A Continuous Derivative for Real-Valued Functions

  • Abbas Edalat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)

Abstract

We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. The values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. This is also the case for the Clarke generalised gradient. The L-derivative, however, is shown to be upper semi continuous with respect to the weak* topology, a result which is not known to hold for the Clarke gradient on infinite dimensional Banach spaces. We also formulate the notion of primitive maps dual to the L-derivative, an extension of Fundamental Theorem of Calculus for the L-derivative and a domain for computation of real-valued functions on a Banach space with a corresponding computability theory.

Keywords

L-derivative Clarke gradient weak* topology upper semi-continuity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abolfathbeigi, S., Mahmoudi, M.: Manuscript in Persian (2003)Google Scholar
  2. 2.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, Clarendon Press, Oxford (1994)Google Scholar
  3. 3.
    Clarke, F.H.: Private communications. Summer (2005)Google Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Chichester (1983)MATHGoogle Scholar
  5. 5.
    Edalat, A.: Dynamical systems, measures and fractals via domain theory. Information and Computation 120(1), 32–48 (1995)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edalat, A.: A continuous derivative for real-valued functions. Invited submission to New Computational Paradigms, 2006. Available at www.doc.ic.ac.uk/~ae/papers/banachf.ps
  7. 7.
    Edalat, A., Krznarić, M., Lieutier, A.: Domain-theoretic solution of differential equations (scalar fields). In: Proceedings of MFPS XIX, volume 83 of Electronic Notes in Theoretical Computer Science (2003), Full paper in www.doc.ic.ac.uk/~ae/papers/scalar.ps
  8. 8.
    Edalat, A., Lieutier, A.: Foundation of a computable solid modelling. Theoretical Computer Science 284(2), 319–345 (2002)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Edalat, A., Lieutier, A.: Domain theory and differential calculus (Functions of one variable). Mathematical Structures in Computer Science 14(6), 771–802 (2004)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Edalat, A., Lieutier, A., Pattinson, D.: A computational model for multi-variable differential calculus. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 505–519. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Edalat, A., Pattinson, D.: A domain theoretic account of Picard’s theorem. In: Proc. ICALP 2004 LNCS vol. 3142, pp. 494–505 (2004), Full paper in www.doc.ic.ac.uk/~ae/papers/picard.icalp.ps
  12. 12.
    Fars, N.: Aspects analytiques dans la mathematique de shraf al-din al-tusi. Historia Sc. 5(1) (1995)Google Scholar
  13. 13.
    Fars, N.: Le calcul du maximum et la ’derive’ selon shraf al-din al-tusi. Arabic Sci. Philos. 5(2), 219–237 (1995)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, UK (2003)MATHGoogle Scholar
  15. 15.
    Hogendijk, J.P.: Shraf al-din al-tusi on the number of positive roots of cubic equations. Fund. Math. 16(1), 69–85 (1989)MATHMathSciNetGoogle Scholar
  16. 16.
    Pour-El, M.B., Richards, J.I.: A computable ordinary differential equation which possesses no computable solution. Annals Math. Logic 17, 61–90 (1979)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Scott, D.S.: Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176 (1970)Google Scholar
  18. 18.
    Weihrauch, K.: Computable Analysis (An Introduction). Springer, Heidelberg (2000)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Abbas Edalat
    • 1
  1. 1.Department of Computing, Imperial College LondonUK

Personalised recommendations