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Vol 212, No 4 (2021)
- Year: 2021
- Articles: 8
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7476
Approximation by simple partial fractions in unbounded domains
Abstract
For unbounded simply connected domains $D$ in the complex plane, bounded by several simple curves with regular asymptotic behaviour at infinity, we obtain necessary conditions and sufficient conditions for simple partial fractions (logarithmic derivatives of polynomials) with poles on the boundary of $D$ to be dense in the space of holomorphic functions in $D$ (with the topology of uniform convergence on compact subsets of $D$). In the case of a strip $\Pi$ bounded by two parallel lines, we give estimates for the convergence rate to zero in the interior of $\Pi$ of simple partial fractions with poles on the boundary of $\Pi$ and with one fixed pole. Bibliography: 16 titles.
Matematicheskii Sbornik. 2021;212(4):3-28
3-28
Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform
Abstract
A first-order quasilinear equation with an odd flux function that has a single point of inflexion at zero is studied. A method for constructing sign-alternating discontinuous entropy solutions of this equation, based on the Legendre transform, is proposed.Bibliography: 18 titles.
Matematicheskii Sbornik. 2021;212(4):29-44
29-44
Locally transitive analytic actions of Lie groups on compact surfaces
Abstract
We investigate locally transitive analytic actions of Lie groups on surfaces. We indicate compact surfaces on which locally transitive actions of Lie groups can be globalized (in particular, with certain conditions on the set of fixed points) and describe the schemes of irreducible locally transitive actions of Lie groups on these surfaces. Bibliography: 13 titles.
Matematicheskii Sbornik. 2021;212(4):45-75
45-75
Logical complexity of induced subgraph isomorphism for certain families of graphs
Abstract
We investigate the problem of the most efficient first-order definition of the property of containing an induced subgraph isomorphic to a given pattern graph, which is closely related to the time complexity of the decision problem for this property.We derive a series of new bounds for the minimum quantifier depth of a formula defining this property for pattern graphs on five vertices, as well as for disjoint unions of isomorphic complete multipartite graphs. Moreover, we prove that for any $\ell\geq 4$ there exists a graph on $\ell$ vertices and a first-order formula of quantifier depth at most $\ell-1$ that defines the property of containing an induced subgraph isomorphic to this graph.Bibliography: 12 titles.
Matematicheskii Sbornik. 2021;212(4):76-90
76-90
Homological dimensions of Banach spaces
Abstract
The purpose of this paper is to lay the foundations for the study of the problem of when $\operatorname{Ext}^n(X, Y)=0$ in Banach spaces. We provide a number of examples of couples $X$, $Y$ such that $\operatorname{Ext}^n(X,Y)$ is (or is not) $0$. We show that $\operatorname{Ext}^n(\mathcal K, \mathcal K)\neq 0$ for all $n\in \mathbb{N}$ when $\mathcal K$ is the Kadec space. Inparticular, both the projective and injective dimensions of $\mathcal K$ are infinite.Bibliography: 48 titles.
Matematicheskii Sbornik. 2021;212(4):91-112
91-112
Birational geometry of singular Fano double spaces of index two
Abstract
We describe the birational geometry of Fano double spaces $V\stackrel{\sigma}{\to}{\mathbb P}^{M+1}$ of index 2 and dimension ${\geqslant 8}$ with at most quadratic singularities of rank ${\geqslant 8}$, satisfying certain additional conditions of general position: we prove that these varieties have no structures of a rationally connected fibre space over a base of dimension ${\geqslant2}$, that every birational map $\chi\colon V\dashrightarrow V'$ onto the total space of a Mori fibre space $V'/{\mathbb P}^1$ induces an isomorphism $V^+\cong V'$ of the blow-up $V^+$ of $V$ along $\sigma^{-1}(P)$, where $P\subset {\mathbb P}^{M+1}$ is a linear subspace of codimension 2, and that every birational map of $V$ onto a Fano variety $V'$ with ${\mathbb Q}$-factorial terminal singularities and Picard number 1 is an isomorphism. We give an explicit lower estimate, quadratic in $M$, for the codimension of the set of varieties $V$ that have worse singularities or do not satisfy the conditions of general position. The proof makes use of the method of maximal singularities and the improved $4n^2$-inequality for the self-intersection of a mobile linear system. Bibliography: 20 titles.
Matematicheskii Sbornik. 2021;212(4):113-130
113-130
On continuous endomorphisms of entire functions
Abstract
The paper is concerned with continuous linear operators on the space of entire functions. The properties of such operators that are related to the definition of convolution-type operators in spaces of analytic functions are investigated. Corollaries refining both the approximation theorem for the kernel of a symmetric convolution operator and the dual definition of a differential operator in a complex domain are stated.Bibliography: 20 titles.
Matematicheskii Sbornik. 2021;212(4):131-158
131-158
159-170