Abstract: The methodological basis is developed and tested for building indicators which shows the main directions of economic development of regions of the Russian Federation. The novelty of the results is determined by the fact that these indicators are built on the basis of a common vector basis. Two groups of indicators that characterize the different directions of the economic development: "production of products and services" and "material welfare" are highlighted in the structure of the main indicators of social and economic development. Two indicators are constructed based on the vector basis, each of which is maximally correlated with the indicator formed on the basis of corresponding group indicators. It is shown that for the considered direction of regional development the vector basis provides a higher consistency of the indexes and ranks of regions than the first major components.
Abstract: In this paper a method is proposed by which any number not equal to zero can have an infinite number of algebraic factorizations. A factorization method of radical solution of polynomial equations is then presented. The method provides radical solution of polynomial equations of degree than 4. As a demonstration the Bring-Jerrard quintic equation formula is derived by factorization.
Abstract: In this research Riemann hypothesis is investigated for a proof. A functional extension of the
Riemann zeta function is proposed for which non trivial zeroes can be generated. It found that non
trivial zeroes can also be generated outside the critical strip. Thus it is found that the Riemann
hypothesis is found to be incomplete.
Abstract: The mathematical model of a fractal financial system that takes into account the effects of
heredity or memory is considered. This mathematical model is a Cauchy problem in which the
model equation is a system of differential equations with derivatives of fractional orders. Using a
numerical algorithm based on the theory of finite-difference schemes, an approximate solution of
the proposed model was obtained. A numerical algorithm was implemented in a computer program
in the language of Scilab, with the help of which the phase trajectories of the fractal financial
system were constructed. It is shown that if fractal properties are taken into account, chaotic
regimes can exist even in dynamical systems of dimension less than three.
Abstract: We propose a method of specification of stochastic production function models for solving
problems related to the ranking of objects by the level of technical efficiency at production-regional
level. The described method takes into account potential dependence between the random
components of the error.
Based on actual data on 80 sub-federal units of the Russian Federation grouped according to
the basic characteristics of the GRP structure, an empirical analysis of the influence of possible
dependence of the error components on the values of regions’ technical efficiency was carried out
in accordance with the developed specification scheme. The main types of problems where it is
possible to use the premise of the independence of the random error components were described.