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Vol 212, No 10 (2021)
3-15
The degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional manifolds or Poincare complexes and their applications
Abstract
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.
Matematicheskii Sbornik. 2021;212(10):16-75
16-75
The regularized asymptotics of a solution of the Cauchy problem in the presence of a weak turning point of the limit operator
Abstract
An asymptotic solution of the linear Cauchy problem in the presence of a ‘weak’ turning point of the limit operator is built using Lomov's regularization method. The major singularities of the problem are written out in an explicit form. Estimates are given with respect to $\varepsilon$, which characterise the behaviour of the singularities as $\varepsilon\to 0$. The asymptotic convergence of the regularized series is proved. The results of the work are illustrated by an example. Bibliography: 8 titles.
Matematicheskii Sbornik. 2021;212(10):76-95
76-95
Asymptotics of the scattering operator for the wave equation in a singularly perturbed domain
Abstract
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.
Matematicheskii Sbornik. 2021;212(10):96-130
96-130
Slide polynomials and subword complexes
Abstract
Subword complexes were defined by Knutson and Miller in 2004 to describe Gröbner degenerations of matrix Schubert varieties. Subword complexes of a certain type are called pipe dream complexes. The facets of such a complex are indexed by pipe dreams, or, equivalently, by monomials in the corresponding Schubert polynomial. In 2017 Assaf and Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes. The slide complexes appearing in such a way are shown to be homeomorphic to balls or spheres. For pipe dream complexes, such strata correspond to slide polynomials. Bibliography: 14 titles.
Matematicheskii Sbornik. 2021;212(10):131-151
131-151
A Littlewood-Paley-Rubio de Francia inequality for bounded Vilenkin systems
Abstract
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.
Matematicheskii Sbornik. 2021;212(10):152-164
152-164