Vol 58, No 4 (2017)

We prove that, in a 6-dimensional Hermitian submanifold of the Cayley algebra, almost contact metric structures are identical on a hypersurface with type number 0 and on a hypersurface with type number 1.

We consider the problem of Subbotin’s parabolic spline interpolation for functions with large gradient domains. In the case of the common piecewise uniform Shishkin’s mesh we obtain two-sided accuracy estimates for the class of functions with exponential boundary layer. The spline interpolation accuracy estimates are not uniform in a small parameter, while the error itself can grow unboundedly as the small parameter vanishes and the number N of nodes remains fixed. We include the results of some simulations.

In 1940, in attempts to solve the Four Color Problem, Henry Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P₅ of 3-polytopes with minimum degree 5. This description depends on 32 main parameters. Very few precise upper bounds on these parameters have been obtained as yet, even for restricted subclasses in P₅. Given a 3-polytope P, denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P by h(P). Jendrol’ and Madaras in 1996 showed that if a polytope P in P₅ is allowed to have a 5-vertex adjacent to four 5-vertices (called a minor (5, 5, 5, 5,∞)-star), then h(P) can be arbitrarily large. For each P^* in P₅ with neither vertices of the degree from 6 to 8 nor minor (5, 5, 5, 5,∞)-star, it follows from Lebesgue’s Theorem that h(P^*) ≤ 17. We prove in particular that every such polytope P^* satisfies h(P^*) ≤ 12, and this bound is sharp. This result is best possible in the sense that if vertices of one of degrees in {6, 7, 8} are allowed but those of the other two forbidden, then the height of minor 5-stars in P₅ under the absence of minor (5, 5, 5, 5,∞)-stars can reach 15, 17, or 14, respectively.

The concepts of weakly injective, fg-weakly injective, and p-weakly injective S-acts generalize that of injective S-act. We study the monoids S over which the classes of weakly injective, fg-weakly injective, and p-weakly injective S-acts are axiomatizable. We prove that the class of p-weakly injective S-acts over a regular monoid is axiomatizable.

The spectrum of a finite group is the set of all its element orders. A finite group G is called critical with respect to a subset ω of natural numbers, if ω coincides with the spectrum of G and does not coincide with the spectrum of any proper section of G. We study the structure of groups isospectral to a simple unitary group PSU(3, 3). In particular, we give a description of the finite groups critical with respect to the spectrum of PSU(3, 3).

Given a homeomorphism ϕ ∈ W_M¹, we determine the conditions that guarantee the belonging of the inverse of ϕ in some Sobolev–Orlicz space W_F¹. We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.

We study the equivalence relations between functional clones which are based on the coincidence of their quantifier-free formal subsets of the corresponding universal algebras.

Given an arbitrary profinite group G and a commutative domain R, we define the notion of permutation RG-module which generalizes the known notion from the representation theory of profinite groups. We establish an independence theorem of such a module as an R-module over a ring of scalars.

Under consideration is the stationary system of equations of electrodynamics relating to a nonmagnetic nonconducting medium. We study the problem of recovering the permittivity coefficient ε from given vectors of electric or magnetic intensities of the electromagnetic field. It is assumed that the field is generated by a point impulsive dipole located at some point y. It is also assumed that the permittivity differs from a given constant ε₀ only inside some compact domain Ω ⊂ R³ with smooth boundary S. To recover ε inside Ω, we use the information on a solution to the corresponding direct problem for the system of equations of electrodynamics on the whole boundary of Ω for all frequencies from some fixed frequency ω₀ on and for all y ∈ S. The asymptotics of a solution to the direct problem for large frequencies is studied and it is demonstrated that this information allows us to reduce the initial problem to the well-known inverse kinematic problem of recovering the refraction index inside Ω with given travel times of electromagnetic waves between two arbitrary points on the boundary of Ω. This allows us to state uniqueness theorem for solutions to the problem in question and opens up a way of its constructive solution.

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclusions. In contrast to assuming that the right-hand side of the inclusion is Lipschitz with respect to the phase variable in the Hausdorff metric, which is traditional in studying this type of questions, we use the (ρ–H) Lipschitz property. Some example is given.