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Showing papers in "Siam Review in 1989"


Journal ArticleDOI
TL;DR: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states, focusing on Weyl–Heisenberg coherent states and affine coherent states.
Abstract: This paper is an expository survey of results on integral representations and discrete sum expansions of functions in $L^2 ({\bf R})$ in terms of coherent states. Two types of coherent states are considered: Weyl–Heisenberg coherent states, which arise from translations and modulations of a single function, and affine coherent states, called ’wavelets,’ which arise as translations and dilations of a single function. In each case it is shown how to represent any function in $L^2 ({\bf R})$ as a sum or integral of these states. Most of the paper is a survey of literature, most notably the work of I. Daubechies, A. Grossmann, and J. Morlet. A few results of the authors are included.

1,121 citations


Journal ArticleDOI
TL;DR: It is shown in Part 1 how conditions on the $c_k $ lead to approximation properties and orthogonality properties of the wavelets, and the recursive algorithms that decompose and reconstruct f.
Abstract: Wavelets are new families of basis functions that yield the representation $f(x) = \sum {b_{jk} W(2^j x - k)} $. Their construction begins with the solution $\phi (x)$ to a dilation equation with coefficients $c_k $. Then W comes from $\phi $, and the basis comes by translation and dilation of W. It is shown in Part 1 how conditions on the $c_k $ lead to approximation properties and orthogonality properties of the wavelets. Part 2 describes the recursive algorithms (also based on the $c_k $) that decompose and reconstruct f. The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms

1,028 citations


Journal ArticleDOI
TL;DR: The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed.
Abstract: The Sherman–Morrison–Woodbury formulas relate the inverse of a matrix after a small-rank perturbation to the inverse of the original matrix. The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed. The Sherman-Morrison-Woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix. This paper surveys the history of these formulas and we examine some applications where these formulas are helpful

1,026 citations


Journal ArticleDOI
TL;DR: It is shown that more comprehensive and simpler error estimates can now be made on the QSSA and this leads to a dimensionless formulation containing a different small parameter from that which is customary.
Abstract: The quasi-steady-state assumption (QSSA) of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ODE systems with an initial fast transient. Simple estimates of the two relevant time scales of the underlying problem are made. These estimates lead to a dimensionless formulation containing a different small parameter from that which is customary. Earlier results on the QSSA are surveyed in the context of the new nondimensionalization. It is shown that more comprehensive and simpler error estimates can now be made. Some general methodological hints are drawn from this example

687 citations


Journal ArticleDOI
TL;DR: This paper describes various enhancements of the minimum degree algorithm, their historical development, and some experiments showing how very effective they are in improving the execution time of the algorithm.
Abstract: Over the past fifteen years, the implementation of the minimum degree algorithm has received much study, and many important enhancements have been made to it. This paper describes these various enhancements, their historical development, and some experiments showing how very effective they are in improving the execution time of the algorithm. A shortcoming is also presented that exists in all of the widely used implementations of the algorithm, namely, that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering. For example, changing the input ordering can lead to an increase (or decrease) of as much as a factor of three in the cost of the subsequent numerical factorization. This sensitivity is caused by the lack of an effective tie-breaking strategy, and the authors’ experiments illustrate the importance of developing such a strategy

417 citations


Journal ArticleDOI
TL;DR: The role of stochastic complementation in the development of the classical Simon–Ando theory of nearly reducible system is presented and some of its properties are developed.
Abstract: A concept called stochastic complementation is an idea which occurs naturally, although not always explicitly, in the theory and application of finite Markov chains. This paper brings this idea to the forefront with an explicit definition and a development of some of its properties. Applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of Markov chains. Furthermore, the role of stochastic complementation in the development of the classical Simon–Ando theory of nearly reducible system is presented

300 citations


Journal ArticleDOI
TL;DR: This paper gives some of the history of the conjugate gradient and Lanczos algorithms and an annotated bibliography for the period 1948-1976.
Abstract: This paper gives some of the history of the conjugate gradient and Lanczos algorithms and an annotated bibliography for the period 1948-1976

216 citations


Journal ArticleDOI
TL;DR: A generalized matrix product is introduced, which inherits some useful algebraic properties from the standard Kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula.
Abstract: Discrete unitary transforms are extensively used in many signal processing applications, and in the development of fast algorithms Kronecker products have proved quite useful. In this semitutorial paper, we briefly review properties of Kronecker products and direct sums of matrices, which provide a compact notation in treating patterned matrices. A generalized matrix product, which inherits some useful algebraic properties from the standard Kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula, is then introduced. The notation is intimately related to sparse matrix factorizations, and examples are included illustrating the utility of the new notation in signal processing applications. Finally, some novel characteristics of Hadamard transforms and polyadic permutations are derived in the framework of Kronecker products.

182 citations




Journal ArticleDOI
TL;DR: A Fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems is presented, which essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions, such as the Dirichlet problem.
Abstract: his paper presents a Fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems. As in the von Neumann stability analysis of hyperbolic and parabolic problems, the approach is easier to apply, reveals more details about convergence properties than about standard techniques, and can be applied in a systematic way to a wide class of numerical methods. Although the analysis is applicable only to periodic problems, the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions, such as the Dirichlet problem. In addition, they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the Dirichlet case

Journal ArticleDOI
TL;DR: The properties of the k-plane transforms P, D, and S and their inverses are discussed, emphasizing relationships and similarities between the operators, and their relation to CT.
Abstract: The mathematics behind Computerized Tomography (CT) is based on the study of the parallel beam transform P and the divergent beam transform D. Both of these map a function f in $\mathbb{R}^n $ into a function defined on the set of all lines in $\mathbb{R}^n $, by integrating f along these lines.The parallel and divergent k-plane transforms are defined in a similar fashion by integration over k-planes (i.e., translates of k-dimensional subspaces) and are also denoted P and D, respectively. A related transform is the spherical k-plane transform S, which maps a function f on the sphere $S^{n - 1} $ into its integrals over k-dimensional great circlesThis paper discusses the properties of the k-plane transforms P, D, and S and their inverses, emphasizing relationships and similarities between the operators, and their relation to CT.Some new results are included. Most notable are more general conditions under which the inversion formulas hold

Journal ArticleDOI
TL;DR: The Stokes Phenomenon is presented as a natural aspect of a well-motivated characterization of functions by approximands of different multivaluedness.
Abstract: The Stokes Phenomenon is known to be a pervasive feature of asymptotics, but its explanation in the literature is obscured by intricate and lengthy technicalities. This article presents a simpler approach to its understanding and treatment as a natural aspect of a well-motivated characterization of functions by approximands of different multivaluedness

Journal ArticleDOI
TL;DR: A unified analysis of reaction-diffusion equations and their finite difference representations is presented, and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems.
Abstract: A unified analysis of reaction-diffusion equations and their finite difference representations is presented. The parallel treatment of the two problems shows clearly when and why the finite difference approximations break down. The approach used provides a general framework for the analysis and interpretation of numerical instability in approximations of dissipative nonlinear partial differential equations Continuous and discrete problems are studied from the perspective of bifurcation theory, and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems. An asymptotic approach, due to Newell (SIAM J. Appl. Math., 33 (1977), 133–160), is used to investigate the instability phenomenon further. In particular, equations are derived that describe the interaction of the dynamics of the partial differential equation with the artefacts of the discretization.

Journal ArticleDOI
TL;DR: This paper shows how the Kelvin transformation (inversion) may be applied to scattering problems of linear acoustics by transforming the exterior problem for the original scatterer into a succession of interior problems for the transformed surface.
Abstract: This paper shows how the Kelvin transformation (inversion) may be applied to scattering problems of linear acoustics. First, the Kelvin transformation and its application to problems in three-dimensional potential theory is reviewed. Then the application to scattering problems is presented. This involves transforming the exterior problem for the original scatterer into a succession of interior problems for the transformed surface. The complete low-frequency expansions of both near and far fields are presented in terms of the solutions of these related interior potential problems. Results are presented for Dirichlet, Neumann, and Robin boundary conditions as well as for the transmission problem.


Journal ArticleDOI
TL;DR: The early motivation for and development of diagonal increments to ease matrix inversion in least squares (LS) problems is discussed and the interplay among factors and the advent of ridge regression are considered in a historical and comparative framework.
Abstract: he early motivation for and development of diagonal increments to ease matrix inversion in least squares (LS) problems is discussed. It is noted that this diagonal incrementation evolved from three major directions: modification of existing methodology in nonlinear LS, utilization of additional information in linear regression, and improvement of the numerical condition of a matrix. The interplay among these factors and the advent of ridge regression are considered in a historical and comparative framework

Journal ArticleDOI
TL;DR: His paper deals with maximum entropy distributions for uncertain parameters that lie between two finite values and shows that these maximally unpresulrlptive distributions depend on the nature of the a priori information available about the uncertain parameter.
Abstract: his paper deals with maximum entropy distributions for uncertain parameters that lie between two finite values. Such parameter uncertainties often arise in the modeling of physical systems. The paper shows that these maximally unpresulrlptive distributions depend on the nature of the a priori information available about the uncertain parameter. In particular, three commonly occurring situations met with in engineering systems are considered: (1) only the interval in which the uncertain parameter lies is known a priori; (2) the interval as well as the mean value of the parameter is known; (3) the interval, the mean value of the parameter, and the parameter’s variance are all known. The nature of the probability distributions is determined and closed form solutions for these three situations are provided.

Journal ArticleDOI
TL;DR: In this sketchy account I will describe the twists and turns as well as the thrusts of applied mathematics in America, where mathematics and the sciences, mainly but by no means only physics, are equal partners, feeding ideas, concepts, problems, and solutions to each other.
Abstract: Mathematicians are notoriously bad historians; they describe the development of an idea as it should logically have unfolded rather than as it actually did, by fits and starts, often false starts, and buffeted by forces outside of mathematics. In this sketchy account I will describe the twists and turns as well as the thrusts of applied mathematics in America. Applied mathematics is alive and well in America today. Looking at the 18 lectures [presented at the Centennial Meeting of the American Mathematical Society] chosen to describe the frontiers of research, we see that topics include physiological modeling, fluid flow and combustion, computer science, and the formation of atoms within the framework of statistical mechanics. We also see that the subject of one lecture and the starting point of several others are physical theories; the conclusions reached are of interest to physicists and mathematicians alike. This was not always so. For a few decades, the late 1930s through the early 1950s, the predominant view in American mathematical circles was the same as Bourbaki's: mathematics is an autonomous abstract subject, with no need of any input from the real world, with its own criteria of depth and beauty, and with an internal compass for guiding further growth. Applications come later by accident; mathematical ideas filter down to the sciences and engineering. Most of the creators of modern mathematics-certainly Gauss, Riemann, Poincare, Hilbert, Hadamard, Birkhoff, Weyl, Wiener, and von Neumann-would have regarded this view as utterly wrongheaded. Today we can safely say that the tide of purity has turned; most mathematicians are keenly aware that mathematics does not trickle down to the applications, but that mathematics and the sciences, mainly but by no means only physics, are equal partners, feeding ideas, concepts, problems, and solutions to each other. Whereas in the not so distant past a mathematician asserting "applied mathematics is bad mathematics" or "the best applied mathematics is pure mathematics" could count on a measure of assent and applause, today a person making such statements would be regarded as ignorant. How did this change come about? Several plausible reasons can be discerned. But first a bit of selective history. World War II, a watershed for our social institutions, concepts and thinking, permanently changed the status of applied mathematics in America. That is not to say that there was no worthwhile applied mathematics in America before 1945; after all, already in the nineteenth century, Gibbs' contributions to statistical mechanics as well as to vector analysis and Fourier series, and Hill's studies of Hill's equation, had put America on the applied mathematics map. The leading American analysts in the 1920s and 1930s were Birkhoff, renowned worldwide for his work in dynamics, and Wiener, a pioneer in the study of physical processes driven by chance influences such


Journal ArticleDOI
TL;DR: A simple proof establishes that, in a stationary population, the average age A is equal to the average expectation of remaining life E, analogous to a result in renewal theory.
Abstract: A simple proof establishes that, in a stationary population, the average age A is equal to the average expectation of remaining life E. This is analogous to a result in renewal theory. An epidemiological model of lifelong infectivity is presented for which E appears in the threshold condition for spread of infection. The substitution of A for E is useful because age is easier to measure than the expectation of remaining life


Journal ArticleDOI
TL;DR: A mathematical model for running includes fatigue as a factor and techniques of numerical approximation and nonlinear programming then determine the best running strategy for a race.
Abstract: A mathematical model for running includes fatigue as a factor. Techniques of numerical approximation and nonlinear programming then determine the best running strategy for a race.

Journal ArticleDOI
TL;DR: An application of bifurcation theory to a simplified model of kite flight bears out this assertion by proving the existence of a limit point biforcation in the wind velocity/kite string angle plane.
Abstract: The careful observation of the dynamics of a kite in flight hints at the possibility of multiple equilibrium states. An application of bifurcation theory to a simplified model of kite flight bears out this assertion by proving the existence of a limit point bifurcation in the wind velocity/kite string angle plane





Journal ArticleDOI
TL;DR: The asymptotic regime discussed here, involving a balance of small amplitudes and small wavelengths for fixed activation energy, illustrates in a simplified context some of the main new phenomena being analyzed in the current theoretical combustion l...
Abstract: Simplified asymptotic equations for the interaction of high-frequency, small-amplitude waves in a chemically reacting gas are developed and analyzed. The equations for nonlinear acoustic simple waves in a reacting mixture are solved explicitly, and reveal substantial wave amplification through combustion for a wide range of activation energies and heat release. The asymptotic equations for the resonant interaction of almost periodic wave trains in a reacting gas are also developed. One prominent new effect of acoustic resonance documented through numerical experiments is the transfer of energy from cold spots to hot spots through resonant interaction; this leads to more dramatic wave amplification through combustion than in the case of nonlinear simple waves. The asymptotic regime discussed here, involving a balance of small amplitudes and small wavelengths for fixed activation energy, illustrates in a simplified context some of the main new phenomena being analyzed in the current theoretical combustion l...