The author had earlier defined the concept of an r-group, generalizing the concept of a rigid (solvable) group. This article proves that every metabelian r-group is equationally Noetherian; i.e., each system of equations in finitely many variables with coefficients in the group is equivalent to some finite subsystem.

We study the compressed zero-divisor graph of a finite associative ring R. In particular, we describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, we find all graphs of order 3 that are the compressed zero-divisor graphs of some finite rings.

The Manturov (2, 3)-group G23 is the group generated by three elements a, b, and c with defining relations a2 = b2 = c2 = (abc)2 = 1. We explicitly calculate the Anick chain complex for G23 by algebraic discrete Morse theory and evaluate the Hochschild cohomology groups of the group algebra \(\mathbb{k}G_3^2\) with coefficients in all 1-dimensional bimodules over a field \(\mathbb{k}\) of characteristic

The Cauchy-Dirichlet problem for the anisotropic parabolic equation with variable exponents in the presence of a nonlinear source and gradient term is considered. We prove the existence and uniqueness of a weak solution that is Lipschitz continuous in the space variables.

A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that \(G^{\mathfrak{U}}=\left(G^{\prime}\right)^{\mathfrak{N}}\). Moreover, if the indices of the subgroups A and B of G are coprime then \(G^{\mathfrak{U}}=\lef

We prove some two- and three-point distortion theorems for rational functions that generalize some recent results on Bernstein-type inequalities for polynomials and rational functions. The rational functions under study have either majorants or restrictions on location of their zeros. The proofs are based on the new version of the Schwarz Lemma and univalence condition for regular functions which was

We find sharp asymptotics of the probability that the trajectory of a compound renewal process crosses (or does not cross) an arbitrary remote boundary. In particular, some limit theorems are obtained for the distribution of the maximum of the process in the domain of large deviations. We also give some applications to the classical ruin probability problem in insurance theory.

We show that the Jordan bracket on an associative commutative superalgebra is extendable to the superalgebra of fractions. In particular, we prove that a unital simple abelian Jordan superalgebra is embedded into a simple superalgebra of a Jordan bracket. We also study the unital simple Jordan superalgebras whose even part is a field. We demonstrate that each of these superalgebras is either a superalgebra

We consider a nonautonomous first-order linear differential equation with several delays and nonnegative coefficients. Some new sufficient conditions for the oscillation of all solutions are obtained in the form of an estimate for the lower limit of the sum of integrals of the coefficients. For the equation with one delay, the obtained oscillation conditions sharpen the classical Koplatadze-Chanturiya

We consider the three-dimensional mixed boundary value problem in elasticity about time harmonic oscillations of a semi-infinite anisotropic cylinder. We show that for certain position and shape of the clamping zone of the surface the elastic wave is trapped; i.e., the problem admits a nontrivial solution with exponential decay at infinity or, conversely, the absence of the trapped wave is guaranteed

Studying the elementary properties of free projective planes of finite rank, we prove that for m > n, an arbitrary ∀∃∀-formula Φ(ȳ) and a tuple ū of elements of the free projective plane \(\mathfrak{F}_{n}\) if Φ(ū) holds on the plane \(\mathfrak{F}_{m}\) then Φ(ū) holds on the plane \(\mathfrak{F}_{n}\) too. This implies the coincidence of the ∀∃-theories of free projective planes of different finite

We consider mixed problems for nonlinear equations of magnetoelasticity. Our main result in the three-dimensional case is the proof of an existence and uniqueness theorem; uniqueness is established under some extra restrictions on the smoothness of solutions. We also manage to prove the existence and uniqueness of a weak solution to the problem in the two-dimensional case; uniqueness is established

We construct an additive basis for the relatively free associative algebra F(5)(K) with the Lie nilpotency identity of degree 5 over an infinite domain K containing \({1 \over 6}\). We prove that approximately half of the elements in F(5)(K) are central. We also prove that the additive group of F(5)(ℤ) lacks the elements of simple degree ≥ 5. We find an asymptotic estimation of the codimension of T-ideal

Let An denote the alternating group of degree n. Consider a group G whose spectrum, i.e. the set of element orders, equals the spectrum of A7. Assume that G has a subgroup H isomorphic to A4 whose involutions are squares of elements of order 4. Then either O2(H) ⊆ O2(G)or G has a nonabelian finite simple subgroup.

We prove that the group of tame automorphisms of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra of rank 3 (as well as of a free anticommutative algebra) over an arbitrary Euclidean ring analogous to the Anick automorphism

Studying the solvable subgroups of 2 × 2 matrix groups over Z, we find a maximal finite order primitive solvable subgroup of GL(2, Z) unique up to conjugacy in GL(2, Z). We describe the maximal primitive solvable subgroups whose maximal abelian normal divisor coincides with the group of units of a quadratic ring extension of Z. We prove that every real quadratic ring R determines h classes of conjugacy

We answer (in the negative) Questions 16.15(a) and 17.3 of The Kourovka Notebook: “Is the set of all restricted Engel elements a subgroup in every group?” and “Is it true that a group is binary nilpotent if any 4-element subset therein contains two elements generating a nilpotent subgroup?”

We prove that the Solovay set Σ is absolutely definable in a sufficiently wide sense; in particular, Σ does not depend on the choice of the ground model.

Let φ be a positive functional on a von Neumann algebra \(\mathscr{A}\) and let \(\mathscr{A}^{\rm{pr}}\) be the projection lattice in \(\mathscr{A}\). Given \(P,Q \in \mathscr{A}^{\rm{pr}}\), put ρφ(P, Q) = φ(∣P − Q∣) and dφ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρφ(P, Q) ≤ dφ(P, Q) and ρφ(P, Q) = dφ(P, Q) provided that PQ = QP. The mapping ρφ (or dφ) meets the triangle inequality if and only if φ is a tracial

We propose a general scheme for the search of a fundamental solution to the hypoelliptic diffusion equation in a “sufficiently good” sub-Riemannian manifold and the small-time asymptotics for the solution, which includes the generalized Fourier transform and the orbit method closely related to it, as well as an application of the perturbative method to the nilpotent approximation, and Trotter’s formula