On some extremal positive definite functions.
Received radar signal power maximizing problems in planetary exploration, analyzing abelian group real line and circle problems
Garsia, A.↗
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Results for “positive definite functions”
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Asymptotic form of the longitudinal correlation function for isotropic homogeneous turbulence
An asymptotic form is derived for the longitudinal correlation function for isotropic homogeneous turbulence in an incompressible fluid governed by the Navier-Stokes equation. The result is obtained from an analysis of the algebraic-differential structure of the two-point correlation tensor contained in the complex-valued Fourier transform of the probability measure over the turbulence ensemble, which reveals that the Hopf characteristic functional (the complex-valued Fourier transform of the probability measure) satisfies the Fourier interference inequality. Consequences of the expression obtained, in which the correlation function is a positive definite function of the inverse cube of the spatial coordinate as it approaches infinity, are shown to include the nonexistence of the Loitsianskii invariant, and the solution is shown to be consistent with empirical formulas.
Rosen, G.↗
Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems
The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.
Straeter, T. A.↗
Generalized invariance principles and the theory of stability.
Description of some recent extensions of the invariance principle to more generalized dynamical systems where the state space is not locally compact and the flow is unique only in the forward direction of time. A sufficient condition for asymptotic stability of an invariant set is obtained which does not require that the Liapunov function be positive-definite. A recently developed generalized invariance principle is described which is applicable to functional differential equations, partial differential equations, and, in particular, to certain stability problems arising in thermoelasticity, viscoelasticity, and distributed nonlinear networks.
Lasalle, J. P.↗
A variable-metric algorithm employing linear and quadratic penalties
A variable-metric algorithm is described that uses both linear and quadratic penalty terms for handling nonlinear constraints. Quadratic penalty coefficients are adjusted in a process which maintains a positive-definite matrix of second partial derivatives of the function without generating the large positive eigenvalues which cause zigzagging and slow convergence. The schemes suggested use inferred second-order properties not only in terms of the variable metric of the Davidson-Fletcher-Powell algorithm (or its relatives) but by estimating of second directional derivatives by fitting cubics to various functions along search directions.
Kelley, H. J.↗
Positive definiteness of gravitational field energy.
Total gravitation field energy functional with one extremum
Brill, D.↗
Digital modeling of human operator dynamics via class of Liapunov functions
A technique is described which is utilized for modeling human operator dynamics. The technique is based upon a model-reference system configuration and a class of Liapunov functions formulated for this purpose which possess variable characteristics. It is shown how such a class of Liapunov functions and their time derivatives is formulated. The crux of the formulation lies in three variable positive definite matrices used for the construction of the class of Liapunov functions. The form and order of these matrices depend upon the form and order of the error differential equation of the human operator model-reference system. These matrices can be modified to include nonlinear functions of the human operator model.
Mekel, R.↗
Algebraic criteria for positive realness relative to the unit circle.
A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.
Siljak, D. D.↗
On the power output of some idealized source configurations with one or more characteristic dimensions
The calculation of power output from a (finite) linear array of equidistant point sources is investigated with allowance for a relative phase shift and particular focus on the circumstances of small/large individual source separation. A key role is played by the estimates found for a twin parameter definite integral that involves the Fejer kernel functions, where N denotes a (positive) integer; these results also permit a quantitative accounting of energy partition between the principal and secondary lobes of the array pattern. Continuously distributed sources along a finite line segment or an open ended circular cylindrical shell are considered, and estimates for the relatively lower output in the latter configuration are made explicit when the shell radius is small compared to the wave length. A systematic reduction of diverse integrals which characterize the energy output from specific line and strip sources is investigated.
Levine, H.↗
Life sciences passive GN2 freezer thermal performance test
Thermal performance tests that were conducted on the life sciences passive GN2 freezer project are summarized as well as the improvements to the freezers to improve the thermal performance of the containers. Procedures were developed, based upon these tests, to initially charge the freezers with LN2 and verify that the freezer performance is adequate for the mission duration. Improvements were made to the corvac sample tube to limit the amount of breakage due to thermal expansion of the liquid during freezing. A method of verifying the freezer vacuum insulative integrity was defined as well as a procedure for refurbishment of the internal vacuum level. Freezer modifications were made to ease the reevacuation of the containers. The orientation of the freezer in a 1-G environment, after being charged, had to remain in a vertical position. The LN2 boiloff rate increased significantly in a horizontal position. This resulted in a stowage definition in the spacecraft prior to launch. Functional testing, using the SL-1 mission timeline showed that the freezer will maintain samples in the frozen state for the duration of the mission.
Belshaw, G. W.↗
Useful extremum principle for the variational calculation of matrix elements. II
Recent work (Gerjuoy et al., 1974) on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Gerjuoy et al. (1974) originally were based.
Gerjuoy, E.↗
A split band-Cholesky equation solving strategy for finite element analysis of transient field problems
The paper describes the split-Cholesky strategy for banded matrices arising from the large systems of equations in certain fluid mechanics problems. The basic idea is that for a banded matrix the computation can be carried out in pieces, with only a small portion of the matrix residing in core. Mesh considerations are discussed by demonstrating the manner in which the assembly of finite element equations proceeds for linear trial functions on a triangular mesh. The FORTRAN code which implements the out-of-core decomposition strategy for banded symmetric positive definite matrices (mass matrices) of a coupled initial value problem is given.
Cooke, C. H.↗
The IRAS radiation environment
Orbital flux integration for three selected mission altitudes and geographic instantaneous flux-mapping for nominal flight-path altitude were used to determine the external charged particle radiation predicted for the Infrared Astronomy Satellite. A current field model was used for magnetic field definitions for three nominal circular trajectories and for the geographic mapping positions. Innovative analysis features introduced include (1) positional fluxes as a function of time and energy for the most severe pass through the South Atlantic Anomaly; (2) total positional doses as a function of time and shield thickness; (3) comparison mapping fluxes for ratios of positional intensities to orbit integrated averages; and (4) statistical exposure-time history of a trajectory as a function of energy indicating, in percent of total mission duration, the time intervals over which the instantaneous fluxes would exceed the orbit integrated averages. Results are presented in tables and graphs.
Stassinopoulos, E. G.↗
Vortex information display system program description manual
A vortex information display system is described which provides flexible control through system-user interaction for collecting wing-tip-trailing vortex data, processing this data in real time, displaying the processed data, storing raw data on magnetic tape, and post processing raw data. The data is received from two asynchronous laser Doppler velocimeters (LDV's) and includes position, velocity, and intensity information. The raw data is written onto magnetic tape for permanent storage and is also processed in real time to locate vortices and plot their positions as a function of time. The interactive capability enables the user to make real time adjustments in processing data and provides a better definition of vortex behavior. Displaying the vortex information in real time produces a feedback capability to the LDV system operator allowing adjustments to be made in the collection of raw data. Both raw data and processing can be continually upgraded during flyby testing to improve vortex behavior studies. The post-analysis capability permits the analyst to perform in-depth studies of test data and to modify vortex behavior models to improve transport predictions.
Conway, R.↗
A BBGKY framework for fluid turbulence
A framework is presented for a systematic kinetic theory of turbulence originating from the Liouville equation for the Fourier coefficients of fluid variables. The real and imaginary parts of these Fourier coefficients play the role that particle coordinates (positions and momenta) play in the BBGKY theory. The basic relations of the problem are the incompressible Navier-Stokes equations in two dimensions with zero viscosity, with the probability distributions of Fourier coefficients rather than moments being the basic variables of the theory. A kinetic equation is derived and shown to possess a number of requirements that any reasonable kinetic equation must have: conservation laws, positive-definite spectral densities, and an H-theorem. The major lack in the theory is any reliable information on the relaxation predicted by the complicated linear operator H. Closure of the hierarchy is achieved by the hypothesis that the five-coefficient correlation function is negligible. Problems associated with inclusion of viscosity and external driving forces are discussed.
Montgomery, D.↗
Free vibration analysis of spinning structural systems.
This article presents an efficient digital computer procedure, along with the complete listing of the associated computer program, which may be conveniently utilized for the accurate solution of a wide range of practical eigenvalue problems. Important applications of the present work are envisaged in the natural frequency analysis of spinning structures discretized by the finite element technique, and in the determination of transfer functions associated with the dynamic blocks of control systems of spacecraft utilizing gas jets or reaction wheels for attitude control, as well as of spin-stabilized and dual-spin-stabilized satellites. The validity of the Sturm sequence property is first established for the related matrix formulation involving Hermitian and real symmetric, positive-definite matrices, both being usually of highly banded configuration. A numerically stable algorithm based on the Sturm sequence method is then developed which fully exploits the banded form of the associated matrices.
Gupta, K. K.↗
The terminal area automated path generation problem
The automated terminal area path generation problem in the advanced Air Traffic Control System (ATC), has been studied. Definitions, input, output and the interrelationships with other ATC functions have been discussed. Alternatives in modeling the problem have been identified. Problem formulations and solution techniques are presented. In particular, the solution of a minimum effort path stretching problem (path generation on a given schedule) has been carried out using the Newton-Raphson trajectory optimization method. Discussions are presented on the effect of different delivery time, aircraft entry position, initial guess on the boundary conditions, etc. Recommendations are made on real-world implementations.
Hsin, C.-C.↗
Definition study of land/sea civil user navigational location monitoring systems for NAVSTAR GPS: User requirements and systems concepts
A low-cost GPS civil-user mobile terminal whose purchase cost is substantially an order of magnitude less than estimates for the military counterpart is considered with focus on ground station requirements for position monitoring of civil users requiring this capability and the civil user navigation and location-monitoring requirements. Existing survey literature was examined to ascertain the potential users of a low-cost NAVSTAR receiver and to estimate their number, function, and accuracy requirements. System concepts are defined for low cost user equipments for in-situ navigation and the retransmission of low data rate positioning data via a geostationary satellite to a central computing facility.
Devito, D. M.↗