Showing papers in "Siam Review in 1981"
TL;DR: In this paper, the inverse of the sum of two matrices, one of them being nonsingular, has been studied and new expressions for the inverse were derived for the non-singular case.
Abstract: Available expressions are reviewed and new ones derived for the inverse of the sum of two matrices, one of them being nonsingular. Particular attention is given to $({\text{{\bf A}}} + {\text{{\bf ...
834 citations
TL;DR: In this article, the existence of positive solutions of semilinear elliptic equations is studied and the results are also interpreted in terms of bifurcation diagrams, and in each case nearly optimal multiplicity results are obtained.
Abstract: In this paper we study the existence of positive solutions of semilinear elliptic equations. Various possible behaviors of nonlinearity are considered, and in each case nearly optimal multiplicity results are obtained. The results are also interpreted in terms of bifurcation diagrams.
693 citations
TL;DR: In this paper, the relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated, and applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behaviour of parasitic waves, and the stability analysis of initial boundary value problems.
Abstract: The relevance of group velocity to the behavior of finite difference models of time-dependent partial differential equations is surveyed and illustrated. Applications involve the propagation of wave packets in one and two dimensions, numerical dispersion, the behavior of parasitic waves, and the stability analysis of initial boundary-value problems.
477 citations
TL;DR: The main feature of row-action methods is that they are iterative procedures which, without making any changes to the original matrix A, use the rows of A, one row at a time as discussed by the authors.
Abstract: This paper brings together and discusses theory and applications of methods, identified and labelled as row-action methods, for linear feasibility problems (find $x \in {\bf R}^n $, such that $Ax \leqq b$), linearly constrained optimization problems (minimize $f(x)$, subject to $Ax \leqq b$) and some interval convex programming problems (minimize $f(x)$, subject to $c \leqq Ax \leqq b$).The main feature of row-action methods is that they are iterative procedures which, without making any changes to the original matrix A, use the rows of A, one row at a time. Such methods are important and have demonstrated effectiveness for problems with large or huge matrices which do not enjoy any detectable or usable structural pattern, apart from a high degree of sparaseness.Fields of application where row-action methods are used in various ways include image reconstruction from projection, operations research and game theory, learning theory, pattern recognition and transportation theory. A row-action method for the ...
472 citations
TL;DR: The authors summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis, including quadrature, approximate evaluation of transforms, and approximate solution of differential and integral equations.
Abstract: This paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis. Described in this paper are methods of interpolation and approximation of functions and their derivatives, quadrature, the approximate evaluation of transforms (Hilbert, Fourier, Laplace, Hankel and Mellin) and the approximate solution of differential and integral equations. The methods have many advantages over classical methods which use polynomials as bases. In addition, all of the methods converge at an optimal rate, if singularities on the boundary of approximation are ignored.
312 citations
312 citations
TL;DR: In this article, a review is presented of various approaches to the generalization of the version of Noether's theorem, which is presented in most textbooks on classical mechanics, and its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations.
Abstract: In this paper, a review is presented of various approaches to the generalization of the version of Noether’s theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an “interrelationship” between symmetries and first integrals there. We then compare different proposed generalizations of Noether’s theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invaria...
285 citations
TL;DR: In this article, the authors review the theory for linear systems where the dynamics are governed by strongly continuous semigrods and show that stability problems for infinite-dimensional systems can be solved in linear systems.
Abstract: In this paper we are concerned with stability problems for infinite dimensional systems. First we review the theory for linear systems where the dynamics are governed by strongly continuous semigro...
159 citations
TL;DR: Boundary value problems in ODE’s arising in various applications are frequently not, in the “standard” form required by the currently existing software, however, many problems can be converted to such a form, thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software.
Abstract: Boundary value problems in ODE’s arising in various applications are frequently not, in the “standard” form required by the currently existing software. However, many problems can be converted to such a form, thus enabling the practitioner to take advantage of the availability and reliability of this general purpose software. Here, various conversion devices are surveyed.
128 citations
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TL;DR: In this article, the authors present a review of the achievements in subsonic and supersonic flow, for bluff bodies, for trailing-edge flows and for internal flows, and discuss the difficulties in unsteady and unstable flows.
Abstract: Since classical inviscid theory leads to the patently absurd conclusion that the resistance experienced by a rigid body moving through a fluid with uniform velocity is zero, great efforts have been made during the last hundred or so years to propose alternate theories and to explain how a vanishingly small frictional force in the fluid can nevertheless have a significant effect on the flow properties. The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction tends to zero. This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved. The lecture will review these achievements in subsonic and supersonic flow, for bluff bodies, for trailing-edge flows and for internal flows. Most of the work has been on steady two-dimensional problems but the special difficulties in unsteady and th...
108 citations
TL;DR: In this article, the numerical solution of the eigenvalue problem is studied, where T is a linear operator in a Bana,ch space and T may represent a bounded integral operator or a closed differential operator with bounded inverse.
Abstract: We are concerned with the numerical solution of the eigenvalue problem $T\varphi = \lambda \varphi ,\varphi
e 0$, where T is a linear operator in a Bana,ch space. T may represent a bounded integral operator or a closed differential operator (with bounded inverse). The linear operator T and its approximation $T_n $ are defined in the same space. Perturbation theory is then a suitable framework for our problem.We present in the first part a systematic study of the various notions of convergence $T_n \to T$, which imply the convergence of the eigenvalues with preservation of the multiplicities. The results are then applied to practical methods for integral and differential operators.In the second part, we present convergence rates. Analytic perturbation theory is used to refine on the computed eigenelements of an integral operator, and to produce localization results on the eigenelements. Finally superconvergence results are discussed, both for integral and differential operators.
TL;DR: The development of numerical solution techniques from the identification of a problem to the never-final preparation of automatic codes for the solution of classes of similar problems is examined.
Abstract: Two decades ago many people thought that there was very little left to occupy the researcher in the numerical initial value problem for ordinary differential equations; but there has since been an enormous increase in the effort devoted to this problem and related software. Is everything finally done so we can turn to more difficult areas? This paper examines the development of numerical solution techniques from the identification of a problem to the never-final preparation of automatic codes for the solution of classes of similar problems. It discusses a number of ODE problems that are only a part of the way along this path of development. These problems include ones with highly oscillatory solutions, ones with frequent discontinuities, and ones in which very little accuracy is needed (and which should be integrated at low cost). There is a spectrum of problems, from ones that can now be solved using very robust software to ones that we still do not know how to solve effectively. The paper then surveys s...
TL;DR: In this paper, the stochastic maximum principle is applied to solve eight different control problems with special structure and the main thrust of the work is to show that some completely observable control problems can be solved quite quickly and easily using the maximum principle.
Abstract: The stochastic maximum principle is applied to solve eight stochastic control problems. Four have been solved previously using much more complicated methods, but the last four are new. The main thrust of the work is to show that some completely observable stochastic control problems with special structure can be solved quite quickly and easily using the maximum principle.
TL;DR: In this paper, the authors review recent work on the stochastic approach of Walsh spectral analysis, focusing mainly on the properties of the finite Walsh transform, which in turn allow them to derive the spectral properties.
Abstract: In this paper we review recent work on the stochastic approach of Walsh spectral analysis. We are mainly interested in the properties of the finite Walsh transform, which in turn allow us to derive...
TL;DR: In this article, a survey of known equilibrium point theorems and proof techniques for non-cooperative n-person games in normal form is given. And various new $\varepsilon $-equilibrium point theoresms are derived, in which one of the players has a big strategy space and where non-bounded, payoff functions are allowed.
Abstract: This paper concentrates on the problem of the existence of $\varepsilon $-equilibrium points in the Nash sense for noncooperative n-person games in normal form. Firstly, a survey of known $\varepsilon $-equilibrium.point theorems and proof techniques is given. Then various new $\varepsilon $-equilibrium point theorems are derived, in which one of the players has a big strategy space and where nonbounded, payoff functions are allowed. To prove these theorems, results of Arzela–Ascoli are extended.
TL;DR: In this article, it was shown that the n-fold partial sums of powers of a complex matrix having spectral radius one or less can be approximated by polynomials, where the approximation is taken in the sense of Cesaro averages.
Abstract: It is shown that the n-fold partial sums of powers of a square (complex) matrix having spectral radius one or less can be approximated by (matrix) polynomials where the approximation is taken in the sense of Cesaro averages The degree of the approximating polynomials and the order of Cesaro averaging is characterized When the matrix is nonnegative the results are strengthened and it is shown that Cesaro limits can be replaced by regular limits of periodic subsequences The periods of these expansions are characterized We obtain the explicit (computable) form of all approximating polynomials, generalizing known results for stochastic and irreducible nonnegative matrices Several applications of the results are discussed
TL;DR: In this article, a differential equation for the position of a trailer relative to the cab of the vehicle pulling it is presented, and the solution is given for two examples and the results are generalized in a theorem.
Abstract: We present a differential equation which is a model for the position of a trailer relative to the cab which is pulling it. The solution is given for two examples, and the results are generalized in a theorem.
TL;DR: In this article, it was shown that the Stormer problem is nonintegrable and ergodic in a suitable region of phase-space, leading to a model for the complete flow.
Abstract: In this paper we present the new and important results obtained for the Stormer problem in the past decade, together with their implications for the existence of the Van Allen radiation belt. Specifically, we prove the existence of quasi-periodic solutions for both small and large values of energy, and we give a complete topological description of all orbits near the dipole singularity. This leads, in turn, to a model for the complete flow. This model, aside from its purely theoretic interest, indicates that the Stormer Problem is nonintegrable, and that the flow is ergodic in a suitable region of phase-space.
TL;DR: In this article, an approximate solution to the wave equation in a time-dependent domain which is moving slowly compared with the speed of the waves is found using the two timing method.
Abstract: In this note we find an approximate solution to the wave equation in a time-dependent domain which is moving slowly compared with the speed of the waves. The solution is found using the two timing method. The validity of the approximate solution is verified finding an appropriate energy inequality with the method of multipliers. Finally the problem is solved using an appropriate averaged Lagrangian.
TL;DR: A survey of results in the study of direct theorems in degree (or order) of "best approximation" of a function $f(x)$ by trigonometric polynomials is given in this paper.
Abstract: The purpose of this paper is to give a survey of results in the study of direct theorems in degree (or order) of “best approximation” $E_n (f)$, of a function $f(x)$, by trigonometric polynomials. We will normally require the function $f(x)$ to be $2\pi $-periodic and integrable in the Lebesgue sense. Further differentiability conditions may be imposed upon f, depending upon the theorems being discussed. Our main purpose is to study approximation by trigonometric series, Fourier series representation of a function and certain summability methods of the Fourier series representation.We will be basically interested in the quantitative aspects of the subject. For a more abstract treatment of the subject, see Shapiro [Topics in Approximation Theory, Springer-Verlag, New York, 1971]. A particularly elegant treatment of “degree of approximation”, up to about 1963, is to be found in Timan’s book [Theory of Approximation of Functions of a Real Variable, Pergamon/Macmillan New York, 1963].We will not study or disc...