Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms
| Autor: | Bannai, Kenichi, Kametani, Yukio, Sasada, Makiko |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 65 pages, 7 figures. Added Appendix A to review the general theory of homology and cohomology of graphs. Final version, to appear in Forum of Mathematics, Sigma |
| Databáze: | arXiv |
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