Problems and solutions by the application of Julia set theory to one-dot and multi-dots numerical methods

In 1977 Hubbard developed the ideas of Cayley (1879) and solved in particular the Newton-Fourier imaginary problem. We solve the Newton-Fourier and the Chebyshev-Fourier imaginary problems completely. It is known that the application of Julia set theory is possible to the one-dot numerical method like the Newton's method for computing solution of the nonlinear equations. The secants method is the two-dots numerical method and the application of Julia set theory to it is not demonstrated. Previously we have defined two one-dot combinations: the Newton's-secants and the Chebyshev's-secants methods and have used the escape time algorithm to analyse the application of Julia set theory to these two combinations in some special cases. We consider and solve the Newton's-secants and Tchebicheff's-secants imaginary problems completely

Moment Identifiability of Homoscedastic Gaussian Mixtures

We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander-Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two component setting we provide a closed form solution for parameter recovery based on moments up to order four, while in the one dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves.Comment: 27 pages, 1 table, 1 figur