The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem. 3. Elementary solutions. We quote Rayleigh 128 who says, "The theory of the free vibrations of a membrane was first successfully considered by Poisson 116]. His theory in the case of the rectangle left little to be desired." For the rectangle 0 _< x _< a, 0 <_ y <_ b, the eigenfunctions are

An on‐line data base consisting of the auroral oval boundaries, structure, and particle fluxes from the DMSP F7 and F9 satellites from December 1983 through the present (with about an 7‐month lag) is announced. The data are divided into distinct regions (e.g., diffuse aurora, discrete aurora, cusp proper, polar rain, etc.). The magnetospheric sources of the various particle precipitation regions are identified using a sophisticated pattern recognition technique (a neural network). For each region the boundaries are specified in both geographic and geomagnetic coordinates, and the average and peak values of the particle fluxes are given, along with the average energies. The DMSP satellites are continuously monitored, so that data gaps are comparatively rare. It is anticipated that the location of the auroral oval (and precipitation intensities) will be of particular interest to ground‐based observers; although anyone interested in the state of the auroral oval may find it of value. SPAN data requests are processed on a completely automated basis, with up to 48 hours of auroral oval parameters provided in response to a single request.

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The main difficulties in reliable automated detection of the K-complex wave in EEG are its close similarity to other waves and the lack of specific characterization criteria. We present a feature-based detection approach using neural networks that provides good agreement with visual K-complex recognition: a sensitivity of 90% is obtained with about 8% false positives. The respective contribution of the features and that of the neural network is demonstrated by comparing the results to those obtained with i) raw EEG data presented to neural networks, and ii) features presented to Fisher's linear discriminant.

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