An extended Generalised Variance, with Applications

We consider a measure $\psi$ k of dispersion which extends the notion of Wilk's generalised variance, or entropy, for a d-dimensional distribution, and is based on the mean squared volume of simplices of dimension k $\le$ d formed by k + 1 independent copies. We show how $\psi$ k can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a n-point sample is used for its estimation, and prove its concavity when raised at a suitable power. Some properties of entropy-maximising distributions are derived, including a necessary and sufficient condition for optimality. Finally, we show how this measure of dispersion can be used for the design of optimal experiments, with equivalence to A and D-optimal design for k = 1 and k = d respectively. Simple illustrative examples are presented.Comment: Corrected references and typos Added figure

Transatmospheric vehicle research

Research was conducted into the alternatives to the supersonic combustion ramjet (scramjet) engine for hypersonic flight. A new engine concept, the Oblique Detonation Wave Engine (ODWE) was proposed and explored analytically and experimentally. Codes were developed which can couple the fluid dynamics of supersonic flow with strong shock waves, with the finite rate chemistry necessary to model the detonation process. An additional study was conducted which compared the performance of a hypersonic vehicle powered by a scramjet or an ODWE. Engineering models of the overall performances of the two engines are included. This information was fed into a trajectory program which optimized the flight path to orbit. A third code calculated the vehicle size, weight, and aerodynamic characteristics. The experimental work was carried out in the Ames 20MW arc-jet wind tunnel, focusing on mixing and combustion of fuel injected into a supersonic airstream. Several injector designs were evaluated by sampling the stream behind the injectors and analyzing the mixture with an on-line mass spectrometer. In addition, an attempt was made to create a standing oblique detonation wave in the wind tunnel using hydrogen fuel. It appeared that the conditions in the test chamber were marginal for the generation of oblique detonation waves

Analytical and experimental investigations of the oblique detonation wave engine concept

Wave combustors, which include the Oblique Detonation Wave Engine (ODWE), are attractive propulsion concepts for hypersonic flight. These engines utilize oblique shock or detonation waves to rapidly mix, ignite, and combust the air-fuel mixture in thin zones in the combustion chamber. Benefits of these combustion systems include shorter and lighter engines which will require less cooling and can provide thrust at higher Mach numbers than conventional scramjets. The wave combustor's ability to operate at lower combustor inlet pressures may allow the vehicle to operate at lower dynamic pressures which could lessen the heating loads on the airframe. The research program at NASA-Ames includes analytical studies of the ODWE combustor using CFD codes which fully couple finite rate chemistry with fluid dynamics. In addition, experimental proof-of-concept studies are being carried out in an arc heated hypersonic wind tunnel. Several fuel injection designs were studied analytically and experimentally. In-stream strut fuel injectors were chosen to provide good mixing with minimal stagnation pressure losses. Measurements of flow field properties behind the oblique wave are compared to analytical predictions

Wave combustors for trans-atmospheric vehicles

The Wave Combustor is an airbreathing hypersonic propulsion system which utilizes shock and detonation waves to enhance fuel-air mixing and combustion in supersonic flow. In this concept, an oblique shock wave in the combustor can act as a flameholder by increasing the pressure and temperature of the air-fuel mixture and thereby decreasing the ignition delay. If the oblique shock is sufficiently strong, then the combustion front and the shock wave can couple into a detonation wave. In this case, combustion occurs almost instantaneously in a thin zone behind the wave front. The result is a shorter, lighter engine compared to the scramjet. This engine, which is called the Oblique Detonation Wave Engine (ODWE), can then be utilized to provide a smaller, lighter vehicle or to provide a higher payload capability for a given vehicle weight. An analysis of the performance of a conceptual trans-atmospheric vehicle powered by an ODWE is given here

Bregman divergences based on optimal design criteria and simplicial measures of dispersion

In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is embedded in the wider theory of divergences and distances between distributions which includes Kullback–Leibler, Jensen–Shannon, Jeffreys–Bregman divergence and Bhattacharyya distance. A general construction is given based on defining a directional derivative of a function ϕ from one distribution to the other whose concavity or strict concavity influences the properties of the resulting divergence. For the normal distribution these divergences can be expressed as matrix formula for the (multivariate) means and covariances. Optimal experimental design criteria contribute a range of functionals applied to non-negative, or positive definite, information matrices. Not all can distinguish normal distributions but sufficient conditions are given. The k-th order simplicial distance is revisited from this aspect and the results are used to test empirically the identity of means and covariances

Extended generalised variances, with applications

We consider a measure ψk of dispersion which extends the notion of Wilk’s generalised variance for a d-dimensional distribution, and is based on the mean squared volume of simplices of dimension k≤d formed by k+1 independent copies. We show how ψk can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a n-point sample is used for its estimation, and prove its concavity when raised at a suitable power. Some properties of dispersion-maximising distributions are derived, including a necessary and sufficient condition for optimality. Finally, we show how this measure of dispersion can be used for the design of optimal experiments, with equivalence to A and D-optimal design for k=1 and k=d, respectively. Simple illustrative examples are presented