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.
K. Shamseddine, M. Berz, Contemporary Mathematics 508 (2010) 215-237
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