多重分形理论在量子微积分中的推广
Generalization of multifractal theory within quantum calculus
摘要：
在量子微积分中变形级数的基础上，推广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熵流形变形有关，而质量指数扩展概括了已知的关系$τ_q=D_q(q-1)$。我们发现方程组的平均值有关的普通，护送，和广义概率的变形膨胀。以与康托二项集、兑换货币序列和多孔表面凝聚相关的多重分形为例。关键词：多重分形集;形变;幂级数.
英文摘要：
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 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 $τ_q=D_q(q-1)$. We find equation for set of averages related to ordinary, escort, and generalized probabilities in terms of the deformed expansion as well. Multifractals related to the Cantor binomial set, exchange currency series, and porous surface condensates are considered as examples. Keywords:Multifractal set; Deformation; Power series.
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