Non-Archimdean Analysis

Our work on Non-Archimedean Analysis is centered on the development of Analysis concepts on totally ordered Non-Archimdean Fields, in particular those first described by Levi-Civita. Those structures have the advantage of being particularly small, and even treatable in computational environments, which is not the case for the structures of the field of Non-Standard Analysis. While in the latter discipline, there is a gerenally valid transfer principle that allows the transformation of known results of conventional analysis, here all relevant calculus theorems are developed separately, which is facilitated by theorems that under certain conditions allow the extension of classical behavior into infinitely small neighborhoods.

We are developing various calculus concepts including new smoothness approaches that allow to formulate intermediate value theorems, theorems about ranges, and local expandability in Taylor series in various topologies. We are also formulating a theory of locally analytic functions.

Besides their intrinsic interest, the methods can be applied for the practical need to compute derivatives of excessively complicated real and complex functions in a computer environment that are intractable in any other way. This is achieved by evaluating their first and higher order difference quotients with infinitely small increments, which is assured to result in infinitely accurate approximations (and for the purposes of classical real analysis, thus exact determination) of the derivative.

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