Vol 60, No 4 (2019)
- Year: 2019
- Articles: 19
- URL: https://journals.rcsi.science/0037-4466/issue/view/10512
Article
Partially Commutative Metabelian Pro-P-Groups
Abstract
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.
The Operator Ln on Quasivarieties of Universal Algebras
Abstract
Let n be an arbitrary natural and let ℳ be a class of universal algebras. Denote by Ln(ℳ) the class of algebras G such that, for every n-generated subalgebra A of G, the coset a/R (a ∈ A) modulo the least congruence R including A × A is an algebra in ℳ. We investigate the classes Ln(ℳ). In particular, we prove that if ℳ is a quasivariety then Ln(ℳ) is a quasivariety. The analogous result is obtained for universally axiomatizable classes of algebras. We show also that if ℳ is a congruence-permutable variety of algebras then Ln(ℳ) is a variety. We find a variety ℘ of semigroups such that L1(℘) is not a variety.
The Partial Clone of Linear Formulas
Abstract
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 Formclonelin(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 Formclonelin(τ, τ′).
Noethericity and Index of a Characteristic Bisingular Integral Operator with Shifts
Abstract
We consider a characteristic bisingular operator with rather arbitrary shifts that decompose into one-dimensional components. We reduce the problem about the Noethericity and index to that about an operator without shifts. The results obtained are straightforwardly applicable to the two-dimensional boundary-value problem with shifts which is a natural generalization of the Haseman and Carleman problems.
Lax Pairs for Linear Hamiltonian Systems
Abstract
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.
Intersections of Three Nilpotent Subgroups of Finite Groups
Abstract
Under study is the conjecture that for every three nilpotent subgroups A, B, and C of a finite group G there are elements x and y such that A ∩ Bx ∩ Cy ≤ F(G), where F(G) is the Fitting subgroup of G. We prove that a counterexample of minimal order to this conjecture is an almost simple group. The proof uses the classification of finite simple groups.
Polyhedral Divisors of Affine Trinomial Hypersurfaces
Abstract
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.
Tests for the Oscillation of Autonomous Differential Equations with Bounded Aftereffect
Abstract
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.
Plane Wave Solutions to the Equations of Electrodynamics in an Anisotropic Medium
Abstract
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.
On Double Wave Type Flows
Abstract
We study the potential double wave equation and the system of spatial double wave equations. In the class of solutions of multiple wave type, these equations are reduced to an ODE and the system of ODEs respectively. We find some exact solutions and obtain formulas for the contact lines of the corresponding double waves with a simple wave, show that in a neighborhood of an arbitrary point in the plane of self-similar variables there exists a special flow of potential double wave type, and construct a spatial double wave type flow around a specified smooth body.
Parametric Control of Solutions to a Linear Evolution Problem in a Neighborhood of an Unstable Equilibrium
Abstract
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.
The Root Class Residuality of the Tree Product of Groups with Amalgamated Retracts
Abstract
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.
The Moduli Space of D-Exact Lagrangian Submanifolds
Abstract
This paper studies the Lagrangian geometry of algebraic varieties. Given a smooth compact simply-connected algebraic variety, we construct a family of finite-dimensional Kähler manifolds whose elements are the equivalence classes of Lagrangian submanifolds satisfying our new D-exactness condition. In connection with the theory of Weinstein structures, these moduli spaces turn out related to the special Bohr-Sommerfeld geometry constructed by the author previously. This enables us to extract from the moduli spaces some stable components and conjecture that they are not only Kähler but also algebraic.
On Strongly Π-Permutable Subgroups of a Finite Group
Abstract
Let σ = {σi | 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 σi-subgroup of G for some σi ∈ Π and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ 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 AHx = HxA for all H ∈ ℋ and x ∈ G; (ii) σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ ⋯ ≤ At = G such that either Ai−1 ≤ Ai or Ai/(Ai−1)Ai is a σk-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.
On Fully Idempotent Homomorphisms of Abelian Groups
Abstract
We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom(A, A) comes down to reduced mixed groups A with dense elementary torsion part.
Finiteness and Infiniteness of 3-Generated Lattices with Distributive Elements Among Generators
Abstract
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.