量子微积分中的多重分形理论
Multifractal theory within quantum calculus
摘要：
在量子微积分的框架内，基于差分$q-1$的变形幂展开，我们表示了多重分形的配分函数和质量指数，以及分布在自相似集上的随机变量的平均值.对于配分函数，这种扩展被证明是由二项式型组合的Tsallis熵有关的流形变形，而质量指数扩展概括已知的关系$τ_q=D_q(q-1)$。我们发现，在变形膨胀的护送概率以及相关的物理平均。证明了在多重分形集的变形过程中，质量指数会出现与多重分形集的相变有关的奇异性。
英文摘要：
Within framework of the quantum calculus, we represent the partition function and the mass exponent of a multifractal, as well as the average of random variables distributed over self-similar set, on the basis of the deformed expansion in powers of the difference $q-1$. 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 known relation $τ_q=D_q(q-1)$. We find the physical average related to the escort probability in terms of the deformed expansion as well. It is demonstrated the mass exponent can acquire a singularity that relates to a phase transition of the multifractal set in the course of its deformation.
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