Showing papers in "Siam Review in 1987"

On the mathematical foundations of nondifferentiable optimization in engineering design

Abstract: It is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions. Such problems are commonly referred to as semi-infinite optimization problems. The bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms, related to phase I - phase II methods of feasible directions, which solve these semi-infinite optimization problems. The applicability of the theory is illustrated with examples that are relevant to engineering design.

Extrapolation methods for vector sequences

Abstract: This paper derives, describes, and compares five extrapolation methods for accelerating convergence of vector sequences or transforming divergent vector sequences to convergent ones. These methods are the scalar epsilon algorithm (SEA), vector epsilon algorithm (VEA), topological epsilon algorithm (TEA), minimal polynomial extrapolation (MPE), and reduced rank extrapolation (RRE). MPE and RRE are first derived and proven to give the exact solution for the right 'essential degree' k. Then, Brezinski's (1975) generalization of the Shanks-Schmidt transform is presented; the generalized form leads from systems of equations to TEA. The necessary connections are then made with SEA and VEA. The algorithms are extended to the nonlinear case by cycling, the error analysis for MPE and VEA is sketched, and the theoretical support for quadratic convergence is discussed. Strategies for practical implementation of the methods are considered.

Inverse scattering for discrete transmission—line models

Abstract: This paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform, discrete transmission-line from its response to a given forcing function. This problem is the prototype for a wealth of one-dimensional inverse scattering problems arising in various fields. A unified, straightforward approach is presented, providing all known inversion procedures and some novel ones. The derivations readily follow from causality of signal propagation on the transmission-line. It is shown that a direct exploitation of the signal propagation model leads to recursive, computationally efficient and reliable inversion algorithms. In particular, the relationships between layer peeling (Schur type, difference equation) and layer-adjoining (Levinson type, integral equation) algorithms are clearly displayed.

A survey of condition number estimation for triangular matrices

Abstract: We survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix, and make recommendations concerning the use of the estimates in applications. Each of the methods is shown to bound the condition number; the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms. For each bound we examine by how much, at worst, it can overestimate or underestimate the condition number. Numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators.

Can one hear the shape of a drum? revisited

Abstract: In a landmark paper, Mark Kac in 1966 [Amer. Math. Monthly, 73, pp. 1–23] showed that geometric properties of regions in $R^2 $ can be obtained by studying the asymptotic properties of the spectrum of the Laplacian. He also conjectured that the shape of a region might be completely determined by the spectrum. We describe recent developments in this field and discuss the status of the Kac conjecture as well as similar conjectures for operators other than the Laplacian.

Perturbation techniques for oscillatory systems with slowly varying coefficients

Abstract: This is an expository paper on perturbation techniques and results for a general system of first order equations that model various weakly nonlinear oscillatory motions with slowly varying parameters. Sample problems from three application areas are derived and then transformed to a standard form. Asymptotic solutions and adiabatic invariants are calculated using the method of near identity averaging transformations. Equivalent results are also derived using the method of multiple scales. When certain divisors in the solution tend to zero, one can isolate the resonant behavior in a reduced problem of lower order. The solution of this reduced problem is discussed in detail for the case of transient resonance, and qualitatively for the case of sustained resonance.

Random set theory and problems of modeling

Abstract: The three- or four-dimensional world in which we live is full of objects to be measured and summarized. Very often a parsimonious finite collection of measurements is enough for scientific investigation into an object’s genesis and evolution. There is a growing need, however, to describe and model objects through their form as well as their size. The purpose of this article is to show the potentials and limitations of a probabilistic and statistical approach. Collections of objects (the data) are assimilated to a random set (the model), whose parameters provide description and/or explanation.

Decoupling and order reduction via the Riccati transformation

Abstract: A program that dates back to Riccati [1724] and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use ...

Stability analysis of finite-difference, pseudospectral and Fourier-Galerkin approximations for time-dependent problems

Abstract: We consider finite-difference, pseudospectral and Fourier—Galerkin methods for the approximate solution of time-dependent problems. The paper provides a unified framework for the stability analysis of all three discrete methods. In particular, the problem of stability for highly accurate stencils is studied in some detail.

Modeling and simulation in agricultural pest management

Abstract: This paper introduces some of the mathematical problems associated with the control of agricultural insect pests. The view advocated here is that since agricultural crops are managed biological systems, much of the applied mathematics developed for biological systems may be used in pest control. The problem is broken into three components (1) strategy selection, (2) tactics selection, and (3) state estimation. The concept of strategy selection is illustrated through a discussion of the Sterile Insect Technique (SIT) as a means of population suppression. Tactics selection is illustrated by a discussion of the scheduling of pesticide applications. The concept of state estimation is illustrated through two examples. The first example is a discussion of sequential sampling for pests to determine if the pest density has crossed an economic threshold. The second example is a discussion of information provided by trapping for pests when trying to determine the extent of an infestation.

Harmonic Analysis on Symmetric Spaces and Applications II (Audrey Terras)

Abstract: harmonic analysis on symmetric spaces and applications ii. citeseerx citation query harmonic analysis and. harmonic analysis on symmetric spaces rakuten kobo. harmonic analysis on symmetric spaces and applications i ??. symmetries and laplacians introduction to harmonic. harmonic analysis on locally symmetric spaces and number. causal symmetric spaces sciencedirect. citeseerx weyl group invariants and application to. mutative and nonmutative harmonic analysis and. pdf weyl group invariants and application to spherical. harmonic analysis on symmetric spaces and applications ii. harmonic analysis on symmetric spaces and applications ii. a selberg harmonic analysis and discontinuous groups in. harmonic analysis on symmetric spaces euclidean space the. harmonic analysis on symmetric spaces rakuten kobo. 9780387961590 harmonic analysis on symmetric spaces and. harmonic analysis on symmetric spaces higher rank. harmonic analysis on symmetric spaces and applications i. harmonic analysis. the harmonic analysis and representation theory group. weakly symmetric space. harmonic analysis and special functions on symmetric spaces. n j wildberger school of mathematics unsw sydney 2052. harmonic analysis and functional analysis chalmers. audrey terras. geometry of pactifications of locally symmetric spaces. terras audrey ucsd mathematics home. overview of harmonic analysis and representation theory. topics in harmonic analysis on combinatorial graphs. harmonic analysis on symmetric spaces euclidean space the. harmonic analysis in rigidity theory. harmonic analysis on symmetric spaces euclidean space. garrett review audrey terras harmonic analysis on. lie theory harmonic analysis on symmetric spaces general. symmetric space. harmonic analysis on symmetric spaces and applications i. harmonic analysis on symmetric spaces higher rank spaces. harmonic analysis on symmetric spaces and applications i. representation theory and harmonic analysis on symmetric. symmetries and laplacians introduction to harmonic. harmonic analysis on symmetric spaces and applications i. harmonic analysis on symmetric spaces and applications ii. harmonic analysis on symmetric spaces and applications i. harmonic analysis on symmetric spaces and applications. harmonic analysis on symmetric spaces max planck. harmonic analysis on symmetric spaces and applications i. upper half plane space the sphere spaces euclidean. harmonic analysis on homogeneous spaces

The Orr-sommerfeld equation on infinite intervals

Abstract: Numerical and asymptotic methods have been developed the last several years to solve the Orr–Sommerfeld (OS) stability equation for boundary layers and other unbounded domain flows. These are surveyed and some of the abstract mathematical questions they raise are mentioned.There has been a conventional wisdom among fluid dynamicists that it has not been “proved” that the eigenvalues of the OS equations are complete on an unbounded domain. There is something to this, since without the continuous spectrum, which occurs for boundary layer flows, the eigenvalues are not complete. However, much has been proved concerning the nature of the spectrum and the expansion properties of the generalized eigenfunctions. The results of Miklavcic and Williams, and of Bakenko and Herron, form the core of the survey.

The carleman type singular integral equations

Abstract: We present a consolidated account of Carleman type singular integral equations. We have started from the most basic equation of this type and have at first attempted to integrate from the existing literature the progress which has been made so far in this field. Many different techniques are presented to solve these types of integral equations both in the real and the complex domains. In the complex domain both closed and open contours for integrals are considered. In the process of this study we find that many known methods need to be polished and extended. We attempt to accomplish this and then interrelate different concepts and solutions. Thereby we obtain solutions of more general singular integral equations of the Carleman type. Many interesting examples are presented to illustrate the general theory.

Note on the Factorization of a Square Matrix into Two Hermitian or Symmetric Matrices

Abstract: This paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices.