On Flexible Sequences

Abstract

In the setting of nonstandard analysis, we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical O(⋅) and o(⋅) notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions, and multiplications. We introduce two forms of convergence and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications.

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References

  1. 1.

    Anishchenko, V.S., Vadivasova, T.E., Strelkova, G.I.: Deterministic Nonlinear Systems. A Short Course. Springer Series in Synergetics, Springer, Cham (2014)

  2. 2.

    Diener, F., Reeb, G.: Analyse Non Standard. Hermann, Paris (1989)

    Google Scholar 

  3. 3.

    Dinis, B., van den Berg, I.: Algebraic properties of external numbers. J. Log. Anal. 3(9), 1–30 (2011)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Dinis, B., van den Berg, I.: Axiomatics for the external numbers of nonstandard analysis. J. Log. Anal. 9(7), 1–47 (2017)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Dinis, B., van den Berg, I.: On the quotient class of non-archimedean fields. Indag. Math. (N.S.) 28(4), 784–795 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Dinis, B., van den Berg, I.: Characterization of distributivity in a solid. Indag. Math. (N.S.) 29(2), 580–600 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Eckhaus, W.: Asymptotic Analysis of Singular Perturbations Studies in Mathematics and Its Applications, vol. 9. North-Holland Publishing Co., Amsterdam-New York (1979)

    Google Scholar 

  8. 8.

    Hardy, G.H.: Orders of Infinity: The ‘Infinitärcalcül’ of Paul Du Bois-Reymond, 2nd ed. Cambridge Univ. Press, Cambridge (1924)

    Google Scholar 

  9. 9.

    Justino, J., van den Berg, I.: Cramer’s rule applied to flexible systems of linear equations. Electron. J. Linear Algebra 24, 126–152 (2012/13)

  10. 10.

    Kanovei, V., Reeken, M.: Nonstandard Analysis, Axiomatically. Springer Monographs in Mathematics. Springer, Berlin (2004)

    Google Scholar 

  11. 11.

    Koudjeti, F.: Elements of External Calculus with an Application to Mathematical Finance. Ph.D. Thesis, Labyrinth Publications, Capelle a/d IJssel, The Netherlands (1995)

  12. 12.

    Koudjeti, F., van den Berg, I.: Neutrices, External Numbers and External Calculus. Nonstandard Analysis in Practice, 145–170. Universitext. Springer, Berlin (1995)

    Google Scholar 

  13. 13.

    Lobry, C., Sari, T., Touhami, S.: On Tykhonov’s theorem for convergence of solutions of slow and fast systems. Electron. J. Differ. Equ. 19, 1–22 (1998)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Nelson, E.: Internal set theory: a new approach to nonstandard analysis. Bull. Am. Math. Soc. 83(6), 1165–1198 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Nelson, E.: The syntax of nonstandard analysis. Ann. Pure Appl. Logic 38(2), 123–134 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Robinson, A.: Non-Standard Analysis. North-Holland Publishing Co., Amsterdam (1966)

    Google Scholar 

  17. 17.

    Tihonov, A.: Systems of differential equations containing a small parameter in the derivatives. Mat. Sbornik N. S. 31(73), 575–586 (1952)

    MathSciNet  Google Scholar 

  18. 18.

    van den Berg, I.: Nonstandard Asymptotic Analysis Lecture Notes in Mathematics, 1249. Springer, Berlin (1987)

    Google Scholar 

  19. 19.

    van den Berg, I.P.: External borders and strongly open sets. In: Benoît, É, Furter, et J.-P. (eds.) Des Nombres Et Des Mondes Editions Hermann, Paris, 69–86 (2012)

  20. 20.

    van den Berg, I.: Asymptotics of families of solutions of nonlinear difference equations. Log. Anal. 1(2), 153–185 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    van den Berg, I.: A decomposition theorem for neutrices. Ann. Pure Appl. Logic 161(7), 851–865 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    van den Berg, I., Diener, M.: Diverses applications du lemme de Robinson en analyse nonstandard. C. R. Acad. Sci. Paris Sé,r. I Math. 293(10), 501–504 (1981)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    van der Corput, J.G.: Introduction to the neutrix calculus. J. Analyse Math. 7, 281–399 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Van Dyke, M.: Perturbation Methods in Fluid Mechanics. Annotated Edition. The Parabolic Press, Stanford (1975)

    Google Scholar 

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Acknowledgements

The first author acknowledges the support of the Centro de Matemática, Aplicações Fundamentais e Investigação Operacional / Fundação da Faculdade de Ciências da Universidade de Lisboa via the grant UID/MAT/04561/2013 and a postdoc-grant from Erasmus Mundus Mobility with Asia–East 14.

The second author acknowledges a PhD-grant of Erasmus Mundus Mobility with Asia–East 14.

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Correspondence to Bruno Dinis.

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Dinis, B., Van Tran, N. & Berg, I.v.d. On Flexible Sequences. Acta Math Vietnam 44, 833–874 (2019). https://doi.org/10.1007/s40306-018-00303-4

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Keywords

  • External numbers
  • Flexible sequences
  • Convergence
  • Nonstandard analysis

Mathematics Subject Classification (2010)

  • 03H05
  • 40A05