Pub Date : 2023-04-09DOI: 10.22405/2226-8383-2023-24-1-203-212
Igor Baskov
We consider the algebra $A^0 (X)$ of polynomial functions on a simplicial complex $X$. The algebra $A^0 (X)$ is the $0$th component of Sullivan's dg-algebra $A^bullet (X)$ of polynomial forms on $X$. Our main interest lies in computing the de Rham cohomology of the algebra $A^0(X)$, that is, the cohomology of the universal dg-algebra $Omega ^bullet _{A^0(X)}$. There is a canonical morphism of dg-algebras $P:Omega ^bullet _{A^0(X)} to A^bullet (X)$. We prove that $P$ is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^0 (X)$ is canonically isomorphic to the cohomology of the simplicial complex $X$ with coefficients in $k$. Moreover, for $k=mathbb{Q}$ the dg-algebra $Omega ^bullet _{A^0 (X)}$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.
{"title":"The de Rham cohomology of the algebra of polynomial functions on a simplicial complex","authors":"Igor Baskov","doi":"10.22405/2226-8383-2023-24-1-203-212","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-203-212","url":null,"abstract":"We consider the algebra $A^0 (X)$ of polynomial functions on a simplicial complex $X$. The algebra $A^0 (X)$ is the $0$th component of Sullivan's dg-algebra $A^bullet (X)$ of polynomial forms on $X$. Our main interest lies in computing the de Rham cohomology of the algebra $A^0(X)$, that is, the cohomology of the universal dg-algebra $Omega ^bullet _{A^0(X)}$. There is a canonical morphism of dg-algebras $P:Omega ^bullet _{A^0(X)} to A^bullet (X)$. We prove that $P$ is a quasi-isomorphism. Therefore, the de Rham cohomology of the algebra $A^0 (X)$ is canonically isomorphic to the cohomology of the simplicial complex $X$ with coefficients in $k$. Moreover, for $k=mathbb{Q}$ the dg-algebra $Omega ^bullet _{A^0 (X)}$ is a model of the simplicial complex $X$ in the sense of rational homotopy theory.","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90717325","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-89-103
O. I. Kuznetsova
{"title":"Application of megastable system with 2-𝐷 strip of hidden chaotic attractors to secure communications","authors":"O. I. Kuznetsova","doi":"10.22405/2226-8383-2023-24-1-89-103","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-89-103","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77517538","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-5-14
Nikos Philipovich Aleksiadis
{"title":"Closed classes in the functional system of polynomials with real coefficients","authors":"Nikos Philipovich Aleksiadis","doi":"10.22405/2226-8383-2023-24-1-5-14","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-5-14","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77655081","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-104-113
Olim Erkinovich Mirzaev
{"title":"Partially-isospectral Sturm-Liouville boundary value problems on the finite segment","authors":"Olim Erkinovich Mirzaev","doi":"10.22405/2226-8383-2023-24-1-104-113","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-104-113","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82336528","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-114-126
V. I. Pan’zhenskii, A. O. Rastrepina
{"title":"The left-invariant Sasakian structure on the group model of the real extension of the Lobachevsky plane","authors":"V. I. Pan’zhenskii, A. O. Rastrepina","doi":"10.22405/2226-8383-2023-24-1-114-126","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-114-126","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"2011 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86344716","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-127-138
Anton Yurievich Popov, V. Sherstyukov
{"title":"Strengthening of Gaisin’s lemma on the minimum modulus of even canonical products","authors":"Anton Yurievich Popov, V. Sherstyukov","doi":"10.22405/2226-8383-2023-24-1-127-138","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-127-138","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73251535","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-194-202
S. Aldashev
{"title":"A criterion for the unique solvability of the spectral Poincare problem for a class of multidimensional hyperbolic equations","authors":"S. Aldashev","doi":"10.22405/2226-8383-2023-24-1-194-202","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-194-202","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79343894","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-182-193
Mirgand Shabozovich Shabozov, G. Yusupov
{"title":"On the best polynomial approximation of functions in the Hardy space 𝐻𝑞,𝑅, (1 ⩽ 𝑞 ⩽ ∞, 𝑅 ⩾ 1)","authors":"Mirgand Shabozovich Shabozov, G. Yusupov","doi":"10.22405/2226-8383-2023-24-1-182-193","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-182-193","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88307536","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-139-181
I. A. Tlyustangelov
{"title":"On symmetries of 3-dimensional algebraic continued fractions","authors":"I. A. Tlyustangelov","doi":"10.22405/2226-8383-2023-24-1-139-181","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-139-181","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80002470","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 : 2023-01-01DOI: 10.22405/2226-8383-2023-24-1-15-26
Cristina Viktorovna Budianskaia, I. B. Kozhukhov
{"title":"Congruences of a free unar","authors":"Cristina Viktorovna Budianskaia, I. B. Kozhukhov","doi":"10.22405/2226-8383-2023-24-1-15-26","DOIUrl":"https://doi.org/10.22405/2226-8383-2023-24-1-15-26","url":null,"abstract":"","PeriodicalId":37492,"journal":{"name":"Chebyshevskii Sbornik","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81476057","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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