2014/12/06 by Benci, Vieri, Baglini, Lorenzo Luperi
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1412.2223
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 \citeBDN2003, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \citekeisler). 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.