Vol 58, No 1 (2017)

Let R be a prime ring of characteristic different from 2, let Q be the right Martindale quotient ring of R, and let C be the extended centroid of R. Suppose that G is a nonzero generalized skew derivation of R and f(x₁,..., x_n) is a noncentral multilinear polynomial over C with n noncommuting variables. Let f(R) = {f(r₁,..., r_n): r_i ∈ R} be the set of all evaluations of f(x₁,..., x_n) in R, while A = {[G (f(r₁,..., r_n)), f(r₁,..., r_n)]: r_i ∈ R}, and let C_R(A) be the centralizer of A in R; i.e., C_R(A) = {a ∈ R: [a, x] = 0, ∀_x ∈ A }. We prove that if A ≠ (0), then C_R(A) = Z(R).

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ⊂ SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.

The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.

Given a nonempty set ω of primes and a nonempty formation F of finite groups, we define the F^ω-normalizer in a finite group and study their properties (existence, invariance under certain homomorphisms, conjugacy, embedding, and so on) in the case that F is an ω-local formation. We so develop the results of Carter, Hawkes, and Shemetkov on the F-normalizers in groups.

We prove that a group whose element orders divide 6 and 7 either is locally finite or an extension of a nontrivial elementary abelian 2-group by a group without involutions.

We prove an analog of the Riemann–Roch Theorem for the Dynnikov–Novikov discrete complex analysis.

Studying the space-like graph surfaces of codimension 2 on the five-dimensional sub-Lorentzian structures with two negative directions of distinct degrees, we determine the differential properties of graph mappings and prove the area formula for the corresponding image surfaces.

We consider the boundary value problem describing the steady barotropic motion of a multicomponent mixture of viscous compressible fluids in a bounded three-dimensional domain. We assume that the material derivative operator is common to all components and is defined by the average velocity of the motion, but keep separate velocities of the components in other terms. Pressure is common and depends on the total density. Beyond that we make no simplifying assumptions, including those on the structure of the viscosity matrix; i.e., we keep all terms in the equations, which naturally generalize the Navier–Stokes model of the motion of one-component media. We establish the existence of weak solutions to the boundary value problem.

We consider a new class of narrow orthogonally additive operators in lattice-normed spaces and prove the narrowness of every C-compact norm-laterally-continuous orthogonally additive operator from a Banach–Kantorovich space V into a Banach space Y. Furthermore, every dominated Urysohn operator from V into a Banach sequence lattice Y is also narrow. We establish that the order narrowness of a dominated Urysohn operator from a Banach–Kantorovich space V into a Banach space with mixed norm W implies the order narrowness of the least dominant of the operator.

We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.

Given a nonempty set π of primes, call a nilpotent group π-bounded whenever it has a central series whose every factor F is such that: In every quotient group of F all primary components of the torsion subgroup corresponding to the numbers in π are finite. We establish that if G is a residually π-bounded torsion-free nilpotent group, while a subgroup H of G has finite Hirsh–Zaitsev rank then H is π’-isolated in G if and only if H is separable in G in the class of all finite nilpotent π-groups. By way of example, we apply the results to study the root-class residuality of the free product of two groups with amalgamation.