Vol 60, No 6 (2019)

We study the modules that are coinvariant under the idempotent endomorphisms of their covers. Some generalizations of discrete and continuous modules are introduced and inspected on using the theory of covers and envelopes of modules. By way of application, we consider the cases of flat covers, injective envelopes, and pure injective envelopes.

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 ρ_φ(P, Q) = φ(∣P − Q∣) and d_φ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρ_φ(P, Q) ≤ d_φ(P, Q) and ρ_φ(P, Q) = d_φ(P, Q) provided that PQ = QP. The mapping ρ_φ (or d_φ) meets the triangle inequality if and only if φ is a tracial functional. If τ is a faithful tracial functional then ρ_τ and d_τ are metrics on \(\mathscr{A}^{\rm{pr}}\). Moreover, if τ is normal then (\(\mathscr{A}^{\rm{pr}}\), ρ_τ) and (\(\mathscr{A}^{\rm{pr}}\), d_τ) are complete metric spaces. Convergences with respect to ρ_τ and d_τ are equivalent if and only if \(\mathscr{A}\) is abelian; in this case ρ_τ = d_τ. We give one more criterion for commutativity of \(\mathscr{A}\) in terms of inequalities.

We study the convergence of interpolation processes by Subbotin polynomial splines of even degree. We prove that the good conditionality of a system of equations for constructing an interpolation spline via the coefficients of the expansion of the kth derivative in B-splines is equivalent to the convergence of the interpolation process for the kth derivative of the spline in the class of functions with continuous kth derivative.

We estimate the complexity of constructing a punctual “online” copy of an algebraic structure. We establish a general upper bound as well as optimal bounds for classes of Boolean algebras, abelian p-groups, and linear orders. Moreover, the methods developed here are applied to solving Montalbán’s open problem (2013) about copyable linear orders.

Considering the level surfaces of the mappings of class C_H¹ which are defined on Carnot manifolds and take values in Carnot—Carathéodory spaces, we introduce some adequate local metric characteristic that bases on a correspondence with a neighborhood of the kernel of the sub-Riemannian differential. Moreover, for the mappings on Carnot groups we construct an adapted basis in the preimage which matches local sub-Riemannian structures on the complement of the kernel of the sub-Riemannian differential (including those meeting the level set) and on the arrival set.

We study the possibilities of defining some operations on fields via the remaining operations. In particular, we prove that multiplication on an arbitrary field can be defined via addition if and only if the field is a finite extension of its prime subfield. We give a sufficient condition for the nondefinability of addition via multiplication and demonstrate that multiplication and addition on the reals and complexes cannot be mutually defined by means of the relations with parameters which are preserved under automorphisms. We also describe the mutual definability of addition, multiplication, and exponentiation via the remaining two operations.

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.

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 in GL(2, Z) of maximal primitive solvable subgroups of GL(2, Z), where h is the number of ideal classes in R.

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?”