A PROOF OF GENERALIZED DAVIS’S2~N-LAW AS A MATHEMATICAL EQUIVALENT OF BECKMANN'S MODEL OF CITY HIERARCHIES: RESEARCHING INTO THE MICRO-MACRO SYMMETRY OF CITY RANK-SIZE RULE
从贝克曼城市等级模型推导出广义戴维斯δ的n次方律，进而得到三参数齐夫模型，并以中国东北城市系统为例分析城市位序-规模法则的微观-宏观对称性。
The δ n -law of city hierarchies, a i=a i+n λ n,f i=f i+n δ -n (λ1,δ1),which was derived by the authors from Davis's 2 n-law, a i=a i+n 2 n,f i=f i+n 2 -n ,is deduced from Beckmann's model on city hierarchies and city-size distribution, P m=RKS m-1 /(1-K) m, and from the δ n -law, the three-parameter Zipf model can be deduced out as folows: P(r)=C(r-α) -dz , where C=P 1[δ/(δ-1)] dz ,α=1/(1-δ),dz= ln λ/ ln δ=1- ln (1-K)/ ln S, and as for the parameters , P 1=a 1,δ=S, dz=1/D, where D is fractal dimension of city-size distributions. Based on the results of the mathematical deduction and the urban systems in Northeast China as analysed examples, a research is made into the micro-macro symmetry of Rank-Size rule of cities.