Pub Date : 2023-12-15DOI: 10.15673/pigc.v16i3.2646
M. Pahirya, Yuliia Myslo
The problem of interpolation of the function of a complex variable at a point of a compact set by the Thiele-Hermite continued fraction is investigated. Formulas for calculating the coefficients of the continued fraction based on values of the function and its derivates at a point are obtained. Several examples of computations are provided.
{"title":"A certain method of construction of Thiele-Hermite continued fraction at a point","authors":"M. Pahirya, Yuliia Myslo","doi":"10.15673/pigc.v16i3.2646","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2646","url":null,"abstract":"The problem of interpolation of the function of a complex variable at a point of a compact set by the Thiele-Hermite continued fraction is investigated. Formulas for calculating the coefficients of the continued fraction based on values of the function and its derivates at a point are obtained. Several examples of computations are provided.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"16 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139000807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.15673/pigc.v16i3.2512
B. I. Hladysh, A. O. Prishlyak
To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighbourhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functions with fore critical points, on a closed 3-manifold, was constructed. A criterion for the topological equivalence of functions with a finite number of critical points on 3-manifolds is given.
{"title":"Topological structure of functions with isolated critical points on a 3-manifold","authors":"B. I. Hladysh, A. O. Prishlyak","doi":"10.15673/pigc.v16i3.2512","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2512","url":null,"abstract":"To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighbourhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functions with fore critical points, on a closed 3-manifold, was constructed. A criterion for the topological equivalence of functions with a finite number of critical points on 3-manifolds is given.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"105 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136352842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.15673/pigc.v16i3.2304
Liliana Jurado, Bruno Scardua
In this paper we study transversely holomorphic foliations of complex codimension one with a transversely homogeneous complex transverse structure. We prove that the only cases are the transversely additive, affine and projective cases. We shall focus on the transversely affine case and describe the holonomy of a leaf which is "at the infinity" with respect to this structure and prove this is a solvable group. Using this we are able to prove linearization results for the foliation under the assumption of existence of some hyperbolic map in the holonomy group. Such foliations will then be given by simple-poles closed transversely meromorphic one-forms.
{"title":"On transversely holomorphic foliations with homogeneous transverse structure","authors":"Liliana Jurado, Bruno Scardua","doi":"10.15673/pigc.v16i3.2304","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2304","url":null,"abstract":"In this paper we study transversely holomorphic foliations of complex codimension one with a transversely homogeneous complex transverse structure. We prove that the only cases are the transversely additive, affine and projective cases. We shall focus on the transversely affine case and describe the holonomy of a leaf which is \"at the infinity\" with respect to this structure and prove this is a solvable group. Using this we are able to prove linearization results for the foliation under the assumption of existence of some hyperbolic map in the holonomy group. Such foliations will then be given by simple-poles closed transversely meromorphic one-forms.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"105 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136352841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.15673/pigc.v16i3.2536
Umed Karimov
In [S. Kermit, Proc. Amer. Math. Soc., 1972, 31(1):271-275] it was proven that if G is compact topological group or field then in the homotopy axiom for Alexander-Spanier-Kolmogoroff cohomology the parameter segment [0;1] can be replaced by any compact connected space T. The purpose of the paper is to show that the parameter space T can not be replaced in general by locally compact connected space.
{"title":"On generalization of homotopy axiom","authors":"Umed Karimov","doi":"10.15673/pigc.v16i3.2536","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2536","url":null,"abstract":"In [S. Kermit, Proc. Amer. Math. Soc., 1972, 31(1):271-275] it was proven that if G is compact topological group or field then in the homotopy axiom for Alexander-Spanier-Kolmogoroff cohomology the parameter segment [0;1] can be replaced by any compact connected space T. The purpose of the paper is to show that the parameter space T can not be replaced in general by locally compact connected space.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"69 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135036881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-12DOI: 10.15673/pigc.v16i3.2576
Irina Kurbatova, Margaret Pistruil, Nadiia Konovenko
In previous papers we studied mappings of pseudo-Riemannian spaces being mutually quasi-geodesic and almost geodesic of the 2nd type. As a result, we arrived at the quasi-geodesic mapping f: (Vn, gij, Fih) → (Vn, gij, Fih) of spaces with an affine structure, which was called generalized-recurrent. Quasi-geodesic mappings are divided into two types: general and canonical. In this article, the fundamental issues of the theory of quasi-geodesic mappings of generalized-recurrent-parabolic spaces are considered. First, the fundamental equations of quasi-geodesic mappings are reduced to a form that allows effective investigation. Then, using a new form of the fundamental equations, we prove theorems that allow for any generalized-recurrent-parabolic space (Vn, gij, Fih) or to find all spaces (Vn, gij, Fih) onto which Vn admits a quasi-geodesic mapping of the general form, or prove that there are no such spaces.
{"title":"Fundamental theorems of quasi-geodesic mappings of generalized-recurrent-parabolic spaces","authors":"Irina Kurbatova, Margaret Pistruil, Nadiia Konovenko","doi":"10.15673/pigc.v16i3.2576","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2576","url":null,"abstract":"In previous papers we studied mappings of pseudo-Riemannian spaces being mutually quasi-geodesic and almost geodesic of the 2nd type. As a result, we arrived at the quasi-geodesic mapping f: (Vn, gij, Fih) → (Vn, gij, Fih) of spaces with an affine structure, which was called generalized-recurrent. Quasi-geodesic mappings are divided into two types: general and canonical. In this article, the fundamental issues of the theory of quasi-geodesic mappings of generalized-recurrent-parabolic spaces are considered. First, the fundamental equations of quasi-geodesic mappings are reduced to a form that allows effective investigation. Then, using a new form of the fundamental equations, we prove theorems that allow for any generalized-recurrent-parabolic space (Vn, gij, Fih) or to find all spaces (Vn, gij, Fih) onto which Vn admits a quasi-geodesic mapping of the general form, or prove that there are no such spaces.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"70 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135036868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-25DOI: 10.15673/pigc.v16i3.2519
M. Pratsiovytyi, Dmytro Karvatskyi
We study properties of the set of subsums for convergent series k1 sin x + ... + km sin x + ... + k1 sin x[(n-1)/m+1] + ... + km sin x[(n-1)/m+1] + ... where k1, k2, k3, ..., km are fixed positive integers and 0
我们研究收敛级数 k1 sin x + ...+ km sin x + ...+ k1 sin x[(n-1)/m+1] + ...+ km sin x[(n-1)/m+1] + ... 其中 k1、k2、k3、...、km 为固定的正整数，0
{"title":"Cantorvals as sets of subsums for a series connected with trigonometric functions","authors":"M. Pratsiovytyi, Dmytro Karvatskyi","doi":"10.15673/pigc.v16i3.2519","DOIUrl":"https://doi.org/10.15673/pigc.v16i3.2519","url":null,"abstract":"We study properties of the set of subsums for convergent series k1 sin x + ... + km sin x + ... + k1 sin x[(n-1)/m+1] + ... + km sin x[(n-1)/m+1] + ... where k1, k2, k3, ..., km are fixed positive integers and 0<x<1. It is proved that depending on the parameter x this set can be a finite union of closed intervals or Cantor-type set or even Cantorval.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139349223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-10DOI: 10.15673/pigc.v13i4.2588
S. Maksymenko, Olexander Prishlyak, Nadiia Konovenko
This note devoted to Volodymyr Vasylyovych Sharko (25.09.1949-07.10.2014)
这篇文章献给弗拉基米尔·瓦西里耶维奇·夏尔科(1949年9月25日- 2014年10月7日)
{"title":"International Conference Morse theory and its applications dedicated to the memory and 70th anniversary of Volodymyr Sharko (25.09.1949-07.10.2014)","authors":"S. Maksymenko, Olexander Prishlyak, Nadiia Konovenko","doi":"10.15673/pigc.v13i4.2588","DOIUrl":"https://doi.org/10.15673/pigc.v13i4.2588","url":null,"abstract":"This note devoted to Volodymyr Vasylyovych Sharko (25.09.1949-07.10.2014)","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83519735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-03DOI: 10.15673/pigc.v16i2.2510
D. Bolotov
The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.
{"title":"On foliations of bounded mean curvature on closed three-dimensional Riemannian manifolds","authors":"D. Bolotov","doi":"10.15673/pigc.v16i2.2510","DOIUrl":"https://doi.org/10.15673/pigc.v16i2.2510","url":null,"abstract":"The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91376016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-03DOI: 10.15673/pigc.v16i2.2336
P. Mormul
The construction of the geometric singularity classes of special multi-flags was exhaustively done in [P. Mormul, SIGMA, 5:Paper 102, 22 pages, 2009] for special 2-flags, i.e. when flag's width, typically denoted by m, was 2. Now analogous singularity classes are being constructed for special flags of all widths m≥2, compatible with and extending by far the construction in the mentioned paper.
{"title":"Singularity classes of special multi-flags, I","authors":"P. Mormul","doi":"10.15673/pigc.v16i2.2336","DOIUrl":"https://doi.org/10.15673/pigc.v16i2.2336","url":null,"abstract":"The construction of the geometric singularity classes of special multi-flags was exhaustively done in [P. Mormul, SIGMA, 5:Paper 102, 22 pages, 2009] for special 2-flags, i.e. when flag's width, typically denoted by m, was 2. Now analogous singularity classes are being constructed for special flags of all widths m≥2, compatible with and extending by far the construction in the mentioned paper.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81846810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cohomology algebra of mapping spaces between quaternion Grassmannians","authors":"O. Maphane","doi":"10.15673/pigc.v16i2.2453","DOIUrl":"https://doi.org/10.15673/pigc.v16i2.2453","url":null,"abstract":"Let Gk,n(ℍ) for 2≤k<n denote the quaternion Grassmann manifold of k-dimensional vector subspaces of ℍn. In this paper we compute, in terms of the Sullivan models, the rational cohomology algebra of the component of the inclusion i: Gk,n(ℍ) → Gk,n+r(ℍ) in the space of mappings from Gk,n(ℍ) to Gk,n+r(ℍ) for r≥1 and, more generally, we show that the cohomology of Map(Gk,n(ℍ),Gk,n+r(ℍ);i) contains a truncated algebra ℚ[x]x4r+n+k^{2}-nk for n≥4.","PeriodicalId":36547,"journal":{"name":"Proceedings of the International Geometry Center","volume":"156 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76447516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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