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Necessary and sufficient conditions on coefficients of real nonnegative polynomial pi/omega/, obtaining algebraic criterion for stability, optimality and passivity of dynamic linear systems
Algebraic criterion for positive realness of rational functions, using Routh algorithm
Automated algebraic manipulation in celestial mechanics, discussing use of Poisson series in perturbation theory problem
Analysis and synthesis of linear optical systems involving polarization effects by Pauli algebra of complex second order matrices
Finite Fourier transforms from finite dimensional algebraic viewpoint, deriving fast Fourier transform algorithm via tensor products induced matrix factorization
Genetic algorithms for function optimization employ genetic operators patterned after those observed in search strategies employed in natural adaptation. Two of these operators, crossover and inversion, are interpreted in terms of their algebraic and geometric properties. Stochastic models of the operators are developed which are employed in Monte Carlo simulations of their behavior.
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
FORTRAN subprograms for the solution of systems of linear algebraic equations are described, listed, and evaluated in this report. Procedures considered are direct solution, iteration, and matrix inversion. Both incore methods and those which utilize auxiliary data storage devices are considered. Some of the subroutines evaluated require the entire coefficient matrix to be in core, whereas others account for banding or sparceness of the system. General recommendations relative to equation solving are made, and on the basis of tests, specific subprograms are recommended.
It is shown that a particular bilinear model is both quite general and easy to work with. A basic structure theory is developed with the aid of previous results. Some preliminary ideas are discussed together with the system interconnection, the canonical form, questions of controllability, aspects of observability, and equivalent realizations. It is pointed out that in actually determining equivalent realizations for systems and in the classification of systems, the results available in the study of Lie algebras are of fundamental importance.
A purely algebraic algorithm is developed for testing positive real character of real rational functions and matrices relative to the unit circle in the complex plane. Since the algorithm is entirely recursive and is performed in finite number of steps, it is suitable for machine computations.
A set of FORTRAN callable subprograms are proposed which will be useful in the development of efficient portable ANSI FORTRAN subprograms and applications programs in the area of linear algebra.
Standing or traveling waves which vary algebraically with the axial distance in uniform ducts with sheared mean velocity profiles are investigated. The results show that such waves are not possible for ducts with uniform cross sections and fully developed mean flows.
The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.
An algorithm is presented for the symbolic integration of a class of algebraic functions. This class consists of functions made up of rational expressions of an integration variable x and square roots of polynomials, trigonometric and hyperbolic functions of x. The algorithm is shown to consist of the following components:(1) the reduction of input integrands to conical form; (2) intermediate internal representations of integrals; (3) classification of outputs; and (4) reduction and simplification of outputs to well-known functions.
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
A theory of Saturn's satellites Enceladus and Dione is discussed which is literal (all constants of integration appear explicitly), canonically invariant (the Hori-Lie method is used), and which correctly handles the eccentricity-type resonance between the two satellites. Algebraic manipulations are designed to be performed using the TRIGMAN formula manipulation language, and computer programs were developed so that, with minor modifications, they can be used on the Mimas-Tethys and Titan-Hyperion systems.
The orbits of the satellites of the outer planets are poorly known, due to lack of attention over the past half century. We have been developing a new theory of Saturn's satellites Enceladus and Dione which is literal (all constants of integration appear explicitly), canonically invariant (the Hori-Lie method is used), and which correctly handles the eccentricity-type resonance between the two satellites. The algebraic manipulations are being performed using the TRIGMAN formula manipulation language, and the programs have been developed so that with minor modifications they can be used on the Mimas-Tethys and Titan-Hyperion systems.