【摘要】 研究了用配分函数法定义的多重分形理论中奇异谱的几何特性 ,通过严格的数学推理证明了广义维数 Dq、质量指数 τ( q)、奇异性指数 α( q)和奇异谱 f( α( q) )的有关性质 ,明确提出了判定合理奇异谱 f( α( q) )的准则 ,给出了在参数 q→±∞时计算相应函数极限的解析算法。更多还原

【Abstract】 To describe multiplicative cascade processes with unequal scales, formalism of multifractals defined via partition function should be adopted. The geometrical characteristics of singularity spectrum of these multifractals are studied. The relevant properties of generalized dimension D q , mass index τ(q) , singularity α(q) and singularity spectrum f(α(q)) are derived rigorously, which are somewhat similar to those obtained from multifractals defined via Renyi information but more general in mathematical form. The curve of generalized dimensions is similar to that of singularity strength when the parameter q tends to infinite. The singularity spectra curve lies in the first quadrant, whose endpoints are not necessary to be nought. The asymptotic behaviors are studied and the analytical algorithms calculating the limits of the relevant functions with q tending to infinity are presented. The criterion based on the fact that, the curve of the multifractal spectrum is tangent to the diagonal of the first quadrant is still valid in determining the proper singularity spectrum. This implies that f≤α for all q and no extreme point in the curve of the generalized dimensions.更多还原