Large Number Of Nonlinear Equations Research Articles

Estimating Components of Univariate Gaussian Mixtures Using Prony's Method

A new technique for estimating the component parameters of a mixture of univariate Gaussian distributions using the method of moments is presented. The method of moments basically involves equating the sample moments to the corresponding mixture moments expressed in terms of component parameters and solving these equations for the unknown parameters. These moment equations, however, are nonlinear in the unknown parameters, and heretofore, an analytic solution of these equations has been obtained only for two-component mixtures [2]. Numerical solutions also tend to be unreliable for more than two components, due to the large number of nonlinear equations and parameters to be solved for. In this correspondence, under the condition that the component distributions have equal variances or equal means, the nonlinear moment equations are transformed into a set of linear equations using Prony's method. The solution of these equations for the unknown parameters is analytically feasible and numerically reliable for mixtures with several components. Numerous examples using the proposed technique for two-, three-, and four-component mixtures are presented.

Saturation of absorption of the powerful radiation of a system of anharmonic oscillators

Recently, the theory of nonequilibrium systems simulated by a set of anharmonic oscillators has received significant development. The investigation of such kinds of systems is especially important in the study of problems associated with the stimulation of chemical reactions and the development of effective molecular lasers. The systematic analysis of the kinetics of anharmonic oscillators assumes the simultaneous solution of a large number of nonlinear equations describing the population balance of the vibrational levels. Realization of this approach is associated with cumbersome numerical calculations and does not permit obtaining a qualitative picture of the behavior of the system as a function of the different parameters (pressure, temperature, etc.). An approximate analytical theory has been formulated in [1, 2] which permits finding the distribution function over the vibrational states with the effects of anharmonicity taken into account. We will employ the approach developed in these papers to describe a system of anharmonic oscillators under conditions of powerful optical pumping. This problem was discussed in [3], where it was found that such a system changes into a saturation mode in the case of high pumping levels. The existence of this mode is explained by the fact that the maximum rate of energy input into a vibrational degree of freedom is determined by the rate of distribution of this energy over all the vibrational levels, i.e., by the constant of V—V-exchange. For sufficiently large pumpings the approximation of the Boltzmann distribution function adopted in [3] in connection with the calculation of the saturation parameters is too crude. The goal of this paper is to derive in explicit form expressions for the vibrational energy supply, the absorbed power, and so on, under saturation conditions without the use of the approximation indicated above [3].