Vol 59, No 2 (2018)

We show that each Ptolemaic semimetric is Möbius-equivalent to a bounded metric. Introducing generalized angles in Ptolemaic Möbius structures, we study the class of multivalued mappings F: X → 2Y with a lower bound on the distortion of generalized angles. We prove that the inverse mapping to the coordinate function of a quasimeromorphic automorphism of ℝ̅ⁿ lies in this class.

Studying the model-theoretic properties that are preserved under expansion of the models of countably categorical weakly o-minimal theories of finite convexity rank with convex unary predicates, we show that countable categoricity and convexity rank are among these properties.

The new class of functions almost periodic at infinity is defined using the subspace of functions with integrals decreasing at infinity. We obtain spectral criteria for almost periodicity at infinity of bounded solutions to differential equations with unbounded operator coefficients. For the new class of asymptotically finite operator semigroups we prove the almost periodicity at infinity of their orbits.

Studying computable representations of projective planes, for the classes K of pappian, desarguesian, and all projective planes, we prove that K^c/≃ admits no hyperarithmetical Friedberg enumeration and admits a Friedberg Δ⁰_α+3-computable enumeration up to a Δ⁰_α-computable isomorphism.

An F-compactum or a Fedorchuk compactum is a Hausdorff compact space that admits decomposition into a special well-ordered inverse system with fully closed neighboring projections. We prove that the square of Aleksandroff’s “double arrow” space is not an F-compactum of countable spectral height. Using this, we demonstrate the impossibility of representing the Helly space as the inverse limit of a countable system of resolutions with metrizable fibers. This gives a negative answer to a question posed by Watson in 1992.

Chigogidze proposed a construction of extending a normal functor from the category Comp to the category Tych. We can apply his scheme to seminormal functors and study the properties of the original functor which are preserved under extension. We introduce the concept of functor having an invariant extension from Comp to Tych because the very existence of this invariance plays a key role in the preservation of the properties of a seminormal functor in its extension. It is proved that the superextension functor λ has an invariant extension. We check that if a seminormal functor has an invariant extension then its extension preserves a point, the empty set, intersection and is a monomorphic functor. If this functor has finite degree then its extension is continuous and hence a seminormal functor in Tych. If the functor is of infinite degree then continuity may be lost. Namely, we show that the extension of λ for Tych is not continuous.

Studying computable representations of projective planes, we prove that the isomorphism problem in the class of free projective planes of finite rank is an m-complete Δ⁰₃-set within the class.

We find the groups of motions of eight three-dimensional maximal mobility geometries. These groups are actions of just three Lie groups SL₂(R)ΔN, SL₂(C)_R, and SL₂(R)⊗SL₂(R) on the space R³, where N is a normal abelian subgroup. We also find explicit expressions for these actions.

We study the overalgebras and the ideals of the Jordan algebras possessing prime (−1, 1)-envelopings. If a Jordan algebra possesses a prime nonassociative (−1, 1)-enveloping then we prove that it is also prime; furthermore, its every ideal is a prime algebra. In particular, the overalgebras and metaideals of Jordan monsters are prime.

We describe the periodic groups whose endomorphism rings satisfy the annihilator condition for the principal left ideals generated by nilpotent elements. We prove that torsion-free reduced separable, vector, and algebraically compact groups have endomorphism rings with the annihilator condition for the principal left (right) ideals generated by nilpotent elements if and only if these rings are commutative. We show that the almost injective groups (in the sense of Harada) are injective, i.e. divisible.