Vol 71, No 4 (2016)

The paper presents a complete description of topologically injective, topologically surjective, isometric and coisometric multiplication operators by a function acting between L_p-spaces of σ-finite measure spaces. It is proved that all such operators are invertible from the right and from the left. As a corollary it is proved that in the category consisting of L_p-spaces for all p ∈ [1,+∞] considered as left Banach modules over the algebra of bounded measurable functions, all objects are metrically and topologically projective, injective, and flat modules.

The paper presents the methods describing the behavior of lines of a three-dimensional vector field near null points of order greater than one assuming that the potential of that field satisfies the Laplace equation. Some general analytic solutions are presented for the second order case.

The problem of completeness of arbitrary systems of monadic predicates defined on finite sets is considered. Completeness criteria are obtained for an arbitrary system of monadic predicates over arbitrary set of Boolean functions.

A topological space X is uniformly normal if the family U of all symmetric neighborhoods of the diagonal Δ ⊂ X × X forms a uniformity on X. A neighborhood of the diagonal is any subset whose interior contains the diagonal. It is proved that the Σ-product of Lindelöf p-spaces of countable tightness is uniformly normal.