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NASA NTRS1979

On certain families of rational functions arising in dynamics

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.

Byrnes, C. I.↗

NASA NTRS1984

A geometric modeler based on a dual-geometry representation polyhedra and rational b-splines

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.

Klosterman, A. L.↗

NASA NTRS1985

Axisymmetric solid elements by a rational hybrid stress method

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.

Tian, Z.↗

NASA NTRS1986

The algorithms for rational spline interpolation of surfaces

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.

Schiess, J. R.↗

NASA NTRS1986

Conditioned invariant subspaces, disturbance decoupling and solutions of rational matrix equations

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.

Li, Z.↗

NASA NTRS1985

Rationalization of some genetic anticodonic assignments

The hydrophobicity of most amino acids correlates well with that of their anticodon nucleotides, with Trp, Tyr, Ile, and Ser being the exceptions to this rule. Using previous data on hydrophobicity and binding constants, and new data on rates of esterification of polyadenylic acid with several N-acetylaminoacyl imidazolides, several of the anticodon assignments are rationalized. Chemical reasons are shown supporting the idea of the inclusion of the Ile in the catalog of biological amino acids late in the evolution, through a mutation of the existing tRNA and its aminoacyl-tRNA-synthetase. It was found that an addition of hexane increases the incorporation of hydrophobic Ac-Phe into poly-A, in support of the Fox (1965) and Oparin (1965) emphasis on the biogenetic importance of phase-separated systems.

Lacey, J. C., Jr.↗

NASA NTRS1987

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗

NASA NTRS1988

Rational-Spline Subroutines

Smooth curves drawn among plotted data easily. Rational-Spline Approximation with Automatic Tension Adjustment algorithm leads to flexible, smooth representation of experimental data. "Tension" denotes mathematical analog of mechanical tension in spline or other mechanical curve-fitting tool, and "spline" as denotes mathematical generalization of tool. Program differs from usual spline under tension, allows user to specify different values of tension between adjacent pairs of knots rather than constant tension over entire range of data. Subroutines use automatic adjustment scheme that varies tension parameter for each interval until maximum deviation of spline from line joining knots less than or equal to amount specified by user. Procedure frees user from drudgery of adjusting individual tension parameters while still giving control over local behavior of spline.

Schiess, James R.↗

NASA NTRS1988

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

Tiffany, Sherwood H.↗

NASA NTRS1986

Rational's experience using Ada for very large systems

The experience using the Rational Environment has confirmed the advantages forseen when the project was started. Interactive syntatic and semantic information makes a tremendous difference in the ease of constructing programs and making changes to them. The ability to follow semantic references makes it easier to understand exisiting programs and the impact of changes. The integrated debugger makes it much easier to find bugs and test fixes quickly. Taken together, these facilites have helped greatly in reducing the impact of ongoing maintenance of the ability to produce a new code. Similar improvements are anticipated as the same level of integration and interactivity are achieved for configuration management and version control. The environment has also proven useful in introducing personnel to the project and existing personnel to new parts of the system. Personnel benefit from the assistance with syntax and semantics; everyone benefits from the ability to traverse and understand the structure of unfamiliar software. It is often possible for someone completely unfamiliar with a body of code to use these facilities, to understand it well enough to successfully with a body of code to use these facilities to understand it well enough to successfully diagnose and fix bugs in a matter of minutes.

Archer, James E., Jr.↗

NASA NTRS1990

Agenda toward the development of a rational noise descriptor system relevant to human annoyance by en route aircraft noise

A rational, internationally consistent, noise descriptor system is needed to express existing and predicted en route aircraft noise levels in terms closely correlated to the annoyance perceived by people and physiologically identifiable in people, to provide guidance for aircraft and powerplant design, flight management, land-use planning, and building codes. Expanding on previous discussions, a new comprehensive statement of the specific questions that must be resolved by needed research, and the nature and quality of proof that must be adduced to justify further steps toward the drafting and adoption of new international en route aircraft-noise standards is sought. The single noise-descriptor system envisioned must be valid for widely varying aircraft-noise frequency spectra, including time-variant components and agreeable and disagreeable discrete tones and combinations of tones. The measures and criteria established by the system must be valid at high and low immission levels, at high and low ambient noise levels, for great and small number of noise events, and outdoors and indoors.

Garbell, Maurice A.↗

NASA NTRS1992

Algebraic grid adaptation method using non-uniform rational B-spline surface modeling

An algebraic adaptive grid system based on equidistribution law and utilized by the Non-Uniform Rational B-Spline (NURBS) surface for redistribution is presented. A weight function, utilizing a properly weighted boolean sum of various flow field characteristics is developed. Computational examples are presented to demonstrate the success of this technique.

Yang, Jiann-Cherng↗

NASA NTRS1992

Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods

Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.

Sidi, Avram↗

NASA NTRS1992

Data reduction using cubic rational B-splines

A geometric method is proposed for fitting rational cubic B-spline curves to data that represent smooth curves including intersection or silhouette lines. The algorithm is based on the convex hull and the variation diminishing properties of Bezier/B-spline curves. The algorithm has the following structure: it tries to fit one Bezier segment to the entire data set and if it is impossible it subdivides the data set and reconsiders the subset. After accepting the subset the algorithm tries to find the longest run of points within a tolerance and then approximates this set with a Bezier cubic segment. The algorithm uses this procedure repeatedly to the rest of the data points until all points are fitted. It is concluded that the algorithm delivers fitting curves which approximate the data with high accuracy even in cases with large tolerances.

Chou, Jin J.↗

NASA NTRS1993

A rational interpolation method to compute frequency response

A rational interpolation method for approximating a frequency response is presented. The method is based on a product formulation of finite differences, thereby avoiding the numerical problems incurred by near-equal-valued subtraction. Also, resonant pole and zero cancellation schemes are developed that increase the accuracy and efficiency of the interpolation method. Selection techniques of interpolation points are also discussed.

Kenney, Charles↗

NASA NTRS1993

Rational trigonometric approximations using Fourier series partial sums

A class of approximations (S(sub N,M)) to a periodic function f which uses the ideas of Pade, or rational function, approximations based on the Fourier series representation of f, rather than on the Taylor series representation of f, is introduced and studied. Each approximation S(sub N,M) is the quotient of a trigonometric polynomial of degree N and a trigonometric polynomial of degree M. The coefficients in these polynomials are determined by requiring that an appropriate number of the Fourier coefficients of S(sub N,M) agree with those of f. Explicit expressions are derived for these coefficients in terms of the Fourier coefficients of f. It is proven that these 'Fourier-Pade' approximations converge point-wise to (f(x(exp +))+f(x(exp -)))/2 more rapidly (in some cases by a factor of 1/k(exp 2M)) than the Fourier series partial sums on which they are based. The approximations are illustrated by several examples and an application to the solution of an initial, boundary value problem for the simple heat equation is presented.

Geer, James F.↗

NASA NTRS1994

Crystals of Human Serum Albumin for Use in Genetic Engineering and Rational Drug Design

This invention pertains to crystals of serum albumin and processes for growing them. The purpose of the invention is to provide crystals of serum albumin which can be studied to determine binding sites for drugs. Form 2 crystals grow in the monoclinic space P2(sub 1), and possesses the following unit cell constraints: a = 58.9 +/- 7, b = 88.3 +/- 7, c = 60.7 +/- 7, Beta = 101.0 +/- 2 degrees. One advantage of the invention is that it will allow rational drug design

Carter, Daniel C.↗

NASA NTRS1994

Inventing and improving ribozyme function: rational design versus iterative selection methods

Two major strategies for generating novel biological catalysts exist. One relies on our knowledge of biopolymer structure and function to aid in the 'rational design' of new enzymes. The other, often called 'irrational design', aims to generate new catalysts, in the absence of detailed physicochemical knowledge, by using selection methods to search a library of molecules for functional variants. Both strategies have been applied, with considerable success, to the remodeling of existing ribozymes and the development of ribozymes with novel catalytic function. The two strategies are by no means mutually exclusive, and are best applied in a complementary fashion to obtain ribozymes with the desired catalytic properties.

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