Vol 60, No 5 (2019)

The authors study the class of finite homogeneous metric spaces and some of its important subclasses that have natural definitions in terms of the metrics and well-studied analogs in the class of Riemannian manifolds. The relationships between these classes are explored. The examples of the corresponding spaces are built, some of which are the vertex sets of the special convex polytopes in Euclidean space. We describe the classes on using the language of graph theory, which enables us to provide some examples of finite metric spaces with unusual properties. Several unsolved problems are posed.

We study finite nonsoluble generalized Frobenius groups; i.e., the groups G with a proper nontrivial normal subgroup F such that each coset Fx of prime order p, as an element of the quotient group G/F, consists only of p-elements. The particular example of such a group is a Frobenius group, given that F is the Frobenius kernel of G, and also the Camina group.

Under study are the systems of nonlinear delay differential equations of neutral type with periodic coefficients of linear terms. We establish sufficient conditions of exponential stability of the zero solution, point out the attraction domains of the zero solution, and provide estimates for solutions characterizing the stabilization rate at infinity.

We give two analogs of Korn’s inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn’s inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.

This paper discusses some results concerning the validity of the Luzin N and N⁻¹ properties for homeomorphisms. Also, we present a counterexample in the form of a C^∞-homeomorphism that violates the N⁻¹ property.

The paper contains two main results that are obtained by using Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices onto the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of AL^p and c₀ (Γ) for the class of the so-called \(\mathbb{B}\)-cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.

Considering regular mappings of Euclidean spaces, we study the distortion of the Hausdorff dimension of a given set under restrictions on the rank of the gradient on the set. This problem was solved for the classical cases of k-smooth and Hölder mappings by Dubovitskii, Bates, and Moreira. We solve the problem for Sobolev and fractional Sobolev classes as well. Here we study the Sobolev case under minimal integrability assumptions that guarantee in general only the continuity of a mapping (rather than differentiability everywhere). Some new facts are found out in the classical smooth case. The proofs are mostly based on our previous joint papers with Bourgain and Kristensen (2013, 2015).