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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.
- 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.
- 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.
- 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.
- ДокументRelative Gottlieb groups of mapping spaces and their rational cohomology(2022) Abdelhadi ZaimLet f:X →Y be a map of simply connected CW-complexes of finite type. Put maxπ★(Y)⊗Q = max{ i | πi(Y)⊗Q≠0 }. In this paper we compute the relative Gottlieb groups of f when X is an F0-space and Y is a product of odd spheres. Also, under reasonable hypothesis, we determine these groups when X is a product of odd spheres and Y is an F0-space. As a consequence, we show that the rationalized G-sequence associated to f splits into a short exact sequence. Finally, we prove that the rational cohomology of map(X,Y;f) is infinite dimensional whenever maxπ★(Y)⊗Q is even.