Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE’s; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of Non-Archimedean mathematics (and of nonstandard analysis) by means of an elementary topological approach; in particular,we construct Non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [21]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.
A topological approach to non-archimedean mathematics / V. Benci, L. Luperi Baglini (SPRINGER PROCEEDINGS IN MATHEMATICS & STATISTICS). - In: Geometric Properties for Parabolic and Elliptic PDE / [a cura di] F.Gazzola, K.Ishige, C.Nitsch, P.Salani. - [s.l] : Springer, 2016. - pp. 17-40 (( Intervento presentato al 4. convegno PDE’s, GPPEPDEs Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic : May, 25th - 29th tenutosi a Palinuro nel 2015.
A topological approach to non-archimedean mathematics
L. Luperi BagliniUltimo
2016
Abstract
Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE’s; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of Non-Archimedean mathematics (and of nonstandard analysis) by means of an elementary topological approach; in particular,we construct Non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [21]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.File | Dimensione | Formato | |
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Revised Topological Approach Non_Archimedean Mathematics.pdf
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