城镇等级体系的Beckmann模型与三参数Zipf定律的数理关系——Beckmann城镇等级 - 规模模型的分形与分维

从基于中心地体系的Beckmann城镇等级－－规模模型Pm=RKSm-1/(1-K)^m出发，通过序列的对称性分析，导出三参数Zipt模型P（N）＝C（N－α）^-dz，证明了参数dz的分维性质（dz=1/D)，以及Beckmann模型与Davis二倍数规律的等价性，进而借助基于Beckmann模型的城镇化水平公式Z＝KS／（K＋S－1）的单调增减性规律论证，中心地的“等级阶梯”必将向Zipt式位序0－规模分布自然演化。

Mathematical relationships between the three-parameter Zipf law and the Beckmann model of city hierarchies

A three-parameter Zipf's model,P(N)=C(N-α)-dz,is deduced out from the well-known Beckmann's model on city hierarchies, Pm=RKSm-1/(1-K)m,where C=P1［S/(S-1)］dz, α=1/(1-S), and dz=1-ln(1-K)/lnS. On the other hand,a formula on level of urbanization based on the Beckmann model, Z=KS/(K+S-1), implies that, S, the number of satellite towns around a city, ‘decreases’ along with the increase of urbanization level of a region, Z, for ( Z/ S)＜0. This paper makes a conclusion as follows: The ‘stairs’ of city-size hierarchies deriving from the central place theory will inevitably transform into the rank-size distribution with fractal nature because of urbanizational dynamics.