will be of benefit to both the individual and the community if the basic reproductive rate of the disease is high, provided that the coverage and efficacy are also high. However, therapy may be of detriment to the community (i.e., produce an increase in the number of AIDS deaths) under a combination of low drug efficacy and high coverage or vice versa, if the basic reproductive rate is low. In a behaviourally heterogeneous population with proportionate mixing, the outcome of treatment is extremely sensitive to the rate of cessation of sexua...

Wavelets: A Tutorial in Theory and Applications (Charles K. Chui, ed.)

Abstract: spectrum of the Hamiltonian are not intrinsically algebraic.\" The conscientiously crafted but unwieldy axioms collected at the end of the book underscore the fact that the authors are working in the vicinity of ideas with more resonance than their formulation. Even they cannot resist referring to these other ideas. The true significance of the rigged Hilbert space formalism, as noted by B6hm [1, p. 22], is in its validation of Dirac’s notion of eigenvectors and its insight into troublesome questions of domain when operators are combined and extended. Rigged Hilbert spaces supplement the von Neumann approach rather than replacing it. To cavil further, their book has a few minor typos here and there (e.g., on p. 159 a convex combination of stationary states and a combination of the associated vectors are accidentally conflated), and an index of symbols is badly needed to help the reader keep track of the notational menagerie. But the book is a fine expository effort. It brings together much recent mathematical research on topics relating to quantum mechanicsmnot only the authors’ contributions but those of many others, a task whose difficulty is easy to underestimate. And it does so while retaining a strong commitment to the motivating physics. For a mathematician or physicist who wants to learn more of the mathematics of quantum mechanics, it offers much. Heavy mathematical topics alternate with readable introductions and informative summaries. The book includes a large bibliography substantively referred to in the text. The authors give insightful asides (e.g., proposing that indistinguishability of identical particles is a consequence of the uncertainty principle since initially separate locations cannot be maintained, or noting that not all unitary transformations in Hilbert space qualify as canonical). The ideas developed here may have their greatest application in advanced parts of quantum theory. Dubin has already given us a treatment of quantum statistical mechanics [2], and we may expect further useful work from these authors.

Stability of Stochastic Differential Equations with Respect to Semi-martingales (X. Mao)

Abstract: the algorithm that computes the orbit structure of a circular CA. This is followed by a 20 page user’s guide to the two accompanying DOS programs, and then a 150 page \"atlas\" of orbits structures of various CA. (In Wolfram’s terminology, these CA include the\"n 3\" and the\"totalistic n 5\" rules with number of sites L < 16; in terms of the notation of the first paragraph, n 2N + is the number of arguments for the update rule F.) The atlas consists ofgraphic representations of the orbit structures ofthese CA.The atlas software appears to be restricted to circular CA with n < 5, attribute set S {0, }, and number of sites L < 32; source code is not included. It seems to me that the audience for this book is largely limited to researchers working on these particular CA. On the one hand, a good part of the 60 pages of text describing CA is devoted to describing the algorithm employed to compute inverse images, so that the book does not serve particularly well as a broad introduction to the theory of CA. On the other hand, the software seems to be limited to the class of examples described above, and doubt that the underlying algorithm can be effectively generalized to higherdimensional CA. The foregoing remarks are not intended to belittle the possibility that the book may be valuable for researchers; the information provided by the atlas and the software may well be important in investigations of these circular CA, and may lead to insights about more general CA.

Ten Lectures on Wavelets (Ingrid Daubechies)

Abstract: The resulting semigroup V is of class AK, and upper bounds are given for the Hausdorff dimension and fractal dimension of a compact invariant set associated with V. One who wishes to study Ladyzhenskaya’s book more carefully will wish to consult other sources. A good new source for Hausdorff dimension and related topics is the elegant text of Evans and Gariepy [3]. Ladyzhenskaya’s bibliography contains only sixteen items. This review has concentrated on the Navier-Stokes equations, but her book is more comprehensive than that alone. There is a huge literature on the ODE approach to PDE, and Ladyzhenskaya’s book is largely in this spirit. The recent books of Hale [4], Ruelle [5], and Temam [6] are good places to get much additional information and references. One should also consult Ladyzhenskaya’s recent paper [7], which gives \"complements and corrections\" to her previous work. These include estimates of Hausdorff and fractal dimensions. The reviewer is indebted to Craig Evans for this reference.