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- ДокументProperties of 2-CNF mutually dual and self-dual T_0 -topologies on a finite set and calculation of T_0-topologies of a certain weight(2022) Anna Skryabina, Polina Stegantseva, Nadia BashovaThe problem of counting non-homeomorphic topologies as well as all topologies on an n-elements set is still open. The topologies with the weight k>2n-1, where k is the number of the elements of the topology on an n-elements set, which are called close to the discrete topology have been studied completely. Moreover R.~Stanley in 1971, M.~Kolli in 2007 and in 2014 have been found the number of T0-topologies on an n-elements set with weights k≥7·2n-4, k ≥3·2n-3, and k≥5·2n-4 respectively. In the present paper we investigate T0-topologies using the topology vector, being an ordered set of the nonnegative integers that define the minimal neighborhoods of the elements of the given finite set, and also using the special form of 2-CNF of Boolean function. In 2021 the authors found the form of the vector of T0-topologies with k≥5·2n-4 and the values k∈[5·2n-4, 2n-1], for which there are no T0-topologies with the weight k. The method of describing of T0-topologies using the special form of 2-CNF of Boolean function is used for the identification of the mutually dual and self-dual T0-topologies, and the properties of such 2-CNF Boolean function are used for counting T0-topologies with the weight 25·2n-6.
- ДокументFlows with minimal number of singularities in the Boy's surface(2022) Luca Di Beo, Alexandr Olegovich PrishlyakWe study flows on the Boy's surface. The Boy's surface is the image of the projective plane under a certain immersion into the three-dimensional Euclidean space. It has a natural stratification consisting of one 0-dimensional stratum (central point), three 1-dimensional strata (loops starting at this point), and four 2-dimensional strata (three of them are disks lying on the same plane as the 1-dimensional strata, and having the loops as boundaries). We found all 342 optimal Morse-Smale flows and all 80 optimal projective Morse-Smale flows on the Boy's surface.
- ДокументA dynamical approach to shape(2022) Martin ShoptrajanovIn this paper we will discuss a dynamical approach to an open problem from shape theory. We will address the problem in compact metric spaces using the notion of Lebesgue number for a covering and the intrinsic approach to strong shape.
- ДокументOn closed weakly m-convexsets(2022) Тетяна ОсіпчукIn the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n>1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n>2 and any m=1,...,n-1.
- ДокументOn symplectic invariants of planar 3-webs(2022) Nadiia KonovenkoThe classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. In this paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [4], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants.