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The cumulative function PðSÞ which gives the number of entities with a size larger or equal (or smaller) than a given S. These two functions DðSÞ and PðSÞ can also be expressed in relative terms. The two relative functions do not give the number of entities but rather the percentage (or probability) of the total number of entities. The rank size representation which is transformed into the logarithm of the size (y ¼ ln Size) and plotted as a function of the logarithm of the rank (x ¼ ln Rank). The function yð xÞ is called the rank-size curve.

8) as constants: the log term because it changes very slowly in its dependence on N and hence on T; and the y 1 term because its argument is larger than 1 (considering the real values of the parameters). The final result suggests that t is linearly dependent on T 1=3 . 9), it can be deduced that the dependence of N on t is indeed the quadratic equation we expected. 3 Macro and Micro Dynamics of the City Size Distribution 41 In the following steps, we consider K as constant, choose an initial state for which the number of cities is roughly 150 (see Fig.

In these cases, the order parameters are straightforward to define. In the urban case, intuition suggests that it is the nature of the configurations of {Wj} that we are concerned with. A dispersed system with many small centres can be considered less ordered than one with a small number of large centres. This suggests that we should examine N[Wj >x] the number of Wj greater than some parameter x. If x is set to zero, this will be a measure of ubiquity of centres and we know that there will be transitions at a¼1.

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