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.

- Analytical
and Computational Methods for the Levi-Civita Field, Martin Berz,
*Lecture Notes in Pure and Applied Mathematics*, Marcel Dekker, ISBN 0-8247-0611-0 (2001), pp. 21-34. - Convergence
on the Levi-Civita Field and Study of Power Series, Khodr Shamseddine and Martin Berz,
*Lecture Notes in Pure and Applied Mathematics*, Marcel Dekker, ISBN 0-8247-0611-0 (2001), pp. 283-299. - The Differential
Algebraic Structure of the Levi-Civita Field and Applications, Khodr Shamseddine and Martin Berz,
*International Journal of Applied Mathematics*,**3**,**4**, 449-464 (2000). - Calculus
and Numerics on Levi-Civita Fields, Martin Berz,
*Computational Differentiation: Techniques, Applications, and Tools*, M. Berz, C. Bischof, G. Corliss, A. Griewank, eds., SIAM, Philadelphia, Penn, 1996, pp. 19-36. - Exception
Handling in Derivative Computation with Non-Archimedean Calculus, Khodr Shamseddine and Martin Berz,
*Computational Differentiation: Techniques, Applications, and Tools*, M. Berz, C. Bischof, G. Corliss, A. Griewank, eds., SIAM, Philadelphia, Penn, 1996, pp. 37-51. - New
Elements of Analysis on the Levi-Civita Field, PhD Dissertation, Khodr Shamseddine,
Department of Mathematics and Department of Physics and Astronomy,
Michigan State University, December 1999, Advisor: Prof. Martin Berz.
- Automatic
Differentiation as Nonarchimedean Analysis, Martin Berz,
*Computer Arithmetic and Enclosure Methods*(Elsevier Science, Amsterdam, 1992), pp. 439-450. - Analysis
on a Nonarchimedean Extension of the Real Numbers, Lecture Notes, 1992
and 1995 Mathematics Summer Graduate Schools of the German National Merit
Foundation, MSUCL-933, Department of Physics and Astronomy, Michigan State
University (1994).

This page is maintained by Jens Hoefkens. Please contact him if there
are any problems with it.