Papers › A topological approach to non-Archimedean Mathematics
A topological approach to non-Archimedean Mathematics
Vieri Benci, Lorenzo Luperi Baglini
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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 (in particular, 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 \cite{BDN2003}, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \cite{keisler}). 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.
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