Showing papers in "Siam Review in 1979"
TL;DR: A survey article concerning recent advances in certain areas of statistical theory, written for a mathematical audience with no background in statistics, can be found in this article, where the authors illustrate how the advent of the high-speed computer has affected the development of statistical theories.
Abstract: This is a survey article concerning recent advances in certain areas of statistical theory, written for a mathematical audience with no background in statistics. The topics are chosen to illustrate a special point: how the advent of the high-speed computer has affected the development of statistical theory. The topics discussed include nonparametric methods, the jackknife, the bootstrap, cross-validation, error-rate estimation in discriminant analysis, robust estimation, the influence function, censored data, the EM algorithm, and Cox’s likelihood function. The exposition is mainly by example, with only a little offered in the way of theoretical development.
784 citations
TL;DR: This paper aims to assist the person who needs to solve stiff ordinary differential equations by identifying the problem area and the basic difficulty and describing the characteristics shared by methods for the numerical solution of stiff problems.
Abstract: This paper aims to assist the person who needs to solve stiff ordinary differential equations.First we identify the problem area and the basic difficulty by responding to some fundamental questions...
314 citations
TL;DR: In this article, the authors discuss the discrete Fourier transform and point out some computational problems in complex analysis where it can be fruitfully applied, such as trigonometric interpolation, conjugate periodic functions, and numerical inversion of Laplace transforms.
Abstract: In this paper we discuss the discrete Fourier transform and point out some computational problems in (mainly) complex analysis where it can be fruitfully applied. We begin by describing the elementary properties of the transform and its efficient implementation, both in the one-dimensional and in the multi-dimensional case, by the reduction formulas of Cooley, Lewis, and Welch (IBM Res, paper, 1967).The following applications are then discussed: Calculation of Fourier coefficients using attenuation factors; solution of Symm’s integral equation in numerical conformal mapping; trigonometric interpolation; determination of conjugate periodic functions and their application to Theodorsen’s integral equation for the conformal mapping of simply and of doubly connected regions; determination of Laurent coefficients with applications to numerical differentiation, generating functions, and the numerical inversion of Laplace transforms; determination of the “density” of the zeros of high degree polynomials. We then...
228 citations
TL;DR: Inverse iteration is one of the most widely used algorithms in practical linear algebra as discussed by the authors, but an understanding of its main numerical properties has developed piecemeal over the last thirty years: a m...
Abstract: Inverse iteration is one of the most widely used algorithms in practical linear algebra but an understanding of its main numerical properties has developed piecemeal over the last thirty years: a m...
211 citations
TL;DR: In this article, four types of convergence for sequences of convex sets are investigated and their interrelationships are explored for convex sequences of sets with respect to convex set sets.
Abstract: Four types of convergence for sequences of convex sets are investigated Their interrelationships are explored
170 citations
TL;DR: In this paper, four well-known methods for the numerical solution of linear discrete ill-posed problems are investigated from a common point of view: namely, the type of algebraic expansion generated for the solu...
Abstract: Four well-known methods for the numerical solution of linear discrete ill-posed problems are investigated from a common point of view: namely, the type of algebraic expansion generated for the solu...
159 citations
148 citations
TL;DR: In this paper, three classes of models of renewable resource economics are discussed: profit-maximizing models, competitive equilibrium models, and cooperative equilibrium models; a variety of complexities, both biological and economic, are also discussed, including irreversibility problems, disaggregated biological models, delay effects, and stochastic models.
Abstract: Three classes of models of renewable resource economics are discussed in this paper: profit-maximizing models, competitive equilibrium models, and cooperative equilibrium models. A variety of complexities, both biological and economic, are also discussed, including: irreversibility problems, disaggregated biological models, delay effects, and stochastic models. Questions deserving further research are raised.
83 citations
61 citations
60 citations
TL;DR: An operator-theoretic overview of discrete-time single-channel signal processing is given, covering such topics as the time and frequency domain structure of weakly stationary stochastic processes.
Abstract: An operator-theoretic overview of discrete-time single-channel signal processing is given, covering such topics as the time and frequency domain structure of weakly stationary stochastic processes,...
TL;DR: In this article, a survey of optimal stopping problems in the case of complete observations is presented, where problems leading to (so-called) variational inequalities and quasi-varianational inequalities are considered.
Abstract: In Part I we survey optimal stopping problems in the case of complete observations. First (§ 1) we consider problems leading to (so-called) variational inequalities, and then (§ 2) we deal with problems leading to quasi variational inequalities.In Part II we look at stopping time problems with incomplete observations. First (§ 3) we deal with a problem arising in quality control, and then (§ 4) we consider problems arising in models with costly observations.
TL;DR: In this paper, the development of the theory of linear wave propagation is described after a brief sketch of what wave propagation was, and the classical techniques of images and separation of variables are cons...
Abstract: The development of the theory of linear wave propagation is described after a brief sketch of what wave propagation is. First the classical techniques of images and separation of variables are cons...
TL;DR: In this paper, it is shown that it is possible to use the theory of generalized gradients to obtain necessary conditions in the form of a Hamiltonian inclusion for optimal control problems.
Abstract: The natural Hamiltonian function in optimal control is generally not differentiable. However, it is possible to use the theory of generalized gradients (which we discuss as a preliminary) to obtain necessary conditions in the form of a “Hamiltonian inclusion”. The advantages of using the true Hamiltonian include the possibility of considering control problems that do not lie within the usual formulation, or problems with nonsmooth data, and the possibility of bringing to bear upon control problems certain Hamiltonian methods from the classical calculus of variations.
TL;DR: In this article, the problem of finding scalars λ and vectors x = 0 for which Ax = λx has been studied, and the form of the matrix affects the way in which the scalars are found.
Abstract: Find scalars λ and vectors x = 0 for which Ax = λx The form of the matrix affects the way in which we solve this problem, and we also have variety as to what is to be found. • A symmetric and real (or Hermitian and complex). This is the most common case. In some cases we want only the eigenvalues (and perhaps only some of them); and in other cases, we also want the eigenvectors. There are special classes of such A, e.g. banded, positive definite, sparse, and others. • A non-symmetric, but with a diagonal Jordan canonical form. This means there is a nonsin-gular matrix P for which P −1 AP = D = diag[λ 1 , ..., λ n ]
TL;DR: The authors make a distinction between the practice of mathematics as it is usually presented, a logical chain of abstract, analytical reasoning from premises to conclusions, and how mathematics seems to be done in actuality, as a series of nonverbal, analog, often kinesthetic or visual insights.
Abstract: In this paper we make a distinction between the practice of mathematics as it is usually presented—a logical chain of abstract, analytical reasoning from premises to conclusions—and how mathematics seems to be done in actuality—as a series of nonverbal, analog, often kinesthetic or visual insights Mathematics in recent years has created a hierarchy with highly abstract,, logical and symbolic material at the peak and with more geometrical, visual, and analog material held to be of lesser worth We argue that humans are known to vary widely in their approaches to cognition and that the areas of the human brain specifically related to language and logical analysis seem to comprise only a part of the machinery of our intellect We suggest that it would be wise for the practitioners of mathematics, and perhaps especially the students of mathematics to be aware of the very important nonverbal elements in mathematics We feel that excessive emphasis on the abstract, analytic aspects of thought may have had dele
TL;DR: In this article, the authors present a survey of the development of inverses of the Cauchy and Hlder inequalities, an approach to establishing bounds for the ratio of weighted means, a proof of equivalence of quoti...
Abstract: This paper presents a survey of the development of inverses of the Cauchy and Hlder inequalities, an approach to establishing bounds for the ratio of weighted means, a proof of equivalence of quoti...
TL;DR: Fourier series solutions for mixed boundary value problems for the Laplace equation in the plane and in the sphere were obtained in this article, which is suitable for presentation in the first (applied) course in partial differential equations.
Abstract: Fourier series solutions are obtained for mixed boundary value problems for the Laplace equation in the plane and in the sphere. The nature of the boundary conditions presents difficulties in obtaining the Fourier coefficients, thus suggesting the use of variational procedures, e.g., least squares, to obtain the coefficients in the separated variable solutions. It is shown that for certain types of mixed boundary conditions numerically useful solutions can be obtained employing a hand held, programmable calculator, whereas other mixed boundary conditions require more computing power, say that needed to invert a $16 \times 16$ matrix. Modern applications of mixed boundary value problems are given, e,g., cryogenic thermal insulation, nuclear power generation, nondestructive testing, transportation of liquified natural gas, etc. This material is suitable for presentation in the first (applied) course in partial differential equations.
TL;DR: In this article, the Gordan-Stiemke Theorem of the alternative is used to demonstrate the similarity of four theorems in combinatorial matrix theory, one of which is the existence in a given pattern of a line-sum-symmetric or constant-line-sum matrix which is semi-positive or strictly positive for the pattern.
Abstract: By use of the Gordan–Stiemke Theorem of the alternative we demonstrate the similarity of four theorems in combinatorial matrix theory. Each theorem contains five equivalent conditions, one of which is the existence in a given pattern of a line-sum-symmetric or constant-line-sum matrix which is semi-positive or strictly positive for the pattern. A generalization of the Gordan–Stiemke Theorem is stated in terms of complementary faces of the positive orthant and combinatorial applications are given. Many of our results are classical, but some may be new.
TL;DR: John Baylis considered the following problem: When making a right hand turn on British roadways one moves as far to the right as possible on one's side of the roadway and then turns, but the rear of the vehicle moves leftward as the right handturn is begun-toward the unsuspecting driver passing on the left.
Abstract: John Baylis [1] considered the following problem: When making a right hand turn on British roadways one moves as far to the right as possible on one's side of the roadway and then turns. Unfortunately, the rear of the vehicle moves leftward as the right hand turn is begun-toward the unsuspecting driver passing on the left. This can be quite noticeable if the turning vehicle is a long bus. We assume familiarity with Baylis's paper and notation. Recall that he defines I = length of wheelbase, h = length of rear overhang, 0 = angle between bus and direction of roadway, 0 = angle of front wheels relative to the bus, V = speed of the bus.
TL;DR: In this article, the existence and continuity of solutions and the shape and smoothness of the free boundary of a dam with time were investigated for the flow of liquid in a dam.
Abstract: Recent results on the existence and continuity of solutions, and on the shape and smoothness of the free boundary are described for the following problems: (a) the flow of liquid in a dam with time...
TL;DR: In this paper, a simple explicit formula is given for the distribution of the number of changes of luck in n Bernoulli trials with probability $p(0 < p < 1)$ for success.
Abstract: In this paper a simple explicit formula is given for the distribution of the number of changes of luck in n Bernoulli trials with probability $p(0 < p < 1)$ for success.
TL;DR: In this article, it was shown that the off-diagonal entries of the coefficient matrix are nonnegative and conditions are established that solutions starting in a certain half-space will eventually wind up in the positive orthant.
Abstract: This paper is an exposition of results achieved by Garrets Birkhoff and coworkers in their study of the linear differential equation ${\bf x}' = A(t){\bf x}$, in which the off-diagonal entries of the coefficient matrix are nonnegative. Of interest is the direction of a solution vector, not its length. Conditions are established guaranteeing that solutions starting in a certain half-space will eventually wind up in the positive orthant, and it is shown that all such solutions are “eventually parallel”. Analogies between length and direction problems are pointed out, including a brief discussion of a directionally stable solution of a perturbed equation ${\bf x}' = A{\bf x} + {\bf f}(t,{\bf x})$.
TL;DR: A broad survey of the theory of ocean waves is presented in this article, with special emphasis on recent developments, and a number of ocean wave problems which involve statistical and probabilistic aspects are also discussed.
Abstract: A broad survey of the theory of ocean waves is presented, with special emphasis on recent developments. After a brief review of the classical wave modes in the ocean, a discussion of several recently discovered modes of oscillation is given. This is followed by a description of (1) finite-amplitude effects on each of these modes and (2) nonlinear interactions between different modes. Various types of wave-media interactions are next reviewed: refraction, scattering and diffraction, critical layer absorption and shear flow instability. A number of ocean wave problems which involve statistical and probabilistic aspects are also discussed. The different mechanisms for wave generation are described, but only the wind-wave generation problem is discussed in detail. The paper concludes with a list of challenging problems that are of current interest to the oceanographic community.