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Vol 211, No 9 (2020)
- Year: 2020
- Articles: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7469
Stable decomposability of matrices over the rational closure of a group algebra of an ordered group
Abstract
Under the assumption that the rational closure of a group algebra of a left-ordered group in the ring of operators of the module of formal Malcev series is a division ring, we find a canonical form of nonsingular matrices of this division ring. Bibliography: 10 titles.
Matematicheskii Sbornik. 2020;211(9):3-23
3-23
Simple tiles and attractors
Abstract
We study self-similar attractors in the space $\mathbb R^d$, that is, self-similar compact sets defined by several affine operators with the same linear part. The special case of attractors when the matrix $M$ of the linear part and the shifts of the affine operators are integer, is well known in the literature due to the many applications in the theory of wavelets and in approximation theory. In this case, if an attractor has measure one it is called a tile. We classify self-similar attractors and tiles in the case when they are either polyhedra or a union of finitely many polyhedra. We obtain a complete description of the matrices $M$ and the digit sets for parallelepiped tiles and for convex tiles in arbitrary dimensions. It is proved that on a two-dimensional plane, every polygonal tile (not necessarily convex) must be a parallelogram. Nontrivial examples of multidimensional tiles which are a finite union of polyhedra are given, and in the case $d=1$ a complete classification is provided for them. Applications to orthonormal Haar systems in $\mathbb R^d$ and to integer univariate tiles are considered. Bibliography: 18 titles.
Matematicheskii Sbornik. 2020;211(9):24-59
24-59
A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients
Abstract
A natural counterpart of Vitushkin's criterion is obtained in the problem of uniform approximation of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficient on compact subsets of $\mathbb R^d$, $d\ge3$. It is stated in terms of a single (scalar) capacity connected with the leading coefficient of the Laurent series. The scheme of approximation uses methods in the theory of singular integrals and, in particular, constructions of certain special Lipschitz surfaces and Carleson measures. Bibliography: 23 titles.
Matematicheskii Sbornik. 2020;211(9):60-104
60-104
Bounded automorphism groups of compact complex surfaces
Abstract
We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kähler manifold of nonnegative Kodaira dimension, always has bounded finite subgroups. Bibliography: 23 titles.
Matematicheskii Sbornik. 2020;211(9):105-118
105-118
Operator $E$-norms and their use
Abstract
We consider a family of equivalent norms (called operator $E$-norms) on the algebra $\mathfrak B(\mathscr H)$ of all bounded operators on a separable Hilbert space $\mathscr H$ induced by a positive densely defined operator $G$ on $\mathscr H$. By choosing different generating operators $G$ we can obtain the operator $E$-norms producing different topologies, in particular,the strong operator topology on bounded subsets of $\mathfrak B(\mathscr H)$.We obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem, which shows that the Stinespring representation of completely positive linear maps is continuous with respect to the energy-constrained norm of complete boundedness on the set of completely positive linear maps and the operator $E$-norm on the set of Stinespring operators.The operator $E$-norms induced by a positive operator $G$ are well defined for linear operators relatively bounded with respect to the operator $\sqrt G$, and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between operator $E$-norms and the standard characteristics of $\sqrt G$-bounded operators. Operator $E$-norms allow us to obtain simple upper bounds and continuity bounds for some functions depending on $\sqrt G$-bounded operators used in applications.Bibliography: 29 titles.
Matematicheskii Sbornik. 2020;211(9):119-152
119-152