Topology in nonlinear extensions of hypernumbers
Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in s...
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| Format: | Article |
| Language: | English |
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Wiley
2005-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/DDNS.2005.145 |
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| author | M. S. Burgin |
| author_facet | M. S. Burgin |
| author_sort | M. S. Burgin |
| collection | DOAJ |
| description | Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis of dynamical systems. It is also proved that construction of extended real
hypernumbers is defined by a definite invariance principle: the space of all extended real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of all real numbers with the topology of conical neighborhoods. In addition,
this topology turns the set of all bounded extended real hypernumbers into a topological algebra. Other topologies in spaces of extended hypernumbers are considered. |
| format | Article |
| id | doaj-art-3473c562871b42b5b258720595edb250 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-3473c562871b42b5b258720595edb2502025-02-03T01:01:36ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2005-01-012005214517010.1155/DDNS.2005.145Topology in nonlinear extensions of hypernumbersM. S. Burgin0Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles 90095, CA, USAModern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending the concept of a hypernumber. In such a way, linear algebras of extended hypernumbers are built. A special topology of conical neighborhoods in these algebras is introduced and studied. It is proved that the space of all extended real hypernumbers is Hausdorff. This provides uniqueness for limits what is very important for analysis of dynamical systems. It is also proved that construction of extended real hypernumbers is defined by a definite invariance principle: the space of all extended real hypernumbers is the biggest Hausdorff factorization of the sequential extension of the space of all real numbers with the topology of conical neighborhoods. In addition, this topology turns the set of all bounded extended real hypernumbers into a topological algebra. Other topologies in spaces of extended hypernumbers are considered.http://dx.doi.org/10.1155/DDNS.2005.145 |
| spellingShingle | M. S. Burgin Topology in nonlinear extensions of hypernumbers Discrete Dynamics in Nature and Society |
| title | Topology in nonlinear extensions of hypernumbers |
| title_full | Topology in nonlinear extensions of hypernumbers |
| title_fullStr | Topology in nonlinear extensions of hypernumbers |
| title_full_unstemmed | Topology in nonlinear extensions of hypernumbers |
| title_short | Topology in nonlinear extensions of hypernumbers |
| title_sort | topology in nonlinear extensions of hypernumbers |
| url | http://dx.doi.org/10.1155/DDNS.2005.145 |
| work_keys_str_mv | AT msburgin topologyinnonlinearextensionsofhypernumbers |