![]() Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities the augmentations are the reciprocals of one another. Number systems that include infinitesimals.The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the " infinity- th" item in a sequence. In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. Vernaeve, Ideals in the ring of Colombeau generalized numbers, Communications in Algebra 38 (2010), 2199–2228.Extremely small quantity in calculus thing so small that there is no way to measure it Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω) 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. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, 1999. Robinson, Non-standard Analysis, Princeton University Press, 1966. Shawagfeh, Generalized Taylor’s formula, Applied Mathematics and Computation 186 (2007), 286–293.Ī. ![]() Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Series 259, Longman Scientific & Technical, Harlow 1992. Reyes, Models for Smooth Infinitesimal Analysis, Springer, Berlin, 1991. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del Regio Istituto Veneto di Scienze, Lettere ed Arti VII (1893), 1765–1815. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1996. ![]() Michor, The Convenient Settings of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997. Michor, Product preserving functors of infinite dimensional manifolds, Archivum Mathematicum (Brno) 32 (1996), 289–306.Ī. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993.Ī. Kock, Synthetic Differential Geometry, Volume 51 London Mathematical Society Lecture Note Series, Cambridge University Press, 1981. Iglesias-Zemmour, Diffeology,, July 9 2012.Ī. Giordano, The ring of Fermat reals, Advances in Mathematics 225 (2010), 2050–2075. Giordano, Order relation and geometrical representation of Fermat reals, American Mathematical Journal, Mathematical Proceedings of the Cambridge Philosophical Society, submitted. Giordano, Infinitesimals without logic, Russian Journal of Mathematical Physics 17 (2010), 159–191. Giordano, Fermat-Reyes method in the ring of Fermat reals, Advances in Mathematics 228 (2011), 862–893. Giordano, Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces, arXiv:0907.1872, July 2009. Ehrlich, An alternative construction of Conway’s ordered field No. 6, Academic Press, London & New York, 1976. Conway, On Numbers and Games, London Mathematical Society Monographs, No. Colombeau, Multiplication of Distributions, Springer, Berlin, 1992. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies 84, North-Holland, Amsterdam, 1984. Wolfenstein, Groupes et anneaux reticulés, Lecture Notes in Mathematics, Vol. ![]() Makino, COSY INFINITY and its Applications to Nonlinear Dynamics, in Chapter Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, Penn, 1966, pp. Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, American Mathematical Society, Providence, RI, 2008. second edition, Springer-Verlag, Berlin, 1988. ![]() Ratiu, Manifolds, Tensors, Analysis and Applications. ![]()
0 Comments
Leave a Reply. |