Showing papers in "Siam Review in 1967"

Random Matrices in Physics

Abstract: Introduction. It has been observed repeatedly that von iNeumann made important contributions to almost all parts of mathematics with the exception of number theory. He had a particular interest in those parts of mathematics which formed cornerstones of other, more empirical sciences, such as physics or economics. A whole new discipline grew out of his theory of games, and it is hard to conjure up a picture of modern United States industry without the computing machines which he espoused. The subject about which I wish to talk to you today is at the crossroads of two of von Neumann's principal interests: it deals with matrices of very large dimensions in which he became interested in connection with his development of computers, and it resembles statistical mechanics, to which he contributed most among the physical theories. This closeness of my subject to von Neumann's interests is also the reason for my choosing it for today's discussion. This discussion will contain very little that is new. As for earlier reviews, there are at least two very good ones: one by Charles Porter, forming an introduction and summary to a collection of papers on the role of random matrices in physics [1], the second a more elaborate one by M. L. Mehta, based on his lectures at the Indian Institute of Technology in Kanpur [2]. There is another reason for my choice of subject. The theory of random matrices, though initiated by mathematicians and in particular statisticians [3], [4], [5], [6], [7], has made large strides in the hands of physicists. The names of Mehta, Gaudin and Dyson come to one's mind most easily. Reading these papers gave me much pleasure-they contain beautiful, though old-fashioned, mathematics. I would like to share some of this pleasure with you. Second, however, a number of problems has turned up, apparently too difficult for us amateur mathematicians. I would like to share these problems with you also. Origin of the problem. For reasons which I hope will become evident in the course of the discussion, I will proceed in my review pretty much in the antihistoric order. However, the reason for the interest of physicists in random matrices should be stated first. This was articulated, most eloquently, by Dyson [8]: "Recent theoretical analyses have had impressive success in interpreting the detailed structure of the low-lying excited states of complex nuclei. Still, there must come a point beyond which such analyses of individual levels cannot usefully go. For example, observations of levels of heavy nuclei in the neutron-capture region give precise information concerning a stretch of levels from number N to number (N + n), where N is an integer

Integer Programming by Implicit Enumeration and Balas’ Method

Abstract: : This memorandum presents a reformulation of the essentials of Balas' algorithm for the zero-one integer linear programming problem, and is based upon the idea of 'elementary tree search' that has also been used by Glover as the basis for his multiphase-dual algorithm. The present reformulation requires considerably less computer storage than the original version, and clarifies the rationale behind the algorithm, thereby leading naturally to variants and extensions.

Seven Kinds of Convexity

Abstract: An Oil-Base Water-Absorbing Fluorescent Dyestuff Carrying Fluid used for testing metal which not only locates surface defects such as cracks but distinguishes between flaw depths and, consequently, flaw relevancy by means of combining different color water-soluble and oil-soluble fluorescent dyestuffs is disclosed in this invention. A process step in the use of this fluid is water washing of the surface to remove fluid which has not become entrapped in a surface discontinuity. During this step, fluid entrapped in shallow defects absorbs a relatively high percentage of water which activates the water-soluble dyestuff while fluid in deep defects remains substantially water free and the water-soluble dyestuff remains substantially inactive. Thus, shallow defects fluoresce the color of the water-soluble dyestuff and deep defects fluoresce the color of the oil-soluble dyestuff.

A Runge-Kutta for all Seasons

Abstract: Traditionally, Runge-Kutta integrations proceed one step at a time, with several function evaluations in each step. Upon proceeding to the next step, one abandons all information about the behavior of the solution that became available in any previous step. It is proposed to proceed in blocks of N steps. The self-starting feature of Runge-Kutta methods is still present, only for a block instead of a single step. If there is any need to prevent the last block from running past the end of the region of integration, one can take its step interval to be half as big, or a fourth as big. In each step of a block one can call on information from the other steps of the block. Thus fewer function evaluations per step are needed, and on this point the method becomes competitive with classic predictor-corrector methods. Also, by analyzing the trends over N steps at once, stimates of the accuracy can be derived which compare in reliability with those for classic predictor-corrector methods. One can adjust the step int...

Transforming Boundary Conditions to Initial Conditions for Ordinary Differential Equations

Abstract: One of the interesting and important classes of transformations in the solution of boundary value problems is the class of transformations from a boundary value to an initial value problem. A classical example is given in Blasius' solution of the steady, two-dimensional, incompressible boundary layer equations with uniform mainstream velocity [1]. This technique, when it applies, simplifies the process of obtaining numerical solutions. Otherwise a trial-and-error technique generally has to be used in order to match the boundary condition at the other point. A recent contribution by Klamkin [2] has greatly extended the range of application of the technique, and as a result, a much wider class of problems, including some simultaneous differential equations, can be solved in this manner. In the present paper, the technique is extended to cases where the boundary conditions are specified over finite intervals, i.e., from zero to a finite value of the independent variable, with one or more boundary conditions required to be specified at the initial point. It also shows that the transformation treated in previous works is not the only type of transformation possible. Similar to the class of problems treated by Blasius [1] and Klamkin [2], the boundary conditions at the initial point have to be homogeneous for the method to apply. It can be homogeneous or nonhomogeneous at the other point. The method given in this paper is developed in terms of a oneor two-parameter group of transformations, which is seen to be as simple here as in its application to the similarity analysis [3]. The concept can best be illustrated by an example. Consider the problem of steady heat conduction with linear heat generation and power-law thermal conductivity. The energy equation can be written as

Today’s Computational Methods of Linear Algebra

Abstract: : This is a survey of selected computational aspects of linear algebra, addressed to the nonspecialist in numerical analysis. Some current methods of solving systems of linear equations, and computing eigenvalues of symmetric and unsymmetric matrices are outlined. A bibliography containing 62 titles is included.

The Mean Convergence of Fourier-Bessel Series

Abstract: can be expressed in terms of Bessel functions and is given by y = ci/VJ, (Xz) + c2N//J-v (Xz), (2) (Z)2n )J(z) = (j2) no n! F(n + ' + 1) The functions J,(z) and J-,(z) are linearly independent if and only if v is not an integer When v is an integer the function J-,(z) in (2) should be replaced by a solution of (1), that is independent of J,(z) It is well known [1] that the function J,(X) will, for v > -1, have an infinite number of real, positive, simple zeros which we denote by 0 < 1Xl < X2 < X3 < There are no complex zeros, and the set { -Xi forms the negative zeros These accumulate at infinity Their asymptotic form can be shown to be [1]

The Zeros of the Hahn Polynomials

Abstract: x = 0, 1, * , n 1. From this it follows when a, d> -1 that, if y is an integer > mn, the zeros of Pm(` 7)(x) are real and simple and lie in the open interval (0, y 1). In the present paper this conclusion is extended to all real -y > mn and also to 7y < -(im + ae + d) with (d + -y, -a 1) as the interval containing the zeros in the latter case. If the inequalities on a, f, -y are relaxed, the conclusion fails. In fact, the zeros need no longer be simple or even real. We also show that under the same conditions consecutive zeros are spaced more than one unit apart. These results are obtained in ?5. In ?6 we prove a number of separation theorems for the zeros of Hahn polynomials of different degrees or with different parameter values. In the earlier sections we derive various formulas partly for later use and partly for their own interest. In particular, in ?3 we establish some symmetry properties and in ?4 derive an expression for the difference APm(`0 ") (x), which we use to give a simplified form to the summation formula of Bartko [13], and to construct from Hahn polynomials an Appell set of the second kind. The Hahn polynomials arose as a limiting form in the treatment of some general systems of orthogonal polynomials by Hahn [1]. Some important particular cases

The product form of inverses of sparse matrices and graph theory.

Abstract: Geometric interpretation of product form of inverse applied to sparse matrices in linear programming, using graph theory

On Certain Fredholm Integral Equations Reducible to Initial Value Problems

Abstract: Nonlinear initial value problems (of a rather unusual nature) will be obtained which have as their solutions so(x) and so(y). From these values Sp(z) may be found by solving another related initial value problem. The equations derived are amenable to numerical treatment and may very well provide advantages over the more customary direct methods of computing $o(z) from (1.1). The representation (1.2) actually encompasses a very large class of kernels. If, for example, a(s) = s, we have the usual Laplace transform. Kernels expressed in terms of such Laplace integrals arise often in transport theory. A problem in gas dynamics analyzed in [10] involves a kernel represenited in this way. Again, if a(s) = -log s, then K is a Mellin transform. We shall also find that a(s) = i log s is admissible in our theory, and this leads to a Fourier integral:

On the Chebyshev Polynomials of the Second Kind

Abstract: (1) Q*() = 1 (1 t) t) = 0), 1, 2, ... Equation (1) defines Qn*(z) as a single-valued analytic function for all z in the plane with the interval -1 ? t 1, maps onto the ellipse &p with foci at z = 41 and semiaxes 2(P ? p') and .1(p -_p). The first lemma gives a simple representation for Qn*(z) on &p. LEMAIA 1. For z E &p,

Propagation of One-Dimensional Waves in Inhomogeneous Elastic Media

Abstract: : Formal progressing wave expansions are applied to problems involving one-dimensional wave propagation through inhomogenous elastic media. Expansions for the stress and particle velocity are obtained in addition to the expansion for the particle displacement which is a special case of previous results. Although one-dimensional problems could be solved with the previously reported asymptotic methods, it is more convenient to use the expansion in terms of the stresses to evaluate the expansion coefficients. The procedure is illustrated by solving several problems in layered and nonlayered inhomogeneous media where the compressional wave speed is subject to power and exponential variations with distance. (Author)