Vol 61, No 5 (2017)

Let p be a prime number. Recall that a group G is said to be a residually finite p-group if for every non-identity element a of G there exists a homomorphism of the group G onto a finite p-group such that the image of a does not coincide with the identity. We obtain a necessary and sufficient condition for the free product of two residually finite p-groups with finite amalgamated subgroup to be a residually finite p-group. This result is a generalization of Higman’s theorem on the free product of two finite p-groups with amalgamated subgroup.

We solve the M. V. Zaicev problem in the following sense: Any Noetherian semiprime special Lie algebra is embedded into algebra of matrices over commutative ring which is the direct sum of fields.

The present paper deals with a linear integral equation of the third kind with fixed singularities in its kernel. We propose and substantiate special generalized methods for its approximate solving in a space of generalized funtions.

We study correctness for a problemwith an analog of Frankl condition on a degeneration line segment and dislocation conditions on parallel characteristics for Gellerstedt equation with singular coefficient. With the help of maximum principle we prove uniqueness of a solution to the problem and with the method of integral equations we prove the existence of a solution to the problem.

We study the stability of the zero solution to a nonlinear system of ordinary differential equations on the base of its Takagi–Sugeno (TS) representation. As is known, the most constructive stability and stabilization conditions for TS systems stated as linear matrix inequalities are established with the help of a general quadratic Lyapunov function (GQLF). However, such conditions are often too rigid. Using a modification of the Lyapunov direct method, we propose asymptotic stability conditions with weaker requirements to GQLF. They allow an application to a wider class of systems. We also give some illustrative examples.

The paper is devoted to the problem of determining of 5-dimensional pseudo-Riemannian manifolds (M, g) admitting projective motions (h-spaces). A similar problem for n-dimensional proper Riemannian and Lorentz spaces was solved by Levi-Civita, Solodovnikov, Petrov and Aminova. For pseudo-Riemannian manifolds of arbitrary signature and dimension the problem of their classification in Lie algebras and Lie groups of projective transformations, set more than a hundred years ago, is still open. In this paper five-dimensional h-spaces of the type {221} are determined using the method of skew-normal frame (Aminova) and necessary and sufficient conditions for the existence of projective motions of the same type are established.