On Complex Matrix-Variate Dirichlet Averages and Its Applications in Various Sub-Domains.

This paper is about Dirichlet averages in the matrix-variate case or averages of functions over the Dirichlet measure in the complex domain. The classical power mean contains the harmonic mean, arithmetic mean and geometric mean (Hardy, Littlewood and Polya), which is generalized to the -mean by de Finetti and hypergeometric mean by Carlson; see the references herein. Carlson's hypergeometric mean averages a scalar function over a real scalar variable type-1 Dirichlet measure, which is known in the current literature as the Dirichlet average of that function.

Entropy Optimization, Generalized Logarithms, and Duality Relations.

Several generalizations or extensions of the Boltzmann-Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sq≡k1-∑ipiqq-1(q∈R;S1=SBG≡-k∑ipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny.

Entropy Optimization, Maxwell-Boltzmann, and Rayleigh Distributions.

In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived.

Space Science and Technology Education, Teaching, Research.

A status report for an attempt to implement a project of education, teaching, and research in space science and technology in the spirit of the United Nations is briefly summarized in this Viewpoint.