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Analysis on the Levi-Civita field, a brief overview. (English) Zbl 1198.26030

Berz, Martin (ed.) et al., Advances in \(p\)-adic and non-Archimedean analysis. Tenth international conference, Michigan State University, East Lansing, MI, USA, June 30–July 3, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4740-4/pbk). Contemporary Mathematics 508, 215-237 (2010).
Summary: We review the algebraic properties of various non-Archimedean ordered structures, extending them in various steps which lead naturally to the smallest non-Archimedean ordered field that is Cauchy-complete and real closed. In fact, the Levi-Civita field is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as “differential quotients” up to very high orders.
We then give an overview of recent research on the Levi-Civita field. In particular, we summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real power series; and we present a Lebesgue-like measure and integration theory on the field. Moreover, based on continuity and differentiability concepts that are stronger than the topological ones, we discuss solutions to one-dimensional and multi-dimensional optimization problems as well as existence and uniqueness of solutions of ordinary differential equations.
For the entire collection see [Zbl 1184.46001].

MSC:

26E30 Non-Archimedean analysis
12J25 Non-Archimedean valued fields
32P05 Non-Archimedean analysis
11D88 \(p\)-adic and power series fields
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis

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