Showing papers in "Siam Review in 1978"
TL;DR: In this article, the exponential of a matrix could be computed in many ways, including approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial.
Abstract: In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...
1,849 citations
TL;DR: The current state of controllability and observability theories for linear PDEs is summarized in this article.However, the state of the art for observability and controllable PDE theories is limited.
Abstract: This paper is an assessment of the current state of controllability and observability theories for linear partial differential equations, summarizing existing results and indicating open problems i...
1,031 citations
TL;DR: In this paper, the authors compared the results obtained in optimal stochastic control with the results derived from a general Pontryagin principal and showed that various maximum principles are derived from the general principal.
Abstract: The purpose of this paper is to compare the results which have been recently obtained in optimal stochastic control. Various maximum principles are shown to derive from a general Pontryagin princip...
420 citations
TL;DR: A comprehensive survey of parallel techniques for problems in linear algebra is given, specific topics include: relevant computer models and their consequences for programs, evaluation of arithmetic expressions, solution of general and special linear systems of equations, and computation of eigenvalues.
Abstract: The existence of parallel and pipeline computers has inspired a new approach to algorithmic analysis. Classical numerical methods are generally unable to exploit multiple processors and powerful vector-oriented hardware. Efficient parallel algorithms can be created by reformulating familiar algorithms or by discovering new ones, and the results are often surprising. A comprehensive survey of parallel techniques for problems in linear algebra is given. Specific topics include: relevant computer models and their consequences for programs, evaluation of arithmetic expressions, solution of general and special linear systems of equations, and computation of eigenvalues.
338 citations
TL;DR: In this article, the authors describe several interconnections between the topics mentioned in the title and show how some previously known formulas for inverting Toeplitz operators in both discrete and contirected setting can be used to obtain the same result.
Abstract: We describe several interconnections between the topics mentioned in the title. In particular, we show how some previously known formulas for inverting Toeplitz operators in both discrete- and cont...
298 citations
TL;DR: A new theory is presented which includes features of most previous theories, but which goes beyond them in being capable of describing the results of initiation-promotion experiments.
Abstract: Various mathematical theories of the genesis of cancer are presented. They involve the transformation of a normal cell and the subsequent proliferation of the transformed cell and its descendants t...
181 citations
TL;DR: In this paper, the authors extend the affine case of a theorem of Cheney and Goldstein on proximity maps of convex sets to show that a generalization of the symmetrization technique of Powell always generates least change updates.
Abstract: In many problems involving the solution of a system of nonlinear equations, it is necessary to keep an approximation to the Jacobian matrix which is updated at each iteration. Computational experience indicates that the best updates are those that minimize some reasonable measure of the change to the current Jacobian approximation subject to the new approximation obeying a secant condition and perhaps some other approximation properties such as symmetry. In this paper we extend the affine case of a theorem of Cheney and Goldstein on proximity maps of convex sets to show that a generalization of the symmetrization technique of Powell always generates least change updates. This generalization has such broad applicability that we obtain an easy unified derivation of all the most successful updates. Furthermore, our techniques apply to interesting new cases such as when the secant condition might be inconsistent with some essential approximation property like sparsity. We also offer advice on how to choose the properties which are to be incorporated into the approximations and how to choose the measure of changes to be minimized.
180 citations
TL;DR: In a finite series of independent success-failure trials, the total number of successes has a binomial probability distribution as mentioned in this paper, and it is a classical result that this probability distribution is subje...
Abstract: In a finite series of independent success-failure trials, the total number of successes has a binomial probability distribution. It is a classical result that this probability distribution is subje...
87 citations
TL;DR: The authors give a description of the basic theorems of elementary catastrophe theory, along with heuristic explanations of why these theoremms are valid and why they can be used.
Abstract: This article is divided into two parts In the first we give a description of the basic theorems of elementary catastrophe theory, along with heuristic explanations of why these theorems are valid
81 citations
TL;DR: In this paper, the authors examine several problems involving a one-dimensional continuum with a free boundary and illuminate the related but far more complex problems which arise in the setting of a single-dimensional space.
Abstract: We examine several problems involving a one-dimensional continuum with a free boundary. Apart from their intrinsic interest, they illuminate the related but far more complex problems which arise in...
68 citations
TL;DR: In this article, a comprehensive treatment of the global qualitative behavior of buckled states of nonuniform, nonlinearly elastic rods under the action of compressive terminal thrusts is presented.
Abstract: This paper furnishes a comprehensive treatment of the global qualitative behavior of buckled states of nonuniform, nonlinearly elastic rods under the action of compressive terminal thrusts. These rods can suffer not only flexure as in the classical elastica theory, but also compression and shear. The governing equations are reduced to a quasilinear ordinary functional-differential equation, which reflects the nonlinearity of the constitutive equations. It is shown that the solution branches of the nonlinear problem inherit their nodal structure from that of the problem linearized about the trivial solution. This nodal structure, which distinguishes the different branches, may be quite complicated, owing to the breadth of physical response permitted. The qualitative behavior of the solutions is particularly sensitive to the nature of the shear response, to the form of the nonhomogeneity, and to the classification of boundary conditions as statically determinate or indeterminate. The analysis is based on mo...
TL;DR: In this article, a comparison method for the stability analysis of nonlinear parabolic equations and systems is described, which is based on comparison functions which are constructed from nonequilibri...
Abstract: A comparison method for the stability analysis of certain nonlinear parabolic equations and systems is described. The method is based on comparison functions which are constructed from nonequilibri...
TL;DR: The Jefferson method of apportionment as discussed by the authors is characterized by three properties; consistency, house monotonicity, and satisfying lower quota, and the method of smallest divisors is characterized similarly by substituting upper quota for lower quota.
Abstract: The Jefferson method of apportionment is characterized by three properties; consistency, house monotonicity, and satisfying lower quota. The method of smallest divisors is characterized similarly by substituting upper quota for lower quota.
TL;DR: In this article, the authors presented sixteen geometric equivalence classes for the integral curves of the associated scalar equation, and then derived conditions on the remaining nonzero coefficients that yield the geometrically equivalent classes for these integral curves.
Abstract: The two-dimensional quadratic differential system (QDS) \[ \begin{gathered} \dot x = a_1 x^2 + b_1 xy + c_1 y^2 , \hfill \\ \dot y = a_2 x^2 + b_2 xy + c_2 y^2 \hfill \\ \end{gathered} \] where $( \cdot ) = {d} / {dt}$ and the coefficients are real constants is considered. Lyagina presented sixteen geometric equivalence classes for the integral curves of the associated scalar equation ${{dy} / {dx}} = {{\dot y} / {\dot x}} $. The application of this classification scheme depends upon making affine transformations so that the linear integral curves through the origin of the resulting equation will lie either along the y axis, the x axis, or on $y = x$. This amounts to transforming the given equation so that certain coefficients are zero. Lyagina then derived conditions on the remaining nonzero coefficients that yield the geometric equivalence classes for the integral curves, exhausting all of the possibilities. This paper classifies the trajectories of a given QDS without first making such an affine transf...
TL;DR: In this article, interval mathematics techniques are used to verify sufficient conditions for existence, uniqueness, and convergence and to construct upper and lower bounds on solutions of nonlinear operator equations with rounding errors.
Abstract: We can compute with bounding sets of numbers, vectors, or functions using the techniques of interval mathematics. The techniques can be used to computationally verify sufficient conditions for existence, uniqueness, and convergence and to construct upper and lower bounds on solutions of nonlinear operator equations. Rounding errors are taken into account. We can compute with set-valued functions and operators on them. The techniques are also useful in search procedures for finding safe starting regions for iterative methods and for constructing natural stopping criteria.
TL;DR: In this article, a theory on M-matrices is derived from simple results on inverse-positive linear operators and some applications to eigenvalue theory and iterative procedures are discussed.
Abstract: A theory on M-matrices is derived from simple results on inverse-positive linear operators. Some applications to eigenvalue theory and iterative procedures are discussed. The concepts occurring in this theory can be generalized. In particular, M-operators are introduced, which share many properties with M-matrices. As special examples, certain boundary value problems are considered.
TL;DR: In this paper, the authors investigated several problems involving differential equations with small parameters, where a multiple-scales approach for finding approximate solutions fails, and the reasons for the failure were investigated.
Abstract: Several problems involving differential equations with small parameters are investigated, where a multiple-scales approach for finding approximate solutions fails. The reasons for the failure are a...
TL;DR: In this article, the first two moments of a random variable for the Cartesian plane were determined for a given constant p = 1,2, ∞, where p is the number of locations on a circular disc.
Abstract: Using the $l^p $ notion of distance in the Cartesian plane, and assuming a uniform density of locations on l circular disc, we consider the resulting distance to any specified point of this domain, and we determine the first two moments of this random variable for $p = 1,2,\infty $. We find the maxima and minima of these average distances and their ratios, hence show their almost exact proportionality over the disc. Situations motivating these results include traffic flow on a rectangular street grid in a circular city and physical design of certain computer systems in two dimensions.
TL;DR: In this essay some muddles caused by not discriminating between the two categories of numerical analysis are pointed out.
Abstract: Developments in numerical analysis fall into two separate categories. The first comprises work on problems which are unsolved in the sense that either no feasible methods are available or else there is no reliable analysis for the methods which are in use. The second category comprises work on solved problems and its aim is to remove the human user from the solution process, in so far as this is possible, and also to improve efficiency in regard to other factors such as execution time, storage requirements or length of code.Shortly after the introduction of modern digital computers and high level programming languages most of numerical analysis fell into the unsolved category. With every success in this category the second one has grown–and vice versa. In order to judge properly the value of the multifarious research activities in numerical analysis it is important to grasp the evolution of this sprawling empire.In this essay we point out some muddles caused by not discriminating between the two categorie...
TL;DR: In this paper, the authors trace the evolution of a mathematical model designed to clarify the relationship between the structure and the function of the cochlea (inner ear) is traced.
Abstract: The evolution of a mathematical model designed to clarify the relationship between the structure and the function of the cochlea (inner ear) is traced. Starting from physical (rather than empirical) considerations, the mathematical description of each model is derived in terms of well-defined physical properties of the cochlea. Initially, the basilar membrane is modeled as a uniform simply supported beam vibrating in a viscous medium, and driven by a concentrated oscillating moment at the basal end. In such a system traveling and standing waves (transients) occur simultaneously. An analysis of the traveling waves reveals a place principle. The model has high and low frequency “thresholds” (i.e., at very high and very low frequencies, frequency discrimination diminishes). The model suggests that the fluid properties play a particularly important role at high frequencies, while the role of cochlear geometry becomes dominant at low frequencies. In the second stage, the beam is enclosed in a rectangular “coch...
TL;DR: In this article, integral representations of several statistical distributions are exploited in a common structure to guarantee significant digit results over a wide range of parameters, where central distributions are treated implicitly by setting noncentrality parameters to zero.
Abstract: Integral representations of several statistical distributions are exploited in a common structure to guarantee significant digit results over wide ranges of parameters. Truncation errors are estimated from computed results and computational problems arising from extreme parameters are identified. Central distributions are treated implicitly by setting noncentrality parameters to zero. Bounds analogous to those for Mill’s ratio are also given for some nonnegative special functions.
TL;DR: Parallel Sturmian comparison theorems for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder, were proved in this article.
Abstract: Parallel Sturmian comparison theorems are proved for hyperbolic and elliptic linear differential equations of second order, with special emphasis on cylindrical domains with the time axis parallel to the axis of the cylinder. These theorems have the following form: If a boundary problem I for a partial differential equation in a bounded domain has a nontrivial solution, then every solution of a second boundary problem II of the same type has a zero in the domain if the coefficient functions and boundary functions of II majorize those of I. The results extend and unify earlier results given for either hyperbolic equations or elliptic equations. In particular, the majorization hypotheses in Theorem 3, in the form of inequalities between the data functions in the differential equations and in the boundary conditions, indicate parallel sets of sufficient conditions for the Sturmian conclusion in the elliptic and hyperbolic cases. Counterexamples are given to show that two known theorems in the elliptic case d...