This paper employs Jacobians of matrix transformations to derive the density function of a matrix-variate generalized gamma distribution, together with its normalizing constant. By applying the inverse Mellin transform, explicit expressions for the density functions of the determinant and the trace are obtained in terms of generalized hypergeometric functions. The characteristic function and the first two moments follow from an associated density generator. Both the real and complex cases are treated, and several important special cases are identified. A simulation study reveals that the proposed model provides a more accurate fit than other distributions that are also defined on the cone of positive definite matrices. Moreover, it is shown to exhibit superior performance when applied to two empirical data sets. Applications involving the modeling of scatter matrices arising in financial studies, biostatistics, and reliability analysis are also discussed.

This paper deals with the interpretation of certain multi-index (multiple) generalized fractional integrals introduced in Kiryakova [4] by using statistical distribution theory and, in particular, the beta distribution and some of its extensions. Then, the fractional integrals are extended to matrix-variate cases in the real and complex domains. Further extensions to rectangular matrix-variate cases in the real and complex domain are also given. It is shown that the approach relying on statistical distribution theory can readily be applied to extend the results to matrix-variate cases which are based on symmetric ratios and symmetric products of matrices whose distribution can also be determined by making use of M-convolutions. Several extensions to the scalar and matrix-variate cases are discussed and a method of obtaining the corresponding differential equation for the corresponding fractional integral is presented.

1930 Wolfgang Pauli hypothesizes the existence of neutrinos to account for the beta decay energy conservation crisis.

Over the past 50 years, radio-chemical and real-time solar neutrino experiments have proven to be sensitive tools to test both astrophysical and elementary particle physics models and principles (Sakurai 2018; Orebi Gann et al. 2021). Solar neutrino detectors (radio-chemical: Homestake, GALLEX + GNO, SAGE, real-time: SuperKamiokande, SNO, Borexino) have demonstrated that the Sun is powered by thermonuclear fusion reactions.

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.

“...some say the solar neutrino problem is still with us. Others say there never was a problem. My purpose is to present this ambiguous situation to you in such a way that you can make your own judgement.” (W.A. FOWLER 1977). It is not the aim of the present chapter to add one more suggestion to the long list of attempts for the solution of the so-called solar neutrino problem (for an updated list cf. HAXTON 1984).

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.

For the sake of ready reference, the basic materials will be restated here. Let $$x_1>0$$ x 1 > 0 and $$x_2>0$$ x 2 > 0 be two real scalar positive variables with the associated functions $$f_1(x_1)$$ f 1 ( x 1 ) and $$f_2(x_2)$$ f 2 ( x 2 ) respectively. Let the joint function of $$x_1$$ x 1 and $$x_2$$ x 2 be $$f_1(x_1)f_2(x_2)$$ f 1 ( x 1 ) f 2 ( x 2 ) , the product. If $$x_1>0$$ x 1 > 0 and $$x_2>0$$ x 2 > 0 are real scalar random variables with the densities $$f_1(x_1)$$ f 1 ( x 1 ) and $$f_2(x_2)$$ f 2 ( x 2 ) , then we say that $$x_1$$ x 1 and $$x_2$$ x 2 are statistically independently distributed when we take the joint density as $$f_1(x_1)f_2(x_2)$$ f 1 ( x 1 ) f 2 ( x 2 ) , the product. Let $$u=x_1x_2$$ u = x 1 x 2 the product. Consider the transformation $$u=x_1x_2,v=x_2$$ u = x 1 x 2 , v = x 2 . Then, we can see that the wedge product of differentials are connected by the relation $$\textrm{d}x_1\wedge \textrm{d}x_2=\frac{1}{v}\textrm{d}u\wedge \textrm{d}v$$ d x 1 ∧ d x 2 = 1 v d u ∧ d v and then the marginal function of u , denoted by g ( u ), is given by the following:

According to the current understanding of the evolution of the Universe the presently observed state of the Universe is the result of expansion from an extremely dense and extremely hot singular origin. Describing that evolution of the Universe by means of the standard cosmological model, based on the ‘big-bang’ hypothesis, cosmological nucleosynthesis occurs at the appropriate temperature in the course of expansion and goes on until the decreasing temperature stops nuclear reactions. No significant cosmological nucleosynthesis beyond helium-4 occurred due to the instability gaps at mass number 5 and mass number 8 as well as the constraints set by the present universal density and temperature. However, it is well-known that ‘big-bang’ nucleosynthesis results in the production of lithium-7 in amounts comparable to the solar system abundance of this nucleus. Some general features of the cosmological nucleosynthesis are listed in Table 2.1 and Figs. 2.1 and 2.2.

Usually, convolution refers to Laplace convolution in the literature, but Mellin convolutions can yield very ueful results. This aspect is illustrated in the coming sections. This study 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 study.Mathematics Subject Classification 2010: 26A33, 44A10, 33C60, 35J10.

The Standard Model (SM) of Particle Physics is the pinnacle of the understanding of neutrino physics Deppisch (2019); Oberauer et al. (2020). It comes with a plethora of parameters, the masses and the flavour mixings, that are seemingly not fixed by any known fundamental principle. In the SM, the neutrino spectrum is simple: all neutrinos are massless. Neutrino oscillations, where neutrinos seemingly change flavour in flight, cannot be accommodated in the SM due to the mass of the neutrinos. Neutrino oscillations thus imply massive neutrino eigenstates, and the SM must be extended. Moreover, neutrino oscillation experimental data suggest that the neutrino spectrum is not hierarchical, with three massive light neutrinos and a mixing matrix exhibiting near-maximal mixing Deppisch (2019); Oberauer et al. (2020).

This paper was prepared for Open-Access-only publication as a guide reporting on Education (all aspects of space science and technology), Teaching (remote sensing and GIS, satellite meteorology and global climate, satellite communication, space and atmospheric sciences, global navigation satellite systems), and Research (solar neutrino problem, formation of structure in the Universe) in astronomy (solar physics, cosmology), physics (nuclear physics, neutrino physics), and mathematics (fractional calculus, special functions of mathematical physics) exercised over 50 years (1974-2024). In this period, more than twenty workshops were held and seven regional centres for space science and technology education were established in all regions of the world: Asia and the Pacific, Latin America and the Caribbean, Africa, Western Asia, and Europe. This effort was undertaken in cooperation with ESA, NASA, JAXA, and 193 member states of the United Nations under the auspices of the UN, also supported by the Committee on Space Research (COSPAR) and the International Astronomical Union (IAU). The paper provides access to most of the documents in the six official languages of the United Nations (Arabic, Chinese, English, French, Russian, and Spanish), proceedings, and published papers and books focusing on education, teaching, and research (listed in Google Scholar and Research Gate).

Shortly after the discovery of the law of conservation of energy by Mayer (1842) and Helmholtz (1847) it was Mayer who raised the question for the origin of the radiative energy emitted by the Sun. J.R. MAYER’s law of conservation of energy (first law of thermodynamics) took into account the energy due to heat.

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.

This paper deals with the extension of principal component analysis, canonical correlation analysis, the Cramer–Rao inequality, and a few other statistical concepts in the real domain to the corresponding complex domain. Optimizations of Hermitian forms under a linear constraint, a bilinear form under Hermitian-form constraints, and similar maxima/minima problems in the complex domain are discussed. Some vector/matrix differential operators are developed to handle the above types of problems. These operators in the complex domain and the optimization problems in the complex domain are believed to be new and novel. These operators will also be useful in maximum likelihood estimation problems, which will be illustrated in the concluding remarks. Detailed steps are given in the derivations so that the methods are easily accessible to everyone.

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.

The paper utilizes data from the SuperKamiokande solar neutrino detection experiment and analyzes them by diffusion entropy analysis and standard deviation analysis to evaluate the scaling exponent of the probability density function. The result indicates that solar neutrinos are subject to Levy flights. Subsequently, the paper derives the probability density function, represented as the Fox H-function, and the governing fractional diffusion equation for solar neutrino Levy flights.

Several extensions of the basic scalar variable logistic density to the multivariate and matrix-variate cases, in the real and complex domains, are given where the extended forms end up in extended zeta functions. Several cases of multivariate and matrix-variate Bayesian procedures, in the real and complex domains, are also given. It is pointed out that there are a range of applications of Gaussian and Wishart-based matrix-variate distributions in the complex domain in multi-look data from radar and sonar. It is hoped that the distributions derived in this paper will be highly useful in such applications in physics, engineering, statistics and communication problems, because, in the real scalar case, a logistic model is seen to be more appropriate compared to a Gaussian model in many industrial applications. Hence, logistic-based multivariate and matrix-variate distributions, especially in the complex domain, are expected to perform better where Gaussian and Wishart-based distributions are currently used.

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.