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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1990

The Design-To-Cost Manifold

Design-to-cost is a popular technique for controlling costs. Although qualitative techniques exist for implementing design to cost, quantitative methods are sparse. In the launch vehicle and spacecraft engineering process, the question whether to minimize mass is usually an issue. The lack of quantification in this issue leads to arguments on both sides. This paper presents a mathematical technique which both quantifies the design-to-cost process and the mass/complexity issue. Parametric cost analysis generates and applies mathematical formulas called cost estimating relationships. In their most common forms, they are continuous and differentiable. This property permits the application of the mathematics of differentiable manifolds. Although the terminology sounds formidable, the application of the techniques requires only a knowledge of linear algebra and ordinary differential equations, common subjects in undergraduate scientific and engineering curricula. When the cost c is expressed as a differentiable function of n system metrics, setting the cost c to be a constant generates an n-1 dimensional subspace of the space of system metrics such that any set of metric values in that space satisfies the constant design-to-cost criterion. This space is a differentiable manifold upon which all mathematical properties of a differentiable manifold may be applied. One important property is that an easily implemented system of ordinary differential equations exists which permits optimization of any function of the system metrics, mass for example, over the design-to-cost manifold. A dual set of equations defines the directions of maximum and minimum cost change. A simplified approximation of the PRICE H(TM) production-production cost is used to generate this set of differential equations over [mass, complexity] space. The equations are solved in closed form to obtain the one dimensional design-to-cost trade and design-for-cost spaces. Preliminary results indicate that cost is relatively insensitive to changes in mass and that the reduction of complexity, both in the manufacturing process and of the spacecraft, is dominant in reducing cost.

Dean, Edwin B.↗
NASA NTRS1988

Continuous Optimization on Constraint Manifolds

This paper demonstrates continuous optimization on the differentiable manifold formed by continuous constraint functions. The first order tensor geodesic differential equation is solved on the manifold in both numerical and closed analytic form for simple nonlinear programs. Advantages and disadvantages with respect to conventional optimization techniques are discussed.

Dean, Edwin B.↗
NASA NTRS1964

On an Approximation Theorem of Kupka and Smale

Simplified and generalized geometrical proof of Kupka and Smale approximation theorem concerning differential equations defined on closed, compact, infinitely differentiable manifold

THEOREM PROVING↗
NASA NTRS1987

Algebra and topology for applications to physics

The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.

Rozhkov, S. S.↗
NASA NTRS1972

System theory on group manifolds and coset spaces.

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

Brockett, R. W.↗
NASA NTRS1979

The geometry of the partial realization problem

It is shown that the space of sequences of length n which have an extrapolation of McMillan degree k, and no extrapolations of lower McMillan degree can be given the structure of a differentiable manifold. This approach makes the proof of certain known results on the partial realization problem quite straightforward and makes it possible to establish some important new results as well. A key tool is the fact, proven here, that the set of n by a real Hankel matrices of rank r is a manifold with r+1 connected components.

Brockett, R. W.↗
NASA NTRS1987

Covariance and the hierarchy of frame bundles

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Estabrook, Frank B.↗
NASA NTRS1975

Fluid manifold design for a solar energy storage tank

A design technique for a fluid manifold for use in a solar energy storage tank is given. This analytical treatment generalizes the fluid equations pertinent to manifold design, giving manifold pressures, velocities, and orifice pressure differentials in terms of appropriate fluid and manifold geometry parameters. Experimental results used to corroborate analytical predictions are presented. These data indicate that variations in discharge coefficients due to variations in orifices can cause deviations between analytical predictions and actual performance values.

Humphries, W. R.↗
NASA NTRS1972

Carbon monoxide oxidation rates computed for automobile thermal reactor conditions

Carbon monoxide oxidation rates in thermal reactors for exhaust manifolds are computed by integrating differential equations for system of twenty-nine reversible chemical reactions. Reactors are noncatalytic replacements for conventional exhaust manifolds and are a system for reducing carbon monoxide and hydrocarbons in automobile exhausts.

Brokaw, R. S.↗
NASA NTRS2000

Any Two Learning Algorithms Are (Almost) Exactly Identical

This paper shows that if one is provided with a loss function, it can be used in a natural way to specify a distance measure quantifying the similarity of any two supervised learning algorithms, even non-parametric algorithms. Intuitively, this measure gives the fraction of targets and training sets for which the expected performance of the two algorithms differs significantly. Bounds on the value of this distance are calculated for the case of binary outputs and 0-1 loss, indicating that any two learning algorithms are almost exactly identical for such scenarios. As an example, for any two algorithms A and B, even for small input spaces and training sets, for less than 2e(-50) of all targets will the difference between A's and B's generalization performance of exceed 1%. In particular, this is true if B is bagging applied to A, or boosting applied to A. These bounds can be viewed alternatively as telling us, for example, that the simple English phrase 'I expect that algorithm A will generalize from the training set with an accuracy of at least 75% on the rest of the target' conveys 20,000 bytes of information concerning the target. The paper ends by discussing some of the subtleties of extending the distance measure to give a full (non-parametric) differential geometry of the manifold of learning algorithms.

Wolpert, David H.↗
NASA NTRS1987

Ensuring Fully Soldered Through Holes

Simple differential-pressure soldering method provides visual evidence that hidden joints are fully soldered. Intended for soldering connector pins in plated through holes in circuit boards. Molten solder flows into plated through holes, drawn by vacuum in manifold over circuit board. Differential-pressure process ensures solder wets entire through hole around connector pin.

Blow, Raymond K.↗
NASA NTRS1998

Constraint-Free Theories of Gravitation

Lovelock actions (more precisely, extended Gauss-Bonnet forms) when varied as Cartan forms on subspaces of higher dimensional flat Riemannian manifolds, generate well set, causal exterior differential systems. In particular, the Einstein- Hilbert action 4-form, varied on a 4 dimensional subspace of E(sub 10) yields a well set generalized theory of gravity having no constraints. Rcci-flat solutions are selected by initial conditions on a bounding 3-space.

Estabrook, Frank B.↗
NASA NTRS1999

Constraint-free Theories of Gravitation

Lovelock actions (more precisely, extended Gauss-Bonnet forms) when varied as Cartan forms on subspaces of higher dimensional flat Riemannian manifolds, generate well set, causal exterior differential systems.

gravitation↗
NASA NTRS1974

Behavior near constant solutions of functional differential equations

Techniques have been developed to determine in a systematic way the local behavior near constant solutions. Local integral manifolds play a very important role in this development, as they have also for ordinary differential equations. An attempt is made to indicate a few more applications of these methods to some problems in bifurcation in the spirit of Sotomayor (to appear) and to a growth model of Cooke and Yorke (to appear). It is also shown how to prove a theorem on stability under constantly acting disturbances using these methods.

Hale, J. K.↗