Topological and algebraic structures on the ring of Fermat reals
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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 by 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.
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- W. Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, American Mathematical Society, Providence, RI, 2008.Google Scholar
- M. Berz, G. Hoffstatter, W. Wan, K. Shamseddine and K. Makino, COSY INFINITY and its Applications to Nonlinear Dynamics, in Chapter Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, Penn, 1966, pp. 363–367.Google Scholar
- P. Giordano, Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces, arXiv:0907.1872, July 2009.Google Scholar
- P. Giordano, Order relation and geometrical representation of Fermat reals, American Mathematical Journal, Mathematical Proceedings of the Cambridge Philosophical Society, submitted.Google Scholar
- P. Iglesias-Zemmour, Diffeology, http://math.huji.ac.il/~piz/documents/Diffeology.pdf, July 9 2012.
- A. Kock, Synthetic Differential Geometry, Volume 51 London Mathematical Society Lecture Note Series, Cambridge University Press, 1981.Google Scholar
- A. Kriegl and P.W. Michor, The Convenient Settings of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997.Google Scholar
- T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del Regio Istituto Veneto di Scienze, Lettere ed Arti VII (1893), 1765–1815.Google Scholar
- A. Robinson, Non-standard Analysis, Princeton University Press, 1966.Google Scholar
- K. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, 1999.Google Scholar