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Rational approximations to solution of second order nonlinear Ricatti equation, using linear transformations - differential equations and operational calculus
Linear transformation to obtain rational approximation to response of physical system described by nonlinear differential equation including Duffing equation as special case
Continued fraction rational approximations to solution of second-order nonlinear equation, including Ricatti equations treated by Merkes, Scott and Fair
Rational approximations to incomplete elliptic integral of first and second kinds derived by main diagonal Pade approximations
Rational approximations and solution of control theory problems involving Ricatti equations
Closed form rational approximations to Tricomi psi function
Computer method for determination of rational functions using sums of exponential approximations
Tabulation of high accuracy gamma function values for some rational arguments, noting applications to Gaussian quadrature formulas, fractional Bessel functions and Airy functions
Rational approximation for coefficients used in calculating hydrogenic photoionization Gaunt factors
Explicit inverse Fourier transformation of rational function with new theorem
Minimization techniques using rational approximations
Realization theory of rational transfer matrices presented as fundamental algebraic objects in theory of finite, constant linear systems
Noniterative method of obtaining rational forms as approximations to functions for power series expressions using Tchebycheff polynomial properties for improving accuracy
It is noted that linear systems, depending on parameters, can occur in diverse situations including families of rational solutions to the Korteweg-de Vries equation or to the finite Toda lattice. The inverse scattering method used by Moser (1975) to obtain canonical coordinates for the finite homogeneous Toda lattice can be used for the synthesis of RC networks. It is concluded that the multivariable RC setting is ideal for the analysis of the periodic Toda lattice.
For speed and data base reasons, solid geometric modeling of large complex practical systems is usually approximated by a polyhedra representation. Precise parametric surface and implicit algebraic modelers are available but it is not yet practical to model the same level of system complexity with these precise modelers. In response to this contrast the GEOMOD geometric modeling system was built so that a polyhedra abstraction of the geometry would be available for interactive modeling without losing the precise definition of the geometry. Part of the reason that polyhedra modelers are effective is that all bounded surfaces can be represented in a single canonical format (i.e., sets of planar polygons). This permits a very simple and compact data structure. Nonuniform rational B-splines are currently the best representation to describe a very large class of geometry precisely with one canonical format. The specific capabilities of the modeler are described.
Four-node axisymmetric solid elements are derived by a new version of hybrid method for which the assumed stresses are expressed in complete polynomials in natural coordinates. The stress equilibrium conditions are introduced through the use of additional displacements as Lagrange multipliers. A rational procedure is to choose the displacement terms such that the resulting strains are also of complete polynomials of the same order. Example problems all indicate that elements obtained by this procedure lead to better results in displacements and stresses than that by other finite elements.
Two algorithms for interpolating surfaces with spline functions containing tension parameters are discussed. Both algorithms are based on the tensor products of univariate rational spline functions. The simpler algorithm uses a single tension parameter for the entire surface. This algorithm is generalized to use separate tension parameters for each rectangular subregion. The new algorithm allows for local control of tension on the interpolating surface. Both algorithms are illustrated and the results are compared with the results of bicubic spline and bilinear interpolation of terrain elevation data.
Conditioned invariant subspaces are introduced both in terms of output injection and in terms of state estimation. Various properties of these subspaces are explored and the problem of disturbance decoupling by output injection (OIP) is defined. It is then shown that OIP is equivalent to the problem of disturbance decoupled estimation as introduced in Willems (1982) and Willems and Commault (1980). Both solvability conditions and a description of solutions for a class of rational matrix equations of the form X(s)M(s) = Q(s) on several ways are given in state-space form. Finally, the problem of output stabilization with respect to a disturbance is briefly addressed.