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On Flexible Sequences

  • Bruno Dinis
  • Nam Van Tran
  • Imme van den Berg
Article
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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.

Keywords

External numbers Flexible sequences Convergence Nonstandard analysis 

Mathematics Subject Classification (2010)

03H05 40A05 

Notes

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.

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)Google Scholar
  2. 2.
    Diener, F., Reeb, G.: Analyse Non Standard. Hermann, Paris (1989)zbMATHGoogle Scholar
  3. 3.
    Dinis, B., van den Berg, I.: Algebraic properties of external numbers. J. Log. Anal. 3(9), 1–30 (2011)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dinis, B., van den Berg, I.: Characterization of distributivity in a solid. Indag. Math. (N.S.) 29(2), 580–600 (2018)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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)Google Scholar
  10. 10.
    Kanovei, V., Reeken, M.: Nonstandard Analysis, Axiomatically. Springer Monographs in Mathematics. Springer, Berlin (2004)CrossRefzbMATHGoogle 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)Google Scholar
  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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nelson, E.: Internal set theory: a new approach to nonstandard analysis. Bull. Am. Math. Soc. 83(6), 1165–1198 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nelson, E.: The syntax of nonstandard analysis. Ann. Pure Appl. Logic 38(2), 123–134 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Robinson, A.: Non-Standard Analysis. North-Holland Publishing Co., Amsterdam (1966)zbMATHGoogle 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)MathSciNetGoogle 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)Google Scholar
  20. 20.
    van den Berg, I.: Asymptotics of families of solutions of nonlinear difference equations. Log. Anal. 1(2), 153–185 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    van den Berg, I.: A decomposition theorem for neutrices. Ann. Pure Appl. Logic 161(7), 851–865 (2010)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  23. 23.
    van der Corput, J.G.: Introduction to the neutrix calculus. J. Analyse Math. 7, 281–399 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Van Dyke, M.: Perturbation Methods in Fluid Mechanics. Annotated Edition. The Parabolic Press, Stanford (1975)zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Bruno Dinis
    • 1
  • Nam Van Tran
    • 2
  • Imme van den Berg
    • 3
  1. 1.Departamento de MatemáticaFaculdade de Ciências da Universidade de LisboaLisboaPortugal
  2. 2.Faculty of Applied SciencesHo Chi Minh City University of Technology and EducationHo Chi Minh CityVietnam
  3. 3.Departamento de MatemáticaUniversidade de ÉvoraÉvoraPortugal

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