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 \(\mathbb {R}\). Our approach is based on the notion of \(\varLambda \)-limit for real functions, and it is called \(\varLambda \)-theory. It can be seen as a topological generalization of the \(\alpha \)-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 \(\varLambda \)-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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- 1.
An ultrafilter \(\mathscr {U}\) is countably incomplete if there exists a family \(\langle A_{n}\mid n\in \mathbb {N}\rangle \) of elements of \(\mathscr {U }\) such that \(\bigcap _{n\in \mathbb {N}}A_{n}=\emptyset \).
- 2.
A superreal non Archimedean field is an ordered field that properly contains \(\mathbb {R}\).
- 3.
\( p^{*}\) is the bounded sentence obtained by changing every constant symbol \(c\in V_{\infty }(\mathbb {R})\) that appears in p with \(c^{*}\).
- 4.
See e.g. [1], where many different applications of hyperfinite objects and other nonstandard tools are developed.
- 5.
Once again, it should be evident to readers expert in NSA that our definition is precisely analogous to the one that is given for ultrapowers.
- 6.
Let us recall that an ultrafilter \(\mathscr {U}\) on \(\mathfrak {L}\) is fine if for every \(\lambda \in \mathfrak {L}\) the set \(\{\mu \in \mathfrak {L}\mid \mu \subseteq \lambda \}\in \mathscr {U}\). We also point out that, for more complicated applications, it would be better to take \(\mathfrak {L}=\mathscr {P }_{fin}\left( V_{\infty }(\mathbb {R})\right) \).
- 7.
Any interested reader can find it in [10].
- 8.
E.g., in [12] a (slightly modified) version of this hypothesis is used to construct an embedding of the space of distributions in a particular algebra of functions constructed by means of ultrafunctions.
- 9.
Let us observe that both the scalar product and the norm take values in \( \mathbb {R}^{*}\).
- 10.
This example has already been studied in greater detail in [11].
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Supported by grants P25311-N25 and M1876-N35 of the Austrian Science Fund FWF.
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Benci, V., Luperi Baglini, L. (2016). A Topological Approach to Non-Archimedean Mathematics. In: Gazzola, F., Ishige, K., Nitsch, C., Salani, P. (eds) Geometric Properties for Parabolic and Elliptic PDE's. GPPEPDEs 2015. Springer Proceedings in Mathematics & Statistics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-41538-3_2
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