Non-Archimedean Analysis

 

Our work on Non-Archimedean Analysis is centered on the development of Analysis concepts on totally ordered Non-Archimedean 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 generally 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.

 

Meetings

MSU hosts and organizes the Tenth International Conference on p-adic and Non-Archimedean Analysis in 2008. Details can be found here.

 

Reprints

The list below contains selected publications, currently up to 2001. Other papers on the Non-Archimedean Analysis can be found on our publication server. 

 

· 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).