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Randomness and Differentiability of Convex Functions

  • Alex GalickiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9136)

Abstract

We study first and second derivatives of computable convex functions on \(\mathbb {R}^n\). The main result of the paper is an effective form of Aleksandrov’s Theorem: we show that computable randomness implies twice-differentiability of computable convex functions.

References

  1. 1.
    Alberti, G., Ambrosio, L.: A geometrical approach to monotone functions in \(\mathbb{R}^n\). Math. Z. 230, 259–316 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aleksandrov, A.D.: Leningrad Univ. Ann. (Math. Ser.) 6, 3–35 (1939)Google Scholar
  3. 3.
    Brattka, V., Miller, J., Nies, A.: Randomness and differentiability. Trans. AMS (forthcoming). http://arxiv.org/abs/1104.4465
  4. 4.
    Dingzhu, D., Ko, K.: Computational complexity of integration and differentiation of convex functions. Syst. Sci. Math. Sci. 2(1), 70–79 (1989)zbMATHGoogle Scholar
  5. 5.
    Downey, R., Hirschfeldt, D.: Algorithmic Randomness and Complexity. Springer, Berlin (2010)zbMATHCrossRefGoogle Scholar
  6. 6.
    Freer, C., Kjos-Hanssen, B., Nies, A., Stephan, F.: Algorithmic aspects of lipschitz functions. Computability 3(1), 45–61 (2014)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Galicki, A., Turetsky, D.: Differentiability and randomness in higher dimensions (2014, Submitted)Google Scholar
  8. 8.
    Minty, G.: Monotone nonlinear operators on a Hilbert space. Duke Math J. 29, 341–346 (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Nies, A.: Computability and Randomness. Oxford Logic Guides. Oxford University Press, Oxford (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Pour-El, M.B., Richards, I.: Computability in Analysis and Physics. Springer, Berlin (1988)Google Scholar
  11. 11.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg (1997)Google Scholar
  12. 12.
    Rute. J.: Computable randomness and betting for computable probability spaces (2012, Submitted)Google Scholar
  13. 13.
    Siksek, S., El Sedy, E.: Points of non-differentiability of convex functions. Appl. Math. Comput. 148, 725–728 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Weihrauch, K.: Computable Analysis. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  15. 15.
    Zhong, N.: Derivatives of computable functions. Math. Log. Q. 44, 304–316 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of AucklandAucklandNew Zealand

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