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This paper presents an approximate method for obtaining truncated balance realizations of systems represented by non-rational transfer functions, that is infinite dimensional systems. It is based on the approximation to the Hankel operator.
We prove known identities for the Khinchin constant and develop new identities for the more general Hoelder mean limits of continued fractions. Any of these constants can be developed as a rapidly converging series involving values of the Riemann zeta function and rational coefficients. Such identities allow for efficient numerical evaluation of the relevant constants. We present free-parameter, optimizable versions of the identities, and report numerical results.
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.
Discussed is an algorithm for smoothing surfaces with spline functions containing tension parameters. The bivariate spline functions used are tensor products of univariate rational-spline functions. A distinct tension parameter corresponds to each rectangular strip defined by a pair of consecutive spline knots along either axis. Equations are derived for writing the bivariate rational spline in terms of functions and derivatives at the knots. Estimates of these values are obtained via weighted least squares subject to continuity constraints at the knots. The algorithm is illustrated on a set of terrain elevation data.
The variational equations of motion of large structures with rotating substructures are derived by the substructure synthesis approach, whereby a discretization procedure akin to the Rayleigh-Ritz method is used to represent the elastic motion of every substructure by a suitable set of admissible functions. Using an inclusion principle for gyroscopic systems, the effects on the predicted system dynamic characteristics of truncating the number of admissible functions used for each substructure can be assessed. Criteria for rational selection of admissible functions are presented.
The use of knowledge-base architecture and planning control; mechanisms to perform an intelligent monitoring task in the flight domain is addressed. The route level, the trajectory level, and parts of the aerodynamics level are demonstrated. Hierarchical planning and monitoring conceptual levels, functional-directed mechanism rationalization, and using deep-level mechanism models for diagnoses of dependent failures are discussed.
A very flexible substrate for printed wiring is disclosed which is composed of a blend of phenoxy resin-polyisocyanate-brominated epoxy resin in which the equivalent ration of the functional groups is hydroxyl grouped: isocyanate group: epoxy group = 1:0.2 to 2:0.5 to 3. The product has outstanding solder resistance and is applied to metal without using adhesives.
A simple procedure was developed and applied for the grid generation around an airplane geometry. This approach is based on a transfinite interpolation with Lagrangian interpolation for the blending functions. A monotonic rational quadratic spline interpolation was employed for the grid distributions.
A simple procedure has been developed and applied for the grid generation around an airplane geometry. This approach is based on a transfinite interpolation with Lagrangian interpolation for the blending functions. A monotonic rational quadratic spline interpolation has been employed for the grid distributions.
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.
An algorithm for weighted least-squares approximation with rational splines is presented. A rational spline is a cubic function containing a distinct tension parameter for each interval defined by two consecutive knots. For zero tension, the rational spline is identical to a cubic spline; for very large tension, the rational spline is a linear function. The approximation algorithm incorporates an algorithm which automatically adjusts the tension on each interval to fulfill a user-specified criterion. Finally, an example is presented comparing results of the rational spline with those of the cubic spline.
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.
Let F(z) be a vector-valued function, F: C yields C(sup N), which is analytic at z = 0 and meromorphic in a neighborhood of z = 0, and let its Maclaurin series be given. In this work we developed vector-valued rational approximation procedures for F(z) by applying vector extrapolation methods to the sequence of partial sums of its Maclaurin series. We analyzed some of the algebraic and analytic properties of the rational approximations thus obtained, and showed that they were akin to Pade approximations. In particular, we proved a Koenig type theorem concerning their poles and a de Montessus type theorem concerning their uniform convergence. We showed how optical approximations to multiple poles and to Laurent expansions about these poles can be constructed. Extensions of the procedures above and the accompanying theoretical results to functions defined in arbitrary linear spaces was also considered. One of the most interesting and immediate applications of the results of this work is to the matrix eigenvalue problem. In a forthcoming paper we exploited the developments of the present work to devise bona fide generalizations of the classical power method that are especially suitable for very large and sparse matrices. These generalizations can be used to approximate simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and invariant subspaces of arbitrary matrices which may or may not be diagonalizable, and are very closely related with known Krylov subspace methods.
A variational principle based on the integrodifferential form of the Boltzmann equation, which allows very general forms of the collision integral and of the boundary operator, has been used to compute the mass flux to a spherical particle. For simplicity the problem has been restricted to the one-speed, constant cross-section approximation, the black sphere problem of neutron transport theory. Using the Hilbert expansion of the solution far from the sphere as a trial function leads to a rational expression in the inverse Knudsen number for the extrapolation distance, which is simply related to the number flux to the sphere. A five-term expansion gives results that are no more than 6 percent in error when compared to an accurate numerical solution. The technique can be applied to other forms of the collision model and other boundary conditions.
Two distinct parametrization procedures of generating free-form surfaces to represent aerospace vehicles are presented. The first procedure is the representation using spline functions such as nonuniform rational b-splines (NURBS) and the second is a novel (geometrical) parametrization using solutions to a suitably chosen partial differential equation. The main idea is to develop a surface which is more versatile and can be used in an optimization process. Unstructured volume grid is generated by an advancing front algorithm and solutions obtained using an Euler solver. Grid sensitivity with respect to surface design parameters and aerodynamic sensitivity coefficients based on potential flow is obtained using an automatic differentiator precompiler software tool. Aerodynamic shape optimization of a complete aircraft with twenty four design variables is performed. High speed civil transport aircraft (HSCT) configurations are targeted to demonstrate the process.
Orifice-to-orifice inconsistencies in data acquired with an electronically-scanned pressure system at the beginning of a wind tunnel experiment forced modifications to the standard, instrument calibration procedures. These modifications included a large increase in the number of calibration points which would allow a critical examination of the calibration curve-fit process, and a subsequent post-test reduction of the pressure data. Evaluation of these data has resulted in an improved functional representation of the pressure-voltage signature for electronically-scanned pressures sensors, which can reduce the errors due to calibration curve fit to under 0.10 percent of reading compared to the manufacturer specified 0.10 percent of full scale. Application of the improved calibration function allows a more rational selection of the calibration set-point pressures. These pressures should be adjusted to achieve a voltage output which matches the physical shape of the pressure-voltage signature of the sensor. This process is conducted in lieu of the more traditional approach where a calibration pressure is specified and the resulting sensor voltage is recorded. The fifteen calibrations acquired over the two-week duration of the wind tunnel test were further used to perform a preliminary, statistical assessment of the variation in the calibration process. The results allowed the estimation of the bias uncertainty for a single instrument calibration; and, they form the precursor for more extensive and more controlled studies in the laboratory.
The paper presents a systematical analysis of the problem of the determination of the unsteady motion about an airfoil moving in an infinite fluid that contains a system of vortices and the determination of the hydrodynamical forces acting on the airfoil. The hydrodynamical problem is reduced to the determination of the function f (xi) which transforms conformally the external region of the airfoil into the interior of a circle. The proposed methods of determining the irrotational motion of a fluid that is produced by any motion of the airfoil are especially simple and effective if the function f (xi) is rational. As an example the flow is determined for the case of an arbitrary motion of an airfoil of the Joukowsky type. The formulas obtained for the determination of the hydrodynamical forces by means of contour integration are similar to those given by S. Chaplygin. These formulas are used to determine the force acting on the airfoil in the cases where the unsteady motion is potential throughout and the circulation about the airfoil is constant and also when the fluid contains a system of vortices. A full discussion is given of the concept of virtual masses together with practical formulas for computing the virtual mass coefficients.
Correlation functions play an essential role in modern data assimilation, where they are used to model covariances given a set of tunable parameters or applied as tapering functions to localize covariances in ensemble-based schemes. One of the most widely-used correlation functions in data assimilation is the Gaspari and Cohn (1999) piecewise-rational, compactly-supported parametric correlation function (hereafter referred to as GC99). The GC99 correlation function is useful due to its tunable cut-off parameter c and Gaussian-like shape achieved when the parameter a is set to one-half. These properties are attractive for tapering functions in data assimilation applications. However, the GC99 correlation function is homogeneous over Euclidean 3-space and isotropic when restricted to the sphere, properties that may be less than ideal for some geophysical applications. GC99 is also compactly-supported on a sphere of fixed radius, which requires tuning of the cut-off parameter c that can depend on the specific application. This work presents a generalization of the GC99 correlation function that allows the cut-off parameter c and shape parameter a to vary over space to gain more flexibility in shape while maintaining its compact support property. The function, which we call the Generalized Gaspari Cohn (GenGC) correlation function, introduces inhomogeneity in Euclidean 3-space and anisotropy when restricted to the sphere by allowing both parameters c and a to vary, as functions, over the spatial domain. The GC99 correlation function is a special case of GenGC where the functions c and a are held constant, as fixed parameters rather than functions. The GenGC correlation function also generalizes the follow-on to the work of Gaspari and Cohn (1999) presented in Gaspari et al. (2006), which allowed a to vary while keeping c fixed. We illustrate through simple one- and two-dimensional examples the variety of inhomogeneous and anisotropic correlation functions GenGC can produce by varying c and a over space, and suggest applications where they may be useful in data assimilation, such as covariance modeling or localization. In particular, we describe how the GenGC correlation function can be used to construct covariances using correlation length and variance fields derived from dynamics. For example, the correlation length field for advective dynamics is governed by a partial differential equation (PDE) in N spatial dimensions, where N is the number of space dimensions of the state. Correlation length fields can be determined from this PDE and used with GenGC to construct the corresponding correlations. We can then approximate the full covariance by rescaling by the variance, which also satisfies a PDE in N spatial dimensions for advective dynamics. Thus we can approximate the full covariance without solving the covariance PDE, which is in 2N spatial dimensions, by solving just two PDEs each in only N spatial dimensions. This approach to evolving the correlation length and variance fields, then reconstructing the correlations using GenGC, is suggested as an alternative to current methods of covariance modeling in data assimilation algorithms.