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Examples of the Usage of Infinities and Infinitesimals in Numerical Computations

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Book cover Advances in Artificial Life and Evolutionary Computation (WIVACE 2014)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 445))

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Abstract

It is well known that traditional computers work numerically with finite numbers only and situations where a use of infinite or infinitesimal quantities is required are studied mainly theoretically by human beings. In this paper, a recently introduced computational methodology that has been proposed with the intention to change this differentiation is discussed. It is based on the principle ‘The part is less than the whole’ applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses as a computational device the Infinity Computer (patented in USA, EU, and Russian Federation) working numerically with infinite and infinitesimal numbers that can be written using a numeral positional system with an infinite base. On a number of examples it is shown that it becomes possible both to execute computations of a new type and to simplify computations where infinity and/or infinitesimals are required.

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References

  1. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  2. Cantor, G.: Contributions to the Founding of the Theory of Transfinite Numbers. Dover Publications, New York (1955)

    Google Scholar 

  3. Conway, J.H., Guy, R.K.: The Book of Numbers. Springer, New York (1996)

    Book  MATH  Google Scholar 

  4. De Cosmis, S., De Leone, R.: The use of grossone in mathematical programming and operations research. Appl. Math. Comput. 218(16), 8029–8038 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. D’Alotto, L.: Cellular automata using infinite computations. Appl. Math. Comput. 218(16), 8077–8082 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gödel, K.: The Consistency of the Continuum-Hypothesis. Princeton University Press, Princeton (1940)

    MATH  Google Scholar 

  7. Gordon, P.: Numerical cognition without words: Evidence from Amazonia. Science 306, 496–499 (2004)

    Article  Google Scholar 

  8. Hardy, G.H.: Orders of Infinity. Cambridge University Press, Cambridge (1910)

    MATH  Google Scholar 

  9. Hilbert, D.: Mathematical problems: Lecture delivered before the International Congress of Mathematicians at Paris in 1900. Bull. Am. Math. Soc. 8, 437–479 (1902)

    Article  MATH  MathSciNet  Google Scholar 

  10. Iudin, D.I., Sergeyev, Y.D., Hayakawa, M.: Interpretation of percolation in terms of infinity computations. Appl. Math. Comput. 218(16), 8099–8111 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Iudin, D.I., Sergeyev, Y.D., Hayakawa, M.: Infinity computations in percolation theory applications. Commun. Nonlinear Sci. Numer. Simul. 20(3), 861–870 (2015)

    Article  Google Scholar 

  12. Leibniz, G.W., Child, J.M.: The Early Mathematical Manuscripts of Leibniz. Dover Publications, New York (2005)

    Google Scholar 

  13. Levi-Civita, T.: Sui numeri transfiniti. Rend. Acc. Lincei (Series 5a) 113, 7–91 (1898)

    Google Scholar 

  14. Lolli, G.: Infinitesimals and infinites in the history of mathematics: A brief survey. Appl. Math. Comput. 218(16), 7979–7988 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lolli, G.: Metamathematical investigations on the theory of grossone. Appl. Math. Comput. (2015) (to appear)

    Google Scholar 

  16. Margenstern, M.: Using grossone to count the number of elements of infinite sets and the connection with bijections. p-Adic Numbers, Ultrametric. Anal. App. 3(3), 196–204 (2011)

    MATH  MathSciNet  Google Scholar 

  17. Margenstern, M.: An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl. Math. Comput. 218(16), 8005–8018 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Newton, I.: Method of Fluxions (1671)

    Google Scholar 

  19. Robinson, A.: Non-standard Analysis. Princeton University Press, Princeton (1996)

    MATH  Google Scholar 

  20. Rosinger, E.E.: Microscopes and telescopes for theoretical physics: How rich locally and large globally is the geometric straight line? Prespacetime J. 2(4), 601–624 (2011)

    Google Scholar 

  21. Sergeyev, Y.D.: Arithmetic of Infinity. Edizioni Orizzonti Meridionali, CS (2003). \(2^{d}\) electronic edn. (2013)

    Google Scholar 

  22. Sergeyev, Y.D.: Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers. Chaos, Solitons Fractals 33(1), 50–75 (2007)

    Article  Google Scholar 

  23. Sergeyev, Y.D.: A new applied approach for executing computations with infinite and infinitesimal quantities. Informatica 19(4), 567–596 (2008)

    MATH  MathSciNet  Google Scholar 

  24. Sergeyev, Y.D.: Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge. Chaos, Solitons Fractals 42(5), 3042–3046 (2009)

    Article  Google Scholar 

  25. Sergeyev, Y.D.: Numerical computations and mathematical modelling with infinite and infinitesimal numbers. J. Appl. Math. Comput. 29, 177–195 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sergeyev, Y.D.: Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains. Nonlin. Anal. Ser. A: Theory Methods Appl. 71(12), e1688–e1707 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Sergeyev, Y.D.: Computer system for storing infinite, infinitesimal, and finite quantities and executing arithmetical operations with them. USA patent 7,860,914 (2010)

    Google Scholar 

  28. Sergeyev, Y.D.: Counting systems and the First Hilbert problem. Nonlinear Anal. Ser. A: Theory, Methods Appl. 72(3–4), 1701–1708 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sergeyev, Y.D.: Lagrange Lecture: Methodology of numerical computations with infinities and infinitesimals. Rendiconti del Seminario Matematico dell’Università e del Politecnico di Torino 68(2), 95–113 (2010)

    MATH  MathSciNet  Google Scholar 

  30. Sergeyev, Y.D.: Higher order numerical differentiation on the infinity computer. Optim. Lett. 5(4), 575–585 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sergeyev, Y.D.: On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function. p-Adic Numbers, Ultrametric Anal. Appl. 3(2), 129–148 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sergeyev, Y.D.: Using blinking fractals for mathematical modelling of processes of growth in biological systems. Informatica 22(4), 559–576 (2011)

    MATH  MathSciNet  Google Scholar 

  33. Sergeyev, Y.D.: Numerical computations with infinite and infinitesimal numbers: Theory and applications. In: Sorokin, A., Pardalos, P.M. (eds.) Dynamics of Information Systems: Algorithmic Approaches, pp. 1–66. Springer, New York (2013)

    Google Scholar 

  34. Sergeyev, Y.D.: Solving ordinary differential equations by working with infinitesimals numerically on the infinity computer. Appl. Math. Comput. 219(22), 10668–10681 (2013)

    Article  MathSciNet  Google Scholar 

  35. Sergeyev, Y.D., Garro, A.: Observability of turing machines: A refinement of the theory of computation. Informatica 21(3), 425–454 (2010)

    MATH  MathSciNet  Google Scholar 

  36. Sergeyev, Y.D., Garro, A.: Single-tape and multi-tape turing machines through the lens of the Grossone methodology. J. Supercomput. 65(2), 645–663 (2013)

    Article  Google Scholar 

  37. Vita, M.C., De Bartolo, S., Fallico, C., Veltri, M.: Usage of infinitesimals in the Menger’s Sponge model of porosity. Appl. Math. Comput. 218(16), 8187–8196 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wallis, J.: Arithmetica infinitorum (1656)

    Google Scholar 

  39. Zhigljavsky, A.A.: Computing sums of conditionally convergent and divergent series using the concept of grossone. Appl. Math. Comput. 218(16), 8064–8076 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  40. Žilinskas, A.: On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl. Math. Comput. 218(16), 8131–8136 (2012)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. N.I. Lobatchevsky State University, Nizhni Novgorod, Russia

    Yaroslav D. Sergeyev

  3. Institute of High Performance Computing and Networking, C.N.R., Rende, Italy

    Yaroslav D. Sergeyev

Authors
  1. Yaroslav D. Sergeyev
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Correspondence to Yaroslav D. Sergeyev .

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Editors and Affiliations

  1. CNR-ICAR, Rende, Italy

    Clara Pizzuti

  2. CNR-ICAR, Rende, Italy

    Giandomenico Spezzano

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Sergeyev, Y.D. (2014). Examples of the Usage of Infinities and Infinitesimals in Numerical Computations. In: Pizzuti, C., Spezzano, G. (eds) Advances in Artificial Life and Evolutionary Computation. WIVACE 2014. Communications in Computer and Information Science, vol 445. Springer, Cham. https://doi.org/10.1007/978-3-319-12745-3_15

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  • DOI: https://doi.org/10.1007/978-3-319-12745-3_15

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12744-6

  • Online ISBN: 978-3-319-12745-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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