Hans Joachim Haubold

The neutrino sector of the seesaw-modified Standard Model is investigated under the anarchy principle. The anarchy principle leading to the seesaw ensemble is studied analytically with tools of random matrix theory. The probability density function is obtained.

The thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields can be described by kappa distributions. The kappa distribution provides a replacement for the Maxwell–Boltzmann distribution, which can be considered as a generalization for describing systems characterized by local correlations among their particles, as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai. The aim of the present paper is to bring to attention the application of various forms of the kappa distribution, its various special cases and its generalizations, which, in scalar-variable and multivariate situations, belong to a general family of distributions known as Mathai’s pathway models, comprising three different families of functions, namely the generalized type-1 beta, type-2 beta and gamma families. Through one parameter, known as the pathway parameter, one will be able to reach all the three families of functions and the stages of transitioning from one family to another. After pointing out the connection of multivariate (vector-variate) kappa distributions to the multivariate pathway model, the multivariate kappa distribution is extended to the real matrix-variate case by working out the various forms and by evaluating the normalizing constants of the various forms of the matrix-variate case explicitly. It is also pointed out that the pathway models are available for the scalar, vector and rectangular matrix-variate cases in the real domain as well as in the complex domain.

Thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields are described by kappa distributions. The kappa distribution provides a replacement for the Maxwell-Boltzmann distribution which can be considered as a generalization for describing systems characterized by local correlations among their particles as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai.

Usually, convolution refers to Laplace convolution in the literature. But Mellin convolutions can yield very ueeful results. This aspect is illustrated in the coming sections. This paper deals with Mellin convolutions of products and ratios. Functions belonging to the pathway family of functions are considered. Several types of integral representations, their equivalent representations in terms of G and H-functions and their equivalent computable series representations are examined in this paper.

From a summer house just outside of Berlin, to the establishment of educational centres across the world —Professors Hans Haubold and Arak Mathai have overseen the remarkable development of space science over the last half-century. Their early research into the curious solar neutrino problem in the 1970s spawned several books and a large volume of academic literature exploring the inner workings of our Sun. Developments in the mathematics of fractional calculus and special functions then helped to facilitate extensive studies of space and our solar system. Over the last 30 years, under the umbrella of the United Nations, space science has boomed, with 7 new centres set up across the globe and over 20 dedicated workshops arranged. A new generation is being trained to tackle the mysteries and challenges presented to us by outer space.

It is shown that a Mittag-Leffler density has interesting properties. The Mittag-Leffler random variable has a structural representation in terms of a positive Levy variable and the power of a gamma variable where these two variables are independently distributed. It is shown that several central limit-type properties hold but the limiting forms are positive Levy variable rather than a Gaussian variable. A path is constructed from a Mittag-Leffler function to the Mathai pathway model which also provides paths to nonextensive statistics and superstatistics.

In a paper [8] the authors classify entropy into three categories, as a thermodynamics quantity, as a measure of information production, and as a means of statistical inference. An entropy measure introduced by Mathai falls into the second and third categories. It is shown that this entropy measure is the same whether the variables involved are real or complex scalar, vector, or matrix variables. If the entropy measure is optimized under some moment-like conditions then one can obtain various types of densities which are applicable in different areas. Unlike Tsallis' entropy [9], it does not need an intermediary escort distribution to yield the desired results. Calculus of variation can be directly applied to obtain the desired results under Mathai's entropy. Tsallis' entropy, which is the basis of the area of non-extensive statistical mechanics, is a modified version of the $α$-generalized entropy of Havrda-Charvat considered in [7]. Various types of distributions that can be obtained through optimization of Mathai's entropy, are illustrated in this paper.

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper.

Thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields are described by kappa distributions. The kappa distribution provides a replacement for the Maxwell-Boltzmann distribution which can be considered as a generalization for describing systems characterized by local correlations among their particles as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai.