Topology in nonlinear extensions of hypernumbers | Zendy

Mark Burgin | Zendy

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Topology in nonlinear extensions of hypernumbers

Author(s) -

Mark Burgin

Publication year - 2005

Publication title -

discrete dynamics in nature and society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.264

H-Index - 39

eISSN - 1607-887X

pISSN - 1026-0226

DOI - 10.1155/ddns.2005.145

Subject(s) - mathematics , hausdorff space , topology (electrical circuits) , bounded function , uniqueness , topological space , differential topology , dynamical systems theory , topological vector space , product topology , nonlinear system , closed set , weak topology (polar topology) , extension topology , compact open topology , space (punctuation) , general topology , pure mathematics , discrete mathematics , computer science , mathematical analysis , topological tensor product , combinatorics , geometry , functional analysis , curvature , operating system , quantum mechanics , ricci flat manifold , physics , scalar curvature , chemistry , biochemistry , gene

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 realhypernumbers 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

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