# Generalization of multifractal theory within quantum calculus

@article{Olemskoi2010GeneralizationOM,
title={Generalization of multifractal theory within quantum calculus},
author={Alexander I. Olemskoi and I. {\A}. Shuda and Vadim Borisyuk},
journal={EPL},
year={2010},
volume={89},
pages={50007}
}`
• Published 1 March 2010
• Mathematics, Physics
• EPL
On the basis of the deformed series in quantum calculus, we generalize the partition function and the mass exponent of a multifractal, as well as the average of a random variable distributed over a self-similar set. For the partition function, such expansion is shown to be determined by binomial-type combinations of the Tsallis entropies related to manifold deformations, while the mass exponent expansion generalizes the known relation τq=Dq(q-1). We find the equation for the set of averages…
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## References

SHOWING 1-10 OF 17 REFERENCES
Erratum: Fractal measures and their singularities: The characterization of strange sets
• Physics, Medicine
Physical review. A, General physics
• 1986
A description of normalized distributions (measures) lying upon possibly fractal sets; for example those arising in dynamical systems theory, focusing upon the scaling properties of such measures, which are characterized by two indices: \ensuremath{\alpha}, which determines the strength of their singularities; and f, which describes how densely they are distributed.
Basic Hypergeometric Series
• Mathematics
• 1990
Foreword Preface 1. Basic hypergeometric series 2. Summation, transformation, and expansion formulas 3. Additional summation, transformation, and expansion formulas 4. Basic contour integrals 5.
Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World
Basics or How the Theory Works.- Historical Background and Physical Motivations.- Learning with Boltzmann-Gibbs Statistical Mechanics.- Generalizing What We Learnt: Nonextensive Statistical
Self-organization of quasi-equilibrium steady-state condensation in accumulative ion-plasma devices
• Physics
• 2009
We consider both theoretically and experimentally self-organization process of quasi-equilibrium steady-state condensation of sputtered substance in accumulative ion-plasma devices. It has been shown
Fractals (Plenum Publishers, New York
• 1988
Quantum Calculus (Springer, New York
• 2002
Symmetric Quantum Calculus
• Physics
• 2002
The q- and h-differentials may be “symmetrized“ in the following way, $$\tilde d_q f(x) = f(qx) - f(q^{ - 1} x),$$ (26.1) $$\tilde d_h g(x) = g(x + h) - g(x - h),$$ (26.2)
Critical Phenomena in Natural Sciences (Springer, New York
• 2001
Phys
• Rev. Lett. 78
• 1997