Topological and algebraic structures on the ring of Fermat reals. (English) Zbl 1275.26049
The ring of Fermat reals is an extension of the field of real numbers that provides an alternative way to perform infinitesimal analysis (alternative to, say, the usual field of hyperreal numbers of nonstandard analysis). In this paper, the authors study two topologies on the ring of Fermat reals arising from two (pseudo) metrics. They then classify the ideals of the ring of Fermat reals and then end by proving that \(p\)th-roots exist (even though there are nilpotent elements).
Reviewer: Isaac Goldbring (Chicago)
Software:
CosyReferences:
| [1] | R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensors, Analysis and Applications. second edition, Springer-Verlag, Berlin, 1988. |
| [2] | W. Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, American Mathematical Society, Providence, RI, 2008. · Zbl 1144.58002 |
| [3] | 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. |
| [4] | A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux reticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1977. · Zbl 0384.06022 |
| [5] | J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies 84, North-Holland, Amsterdam, 1984. · Zbl 0532.46019 |
| [6] | J. F. Colombeau, Multiplication of Distributions, Springer, Berlin, 1992. · Zbl 0815.35002 |
| [7] | J. H. Conway, On Numbers and Games, London Mathematical Society Monographs, No. 6, Academic Press, London & New York, 1976. · Zbl 0334.00004 |
| [8] | P. Ehrlich, An alternative construction of Conway’s ordered field No. Algebra Universalis 25 (1988), 7–16. · Zbl 0654.03023 · doi:10.1007/BF01229956 |
| [9] | P. Giordano, Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces, arXiv:0907.1872, July 2009. |
| [10] | P. Giordano, Fermat-Reyes method in the ring of Fermat reals, Advances in Mathematics 228 (2011), 862–893. DOI: 10.1016/j.aim.2011.06.008. · Zbl 1238.03052 · doi:10.1016/j.aim.2011.06.008 |
| [11] | P. Giordano, Infinitesimals without logic, Russian Journal of Mathematical Physics 17 (2010), 159–191. · Zbl 1264.26035 · doi:10.1134/S1061920810020032 |
| [12] | P. Giordano, Order relation and geometrical representation of Fermat reals, American Mathematical Journal, Mathematical Proceedings of the Cambridge Philosophical Society, submitted. |
| [13] | P. Giordano, The ring of Fermat reals, Advances in Mathematics 225 (2010), 2050–2075. DOI: 10.1016/j.aim.2010.04.010. · Zbl 1205.26051 · doi:10.1016/j.aim.2010.04.010 |
| [14] | P. Iglesias-Zemmour, Diffeology, http://math.huji.ac.il/\(\sim\)piz/documents/Diffeology.pdf , July 9 2012. |
| [15] | A. Kock, Synthetic Differential Geometry, Volume 51 London Mathematical Society Lecture Note Series, Cambridge University Press, 1981. · Zbl 0466.51008 |
| [16] | I. Kolár, P.W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993. · Zbl 0782.53013 |
| [17] | A. Kriegl and P.W. Michor, Product preserving functors of infinite dimensional manifolds, Archivum Mathematicum (Brno) 32 (1996), 289–306. · Zbl 0881.58010 |
| [18] | A. Kriegl and P.W. Michor, The Convenient Settings of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997. · Zbl 0889.58001 |
| [19] | R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1996. · Zbl 0866.58001 |
| [20] | 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. |
| [21] | I. Moerdijk and G.E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, Berlin, 1991. · Zbl 0715.18001 |
| [22] | M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Series 259, Longman Scientific & Technical, Harlow 1992. · Zbl 0818.46036 |
| [23] | Z. M. Odibat and N. T. Shawagfeh, Generalized Taylor’s formula, Applied Mathematics and Computation 186 (2007), 286–293. · Zbl 1122.26006 · doi:10.1016/j.amc.2006.07.102 |
| [24] | A. Robinson, Non-standard Analysis, Princeton University Press, 1966. · Zbl 0151.00803 |
| [25] | K. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, 1999. |
| [26] | K. Shamseddine and M. Berz, Intermediate value theorem for analytic functions on a Levi-Civita field, The Bulletin of the Belgian Mathematical Society Simon Stevin 14 (2007), 1001–1015. · Zbl 1181.26044 |
| [27] | H. Vernaeve, Ideals in the ring of Colombeau generalized numbers, Communications in Algebra 38 (2010), 2199–2228. · Zbl 1198.13022 · doi:10.1080/00927870903055222 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
