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- ДокументFlows with collective dynamics on a sphere(2021) Андрій Прус, Олександр Пришляк, Софія ГуракаIn this article different properties of flow codes are studied and a diagram is constructed as a whole topological invariant of them. In particular, flows with no more than 6 saddles are described. Two types of simple bifurcations: positive and negative – are considered as well. Summarizing the results on compact surfaces with boundary remains an interesting question for future works.
- ДокументSpecial semi-reducible pseudo-Riemannian spaces(2021) Юлія Степанівна Федченко, Олександр Васильович ЛесечкоThe paper contains necessary conditions allowing to reduce matrix tensors of pseudo-Riemannian spaces to special forms called semi-reducible, under assumption that the tensor defining tensor characteristic of semireducibility spaces, is idempotent. The tensor characteristic is reduced to the spaces of constant curvature, Ricci-symmetric spaces and conformally flat pseudo-Riemannian spaces. The obtained results can be applied for construction of examples of spaces belonging to special types of pseudo-Riemannian spaces. The research is carried out locally in tensor shape, without limitations imposed on a sign of a metric.
- ДокументGeodesic mappings of compact quasi-Einstein spaces, II(2021) V. Kiosak, A. Savchenko, O. LatyshThe paper treats geodesic mappings of quasi-Einstein spaces with gradient defining vector. Previously the authors defined three types of these spaces. In the present paper it is proved that there are no quasi-Einstein spaces of special type. It is demonstrated that quasi-Einstein spaces of main type are closed with respect to geodesic mappings. The spaces of particular type are proved to be geodesic $D$-symmetric spaces.
- ДокументOn the geometry of $Diff(S^1)$-pseudodifferential operators based on renormalized traces.(2021) Jean-Pierre MagnotIn this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the corresponding groups with formal pseudodifferential operators are defined. We investigate the relationship of this group with the restricted general linear group $GL_{res}$, we define a right-invariant pseudo-Riemannian metric on it that extends the Hilbert-Schmidt Riemannian metric by the use of renormalized traces of pseudo-differential operators, and we describe classes of remarkable connections.
- ДокументOn inverse problem for tree of Stieltjes strings(2021) Анастасія Ігорівна Дудко, Vyacheslav PivovarchikFor a given metric tree and two strictly interlacing sequences of numbers there exits a distribution of point masses on the edges (which are Stieltjes strings) such that one of the sequences is the spectrum of the spectral problem with the Neumann condition at the root of the tree while the second sequence is the spectrum of the spectral problem with the Dirichlet condition at the root.