Abstract The author previously defined the divisible completion of the solvable Baumslag–Solitar group, described the groups elementarily equivalent to it, constructed an axiomatization of the corresponding theory, and proved its decidability. In the present work, elementary submodels of models of this theory are described, and $ \omega $ -stability is established.

This paper studies the stabilization of a linear delay system. We show that the system is stabilizable if and only if there exists a stabilizing controller with lumped-delay terms. Then a numerical algorithm from the literature can be applied to design such a controller.

We study electrodynamic equations in which the dielectric permittivity and the conductivity of the medium possess “memory.” Owing to this property, the solution to the equations depends on the entire history of the wave propagation process. It is assumed that the kernels of the integral operators modeling the memory effect depend on both spatial and temporal variables, and these kernels can be represented as products of two functions, one depending on the spatial variables and the other on time. The functions depending on the temporal variable are assumed to be known, while those depending on the spatial variables are unknown and have to be determined. We assume that the functions $ p({\mathbf{x}}) $ and $ q({\mathbf{x}}) $ , which correspond to the kernels describing the memory properties of the dielectric permittivity and the conductivity, respectively, are compactly supported, and their supports lie inside a ball $ B $ of finite radius. To solve the inverse problem, we consider the direct problem with completely known kernels and its special solution for a homogeneous medium corresponding to a running delta-shaped wave propagating in the direction $ \nu $ . This wave impinges on an inhomogeneity localized in $ B $ , and on the boundary of this ball we measure the amplitude of the singular part of the solution and the amplitude of the first time derivative of its regular part on the wave front. The corresponding data recorded for various directions $ \nu $ constitute the input information for solving the inverse problem. It is shown that the problems of determining the functions $ p({\mathbf{x}}) $ and $ q({\mathbf{x}}) $ reduce to successive solutions of the well-known problem of X-ray tomography. Hence, the solution to the inverse problem under consideration is unique and can be efficiently found both analytically and numerically.

For a complex polynomial of degree at least two that fixes the origin and has all its critical points on the positive or negative real half-axis, we establish an exact lower bound for the largest modulus of its critical values. This estimate involves the Schwarzian derivative of the polynomial at the origin and is independent of the degree of the polynomial. A similar estimate is given for the case when all the critical points of the polynomial are real and located on opposite sides of the origin.

We prove that the classes of exponentially dense, Cartesian dense, and recursively dense vector subspaces of $ {𝕉}^{𝕅} $ are pairwise distinct.

Let $ \pi $ be a set of primes. A group $ G $ is said to be $ \pi $ -potent if, given an element $ a\in G $ and a positive integer $ \pi $ -number $ n $ dividing the order of $ a $ , there exists a homomorphism from $ G $ onto a finite group that maps $ a $ to an element of order $ n $ . It is proved that if $ G $ is a finitely generated metabelian group (or a residually finite abelian group, or a residually finite nilpotent group of finite rank, or a residually finite metabelian FATR group), then the following statements are equivalent for $ G $ : (1) $ G $ is $ \pi $ -potent; (2) $ G $ has no $ p $ -radicable elements of infinite order for any $ p\in\pi $ ; (3) $ G $ is virtually $ \pi $ -potent. It is also proved that if $ G $ is a residually finite soluble FATR group (or a finitely generated abelian-by-polycyclic group), then conditions (2) and (3) are equivalent for $ G $ .

A linear functional on a real vector space is determined up to a scalar by its zero hyperplane. A linear operator is recovered from its kernel up to a simple multiplier in rather special cases. Nevertheless, some operator analog of the functional case is valid: each operator with target a Kantorovich space, a Dedekind complete vector lattice, is determined up to some orthomorphism by the kernels of its strata. We call this idea Kutateladze’s stratification principle. Two approaches are outlined: a standard approach, based on the concept of a local Hamel basis, and a nonstandard approach, involving Boolean valued analysis. It is clarified under what conditions the principle works. Some new applications to simultaneous inequalities with linear and multilinear operators are also given.

We consider two-step Carnot groups and their model cases and study the existence of graph mappings on them that are Lipschitz in the intrinsic sense. For the general case of two-step Carnot groups, a criterion for the Lipschitz property of a graph mapping is derived in terms of the mapping that models this graph. Examples are given that demonstrate the specificity of the necessary and sufficient conditions established.

Studying finite systems of Diophantine equations over finite configurations, we prove that the consistency problem for such systems is NP-complete.

Given a continuous open finite-to-one mapping $ f $ of a domain $ G $ including a closed set $ E $ , for each natural $ k $ we consider the set $ E(k) $ (possibly empty) of all points in $ E $ at which $ f $ attains a value with multiplicity $ k $ over $ G $ . Suppose that each point of $ E(k) $ has a neighborhood where the restriction of $ f $ to $ E(k) $ is injective, and its inverse mapping is weakly $ (h,H) $ -quasisymmetric. If, moreover, $ f $ is quasiregular outside $ E $ , then it is quasiregular on the entire domain $ G $ . This theorem generalizes the sufficient condition for the removability of closed sets in the class of quasiconformal mappings obtained by Väisälä in 1990.