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1997

Volume 39, Issue 4, pp. 575-809


Of Stable Marriages and Graphs, and Strategy and Polytopes

Michel Balinski and Guillaume Ratier

SIAM Rev. 39, pp. 575-604 (30 pages) | Cited 7 times

Online Publication Date: August 04, 2006

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This expository paper develops the principal known results (and some new ones) on the stable matchings of marriage games in the language of directed graphs. This both unifies and simplifies the presentation and renders it more symmetric. In addition, it yields a new algorithm and a new proof for the existence of stable matchings, new proofs for many known facts, and some new results (notably concerning players' strategies and the properties of the stable matching polytope).

A Survey of Combinatorial Gray Codes

Carla Savage

SIAM Rev. 39, pp. 605-629 (25 pages) | Cited 42 times

Online Publication Date: August 04, 2006

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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing n-bit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960s and 1970s on minimal change listings for other combinatorial families, including permutations and combinations.
The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Conference on Discrete Mathematics in 1988 and his subsequent SIAM monograph [Combinatorial Algorithms: An Update, 1989] in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area, and most of the problems posed by Wilf are now solved.
In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems.

Interference Effects in Computation

Willard L. Miranker

SIAM Rev. 39, pp. 630-643 (14 pages)

Online Publication Date: August 04, 2006

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The law of addition of probabilities by means of complex probability amplitudes with its powerful interference consequences, unique to quantum mechanics, is shown to apply elsewhere. The two parts of quantum mechanics: an unvisualizable dynamics and a visualizable concretization (the latter corresponding to reduction of the state vector when a measurement is made) are shown to have a correspondence in computation. Namely, a conceptual structure called a field (of the real numbers, say), its operations, and the concretization of that structure is done by means of a (digital) computer (the latter is interpreted as playing the role of a measuring apparatus). The probabilistic state-reduction operator of quantum mechanics is replaced by a deterministic operation, an extension of rounding. State, wave function, dynamics, observation, uncertainty, and nonlocality are shown to have their counterparts in the new model. The algorithmic counterpart of the double slit experiment to validate the existence of interference in the new framework is defined and performed. A notion of spin in computation is introduced. In an appendix, we comment on how this model impacts a speculative theory of the mind by Penrose.

On the Gibbs Phenomenon and Its Resolution

David Gottlieb and Chi-Wang Shu

SIAM Rev. 39, pp. 644-668 (25 pages) | Cited 48 times

Online Publication Date: August 04, 2006

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The nonuniform convergence of the Fourier series for discontinuous functions, and in particular the oscillatory behavior of the finite sum, was already analyzed by Wilbraham in 1848. This was later named the Gibbs phenomenon.
This article is a review of the Gibbs phenomenon from a different perspective. The Gibbs phenomenon, as we view it, deals with the issue of recovering point values of a function from its expansion coefficients. Alternatively it can be viewed as the possibility of the recovery of local information from global information. The main theme here is not the structure of the Gibbs oscillations but the understanding and resolution of the phenomenon in a general setting.
The purpose of this article is to review the Gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function, with the same order of accuracy as in the smooth case. This is done by using the finite expansion series to construct a different, rapidly convergent, approximation.

Engineering and Economic Applications of Complementarity Problems

M. C. Ferris and J. S. Pang

SIAM Rev. 39, pp. 669-713 (45 pages) | Cited 184 times

Online Publication Date: August 04, 2006

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This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

Case Study from Industry:Process Modeling in Resin Transfer Molding as a Method to Enhance Product Quality

W. K. Chui, J. Glimm, F. M. Tangerman, A. P. Jardine, J. S. Madsen, T. M. Donnellan, and R. Leek

SIAM Rev. 39, pp. 714-727 (14 pages) | Cited 7 times

Online Publication Date: August 04, 2006

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Resin transfer molding (RTM) has drawn interest in recent years as an attractive technique for the manufacture of advanced fiber-reinforced composite materials. A major issue in this new manufacturing process is the reduction of voids during the resin fill process so that products with high quality are manufactured. Process modeling is particularly useful in understanding, designing, and optimizing the process conditions. The purpose of this paper is to illustrate the importance of mathematical and numerical modeling to this industrial problem. First, an overview of the RTM process, its manufacturing problems, and related background issues is given. A survey of various RTM models developed in recent years by researchers in this field are then presented. Finally, as an application, a novel two-phase flow model, developed recently by the authors, is proposed to study the formation and migration of the macrovoids, a major manufacturing problem. The unique feature of this model is the identification of local pressure as a major mobilization factor of these macrovoids. It is demonstrated that the model is in good agreement with experimental results.

Classroom Note:Geometry and Convergence of Euler's and Halley's Methods

A. Melman

SIAM Rev. 39, pp. 728-735 (8 pages) | Cited 10 times

Online Publication Date: August 04, 2006

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We investigate the global convergence of Euler's and Halley's methods by using their geometric interpretation. A generalization of these methods is also briefly discussed.

Classroom Note: Initialization of the Simplex Algorithm: An Artificial-Free Approach

H. Arsham

SIAM Rev. 39, pp. 736-744 (9 pages) | Cited 4 times

Online Publication Date: August 04, 2006

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The simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point. We present a new general purpose solution algorithm which obviates the use of artificial variables. The algorithm searches for a feasible segment of a boundary hyperplane (a face of feasible region or an intersection of several faces) by using rules similar to the ordinary simplex.

Classroom Note:Finding the Center of a Circular Starting Line in an Ancient Greek Stadium

Chris Rorres and David Gilman Romano

SIAM Rev. 39, pp. 745-754 (10 pages) | Cited 6 times

Online Publication Date: August 04, 2006

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Two methods for finding the center and radius of a circular starting line of a racetrack in an ancient Greek stadium are presented and compared. The first is a method employed by the archaeologists who surveyed the starting line and the second is a least-squares method leading to a maximum-likelihood circle. We show that the first method yields a circle whose radius is somewhat longer than the radius determined by the least-squares method and propose reasons for this difference. A knowledge of the center and radius of the starting line is useful for determining units of length and angle used by the ancient Greeks, in addition to providing information on how ancient racetracks were laid out.

Classroom Note:Stability Considerations for Numerical Methods

Johnny Snyder

SIAM Rev. 39, pp. 755-760 (6 pages) | Cited 1 time

Online Publication Date: August 04, 2006

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In this paper we address the issue of stability in numerical methods for partial differential equations (PDEs) through normal mode analysis of a particular scheme. It is shown that when numerical methods appear to be stable, there are subtleties one must be aware of before conclusions may be drawn. This is illustrated through the analysis of the one-way wave equation using the leap-frog scheme. A bifurcation from stability to instability is illustrated along with numerical simulations illustrating the growing modes.

Problems and Solutions

Cecil C. Rousseau and Otto G. Ruehr, Editors

SIAM Rev. 39, pp. 761-789 (29 pages)

Online Publication Date: August 04, 2006

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Book Reviews

R. B. Kellogg, Editor

SIAM Rev. 39, pp. 790-809 (20 pages)

Online Publication Date: August 04, 2006

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