This paper develops a set of quantitative variables to characterise urban forms at the metropolitan level and, in particular, to distinguish compactness from ‘sprawl’. It first reviews and analyses past research on the definitions of urban form, compactness and sprawl, and corresponding quantitative variables. Four quantitative variables are developed to measure four dimensions of urban form at...

We consider a multi-type Moran model (in continuous time) with selection and typedependent mutation. This paper is concerned with the evolution of genealogical information forward in time. For this purpose we define and analytically characterize a pathvalued Markov process that contains in its state at time t the extended ancestral lines (adding genealogical distances) of the population alive a...

By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit. We show further that the invariant probability measure of the Moran model converges to the unique stationary solution of Eigen's quasispecies model.

A fractal, as originally described by Mandelbrot, is a set with an irregular and fragmented shape. Many fractals that have been extensively studied, such as self-similar sets, have the same degree of irregularity and fragmentation at all length scales. In contrast to this, an equi-homogeneous set is an irregular and fragmented shape that at each fixed scale is identical at every point. We show ...