Topological structures of large-scale interacting systems via uniform functions and forms
In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uni...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000616/type/journal_article |
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| author | Kenichi Bannai Yukio Kametani Makiko Sasada |
| author_facet | Kenichi Bannai Yukio Kametani Makiko Sasada |
| author_sort | Kenichi Bannai |
| collection | DOAJ |
| description | In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed
$L^2$
-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan’s decomposition theorem for a general class of large-scale interacting systems. |
| format | Article |
| id | doaj-art-e8ca2c15bbcd4ba194c2e2074ab53e1d |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-e8ca2c15bbcd4ba194c2e2074ab53e1d2024-11-26T08:01:11ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.61Topological structures of large-scale interacting systems via uniform functions and formsKenichi Bannai0https://orcid.org/0000-0001-9493-0265Yukio Kametani1Makiko Sasada2https://orcid.org/0000-0002-4184-108XDepartment of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan; E-mail: , . Mathematical Science Team, Center for Advanced Intelligence Project (AIP), RIKEN, 1-4-1 Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan.Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan; E-mail: , .Department of Mathematics, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-0041, Japan Mathematical Science Team, Center for Advanced Intelligence Project (AIP), RIKEN, 1-4-1 Nihonbashi, Chuo-ku, Tokyo, 103-0027, Japan.In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$ -forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan’s decomposition theorem for a general class of large-scale interacting systems.https://www.cambridge.org/core/product/identifier/S2050509424000616/type/journal_article82C2205C6355N9160J6070G40 |
| spellingShingle | Kenichi Bannai Yukio Kametani Makiko Sasada Topological structures of large-scale interacting systems via uniform functions and forms Forum of Mathematics, Sigma 82C22 05C63 55N91 60J60 70G40 |
| title | Topological structures of large-scale interacting systems via uniform functions and forms |
| title_full | Topological structures of large-scale interacting systems via uniform functions and forms |
| title_fullStr | Topological structures of large-scale interacting systems via uniform functions and forms |
| title_full_unstemmed | Topological structures of large-scale interacting systems via uniform functions and forms |
| title_short | Topological structures of large-scale interacting systems via uniform functions and forms |
| title_sort | topological structures of large scale interacting systems via uniform functions and forms |
| topic | 82C22 05C63 55N91 60J60 70G40 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000616/type/journal_article |
| work_keys_str_mv | AT kenichibannai topologicalstructuresoflargescaleinteractingsystemsviauniformfunctionsandforms AT yukiokametani topologicalstructuresoflargescaleinteractingsystemsviauniformfunctionsandforms AT makikosasada topologicalstructuresoflargescaleinteractingsystemsviauniformfunctionsandforms |