Topological and algebraic structures on the ring of Fermat reals

Details Topological and algebraic structures on the ring of Fermat reals Topological and algebraic structures on the ring of Fermat reals

2011-04-08

arxiv.org/abs/1104.1492v1

Mathematics

Commutative Algebra

43 pages

Scientific paper

The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring from using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.

Affiliated with

Also associated with

No associations

Rating

Topological and algebraic structures on the ring of Fermat reals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Topological and algebraic structures on the ring of Fermat reals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topological and algebraic structures on the ring of Fermat reals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-354025