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The electrons of a pure electron plasma may be in thermal equilibrium with each other and still be confined by static magnetic and electric fields. Since the electrons make a significant contribution to the electric field, only certain density profiles are consistent with Poisson’s equation. The class of such distributions for a finite length cylindrical column is investigated. In the limit where the Debye length is small compared with the dimensions of the column, the density is essentially constant out to some surface of revolution and then falls off abruptly. The falloff in density is a universal function when measured along the local normal to the surface of revolution and scaled in terms of the Debye length. The solution for the shape of the surface of revolution is simplified by passage to the limit of zero Debye length.

Abstract-In this paper, we present a method of parameter estimation for a class of problems where the desired signal is embedded in colored noise with unknown covariance. The new algorithm is a variation of the covariance differencing scheme proposed by Paulraj and Kailath. Unlike the previous method, however, the proposed algorithm does not require multiple estimates of the signal covariance matrix. Instead, it uses a priori knowledge of the structure of the noise covariance matrix to transform the array covariance matrix in a way that leaves its noise component unchanged while transforming the signal component in some appropriate manner. We can then eliminate the noise component by forming the difference of the original and transformed covariance matrices. This unique feature of the proposed method allows covariance differencing methods to be applied to a wider class of problems than was previously possible. To illustrate this, we apply the new covariance differencing algorithm to the problems of bearing estimation, resolution of overlapping echos, and transient response analysis. Simulation results are presented for each problem, and the new method's performance is compared to that of conventional methods for solving each respective problem. It is interesting to note that our algorithm eliminates the need for array rotation or translation which is often required for the conventional covariance differencing technique.