Vol 60, No 4 (2019)

We prove two theorems about a partially commutative metabelian pro-p-group. The first theorem concerns the structure of annihilators for the commutators of nonadjacent vertices of the defining graph, and the second discusses the centralizers of the vertices of the defining graph.

A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclone_lin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclone_lin(τ, τ′).

We construct Lax pairs for linear Hamiltonian systems of differential equations. In particular, the Gröbner bases are used for computations. It is proved that the mappings in the construction of Lax pairs are Poisson. Under study are the various properties of first integrals of the system which are obtained from Lax pairs.

We find the general form of the polyhedral divisors corresponding to the natural torus action of complexity 1 on affine trinomial hypersurfaces. Some explicit computations of the divisors for the particular classes of the hypersurfaces are given.

Considering autonomous delay functional differential equations, we establish some oscillation criterion that reduces the oscillation problem to computing the only root of the real-valued function defined by the coefficients of the initial equation. Using the criterion, we obtain effectively verifiable oscillation tests for equations with various aftereffects.

Under examination is the system of equations of electrodynamics for a nonconducting nonmagnetic medium with the simplest anisotropy of permittivity. We assume that permittivity is characterized by the diagonal matrix ϵ = diag(ε₁, ε₁, ε₂), with the functions ε₁ and ε₂ equal to positive constants beyond a bounded convex domain Ω ⊂ ℝ³. Two modes of traveling plane waves exist in a homogeneous anisotropic medium. The structure is studied of the solutions related to the traveling plane waves incident from infinity on an inhomogeneity located in Ω.

Under study is some control problem for a linear system of ordinary differential equations with unstable equilibria. We construct the control under which the solution remains in a neighborhood of an unstable equilibrium however long.

Given a root class \(\mathscr{K}\) of groups, we prove that the tree product of residually \(\mathscr{K}\)-groups with amalgamated retracts is a residually \(\mathscr{K}\)-group. This yields a criterion for the \(\mathscr{K}\)-residuality of Artin and Coxeter groups with tree structure. We also prove that the HNN-extension X of a residually \(\mathscr{K}\)-group B is a residually \(\mathscr{K}\)-group provided that the associated subgroups of X are retracts in B and \(\mathscr{K}\) contains at least one nonperiodic group.

Let σ = {σ_i | i ∈ I} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ σ, and let G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall Π-set of G if each member ≠ 1 of ℋ is a Hall σ_i-subgroup of G for some σ_i ∈ Π and ℋ has exactly one Hall σ_i-subgroup of G for every σ_i ∈ Π such that σ_i ∩ π(G) ≠ ∅. A subgroup A of G is called (i) Π-permutable in G if G has a complete Hall Π-set ℋ such that AH^x = H^xA for all H ∈ ℋ and x ∈ G; (ii) σ-subnormal in G if there is a subgroup chain A = A₀ ≤ A₁ ≤ ⋯ ≤ A_t = G such that either A_i−1 ≤ A_i or A_i/(A_i−1)A_i is a σ_k-group for some k for all i = 1,…,t; and (iii) strongly Π-permutable if A is Π-permutable and σ-subnormal in G. We study the strongly Π-permutable subgroups of G. In particular, we give characterizations of these subgroups and prove that the set of all strongly Π-permutable subgroups of G forms a sublattice of the lattice of all subgroups of G.

We consider 3-generated lattices whose generators are distributive, dually distributive, right modular, dually right modular elements, or elements possessing a combination of these properties. For these lattices, we find all triples of generators that suffice for the generated lattice to be finite.