No 3 (2015)

Methods of analytically theory of ordinary differential equations are based on the analysis of singularities, but the most popular method for the numerical solving, that is the method of finite differences does not work in neighborhood of singularities. However Painlevé gave an algebraic method for the solution of the differential equations in finite terms and general solutions of this equations depend algebraically on a constant of integration. This approach which was presented as Galois theory on the contrary can be well combined with method of finite differences. It is well known, the ordinary differential equation of form y′ = f(x,y) with this property can be algebraically transformed by substitution to Riccati equation. Euler scheme yn+1 = yn + f(xn,yn)Δx always determines (1,k)-correspondence between neighboring layers. But exact solution of Riccati equation determines (1,1)-correspondence between any layers and thus we can write a scheme which determines (1,1)-correspondence between neighboring layers. In this case anharmonic ratio of 4 points does not change from layer to layer not only for exact solution but also for approximate solution. Thus if an exact solution has a pole then the approximate solution passes through infinity without accumulation of an error. In the presented article this property of (1,1)-scheme will be illustrated by two examples: with and without solution in elementary functions. So the cause of destruction of the approximate solution near a pole is put in Euler scheme itself. In more general case when exact solution of ordinary differential equation depends algebraically on an integration constant we can write a scheme which determines (l,l)-correspondence between neighboring layers. Approximate solution which is found on this way passes through movable algebraic singularities without accumulation of an error.

The three-dimensional model of interaction of populations taking into account the competition and diffusion of species is considered. For research of model the combination of known methods of synthesis and the analysis of models, the principle of a reduction and the developed method of construction of the stochastic self-consistent models is used. Existence conditions of equilibrium states are obtained and the analysis of stability is made. Stability conditions on the basis of the principle of a reduction of a problem about stability of solutions of differential inclusion to a problem on stability of other types of the equations are offered. The specified principle assumes transition from the vector ordinary differential equations to vector differential inclusion and the fuzzy differential equation, taking into account change of parameters of different types in the studied models. For the considered model of population dynamics synthesis of the corresponding stochastic model on the basis of application of a method of construction of the stochastic self-consistent models is carried out. The structure of stochastic model is described, Fokker-Planck equation is written out, and the rule of transition to the stochastic differential equation in the form of Langevin is formulated. The offered approach allowed to carry out the comparative analysis of qualitative properties of the models considering the competition and diffusion of species in deterministic and stochastic cases. Stability conditions can be used for studying of dynamic behavior of models of population dynamics. The received results are aimed at the further development of methods of construction and the analysis of stability of nondeterministic mathematical models of natural sciences.

The article traces the evolution of free roaming hypothesis, which plays a prominent role in many areas of human knowledge: mathematics, molecular physics, hydro and gas dynamics, cosmology, chemistry, and biology. This hypothesis is the foundation for the concept of an efficient market. Also it is the source of many modern theories, as well as methods for the financial markets analysis and forecasting. Development of free roaming hypothesis was one of the sources which at the end of the 90s. of the last century led to the emergence of a new field of knowledge - econophysics, a scientific discipline that has emerged at the intersection of economics, physics and mathematics.

In classical mechanics such notion as nonholonomicity is applied only to constraints put on a dynamical system. Besides, Pfaffian nonholonomic constraints might be associated with vector fields. The Nonholonomicity value is one of the principal characteristics of such fields, which determines properties of geometry of these vector fields. However, the application of this characteristic in the geometry of vector fields was restricted only to fields in Euclidean spaces. Some generalization of nonholonomicity value of vector fields in non-Euclidean spaces is proposed in this paper. For this purpose the nonholonomicity value is considered as a trilinear form. It is obvious that the coefficients of this form are connected with the components of the metric tensor of the space, where a vector field is defined. So generalization of metric tensor on non-Euclidean spaces generates the generalization of the coefficients of trilinear form, which in its turn generates the generalization of nonholonomicity value. As an example, the nonholonomicity values of Hamiltonian vector fields in sympletic spaces are analyzed in this article. Also it is important to find out whether a mechanical interpretation of the received results exists and can we actually apply this method to Hamiltonian fields.

In November 2014 the 95... anniversary of Abdelhaq Safiullovich Galiullin, Doctor of Technical Sciences, Honored Worker of Science and Technology of the Russian Federation, Academician of the International Academy of High School, Honorary Academician of the Academy of Sciences of the Republic of Tatarstan was marked. A.S. Galiullin was born on November 26, 1919 in the village of Old Arnash under the old town Arsk. He studied in tatar secondary school No 13. In 1938 he began studies at the Faculty of Mathematics and Physics of Kazan State University. From September 1941 to 1944 he was a student at the Air Force Engineering Academy named after N.E. Zhukovsky. In 1942 he hold a front-line training on the Volkhov front as a mechanic for armaments. From 1946 to 1969 he worked at the Kazan Aviation Institute. On the advice of the scientific supervisor G.V. Kamenkov he started researches in the area of stability and control theory. In 1950 he defended his PhD thesis, and in 1958 - his doctoral dissertation. From 1961 to 1964 A.S. Galiullin worked as Dean of the Faculty of Physics, Mathematics and Natural Sciences at Peoples’ Friendship University named after Patrice Lumumba and, until 1988, as Head of the Department of Theoretical Mechanics. He organized the work of the scientific seminar, and first scientific conferences. He was the Chairman of the Academic Council for master’s and doctoral theses. Abdelhak Safiullovich was a renowned scientist in the field of analytical mechanics, the theory of stability and motion control. He has published more than 90 scientific works, including 8 monographs and textbooks which were widely recognized. The result of his investigations were new, non-traditional methods of analytical dynamics, sections related to the symmetry of the dynamics, with generalizations of Hamiltonian mechanics, systems of Helmholtz, Birkhoff and Nambu. A.S. Galiullin died on April 17, 1999.