城市位序-规模法则的理论难题:数理本质与解决方案  

Zhou′s Paradox on Rank-Size rule of cities: mathematical essence and solution

作　　者：陈彦光[1] 

机构地区：[1]北京大学城市与环境学系,北京100871

出　　处：《陕西师范大学学报（自然科学版）》2001年第2期98-102,共5页Journal of Shaanxi Normal University：Natural Science Edition

基　　金：国家自然科学基金!资助项目 ( 4 97710 32 ) ;河南省自然科学基础研究资助!项目 ( 0 0 4 0 712 0 0 )

摘　　要：周一星在研究京津冀地区的城市规模分布时发现Zipf模型P(r) =P1r-q的参数P1的计算值与实际值存在误差 ,即最大城市的人口数的理论值与实际数有不容忽视的差距 ,从而出现了理论与实验的矛盾 .为了解决这一问题 ,从一对具有普遍意义的奇异对称序列出发 ,导出了城市位序 规模法则的精确表达式 ,从而揭示了“周一星难题”的数理本质 .It was found by Y.X. Zhou(1995) in his studying the city-size distribution of the Beijing area, the Tianjin area, and Hebei Province in North China that there is a great difference between the calculated value of the parameter, P 1, in Zipfs model, P(r)=P 1r -q , and the real population value of the first-ranking city in a region. The contradiction between theory and reality is named ‘the Zhous paradox′ by the author of this paper in which the mathematical essence of the `paradox′ is brought to light. The Zipf model is precisely derived out as follows, P(r)=C mr -q =τ qP 1r -q ,where τ=(δ-δ 1-m )/(δ-1), and q>0,δ>1,m=1,2,…,N. When and only when τ q=1,C m=τ qP 1=P 1, however, it is obvious that τ q≥1, so C m≥P 1, and moreover, C m is not a constant in a strict sense. Because C m used to be considered to be equal to P 1, a paradox emerged. The solution to the problem is to substitute a three-parameter model, P(r)=C(r-α) -dz , for the traditional two-parameter Zipf′s model, where C, α, and dz are parameters. It can be proved that dz=q=1/D, and D is a kind of fractal dimension.

关 键 词：城市体系 城市规模分布 ZiPf定律 分维 "同一星难题" 城市位序-规模法则 

分 类 号：C912.81[经济管理]