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We consider the Landau-Lifshitz equation on a three-dimensional torus. The equation is reduced to the form of the Euler equation for the geodesic left-invariant metric on the infinite-dimensional Lie algebra of the current group. The group of currents is given by a pointwise mapping of the three-dimensional torus into a three-dimensional orthogonal group. In Lie algebra we use the non-standard commutator introduced earlier. The solutions of the Landau-Lifshitz equation can be expanded in terms of the orthonormal basis of the left-invariant metric in the currents algebra. For the expansion coefficients of the solution of the Landau-Lifshitz equation, the explicit form of the evolution equations is deduced in the framework of the constructed model. To do this, we use the expressions obtained earlier for the sums of the adjoint and coadjoint action operators in an infinite-dimensional Lie algebra of currents with nonstandard commutator. The compactness property of the indicated sum operators makes it possible to obtain the asymptotic form of the Landau-Lifshitz equation on a threedimensional torus. Evolution equations are found on the subspace of flows consisting of vector fields whose Fourier expansions contain only simple harmonics of the form cos (kO) Such vector fields form a subalgebra of the currents algebra which is also closed under the action of coadjoint operators. In this case, an arbitrary Landau-Lifshitz equation for which the vector of initial conditions lies in this subalgebra remains in it for all t for which this solution is defined. We note that to study the Landau-Lifshitz equation the currents algebra with the standard commutator turned out to be ineffective: in particular, the Landau-Lifshitz equation is not an Euler equation on the current algebra with a standard commutator. Thus, for the Landau-Lifshitz equation on the three-dimensional torus, the explicit form of the evolution equations for the coefficients of the Fourier expansion of its solutions by means of operators representing the sum of the operators of the adjoint and co-adjoint action of the current algebra on a three-dimensional torus with nonstandard commutator is obtained. Moreover, it is the property of compactness of the indicated sum operators (while, separately, their components, the operator of the adjoint action operator as well as the coadjoint one are not even continuous) made it possible to obtain the indicated asymptotic form.

This paper continues the author’s series of works on the modeling of one-sided surfaces. There is a closed curve (disorienting contour) on a one-sided surface, and this curve has the property of changing the sign when it is bypassed local orientation in tangent space. A one-sided surface is a Klein bottle. Two smooth vector functions are considered. It is assumed that one of them is 2 pi — periodic, the other is 2 pi — antiperiodic. The Klein bottle equations, disorienting contours, and the equations of two Mobius sheets into which a Klein bottle is cut are determined using the obtained functions. In this paper, the inversion of the Klein bottle is investigated. We prove that if a Klein bottle does not pass through the center of inversion, then the inversion of the Klein bottle is also the Klein bottle. It is also proved that if the Klein bottle does not pass through the center of inversion, then the disorienting contours of the Klein bottle under the inversion will pass into the disorienting contours. Models of the considered surfaces are built with the help of the computer mathematics system., Данная публикация продолжает серию работ автора о моделировании односторонних поверхностей. На односторонней поверхности существует замкнутая кривая (дезориентирующий контур), обладающая тем свойством, что при обходе локальная ориентация в касательном пространстве меняет знак. Односторонней поверхностью является бутылка Клейна. Рассматриваются две гладкие вектор-функции. Предполагается, что одна из них есть 2π периодическая, другая 2π — антипериодическая. С использованием найденных функций определяются уравнения бутылки Клейна, дезориентирующие контуры и уравнения двух листов Мебиуса, на которые разрезается бутылка Клейна. В работе исследуется инверсия бутылки Клейна. Доказывается, что если бутылка Клейна не проходит через центр инверсии, то инверсия бутылки Клейна есть бутылка Клейна. Доказывается также, что если бутылка Клейна не проходит через центр инверсии, то дезориентирующие контуры бутылки Клейна при инверсии перейдут в дезориентирующие контуры. С помощью системы компьютерной математики строятся исследуемые поверхности.

Представленная статья посвящена проблеме термоупругопластического деформирования полого тора. Основные уравнения теории температурных напряжений представлены в тороидальной системе координат. Рассчитаны и графически проанализированы остаточные напряжения в полом упругопластическом торе, подверженном неравномерному радиальному нагреву. Приближенное аналитическое решение указанной задачи получено в рамках подхода обобщенного плоского напряженного состояния. Поля напряжений и смещений вычислены в областях термоупругого деформирования и пластического течения. Обсуждаются корректность и применимость полученных решений для тора произвольного размера в условиях осесимметричного теплового воздействия., The present paper is devoted to problem of thermoelastoplastic deformation of hollow torus. The requisite equation of the temperature stresses theory are furnished in toroidal coordinate frame. The residual stresses in a hollow elastoplastic torus subject to non-uniform radial heating are calculated and graphically analyzed. The aproximate analytical solution of the noted problem is obtained in the framework of the generalized plane stress state approach. The stress and displacement fields are computed in the thermoelastic and plastic flow domains. The correctness and apliability of the obtained solutions for a torus of arbitrary size under conditions of axisymmetric thermal effect are discussed., №2(40) (2019)

The conditions under which the chaotic dynamics is formed during the materials processing by cutting are analyzed. In the previous studies, in case of loss of sta-bility of the cutting process, limit cycles or invariant tori are generated in the neighborhood of the equilibrium system. Contrastingly to these studies, the case when the tool properties are such that a nonlinear positive feed-back is formed by flexural deformations is considered. A mathematical system model is provided for this case. On the basis of the numerical simulation, using the MATLAB application program package, the dynamic model parameters effect is explored under the conditions of the chaotic dynamics formation. The research results show that with increasing the parameters characterizing the formation of a positive feedback, the system undergoes a series of period-doubling bifurcations in the system of strange attractors. They are located in the vicinity of the equilibrium points and have a limited area. It is shown that the tool chaotic oscillations lead to the chaotic work surface forming, therefore, in the application sector, it is necessary to choose the parameters under which the chaotic dynamics is not formed. Although the considered examples relate to the cutting process, the obtained results are of general validity for the dynamic systems interacting with various environments, for example, with a tribological environment.

The conditions for convergence of square and rectangular Fejer means of functions on the infinite dimensional torus were obtained, also a generalization of the results for the case of abstract measure spaces was formulated., In Russian

This paper is devoted to the study of topological quotients of compact linear Lie groups. More precisely, it investigates the question of when such a quotient is a topological or a smooth manifold. The topological quotient of a finite linear group was studied by Mikhailova in 1984. Here the connected component of the original Lie group G is assumed to be a torus of positive dimension., in Russian