Abstract: A Cobb-Douglas-function-type model on cities as systems is derived from general expressions on urban dynamic systems, y=kf_i(x₁, x₂,..., x_n), the model can be named C-D function of cities and written as y=μ∏x_ii^σ_i,where x_i(i=1, 2,..., n) represents some kind of measure on cities, y is output in some sense, k and μ both proportionality coefficients. Throught the allometric relationship between two elements of cities, x_i～x_j^α_ij, the C-D function can be transformed into the two-variable expression, y=η(x_i^b_i) (x_j^b_j). A number of equations of fractal dimension are deduced out such as α_ij=D_i/D_j=σ_j/σ_i=b_j/b_i, σ_iD_i=σ_jD_j. It is proved that the equation. σ_i/(w_ix_i)=σ_j/(w_jx_j), should be met in order to optimize urban structure and function. Taking the city of Zhengzhou as example, the models presented and advanced are verified and vindicated.

Key words: urban system, allometric growth, C-D function, fractals, Zhengzhou city, urban structure

摘要： 从一般城市动力系统出发，推导出城市和城市体系的异速生长方程和Cobb-Douglas函数(即C-D函数)，建立了二者之间的数理关系并揭示了其隐含的分形性质；进而证明城市C-D函数中的系数包含有其他各种要素的有关信息，这为借助单要素分析复杂地理动力系统提供了理论依据和实用方法。利用异速生长关系和要素-产出的弹性性质可将C-D函数化为二要素形式，据此发展了城市结构的广义维数方程。基于分形优化思想，运用数学规划原理，从上述模型中导出城市地理系统结构与功能的优化条件表达式。以郑州市为实证对象，对文中有关理论模型进行了实证分析，最后将结论推广到一般地理学领域。

关键词: 城市体系, 异速生长, C-D函数, 分维, 郑州市, 城市结构