Yu. I. Sapronov

Global comparison of finite-dimensional reduction schemes in smooth variational problems

A new criterion for global smooth equivalence of a pair of key functions corresponding to a smooth functional in the calculus of variations for a given pair of finite-dimensional reduction schemes is established. The statement is presented in abstract form (we consider a functional on a Banach space with a Fredholm gradient). The main condition is the possibility to deform the reduction schemes into each other preserving the coercivity of the key functions. As a corollary, we obtain the theorem concerning global smooth equivalence of the key functions calculated by the Lyapunov-Schmidt and Morse-Bott reduction schemes in the two-point boundary value problem for a natural mechanical system of sufficiently general form.

Application of normalized key functions in a problem of branching of periodic extremals

In this paper we construct a procedure of approximate calculation and analysis of branches of bifurcating solutions to a periodic variational problem. The goal of the work is a study of bifurcation of cycles in dynamic systems in cases of double resonances 1: 2: 3, 1: 2: 4, p: q: p + q and others. An ordinary differential equation (ODE) of the sixth order is considered as a general model equation. Application of the Lyapunov–Schmidt method and transition to boundary and angular singularities allow to simplify a description of branches of extremals and caustics. Also we list systems of generating algebraic invariants under an orthogonal semi-free action of the circle on ℝ6 and normal forms of the principal part of the key functions.

Phase transitions in crystals characterized by polarization and deformation components of the order parameter

Abstract By the method of Landau thermodynamics a possibility of realization of phase transitions characterized by a vector of polarization and a tensor of deformation in crystals of all classes of point group of symmetry is investigated. The analysis is carried out by drawing up of the lists of invariants, constructed from a component of the order parameter. It is shown, that such phase transitions are possible in 14 crystallographic classes. The orientations of domain boundaries in low symmetrical phases and nonlinear effects are specified

Cellular Complexes for Thermodynamic Potential of Ferroelectrics

The topological method of investigation of regularities at phase transitions in ferroelectric crystals and the nonlinear phenomena, based on representations about cellular complexes of system of special points of non-equilibrium thermodynamic potential have been stated. Full set of possible cellular complexes for crystals of cubic structure, for which thermodynamic potential can be presented as polynomial of the sixth degree, is constructed.

Formation of nonsmoothness lines in optimization of screw pairs in screw pumps

Computer modeling of screw pairs in screw pumps reveals the formation of nonsmoothness lines on the screw surface (with the growth of the height of the helical tooth) conjugate to the smooth helical surface. The mathematical explanation of the appearance of such lines represents considerable interest for the theory and practice of pumps manufacturing. In this paper we show that one can explain the mentioned phenomenon by studying a wider problem, namely, the bifurcation of regressive points on a plane contour conjugate to the smooth one.

Linear classification of quadratic mappings in the space C3

The action of the group G1(n, C)2 on the manifolds of the regular quadratic mappings F:Cn → Cn is considered. For the case n=3 all the normal forms are listed.