Volume 212, Nº 10 (2021)

In this paper, using homotopy theoretical methods we study the degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional Poincare complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincare complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree. As an application of mapping degrees, we consider maps between ${(n-1)}$-connected $(2n+1)$-dimensional Poincare complexes with degree $\pm 1$, and give a sufficient condition for these to be homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov's question: when is a map of degree $1$ between manifolds a homeomorphism? For low $n$, we classify, up to homotopy, torsion free $(n-1)$-connected $(2n+1)$-dimensional Poincare complexes. Bibliography: 29 titles.

A family of Cauchy-Dirichlet problems for the wave equations in unbounded domains $\Lambda_{\varepsilon}$ is considered (here $\varepsilon\ge 0$ is a small parameter); a scattering operator $\mathbb{S}_{\varepsilon}$ is associated with each domain $\Lambda_\varepsilon$. For $\varepsilon>0$ the boundaries of $\Lambda_{\varepsilon}$ are smooth, whilw the boundary of the limit domain $\Lambda_{0}$ contains a conical point. The asymptotics of $\mathbb{S}_{\varepsilon}$ as $\varepsilon\to 0$ is determined. Bibliography: 11 titles.

Rubio de Francia proved a one-sided Littlewood-Paley inequality for the square function constructed from an arbitrary system ofdisjoint intervals. Later, Osipov proved a similar inequality for Walsh systems. We prove a similar inequality for more general Vilenkin systems.Bibliography: 11 titles.