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Vol 213, No 5 (2022)
- Year: 2022
- Articles: 7
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7489
On tensor fractions and tensor products in the category of stereotype spaces
Abstract
We prove two identities that connect some natural tensor products in the category $\operatorname{LCS}$ of locally convex spaces with tensor products in the category $\operatorname{Ste}$ of stereotype spaces. In particular, we give sufficient conditions under which the identity $$X^\vartriangle\odot Y^\vartriangle\cong (X^\vartriangle\cdot Y^\vartriangle)^\vartriangle\cong (X\cdot Y)^\vartriangle$$holds, where $\odot$ is the injective tensor product in the category $\operatorname{Ste}$, $\cdot $ is the primary tensor product in $\operatorname{LCS}$, and $\vartriangle$ is the pseudosaturation operation in $\operatorname{LCS}$. The study of relations of this type is justified by the fact that they turn out to be important instruments for constructing duality theory based on the notion of an envelope. In particular, they are used in the construction of the duality theory for the class of (not necessarily Abelian) countable discrete groups. Bibliography: 15 titles.
Matematicheskii Sbornik. 2022;213(5):3-29
3-29
Strong convexity of reachable sets of linear systems
Abstract
The reachable set on some time interval of a linear control system $x'\in Ax {+} U$, $x(0)=0$, is considered. A number of cases is examined when the reachable set is the intersection of some balls of fixed radius $R$ (that is, a strongly convex set of radius $R$). In some cases the radius $R$ is estimated from above. It turns out that strong convexity is fairly typical for this class of reachable sets in a certain sense. Among possible applications of this result are the possibility of constructing outer polyhedral approximation of reachable sets with better accuracy in the Hausdorff metric than in the general case, and applications to linear differential games and some optimization problems. Bibliography: 23 titles.
Matematicheskii Sbornik. 2022;213(5):30-49
30-49
Asymptotic behaviour of the sphere and front of a flat sub-Riemannian structure on the Martinet distribution
Abstract
The sphere and front of a flat sub-Riemannian structure on the Martinet distribution are surfaces with nonisolated singularities in three-dimensional space. The sphere is a subset of the front; it is not subanalytic at two antipodal points (the poles). The asymptotic behaviour of the sub-Riemannian sphere and Martinet front are calculated at these points: each surface is approximated by a pair of quasihomogeneous surfaces with distinct sets of weights in a neighbourhood of a pole. Bibliography: 13 titles.
Matematicheskii Sbornik. 2022;213(5):50-67
50-67
Geometry of the Gromov-Hausdorff distance on the class of all metric spaces
Abstract
The geometry of the Gromov-Hausdorff distance on the class of all metric spaces considered up to isometry is studied. The concept of a class in the sense of von Neumann-Bernays-Gödel set theory is used. As in the case of compact metric spaces, continuous curves and their lengths are defined, and the Gromov-Hausdorff distance is shown to be intrinsic on the entire class. As an application, metric segments (classes of points lying between two given points) are considered and their extendability beyond endpoints is examined. Bibliography: 13 titles.
Matematicheskii Sbornik. 2022;213(5):68-87
68-87
On the universality of the zeta functions of certain cusp forms
Abstract
We consider a certain Dirichlet series associated with the zeta function of a normalized Hecke cusp form. It is absolutely convergent on the right of the critical strip. We obtain universality theorems on the approximation of a wide class of analytic functions by shifts of this series. Bibliography: 9 titles.
Matematicheskii Sbornik. 2022;213(5):88-100
88-100
Central extensions and the Riemann-Roch theorem on algebraic surfaces
Abstract
We study canonical central extensions of the general linear group over the ring of adeles on a smooth projective algebraic surface $X$ by means of the group of integers. Via these central extensions and the adelic transition matrices of a rank $n$ locally free sheaf of $\mathcal{O}_X$-modules we obtain a local (adelic) decomposition for the difference of Euler characteristics of this sheaf and the sheaf $\mathcal{O}_X^n$. Two distinct calculations of this difference lead to the Riemann-Roch theorem on $X$ (without Noether's formula). Bibliography: 21 titles.
Matematicheskii Sbornik. 2022;213(5):101-125
101-125
Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic
Abstract
Integral inequalities for integrals of differences of subharmonic functions against Borel measures on balls in multidimensional Euclidean spaces are obtained. These integrals are estimated from above in terms of the product of the Nevanlinna characteristic of the function and various characteristics of the Borel measure and its support. The main theorem, which is a criterion concerning such estimates, presents several equivalent statements of different character. All results are new for the logarithms of the moduli of meromorphic functions on discs in the complex plane. They cover all preceding results, which go back to the classical small arcs lemma of Edrei and Fuchs, as special cases. Integrals against Borel measures with support on fractal sets are also allowed; in this case estimates are in terms of the Hausdorff measure and Hausdorff content of the support of the measure. Special cases of functions on the whole complex plane or space, or in the unit disc or ball, which are important for applications are distinguished, as well as cases involving integration against arc length over subsets of Lipschitz curves and against surface area over subsets of Lipschitz hypersurfaces. Bibliography: 42 titles.
Matematicheskii Sbornik. 2022;213(5):126-166
126-166