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

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

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 Zipf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 Zipf式位序 -规模分布自然演化 .

A three parameter Zipf's model,P(N)=C(N-α) - d z ,is deduced out from the well known Beckmann's model on city hierarchies, P m=RKS m-1 /(1-K) m,where C=P 1[S/(S-1)] d z , α=1/(1-S), and d z=1- ln (1-K)/ ln S. 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.