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On some nonlinear nonisotropic quasi-self-similar functions

Abstract

Nonlinear nonisotropic quasi-self-similar functions are important because of their relation with many physical phenomena such as fully developed turbulence or diffusion limited aggregates. Such functions are superpositions of “similar" structures at different scales, reminiscent of some modelization of turbulence. In this paper we continue to study such a class of functions. We extend our results in [Ben Mabrouk. A., Far East J. Dynam. Syst. 7(1), 23–63 (2005)] to some nonlinear cases and where some separation condition is not satisfied.

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References

  1. 1.

    Antoine, J.P., Vandergheynst, P.: Wavelets on the n-sphere and related manifolds. J. Math. Phys. 39(8), 3987–4008 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    Antoine, J.P., Vandergheynst, P.: Wavelets on the 2-sphere: a group-theoretical approach, Appl. Comp. Harmonic Anal. 7, 262–291 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    Arneodo, A., Bacry, E., Muzy, J.F.: Random cascades on wavelet dyadic trees. J. Math. Phys. 39(8), 4142–4164 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    Ben Mabrouk, A.: Spectre Multifractal de Mesure. Faculté des Sciences de Monastir. Mémoire de DEA (1998)

  5. 5.

    Ben Mabrouk, A.: Multifractal analysis of some nonisotropic quasi-self-similar functions. Far East J. Dynam. Syst. 7(1), 23–63 (2005)

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Ben Mabrouk, A.: Wavelet analysis of some nonisotropic quasi-self-similar functions. Thesis in Mathematics, Faculty of Sciences, Monastir (2005)

  7. 7.

    Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775–779 (1992)

    MATH  Article  Google Scholar 

  8. 8.

    Frisch, U., Parisi, G.: Fully developed turbulence and intermittency. In: Proceedings of the International Summer School of Physics Enrico Fermi, North Holland, pp. 84–88 (1985)

  9. 9.

    Hunt, B.R., Kaloshin, V.Y.: How projections affect the dimension spectrum of fractal measures. Nonlinearity 10, 1031–1046 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    Jaffard, S.: Multifractal formalism for functions. Part 2: selfsimilar functions. SIAM J. Math. Anal. 28(4), 971–998 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Mauroy, B.: Hydrodynamique dans le poumon. Relations entre flux et geometrie. (In French). Thesis in Mathematics, Ecole normale supérieure de Cachan (2004)

  12. 12.

    Pesin, Y.B.: Dimension Theory in Dynamical Systems, Contemporary views and Applications. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1996)

    Google Scholar 

  13. 13.

    Soares, J.V.B., Leandro, J.J.G., Cesar-Jr, R.M., Jelinek, H.F., Cree, M.J.: Retinal vessel segmentation using the 2-D Morlet wavelet and supervised classification. IEEE Trans. Med. Imag. 25(9), 1214–1222 (2006)

    Article  Google Scholar 

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Correspondence to Anouar Ben Mabrouk.

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Mathematics subject classification (2000): 42C40, 28A80, 76M55

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Mabrouk, A.B. On some nonlinear nonisotropic quasi-self-similar functions. Nonlinear Dyn 51, 379–398 (2008). https://doi.org/10.1007/s11071-007-9218-1

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  • Wavelets
  • Quasi-self-similarity
  • Multifractal analysis
  • Regularity
  • Multifractal formalism