Point configurations that are asymmetric yet balanced
Abstract: A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, bu...
| Auteurs principaux: | , , , |
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| Format: | Article |
| Langue: | en_US |
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American Mathematical Society
2011
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| Accès en ligne: | http://hdl.handle.net/1721.1/60891 https://orcid.org/0000-0001-9261-4656 |
| _version_ | 1826198926749859840 |
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| author | Cohn, Henry Elkies, Noam D. Kumar, Abhinav Shurmann, Achill |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Cohn, Henry Elkies, Noam D. Kumar, Abhinav Shurmann, Achill |
| author_sort | Cohn, Henry |
| collection | MIT |
| description | Abstract: A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in $ \mathbb{R}^3$, and his classification is equivalent to the converse for $ \mathbb{R}^3$. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group. |
| first_indexed | 2024-09-23T11:12:04Z |
| format | Article |
| id | mit-1721.1/60891 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:12:04Z |
| publishDate | 2011 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | mit-1721.1/608912022-10-01T01:58:48Z Point configurations that are asymmetric yet balanced Cohn, Henry Elkies, Noam D. Kumar, Abhinav Shurmann, Achill Massachusetts Institute of Technology. Department of Mathematics Cohn, Henry Cohn, Henry Kumar, Abhinav Abstract: A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in $ \mathbb{R}^3$, and his classification is equivalent to the converse for $ \mathbb{R}^3$. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group. National Science Foundation (U.S.) (Grant No. DMS-0757765) National Science Foundation (U.S.) (Grant No. DMS-0501029) Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 1503/4-2) 2011-02-04T13:12:56Z 2011-02-04T13:12:56Z 2010-03 2009-03 Article http://purl.org/eprint/type/JournalArticle 0002-9939 1088-6826 http://hdl.handle.net/1721.1/60891 Cohn, Henry. et al. "Point configurations that are asymmetric yet balanced ." Proc. Amer. Math. Soc. 138 (2010): 2863-2872. https://orcid.org/0000-0001-9261-4656 en_US http://dx.doi.org/10.1090/S0002-9939-10-10284-6 Proceedings of the American Mathematical Society Attribution-Noncommercial-Share Alike 3.0 Unported http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf American Mathematical Society MIT web domain |
| spellingShingle | Cohn, Henry Elkies, Noam D. Kumar, Abhinav Shurmann, Achill Point configurations that are asymmetric yet balanced |
| title | Point configurations that are asymmetric yet balanced |
| title_full | Point configurations that are asymmetric yet balanced |
| title_fullStr | Point configurations that are asymmetric yet balanced |
| title_full_unstemmed | Point configurations that are asymmetric yet balanced |
| title_short | Point configurations that are asymmetric yet balanced |
| title_sort | point configurations that are asymmetric yet balanced |
| url | http://hdl.handle.net/1721.1/60891 https://orcid.org/0000-0001-9261-4656 |
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