A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics
Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks –...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2013 |
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| Language: | English |
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Інститут математики НАН України
2013
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/149229 |
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| Cite this: | A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics / M. Gopalkrishnan, E. Miller, A. Shiu // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 22 назв. — англ. |
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Gopalkrishnan, M. Miller, E. Shiu, A. 2019-02-19T19:02:47Z 2019-02-19T19:02:47Z 2013 A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics / M. Gopalkrishnan, E. Miller, A. Shiu // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34A60; 80A30; 92C45; 37B25; 34D23; 37C10; 37C15; 92E20; 92C42; 54B30; 18B30 DOI: http://dx.doi.org/10.3842/SIGMA.2013.025 https://nasplib.isofts.kiev.ua/handle/123456789/149229 Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks – including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks – that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates. MG was supported by a Ramanujan fellowship from the Department of Science and Technology, India, and, during a semester-long stay at Duke University, by the Duke MathBio RTG grant NSF DMS-0943760. EM had support from NSF grant DMS-1001437. AS was supported by an NSF postdoctoral fellowship DMS-1004380. The authors thank David F. Anderson, Gheorghe Craciun, and Casian Pantea for helpful discussions, and Duke University where many of the conversations occurred. The authors also thank the two referees, whose perceptive and insightful comments improved this work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics |
| spellingShingle |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics Gopalkrishnan, M. Miller, E. Shiu, A. |
| title_short |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics |
| title_full |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics |
| title_fullStr |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics |
| title_full_unstemmed |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics |
| title_sort |
projection argument for differential inclusions, with applications to persistence of mass-action kinetics |
| author |
Gopalkrishnan, M. Miller, E. Shiu, A. |
| author_facet |
Gopalkrishnan, M. Miller, E. Shiu, A. |
| publishDate |
2013 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks – including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks – that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149229 |
| citation_txt |
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics / M. Gopalkrishnan, E. Miller, A. Shiu // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 22 назв. — англ. |
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2025-12-07T17:21:35Z |
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2025-12-07T17:21:35Z |
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