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NASA NTRS2002

Elliptic Grid Generation of Spiral-Bevel Pinion Gear Typical of OH-58 Helicopter Transmission

This paper discusses the source term treatment in the numerical solution of elliptic partial differential equations for an interior grid generation problem in generalized curvilinear coordinates. The geometry considered is that of a planar cross-section of a generic spiral-bevel gear tooth typical of a pinion in the OH-58 helicopter transmission. The source terms used are appropriate for an interior grid domain where all the boundaries are prescribed via a combination of Dirichlet and Neumann boundary conditions. New constraints based on the Green's Theorem are derived which uniquely determine the coefficients in the source terms. These constraints are designed for boundary clustered grids where gradients in physical quantities need to be resolved adequately. However, it is seen that the present formulation works satisfactorily for mild clustering also. Thus, a fully automated elliptic grid generation technique is made possible where there is no need for a parametric study of these parameters since the new relations fix these free parameters uniquely.

Kaul, Upender K.↗

NASA NTRS1980

On the existence, uniqueness, and asymptotic normality of a consistent solution of the likelihood equations for nonidentically distributed observations: Applications to missing data problems

A general theorem is given which establishes the existence and uniqueness of a consistent solution of the likelihood equations given a sequence of independent random vectors whose distributions are not identical but have the same parameter set. In addition, it is shown that the consistent solution is a MLE and that it is asymptotically normal and efficient. Two applications are discussed: one in which independent observations of a normal random vector have missing components, and the other in which the parameters in a mixture from an exponential family are estimated using independent homogeneous sample blocks of different sizes.

Peters, C.↗

NASA NTRS1978

Forced oscillations in quadratically damped systems

Bayliss (1975) has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations. In the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear. The properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution. By using the Lipschitz continuity of the mapping and the contracting mapping principle, almost periodic solutions can be found for perturbations of the considered equation. Attention is also given to the Lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory.

Bayliss, A.↗

NASA NTRS2004

An Empirical Evaluation of Automated Theorem Provers in Software Certification

We describe a system for the automated certification of safety properties of NASA software. The system uses Hoare-style program verification technology to generate proof obligations which are then processed by an automated first-order theorem prover (ATP). We discuss the unique requirements this application places on the ATPs, focusing on automation, proof checking, and usability. For full automation, however, the obligations must be aggressively preprocessed and simplified, and we demonstrate how the individual simplification stages, which are implemented by rewriting, influence the ability of the ATPs to solve the proof tasks. Our results are based on 13 certification experiments that lead to more than 25,000 proof tasks which have each been attempted by Vampire, Spass, e-setheo, and Otter. The proofs found by Otter have been proof-checked by IVY.

Denney, Ewen↗

NASA NTRS1975

Almost periodic solutions to difference equations

The theory of Massera and Schaeffer relating the existence of unique almost periodic solutions of an inhomogeneous linear equation to an exponential dichotomy for the homogeneous equation was completely extended to discretizations by a strongly stable difference scheme. In addition it is shown that the almost periodic sequence solution will converge to the differential equation solution. The preceding theory was applied to a class of exponentially stable partial differential equations to which one can apply the Hille-Yoshida theorem. It is possible to prove the existence of unique almost periodic solutions of the inhomogeneous equation (which can be approximated by almost periodic sequences) which are the solutions to appropriate discretizations. Two methods of discretizations are discussed: the strongly stable scheme and the Lax-Wendroff scheme.

Bayliss, A.↗

NASA NTRS1993

Estimating the instabilities of N clocks by means of comparison measurements

The estimation of individual instabilities of N clocks, compared by measuring the differences of their readings, is considered without assuming a priori any hypotheses on their uncorrelation. Instabilities of the N clocks are described by a complete (non-diagonal) N x N covariance matrix R. Only differences of clock readings are available in order to estimate R. Statistical processing of these data allows one to calculate the (N-1)x(N-l) covariance matrix S of the differences relative to the N-th(reference) clock. By analyzing the relationships tying R and S, several pieces of information can be inferred and, in particular, the conditions for the validity of the uncorrelation hypothesis are established. The estimation of R from S is not unique: in any case R must be positive definite. A theorem states that R is positive definite if and only if its determinant is positive. Nevertheless infinitely many acceptable choices of R still fulfill the condition of positive definiteness. This paper shows that, by increasing the number N of compared clocks, the amount of arbitrariness in estimating R is reduced. The analysis of some experimental data illustrates the capability of the method.

Premoli, Amedeo↗

NASA NTRS2018

The Chaotic Long-Term X-Ray Variability of 4U 1705-44

The low-mass X-ray binary 4U1705-44 exhibits dramatic long-term X-ray time variability with a time-scale of several hundred days. The All-SkyMonitor (ASM) aboard the Rossi X-ray Timing Explorer (RXTE) and the Japanese Monitor of All-sky X-ray Image (MAXI) aboard the International Space Station together have continuously observed the source from 1995 December through 2014 May. The combined ASM-MAXI data provide a continuous time series over 50 times the length of the time-scale of interest. Topological analysis can help us identify 'fingerprints' in the phase space of a system unique to its equations of motion. The Birman-Williams theorem postulates that if such fingerprints are the same between two systems, then their equations of motion must be closely related. The phase-space embedding of the source light curve shows a strong resemblance to the double-welled non-linear Duffing oscillator. We explore a range of parameters for which the Duffing oscillator closely mirrors the time evolution of 4U1705-44. We extract low period, unstable periodic orbits from the 4U1705-44 and Duffing time series and compare their topological information. The Duffing and 4U1705-44 topological properties are identical, providing strong evidence that they share the same underlying template. This suggests that we can look to the Duffing equation to help guide the development of a physical model to describe the long-term X-ray variability of this and other similarly behaved X-ray binary systems.

Rossi X-ray Timing Explorer (RXTE)↗

NASA NTRS1978

Unified connected theory of few-body reaction mechanisms in N-body scattering theory

A unified treatment of different reaction mechanisms in nonrelativistic N-body scattering is presented. The theory is based on connected kernel integral equations that are expected to become compact for reasonable constraints on the potentials. The operators T/sub +-//sup ab/(A) are approximate transition operators that describe the scattering proceeding through an arbitrary reaction mechanism A. These operators are uniquely determined by a connected kernel equation and satisfy an optical theorem consistent with the choice of reaction mechanism. Connected kernel equations relating T/sub +-//sup ab/(A) to the full T/sub +-//sup ab/ allow correction of the approximate solutions for any ignored process to any order. This theory gives a unified treatment of all few-body reaction mechanisms with the same dynamic simplicity of a model calculation, but can include complicated reaction mechanisms involving overlapping configurations where it is difficult to formulate models.

Polyzou, W. N.↗

NASA NTRS1982

Sampling theory for asynoptic satellite observations. I Space-time spectra, resolution, and aliasing. II - Fast Fourier synoptic mapping

An evaluation of the information content of asynoptic data taken in the form of nadir sonde and limb scan observations is presented, and a one-to-one correspondence is established between the alias-free data and twice-daily synoptic maps. Attention is given to space and time limitations of sampling and the orbital geometry is discussed. The sampling pattern is demonstrated to determine unique space-time spectra at all wavenumbers and frequencies. Spectral resolution and aliasing are explored, while restrictions on sampling and information content are defined. It is noted that irregular sampling at high latitudes produces spurious contamination effects. An Asynoptic Sampling Theorem is thereby formulated, as is a Synoptic Retrieval Theorem, in the second part of the article. In the latter, a procedure is developed for retrieving the unique correspondence between the asymptotic data and the synoptic maps. Applications examples are provided using data from the Nimbus-6 satellite.

Salby, M. L.↗

NASA NTRS1987

The existence and stability of steady circulations in a conditionally symmetrically unstable basic flow

The existence of steady, nonlinear circulations in a flow susceptible to conditionally symmetric instability is studied, treating the latent heating as an energy source which is implicitly related to the motion field. The viscous nonlinear circulations of symmetrical instability are briefly discussed, and an existence theorem for steady, nonlinear symmetric circulations with bounded rates of latent heat release is given. The uniqueness and stability of these circulations are discussed, and some physical interpretations are given.

Xu, Qin↗

NASA NTRS1982

Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms

Various aspects of the convergence, uniqueness, and existence properties associated with solutions generated via the elliptically constrained incremental Newton-Raphson (ECINR) algorithm are analyzed. Several theorems are developed, and the formal behavior of the elliptically constrained scheme developed by Padovan (1981) is discussed in detail. Consideration is given to global and local rates of convergence, to the determination of the occurrence of safety zones wherein the algorithm yields inherently convergent results, to formal limitations on the class of functions which the scheme can be applied to solve, and to single and multidimensional formalisms on existence uniqueness and convergence. Special attention is given to functions whose Jacobian matrix exhibit positive, negative, semi and indefinite properties. Several significant advantages of ECINR over the classical INR are mentioned.

Padovan, J.↗

NASA NTRS2004

Using Automated Theorem Provers to Certify Auto-Generated Aerospace Software

We describe a system for the automated certification of safety properties of NASA software. The system uses Hoare-style program verification technology to generate proof obligations which are then processed by an automated first-order theorem prover (ATP). For full automation, however, the obligations must be aggressively preprocessed and simplified We describe the unique requirements this places on the ATP and demonstrate how the individual simplification stages, which are implemented by rewriting, influence the ability of the ATP to solve the proof tasks. Experiments on more than 25,000 tasks were carried out using Vampire, Spass, and e-setheo.

Denney, Ewen↗

NASA NTRS1972

A reciprocal theorem for a mixture theory

A dynamic reciprocal theorem for a linearized theory of interacting media is developed. The constituents of the mixture are a linear elastic solid and a linearly viscous fluid. In addition to Steel's field equations, boundary conditions and inequalities on the material constants that have been shown by Atkin, Chadwick and Steel to be sufficient to guarantee uniqueness of solution to initial-boundary value problems are used. The elements of the theory are given and two different boundary value problems are considered. The reciprocal theorem is derived with the aid of the Laplace transform and the divergence theorem and this section is concluded with a discussion of the special cases which arise when one of the constituents of the mixture is absent.

Martin, C. J.↗

NASA NTRS1985

Steady flows at the top of the core from geomagnetic field models - The steady motions theorem

It is demonstrated that the steady tangential velocity at the closed surface of a perfect-fluid conductor bounded by a rigid impenetrable exterior can be uniquely determined from knowledge of the normal component of the time-varying magnetic-flux density on the surface. In the context of a simple earth model consisting of an electrically insulating mantle surrounding a perfectly conducting core, the assumption of steady flow provides enough extra information to eliminate the toroidal ambiguity and to allow derivation of a unique global flow at the top of the core from a model of the geomagnetic field.

Voorhies, C. V.↗

NASA NTRS1973

The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary

Development of a complete theory of the two-dimensional Dirichlet problem for an open boundary. It is shown that the solution of the Dirichlet problem for an open boundary requires the solution of a Fredholm integral equation of the first kind. Although a Fredholm integral equation of the first kind usually has no solution if the kernel is continuous, owing to the logarithmic singularity of the kernel, the equation in this case is converted to a singular integral equation with a Cauchy kernel. It is proven that the homogeneous adjoint equation of the singular integral equation has no nonzero solution. By virtue of this result, and with the aid of an existence theorem known in the theory of singular integral equations, the existence of solutions of the singular integral equation, and then of the unique solution of the Fredholm integral equation of the first kind is proved.

Hayashi, Y.↗

NASA NTRS1977

Active flutter control using generalized unsteady aerodynamic theory

This paper describes the application of generalized unsteady aerodynamic theory to the problem of active flutter control. The controllability of flutter modes is investigated. It is shown that the response of aeroelastic systems is composed of a portion due to a rational transform and a portion due to a nonrational transform. The oscillatory response characteristic of flutter is due to the rational portion, and a theorem is given concerning the construction of a linear, finite-dimensional model of this portion of the system. The resulting rational model is unique and does not require state augmentation. Active flutter control designs using optimal regulator synthesis are presented.

Edwards, J. W.↗

NASA NTRS1988

Range data description based on multiple characteristics

An algorithm for describing range images based on Mean curvature (H) and Gaussian curvature (K) is presented. Range images are unique in that they directly approximate the physical surfaces of a real world 3-D scene. The curvature parameters are derived from the fundamental theorems of differential geometry and provides visible invariant pixel labels that can be used to characterize the scene. The sign of H and K can be used to classify each pixel into one of eight possible surface types. Due to the sensitivity of these parameters to noise the resulting HK-sing map does not directly identify surfaces in the range images and must be further processed. A region growing algorithm based on modeling the scene points with a Markov Random Field (MRF) of variable neighborhood size and edge models is suggested. This approach allows the integration of information from multiple characteristics in an efficient way. The performance of the proposed algorithm on a number of synthetic and real range images is discussed.

Al-Hujazi, Ezzet↗

NASA NTRS1984

Prospects for three-dimensional projective and tomographic imaging radar networks

Results of a study demonstrating the feasibility of broadband and speckle-free projective imaging of a complex shaped scattering object in the 6-17 GHz range with centimeter resolution are presented. It is shown how angular, spectral, and polarization diversities can be combined in the data acquisition process with a unique target-derived reference technique to access the three-dimensional Fourier space of the scatterer cost-effectively. Imagea retrieval algorithms, based on the projection slice theorem and knowledge of object symmetry, are utilized in obtaining images of a model aircraft with near optical resolution. The implications for high-resolution three-dimensional tomographic imaging radar networks are discussed.

Farhat, N. H.↗