Pub Date : 1900-01-01DOI: 10.32523/2616-7182/bulmathenu.2022/1.3
A. Iosevich, S. Mkrtchyan, Tao Shen
We prove that if the Hausdorff dimension of a compact subset E of Rd with d≥2 is sufficiently large, and if G is a star-like graph with two parts, and each of its parts is arigid graph, then the Lebesgue measure in the appropriate dimension, of the set of distances in E specified by the graph is positive. We also prove that ifdimH(E)is sufficiently large, then∫νG(r~t)dνG(~t)>0,whereνGis the measure on the space of distances specified by G which is induced by a Frostman measure. In particular, this means that for any r>0 there exist many configurations encoded by ~t with vertices in E such that the vertices of r~t are also in E.
{"title":"Pinned point configurations and Hausdorff dimension","authors":"A. Iosevich, S. Mkrtchyan, Tao Shen","doi":"10.32523/2616-7182/bulmathenu.2022/1.3","DOIUrl":"https://doi.org/10.32523/2616-7182/bulmathenu.2022/1.3","url":null,"abstract":"We prove that if the Hausdorff dimension of a compact subset E of Rd with d≥2 is sufficiently large, and if G is a star-like graph with two parts, and each of its parts is arigid graph, then the Lebesgue measure in the appropriate dimension, of the set of distances in E specified by the graph is positive. We also prove that ifdimH(E)is sufficiently large, then∫νG(r~t)dνG(~t)>0,whereνGis the measure on the space of distances specified by G which is induced by a Frostman measure. In particular, this means that for any r>0 there exist many configurations encoded by ~t with vertices in E such that the vertices of r~t are also in E.","PeriodicalId":286555,"journal":{"name":"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116482585","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 : 1900-01-01DOI: 10.32523/2616-7182/bulmathenu.2022/2.1
D. Kozybaev, A. Naurazbekova
In the present paper, necessary and sufficient conditions are found under which a coalgebra is a left-symmetriccoalgebra. A method for constructing coalgebras is described. An example of a left-symmetric coalgebra is constructed.Necessary and sufficient conditions are given under which a left-symmetric coalgebra is a Novikov coalgebra. An exampleof a non-locally finite Novikov coalgebra is constructed.
{"title":"The Novikov coalgebras","authors":"D. Kozybaev, A. Naurazbekova","doi":"10.32523/2616-7182/bulmathenu.2022/2.1","DOIUrl":"https://doi.org/10.32523/2616-7182/bulmathenu.2022/2.1","url":null,"abstract":"In the present paper, necessary and sufficient conditions are found under which a coalgebra is a left-symmetriccoalgebra. A method for constructing coalgebras is described. An example of a left-symmetric coalgebra is constructed.Necessary and sufficient conditions are given under which a left-symmetric coalgebra is a Novikov coalgebra. An exampleof a non-locally finite Novikov coalgebra is constructed.","PeriodicalId":286555,"journal":{"name":"BULLETIN of the L N Gumilyov Eurasian National University MATHEMATICS COMPUTER SCIENCE MECHANICS Series","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129734464","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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