Abstract
An extension of the ring of scalar quantities, from the usual field of real numbers to a non-Archimedean, sometimes permits to simplify some problems which, at a first sight, may seem not correlated with infinitesimal and infinite numbers. We present four simple cases, each one at the level of possibility for the creativity of a motivated student. The ring of Fermat reals and its applications to physics and differential geometry, the ring of Colombeau generalized numbers and its applications to the foundations of generalized functions, the Levi-Civita field and the derivation of complicated computer functions and the Surreals numbers as a universal non-Archimedean ring. The definition of each one of these rings is strongly motivated at elementary level and some open problems and ideas are introduced in the first two cases.
The author has been supported by FWF grants M1247-N13 and P25116-N25.
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Notes
- 1.
The notation with square brackets \(x[q]\) permits to avoid confusion when one consider functions defined on the LCF \(\mathcal{R}\).
- 2.
- 3.
That is \(L_{x}<R_{x}\) and \(L_{y}<R_{y}\). Let us note that using a notation like \(x=\{L_{x}\,|\, R_{x}\}\) we do not mean that a number \(x\in{\bf No}\) uniquely determines the subsets L x and R x .
- 4.
Of course, at this stage of development and using this not-strictly formal point of view, our use of the notion of “simplicity” is only informal and it is natural to ask for a more formal definition, considering, moreover, its uniqueness. This will be done in the next section.
- 5.
From this point of view the name “surreal numbers” is less meaningful than the original Conway’s “numbers” without any adjective.
References
Albeverio, S., Fenstad, J.E., Hø egh-Krohn, R., Lindstrø m, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics. Pure and applied mathematics, 2nd ed. Academic Press, Orlando (1988). Reprint: Dover (2009)
Alling, N.L.: Foundation of Analysis Over Surreal Number Fields, vol. 141 of North-Holland Mathematics Studies. North-Holland Publishing Co., Holland (1987)
Bell, J.L.: A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge (1998)
Benci, V., Di Nasso, M.: A ring homomorphism is enough to get nonstandard analysis. Bull. Belg. Math. Soc.–S. Stevin. 10, 481–490 (2003)
Benci, V., Di Nasso, M.: A purely algebraic characterization of the hyperreal numbers. Proc. Am. Math. Soc. 133(9), 2501–2505 (2005
Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications, Encyclopedia of Mathematics and its Applications, vol. 137. Cambridge University Press, Cambridge (2011)
Bertram, W.: Differential Geometry, Lie Groups and Symmetric Spaces Over General Base Fields and Rings. American Mathematical Society, Providence (2008)
Berz, M.: Automatic differentiation as nonarchimedean analysis, Computer arithmetic and enclosure methods. p. 43–9. Elsevier Science Publisher, Amsterdam (1992)
Berz, M.: Analysis on a nonArchimedean extension of the real numbers. Mathematics Summer Graduate School of the German National Merit Foundation, MSUCL-933, Department of Physics, Michigan State University, 1992 and 1995 edition, (1994)
Berz, M., Hoffstatter, G., Wan, W., Shamseddine, K., Makino, K.: COSY INFINITY and its applications to nonlinear dynamics. In: Berz, M., Bischof, C., Corliss, G., Griewank, A. (eds.) Computational Differentiation: Techniques, Applications, and Tools, pp. 363–367. SIAM, Philadelphia (1996)
Colombeau, J.F.: Multiplication of Distributions. Springer, Berlin (1992)
Conway, J.H.: On Numbers and Games, Number 6 in L.M.S. Monographs. Academic Press, London & New York (1976)
Conway, J.: Infinitesimals vs. indivisibles replies: 20. The Math Forum Drexel, Feb. 17 1999. URL http://mathforum.org/kb/message.jspa? messageID=1381465&tstart=0. (1999)
Cuesta Dutari, N.: Algebra ordinal, Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. XLVIII(2), 79–160 (1954)
Delcroix, A.: Topology and functoriality in (\(\mathcal C,\mathcal E,\mathcal P\))-algebras: application to singular differential problems. J. Math. Anal. Appl. 359, 394–403 (2009)
Delcroix, A., Scarpalezos, D.: Asymptotic scales-asymptotic algebras. Integr. Trans. Spec. Funct. 6, 1–4, 181–190 (1998)
Dirac, P.A.M.: The physical interpretation of the quantum dynamics. Proc. R. Soc. Lond. A(113), 621–641 (1926–1927)
Dirac, P.A.M.: General Theory of Relativity. Wiley, Hoboken (1975)
Ehrlich, P.: An alternative construction of Conway’s ordered field No. Algebra Uni. 25, 7–16 (1988)
Ehrlich, P.: The rise of non-Archimedean mathematics and the roots of a misconception I: The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60(1), 1–121 (2006)
Ehrlich, P.: Conway names, the simplicity hierarchy and the surreal number tree. J. Logic Anal. 3(1), 1–26 (2011)
Ehrlich, P.: The absolute arithmetic continuum and the unification of all numbers great and small. Bull. Symbol. Logic. 18(1), 1–45 (2012)
Einstein, A.: Investigations on the Theory of the Brownian Movement. Dover, New York (1926)
Fornasiero, A.: Integration on surreal numbers. PhD thesis, University of Edinburgh. www.dm.unipi.it/˜fornasiero/phd_thesis/thesis_fornasiero_linearized.pdf (2004)
Garth Dales, H., Woodin, W. H.: Super-real fields, London Mathematical Society Monographs, New Series, vol. 14. The Clarendon Press Oxford University Press, New York, 1996, Totally ordered fields with additional structure; Oxford Science Publications (1996)
Giordano, P.: Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces. arXiv:0907.1872, July (2009)
Giordano, P.: The ring of Fermat reals. Adv. Math. 225(4), 2050–2075 (2010)
Giordano, P.: Infinitesimals without logic. Russian J. Math. Phys. 17(2), 159–191 (2010)
Giordano, P.: Infinite dimensional spaces and cartesian closedness. J. Math. Phys. Anal. Geomet. 7(3), 225–284 (2011)
Giordano, P.: Fermat–Reyes method in the ring of Fermat reals. Adv. Math. 228, 862–893 (2011)
Giordano, P., Katz, M.: Potential and actual infinitesimals in models of continuum, www.mat.univie.ac.at/˜giordap7/#preprints (2012)
Giordano, P., Kunzinger, M.: Topological and algebraic structures on the ring of Fermat reals. Israel J. Math. DOI: 10.1007/s11856-012-0079-z. (2011)
Giordano, P., Kunzinger, M.: Generalized functions as smooth set-theoretical maps, article in preparation, www.mat.univie.ac.at/~giordap7/#preprints
Giordano, P., Kunzinger, M., Vernaeve, H.: Strongly internal sets and generalized smooth functions. Submitted to Journal of Mathematical Analysis and Applications on February 2014. See www.mat.univie.ac.at/~giordap7/#preprints
Goldblatt, R.: Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Springer, New York, (1998)
Grosser, M., Kunzinger, M., Oberguggenberger, M., Steinbauer, R.: Geometric Theory of Generalized Functions. Kluwer, Dordrecht (2001)
Harzheim, E.: Beiträge zur Theorie der Ordnungstypen, insbesondere der a \(\eta\alpha\)-Mengen. Math. Annalen. 154, 116–134 (1964)
Henle, M.: Which Numbers are Real? The Mathematical Association of America, Wasington, DC (2012)
Iglesias-Zemmour, P.: Diffeology, AMS, Mathematical Surveys and Monographs 185, to appear on April (2013)
Katz, M.G., Tall, D.: A Cauchy–Dirac delta function: foundations of science. http://dx.doi.org/10.1007/s10699-012-9289-4 and http://arxiv.org/abs/1206.0119 (2012)
Knuth, D.E.: Surreal Numbers: How Two Ex-students Turned On to Pure Mathematics and Found Total Happiness: A Mathematical Novelette. Addison-Wesley, Massacheuts (1974)
Kock, A.: Synthetic Differential Geometry. London Math, Society Lecture Note Series, vol. 51. Cambridge University Press, Cambridge (1981)
Kolár, I., Michor, P.W. Slovák, J.: Natural Operations in Differential Geometry. Springer, Berlin (1993)
Kriegl, A., Michor, P.W.: Product preserving functors of infinite dimensional manifolds. Archivum Mathematicum (Brno). 32(4), 289–306 (1996)
Kriegl, A., Michor, P.W.: The Convenient Settings of Global Analysis. Mathematical Surveys and monographs, vol. 53. American Mathematical Society, Providence (1997)
Laugwitz, D.: Tullio Levi-Civita’s work on nonarchimedean structures (with an appendix: Properties of Levi-Civita fields). In Atti dei Convegni Lincei 8: Convegno Internazionale Celebrativo del Centenario della Nascita di Tullio Levi-Civita, Roma, 1975, Accademia Nazionale dei Lincei (1975)
Laugwitz, D.: Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39(3), 195–245 (1989)
Lavendhomme, R.: Basic Concepts of Synthetic Differential Geometry. Kluwer Academic Publishers, Dordrecht (1996)
Levi-Civita, T.: Sugli infiniti ed infinitesimi attuali quali elementi analitici. Atti del Regio Istituto Veneto di Scienze Lettere ed Arti. VII(4), 1765–1815 (1893)
Levi-Civita, T.: Sui numeri transfiniti. Rendiconti della Reale Accademia dei Lincei. VI(1 Sem.), 113–121 (1898)
Mamane, L.E.: Surreal numbers in Coq. In: Types for Proofs and Programs. Lecture Notes in Computer Science, vol. 3839, pp. 170–185. Springer, Berlin (2006)
Moerdijk, I., Reyes, G.E.: Models for Smooth Infinitesimal Analysis. Springer, Berlin (1991)
Palmgren, E.: Developments in constructive nonstandard analysis. Bull. Symbol. Logic. 4(3), 233–272 (1998)
Palmgren, E.: A note on [AQ1] Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis. J. Logic Anal. 3 pp. 1–7 (2012)
Robinson, A.: Non-standard Analysis. North-Holland Publishing Co., Amsterdam (1966)
Schmieden, C., Laugwitz, D.: Eine Erweiterung der Infinitesimalrechnung. Math. Zeitschr. 69, l–39 (1958)
Schwartz, L.: Théorie des distributions, vol. 1–2. Hermann, Paris (1950)
Schwartz, L.: Sur l’impossibilité de la multiplications des distributions. C.R. Acad. Sci. Paris. 239, 847–848 (1954)
Sebasti˜ao e Silva, J.: Sur une construction axiomatique de la theorie des distributions, Rev. da Fac. de Ciências de Lisboa, 2⍶ série-A, vol. 4, pp. 79–186 (1954–1955)
Shamseddine, K.: One-variable and multi-variable calculus on a non-Archimedean field extension of the real numbers, p-Adic numbers, ultrametric analysis, and applications. In print
Shamseddine, K.: On the topological structure of the Levi-Civita field. J. Math. Anal. Appl. 368, 281–292 (2010)
Shamseddine, K.: Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a Levi-Civita field. In: Jesus A.-G., Diarra, B., Escassut, A. (eds.) Contemporary Mathematics, Advances in Non-Archimedean Analysis, vol. 551, pp. 257–268. American Mathematical Society, Providence (2011)
Shamseddine, K.: Nontrivial order preserving automorphisms of non-Archimedean fields. In: Jarosz, K. (ed.) Contemporary Mathematics. Function Spaces in Modern Analysis, vol. 547, pp. 217–225. American Mathematical Society, Providence (2011)
Shamseddine, K.: New results on integration on the Levi-Civita field. Indagationes Math. 24(1), 199–211 (2013)
Shamseddine, K., Berz, M.: Intermediate value theorem for analytic functions on a Levi-Civita field. Bull. Belg. Math. Soc. Simon Stevin. 14, 1001–1015 (2007)
Shamseddine, K., Berz, M.: Analysis on the Levi-Civita field: a brief overview. In: Berz, M., Shamseddine, K. (eds.) Contemporary Mathematics. Advances in p-Adic and Non-Archimedean Analysis, vol. 508, pp. 215–237. American Mathematical Society, Providence (2010). ISBN 978-0-8218-4740-4
Shamseddine, K., Grafton, W.: Preliminary notes on Fourier series for functions on the Levi-Civita field. Int. J. Math. Anal. 6(19), 941–950 (2012)
Shamseddine, K., Sierens, T.: On locally uniformly differentiable functions on a complete non-Archimedean ordered field extension of the real numbers. ISRN Math. Anal. 2012: Article ID 387053 (2012), http://www.hindawi.com/journals/isrn.mathematical.analysis/2012/387053/abs/ they say only the additional information: 20 pages http://dx.doi.org/10.5402/2012/387053
Shamseddine, K., Rempel, T., Sierens, T.: The implicit function theorem in a non-Archimedean setting. Indagationes Math. 20(4), 603–617 (2009)
Tall, D.: Looking at graphs through infinitesimal microscopes, windows and telescopes. Math. Gaz. 64, 22–49 (1980)
Vernaeve, H.: Generalized analytic functions on generalized domains, arXiv:0811.1521v1, (2008)
Weil, A.: Théorie des points proches sur les variétés différentiables, Colloque de Géometrie Différentielle, pp. 111–117, C.N.R.S. (1953)
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Giordano, P. (2014). Which Numbers Simplify Your Problem?. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_7
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