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H E Rauch

Publications and source records attributed to H E Rauch.

10 recordsLinked to original sources

Neural [correction of Neutral] networks for control, identification and diagnosis.

Advances in the theory and technology of artificial neural networks provide the potential for new approaches to the problems of control, identification, and diagnosis for large, complex systems. However, these approaches must be validated for specific applications before they can be exploited effectively. Because of the unique capabilities they offer, neural networks should play an important role in space exploration systems operations. After a brief introduction to neural networks is presented, some applications of neural networks to identification and control of space systems are described and discussed. They span the spectrum of relatively straightforward to rather complex applications. An explanation of how neural networks can be applied to such important tasks as fault diagnosis and accommodation is presented. Neural networks are shown to be part of the hierarchy of intelligent control where a higher order decision element monitors and supervises lower order elements for sensing and actuation.

Algorithms↗

Holomorphic embedding of complex curves in spaces of constant holomorphic curvature.

A special case of Wirtinger's theorem asserts that a complex curve (two-dimensional) holomorphically embedded in a Kaehler manifold is a minimal surface. The converse is not necessarily true. Guided by considerations from the theory of moduli of Riemann surfaces, we discover (among other results) sufficient topological and differential-geometric conditions for a minimal (Riemannian) immersion of a 2-manifold in complex projective space with the Fubini-Study metric to be holomorphic.

Journal Article↗

Two kinds of theta constants and period relations on a riemann surface.

It was recognized in Riemann's work more than one hundred years ago and proved recently by Rauch (cf. Bull. Am. Math. Soc., 71, 1-39 (1965) that the g(g + 1)/2 unnormalized periods of the normal differentials of first kind on a compact Riemann surface S of genus g >/= 2 with respect to a canonical homology basis are holomorphic functions of 3g - 3 complex variables, "the" moduli, which parametrize the space of Riemann surfaces near S and, hence, that there are (g - 2)(g - 3)/2 holomorphic relations among those periods. Eighty years ago, Schottky exhibited the one relation for g = 4 as the vanishing of an explicit homogeneous polynomial in the Riemann theta constants. Sixty years ago, Schottky and Jung conjectured a result which implies Schottky's earlier one and some generalizations for higher genera.Here, we formulate Schottky and Jung's conjecture precisely and, on the basis of a recent result of Farkas (these PROCEEDINGS, 62, 320 (1969)), prove it. We then derive Schottky's result (we believe for the first time correctly) and exhibit a typical relation of this kind for g = 5 (we can do this for any genus). We do not prove that our relations imply all relations, but there are some indications that they do.

Journal Article↗