Summary: The structure group of an involutive set-theoretic solution to the Yang-Baxter equation is a generalized radical ring called a brace . The concept of brace is extended to that of a quasiring where the adjoint group is just a monoid. It is proved that a special class of lattice-ordered quasirings characterizes the divisor group A of a smooth non-commutative curve X . The multiplicative monoid A ◦ of A is related to the additive group by a bijective 1-cocycle. Extending previous results on non-commutative arithmetic, the elements of A are represented as a class Φ ( X ) of self-maps of a universal cover of X . For affine subsets U of X , the regular functions on U form a hereditary order such that the monoid of fractional ideals embeds into A ◦ as the class of monotone functions in Φ ( U ) . The unit group of A is identified with the annular symmetric group , which occurred in connection with quasi-Garside groups of Euclidean type. The main part of the paper is self-contained and provides a quick approach to non-commutative prime factorization and its relationship to braces.
{"title":"Bijective 1-cocycles, braces, and non-commutative prime factorization","authors":"W. Rump","doi":"10.4064/cm8684-2-2022","DOIUrl":"https://doi.org/10.4064/cm8684-2-2022","url":null,"abstract":"Summary: The structure group of an involutive set-theoretic solution to the Yang-Baxter equation is a generalized radical ring called a brace . The concept of brace is extended to that of a quasiring where the adjoint group is just a monoid. It is proved that a special class of lattice-ordered quasirings characterizes the divisor group A of a smooth non-commutative curve X . The multiplicative monoid A ◦ of A is related to the additive group by a bijective 1-cocycle. Extending previous results on non-commutative arithmetic, the elements of A are represented as a class Φ ( X ) of self-maps of a universal cover of X . For affine subsets U of X , the regular functions on U form a hereditary order such that the monoid of fractional ideals embeds into A ◦ as the class of monotone functions in Φ ( U ) . The unit group of A is identified with the annular symmetric group , which occurred in connection with quasi-Garside groups of Euclidean type. The main part of the paper is self-contained and provides a quick approach to non-commutative prime factorization and its relationship to braces.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70142180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An improvement of the pinned distance set problem\u0000in even dimensions","authors":"Zijian Wang, Jiqiang Zheng","doi":"10.4064/cm8632-10-2021","DOIUrl":"https://doi.org/10.4064/cm8632-10-2021","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70141836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A simple proof of the real Riesz–Thorin interpolation theorem in the lower triangle","authors":"L. Maligranda","doi":"10.4064/cm8942-10-2022","DOIUrl":"https://doi.org/10.4064/cm8942-10-2022","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70144390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We study rigidity of critical metrics for quadratic curvature functions F t,s ( g ) involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor. In particular, when s = 0 , we give new characterizations by pointwise inequali-ties involving the Weyl curvature and the traceless Ricci tensor for critical metrics with divergence-free Cotton tensor. We also provide a few rigidity results for locally conformally flat critical metrics. Secondary 53C21.
{"title":"Rigidity of critical metrics for quadratic curvature functions on closed Riemannian manifolds","authors":"B. Ma, Guangyue Huang","doi":"10.4064/cm8236-6-2021","DOIUrl":"https://doi.org/10.4064/cm8236-6-2021","url":null,"abstract":". We study rigidity of critical metrics for quadratic curvature functions F t,s ( g ) involving the scalar curvature, the Ricci curvature and the Riemannian curvature tensor. In particular, when s = 0 , we give new characterizations by pointwise inequali-ties involving the Weyl curvature and the traceless Ricci tensor for critical metrics with divergence-free Cotton tensor. We also provide a few rigidity results for locally conformally flat critical metrics. Secondary 53C21.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70137016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Capulín, E. Castañeda-Alvarado, L. Juárez-Villa, David Maya
. We introduce the concept of g-growth hyperspace: if X is a continuum, then a non-empty subset H of 2 X is a g-growth hyperspace of X provided that if A is a subcontinuum of 2 X and A∩H (cid:54) = ∅ , then (cid:83) A ∈ H . We study pseudo-homotopies between maps of hyperspaces of continua. As a consequence, we show that pseudo-contractibility and contractibility are equivalent in g-growth hyperspaces.
{"title":"Pseudo-homotopies between maps on\u0000g-growth hyperspaces of continua","authors":"F. Capulín, E. Castañeda-Alvarado, L. Juárez-Villa, David Maya","doi":"10.4064/cm8254-7-2021","DOIUrl":"https://doi.org/10.4064/cm8254-7-2021","url":null,"abstract":". We introduce the concept of g-growth hyperspace: if X is a continuum, then a non-empty subset H of 2 X is a g-growth hyperspace of X provided that if A is a subcontinuum of 2 X and A∩H (cid:54) = ∅ , then (cid:83) A ∈ H . We study pseudo-homotopies between maps of hyperspaces of continua. As a consequence, we show that pseudo-contractibility and contractibility are equivalent in g-growth hyperspaces.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70137077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a certain determinant for a U.F.D.","authors":"Guangyan Zhu","doi":"10.4064/cm8722-1-2022","DOIUrl":"https://doi.org/10.4064/cm8722-1-2022","url":null,"abstract":"","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70142669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Erdős proved in 1946 that if a set E ⊂ R n is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in R n with the property that the nearest point in E is not unique, can be covered by countably many surfaces, each of finite ( n − 1) -dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and C 2 regularity.
{"title":"On an old theorem of Erdős about ambiguous locus","authors":"P. Hajłasz","doi":"10.4064/cm8460-9-2021","DOIUrl":"https://doi.org/10.4064/cm8460-9-2021","url":null,"abstract":". Erdős proved in 1946 that if a set E ⊂ R n is closed and non-empty, then the set, called ambiguous locus or medial axis, of points in R n with the property that the nearest point in E is not unique, can be covered by countably many surfaces, each of finite ( n − 1) -dimensional measure. We improve the result by obtaining a new regularity result for these surfaces in terms of convexity and C 2 regularity.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"33 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70140735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：