All the contents are developed around a set of scaling laws taking the form of exponentials which relate to almost all the issues of complexity including fractals , chaos , strange attractors , localization , and symmetry breaking , etc . the main work can be summarized as follows : starting from the law of allmetric growth three fractal dimensions in a broad sense are derived , and according to these dimensions , geographical space is divided into three levels , i . e . , real space , phase space , and order space , each of which corresponds to a kind of dimension . based on the idea of spatial disaggregation and using the rmi ( relationship - mapping - reversion ) principle , the urban system is formulated as three scaling laws of the three spaces , including number law , size law , and area law , which can be transformed into a set of power laws such as allometric law and zipf ’ s law associated with fractal structure 从异速生长律的纵向、横向和切向三个角度将地理空间划分为实空间、相空间和序空间,分别对应于空间系列、时间序列和等级序列三个层面,每个层面的测度各有自己的空间维度。基于“空间循环细分-等级体系-网络结构”的数理等价关系,利用rmi (关系-映射-反演)原则,成功地实现了城市系统宏观模型的理论抽象,将空间复杂性问题表征为简单的指数式标度定律(包括数量律、规模律和尺度律) ,这一组标度律可以与一组幂次定律(包括具有分形性质的规模-数目律、异速生长定律和三参数zipf定律)互为变换。
In the field of polymer solutions , a series of high precision measurements of the critical properties of polymer solutions near their critical points by his group has established scaling laws and resolved several long - standing disputes about the exact values of the associated exponents and the relations among them . the results are now benchmark data sets against which various theoretical models are tested 在聚合物溶液的研究领域中，他领导的研究小组完成了一系列有关聚合物在临界点附近的高精度测量。这些工作导致了一系列标度律的建立并且结束了对有关标度指数数值的长期争论。这些实验结果现已成为验证多种理论模型的基准资料。
Thus scale - measurement - dimension constitutes the fundamental concepts of this project . the scaling laws are derived from different perspectives , using entropy - maximizing methods in macro - level and utility - maximizing method in micro - level , consequently a pair of dual models based on the idea from nonlinear programming are built to interpret both the scaling laws and thus power laws . a discovery is that the energy and information can be transformed into each other in urban systems 借助熵最大化方法从宏观上推导了标度律,利用效用最大化原理从微观上导出了标度律,然后建设了熵与效用最大化的对偶转换模型;通过对简单的网络模型的研究,论证熵与效用最大化过程本质上是追求系统整体上的最高效率与局部公平合理。
In order to analytically deduce the characteristic scaling law , we have constructed a simplified piecewise linear model that describes the characteristic phenomenon so that we can quantitatively and analytically deduce the sudden change of the rules of the fractal dimension of the strange repeller and the averaged lifetime in the region occupied by the original attractor at a critical parameter value when the repeller disappears 为了解析地导出这种激变的特征标度律，我们构造了一个描述这种特征现象的简化分段线性模型，并借助它定量地解析描绘了当奇异排斥子在临界参数值消失时，奇异排斥子的分数维和在原混沌吸引子区域迭代的平均生存时间的突变。
The braess paradox model is taken as an example to reveal the essence of entropy - maximization and utility - maximization which lies in the unity of the most efficient for the whole and the fairest for local parts and individuals . the internal symmetry of the scaling laws are bring to light based on the dual model of entropy - and utility - maximization . the internal symmetry including translational symmetry , dilation symmetry , and bilateral symmetry originate from relative energy - conservation and information conservation of urban systems reflected by the dual models , and parity conservation of scaling laws 揭示标度律及其等价模型各种对称性质,发现城市系统具有平移对称、镜像对称和标度对称等内部对称性,这些对称性都对应着城市系统的不可观测量,其本质在于系统内部隐含的守恒律,包括标度变换的奇偶守恒、能量和信息的相对守恒等。
