Features of the Module «Probability and Statistics» as Part of the Higher Mathematics Course

The article deals with the problem of providing a natural and logically grounded “embeddedness” of the stochastic module in the general course of higher mathematics. Its solution is proposed both by structural means of constructing mathematical theories (primary concepts, axioms, definitions, theorems), and by means of establishing intra-subject connections. In particular, the connections of standard distributions with the theory of summation of series are demonstrated. It is argued that the practice-oriented orientation of stochastic knowledge and methods is achieved by a “model” approach, that is, by bringing the process of solving problems in accordance with the standard stages of mathematical modeling. Using the example of empirical power-law moments, it is proposed to familiarize students with some modern ideas of mathematical analysis. In particular, an algorithm for reconstructing continuous theoretical distributions based on convergence theorems for mean Fourier–Chebyshev expansions is presented.

Fractional Integration of the Product of two Multivariable Gimel-Functions and a General Class of Polynomials

A significantly large number of earlier works on the subject of fractional calculus give the interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, the summation of series, etc.). The object of the present paper is to study and develop the Saigo-Maeda operators. First, we establish four results that give the images of the product of two multivariable Gimel-functions and a general class of multivariable polynomials in Saigo- Maeda operators. On account of the general nature of the Saigo-Maeda operators, multivariable Gimel-functions and a class multivariable polynomials a large number of new and known theorems involving Riemann-Liouville and Erdelyi- Kober fractional integral operators and several special functions.

Contribution of Aryabhatta in world Mathematics

Great mathematician Aryabhatta has immense contribution not only in mathematics but also in physics, astronomy and other fields. Aryabhatta has initiated to find a solution of quadratics equation, summation of series etc. He has written 108 shlokas in eight pages which play a vital role in mathematics even today.