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Counterexamples to theorems for zeros of solutions of second order linear differential equations

This paper investigates various aspects of the application of the Jordan canonical form of a matrix in system theory and develops a computational approach to determining the Jordan form for a given matrix. Applications include pole placement, controllability and observability studies, serving as an intermediate step in yielding other canonical forms, and theorem proving. The computational method developed in this paper is both simple and efficient. The method is based on the definition of a generalized eigenvector and a natural extension of Gauss elimination techniques. Examples are included for demonstration purposes.

This paper deals with observed processes in situations in which observations are available only when the state vector lies in certain regions. For linear autonomous observed processes, necessary and sufficient conditions are obtained for half-space observation regions. These results are shown to contain a theorem dual to a controllability result proved by the author for a linear autonomous control system whose control restraint set does not contain the origin as an interior point. Observability results relating to continuous observation systems and sampled data systems are presented, and an example of observing the state of an electrical network is given.

Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.

A series of research projects is briefly summarized which includes investigations in the following areas: (1) mathematical programming problems for large system and infinite-dimensional spaces, (2) bounded-input bounded-output stability, (3) non-parametric approximations, and (4) differential games. A list of reports and papers which were published over the ten year period of research is included.

This paper presents the best obtainable random coding and expurgated upper bounds on the probabilities of undetectable error, of t-order failure (advance to depth t into an incorrect subset), and of likelihood rise in the incorrect subset, applicable to sequential decoding when the metric bias G is arbitrary. Upper bounds on the Pareto exponent are also presented. The G-values optimizing each of the parameters of interest are determined, and are shown to lie in intervals that in general have nonzero widths. The G-optimal expurgated bound on undetectable error is shown to agree with that for maximum likelihood decoding of convolutional codes, and that on failure agrees with the block code expurgated bound. Included are curves evaluating the bounds for interesting choices of G and SNR for a binary-input quantized-output Gaussian additive noise channel.

This paper discusses the power spectral effects of spline interpolators. A general technique is given for finding the steady-state spectral effects of splines of all orders, when applied following uniform sampling of the input function. The following observations are made: (1) the even order splines that were examined (second and fourth order) possessed divergent steady-state frequency transfer functions, (2) the degree of preservation of the power spectral density of the input process increased with the order of the (odd order) spline used for interpolation, and (3) the reconstruction of a stationary random process over a finite record length will, on the average, have less power than indicated by the steady-state transfer function.

This paper presents a technique for the partial-fraction expansion of functions which are ratios of polynomials with real coefficients. The expansion coefficients are determined by writing the polynomials as Taylor's series and obtaining the Laurent series expansion of the function. The general formula for the inverse Laplace transform is also derived.

This paper examines the robustness of a Bayes estimator with respect to the assigned prior distribution. A Bayesian analysis for a stochastic scale parameter of a Weibull failure model is summarized in which the natural conjugate is assigned as the prior distribution of the random parameter. The sensitivity analysis is carried out by the Monte Carlo method in which, although an inverted gamma is the assigned prior, realizations are generated using distribution functions of varying shape. For several distributional forms and even for some fixed values of the parameter, simulated mean squared errors of Bayes and minimum variance unbiased estimators are determined and compared. Results indicate that the Bayes estimator remains squared-error superior and appears to be largely robust to the form of the assigned prior distribution.

In this paper, a class of nonlinear nonautonomous systems with multiple nonlinearities is considered. Sufficient conditions are developed for a type of practical stability with specified settling time. The conditions are independent of the actual form of nonlinear characteristics so that they can be interpreted as conditions for 'absolute' practical stability. The stability test is reduced to verification of the Hurwitz property of a constant matrix. This makes the stability analysis of the considered class of nonlinear systems convenient for machine computations. The proposed practical stability analysis is applied to a third-order system with several nonlinearities.

This paper considers discrete-time systems of the Lur'e-Postnikov class where the linear part is not asymptotically stable and the nonlinear characteristic satisfies only partially the usual sector condition. Estimates of the resulting finite regions of absolute ultimate boundedness are calculated by means of a quadratic Liapunov function.

The complementary variational principle developed in a Hilbert space setting provides a duality principle in the calculus of variations with dynamic constraints. This concept is adopted in this paper to investigate dual characterizations of optimal control systems. Systems under consideration include those with dynamics governed by linear ordinary differential equations, linear partial differential equations and non-linear ordinary differential equations.

The present paper takes the basic mathematical approach to analyze the stability of Riccati equations. First, a simple stability criterion like that of the linear constant coefficient system is sought for the nonlinear constant Riccati Equation. Then, a complete analysis on the limiting covariance P follows. The analysis is focussed on the main theorem. Finally, the computational aspects of the Riccati equations are analyzed with examples. The application of this analysis is to provide a simple stability criterion for the Kalman filter computation to test and predict filter convergence and thus modify filter parameters accordingly.

In this paper, we attempt to provide a simple taxonomy for incomplete-data problems and at the same time develop unified methods of analysis. The emphasis is on techniques which are natural extensions of the complete-data analysis and which will handle rather general classes of incomplete-data problems as opposed to custom-made techniques for special problems. The principle of estimation is either maximum likelihood or is at least based on maximum likelihood.

A technique to generate multidimensional random hypersurfaces with given statistical properties numerically on a digital computer is discussed in this paper. The technique is based on the linear filter theory. Using this technique, two-dimensional random surfaces are generated to illustrate the procedure. Statistics of these two-dimensional random surfaces are presented. Possible applications of these random surfaces in various physical problems are discussed.

Compilation of papers on applications of mathematics and computer programs for industry related problems

Transformation of control problems to ordinary differential equations

High order time derivatives of powers of radius vector and power series expansions for celestial mechanics

Restricted minima of functions of control, assuming Euclidean topological space

Perturbation of solutions of system of nonlinear differential equations under general linear boundary conditions

Multivariate functional models for least squares approximation of error tolerance

Mathematical control theory problems involving solutions of certain partial differential equations, nonadditive set functions, or other functionals - approximation and existence theorems

This paper presents some exact solutions of problems in random function theory for the purpose of testing the validity of an approximate method known variously in the many different fields of its application as first order smoothing theory, first order cumulant discard, quasilinear theory, or the adiabatic approximation. The hydromagnetic dynamo equations are used here, as particularly appropriate for such an investigation. The calculations show that in one case the exact and approximate solutions agree. In the other case the approximate solution is wrong. Hence, in the absence of a general criterion for validity, a result based on first order smoothing theory is a conjecture rather than a fact. This impacts strongly on much of the recent work on hydromagnetic dynamos.

This paper discusses how to numerically test a subroutine for the solution of ordinary differential equations. Results obtained with a variable order Adams method are given for eleven simple test cases.

This paper presents two new algorithms belonging to the family of dual methods of centers. The first can be used for solving fixed time optimal control problems with inequality constraints on the initial and terminal states. The second one can be used for solving fixed time optimal control problems with inequality constraints on the initial and terminal states and with affine instantaneous inequality constraints on the control. Convergence is established for both algorithms. Qualitative reasoning indicates that the rate of convergence is linear.

Linear programming and linear programming like techniques as applied to pattern recognition problems are discussed. Three relatively recent research articles on such applications are summarized. The main results of each paper are described, indicating the theoretical tools needed to obtain them. A synopsis of the author's comments is presented with regard to the applicability or non-applicability of his methods to particular problems, including computational results wherever given.

This paper deals with the rate of convergence of four methods of feasible directions: the Zoutendijk procedures 1 and 2 and two modifications of these procedures due to the authors. It is shown that of these methods, the two due to the authors converge linearly under convexity assumptions, that the Zoutendijk procedure 2 converges sublinearly under these assumptions, and that the Zoutendijk procedure 1 converges linearly provided the solution of the problem is a vertex of the constraint set.

In this paper a new approach to nonlinear system analysis based on the Fokker-Planck equation is developed. The development begins as a modification of the parametrix method of partial differential equation theory and provides a sequence solution to the multidimensional time-varying Fokker-Planck equation. This sequence is simplified for the case where only a steady-state solution or a solution for the variance of a particular combination of states is desired. Equations are presented in a form suitable for obtaining numerical results using a digital computer.

Many problems in analysis are concerned with the dependence upon parameters of fixed points of maps. For contraction mappings, criteria are relatively easy to obtain and have been known for some time. In the study of solutions of functional differential equations, more general results were needed. It is the purpose of this paper to give a rather general fixed-point theorem for condensing maps depending on a parameter, to prove continuous dependence and to indicate how many of the previous results are special cases.

Seven filter algorithms were presented in a recent survey paper (Kaminski, 1971), and were compared computationally (operations count) when relatively few observations were to be processed. These algorithms are now elaborated further. Details of the computations are presented, and it is shown that for problems with even moderately large amounts of data, the information matrix and square-root information matrix formulations are computationally more efficient than the other methods considered (conventional Kalman, stabilized Kalman, and square-root covariance mechanizations). It is pointed out that Schmidt's matrix factorization-Householder transformation technique leads to the same equations as those obtained via Potter's method. Several improvements in the equation mechanization are given.

A variety of techniques are available for estimating the states of nonlinear dynamic systems from noisy data. These procedures are generally equivalent when applied to linear systems. This paper investigates the difference between several of these procedures in the presence of small dynamic and observational nonlinearities. In particular, it examines one least square batch processing algorithm, and three recursive algorithms similar to the Kalman filter. To first order, all the estimators have the same covariance. Expressions for the means, however, show that each estimator has a different bias. The examples presented show that the biases can be a strong function of such parameters as initial covariances and number of data points being considered.

The progressive realization of the consequences of nonuniqueness imply an evolution of both the methods and the centers of interest in inverse problems. This evolution is schematically described together with the various mathematical methods used. A comparative description is given of inverse methods in scientific research, with examples taken from mathematics, quantum and classical physics, seismology, transport theory, radiative transfer, electromagnetic scattering, electrocardiology, etc. It is hoped that this paper will pave the way for an interdisciplinary study of inverse problems.

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

Note is made of an earlier paper which defined finite difference operators for the Hilbert space L2(m), and gave the eigenvalues for these operators. The present work examines eigenvalues for higher order singular differential operators by using finite difference methods. The two self-adjoint operators investigated are defined by a particular value in the same Hilbert space, L2(m), and are strictly positive with compact inverses. A class of finite difference operators is considered, with the idea of application to the theory of Toeplitz matrices. The approximating operators consist of a good approximation plus a perturbing operator.

Most of the known results concerning convergence of iterative methods for solving linear systems involve either positive definiteness or monotonicity. In this paper a new concept, called K-semipositivity, is introduced, which provides a link between convergence theory, monotonicity, and positive definiteness. By using this concept, together with partial orderings in Euclidean n-space, several new convergence theorems are proved. Application to Jacobi's methods and the theory of regular splittings shows the usefulness of these new results.