DOES THE SIZE MATTER? Zipf's Law for cities Revisited

Several authors (Berry 1970, Krugman 1996 or Eaton and Eckstein 1997, among many others) have experienced amazement about the accurate functioning of the law of “least effort†established by Zipf (1949) in most places. Cities, ranked by population, seem to follow almost exactly a log/log function, in which the logarithm of the 'mass' (population, density, etc.) correlates almost perfectly with the logarithm of the order of that mass. This log/log function, advanced by Pareto in the nineteenth century, has seduced quite a number of researchers, for its presence, hypothetically, both in natural phenomena (earthquakes, meteorites, living species) and in the ones which derive from society (language, or cities), which has led to investigate its theoretical basis (Simon 1955, Brakmar et al. 1999, Gabaix 1999). While some authors (Rosen and Resnick 1980, Fan and Casetti 1994) have discussed the linear validity of Zipf's Law, introducing nonlinear models, technical literature has focused on the 'upper tail' of the urban hierarchy, large metropolitan areas, tend to silence the fact that the log / log function does not appear to be a general model. This paper attempts to show that when taking into account all the cases (ie, all populated localities in a particular territory), the log/log model seems to be only a special case of 'the big' ones. In fact it shows that a log/lin model tends to be more efficient, even with 'folded tails'. This has led to the hypothesis which was tested in this study, that the logarithm of the urban mass tends to have a 'normal distribution', leading its cumulative distribution (and ordered by rank) to be spread in a logistical structure, in 'S'. In this sense, the repeated observation of fulfillment of the Law of Zipf in the size of the cities would be just “the tip of the iceberg“, in which small and medium cities also take their part, and where a “law†of a higher level appears. The presented research questions if this “normal†appearance of the logarithm of the mass could be shaped in a simple and elegant form, and makes some experiments in this regard.

A model to evaluate the environmental and energetic efficiency of the territorial functionality (transport and activity location)

The main objective of the study is the development of a model for the evaluation of the environmental and energy efficiency of the interaction (mobility) and land use structure (called as 'territorial functionality'), applied to the metropolitan area of Barcelona . In particular there are four points that underpin the model: a) a basic land use-transportation model (LUTM), b) a model for the energy consumption and environmental emissions produced by the territorial functionality, c) a model for the consumption of land produced by the territorial functionality (developed land), and d) assessment of social equity in access to urban activities, and exposure to environmental effects. The goodness of the proposed model is that is composed of empirical models (econometric), robust in their specific topic, but spatially disaggregated (municipality). With this structure is possible to evaluate the effects of functional changes (transport projects or urban planning) in the spatial structure of energy consumption, environmental emissions, and consumption of land, and also identifying the participation (responsibility) of different territories in these effects. It is a systemic and spatial view of the role that each territory plays in the functionality, and their responsibility in the environmental effects. The model is currently under construction, joining the calibrated models of consumption and environmental emissions to an existing transport model in the metropolitan area of Barcelona. This paper presents the mathematical model, and the indicators defined for characterizing the state of the territorial system, and a simple simulation of the possible analysis and results.

Towards a New Methodology to evaluate the Urban structure of the Metropolitan Systems; Chicago and Barcelona Metropolitan Areas as Examples.

The specialized researches have proposed various approaches to the delimitation of urban and metropolitan systems. Administrative aspects (administrative units historically inherited), morphological (urban continuum), some linked to the economies of agglomeration (population and employment density, urban economic activities, …) or functional interaction (home-work commuting) have been used for the definition of a metropolitan area. At the same time it has developed in recent years, an extensive research that has come to reveal the progressive trend of the metropolitan areas to the polycentrism. There is no doubt that the monocentric city paradigm, structured around a single CBD, is broken. The contemporary metropolises have seen the emergence of the phenomenon of the sub centers. This changes in the internal structure of many cities has extended the hypothesis that the polycentric city is more efficient than traditional monocentric city, from an environmental perspective (ETE, 1999), representing usually shorter home-work trips, thus contributing to more sustainable mobility. However, few efforts have been directed to address the three aspects together: a) the metropolitan boundaries, b) analysis of its internal structure, and c) contrasting the hypothesis of the environmental efficiency of polycentric systems. This paper addresses the challenge of defining an integrated way of both metropolitan areas and their internal composition, structured or not around different sub centers. The methodology developed of the Interaction Value simultaneously allows delimiting these two levels of urban structure: the metropolitan system as a whole and the subsystems articulated around the emerging sub centers, by measuring the functional relationships between housing and workplaces. At the same time, the Interaction Value assesses the degree of polycentrism beyond the simple identification of sub centers developed in the literature. And, so, support the hypothesis that the polycentric city structure is more efficient, from an environmental perspective, than the monocentric. In this way, and taking the cases of Chicago and Barcelona metropolitan areas as examples of these types of organizations, the efficiency of metropolitan structures is evaluated from the dual perspective of land consumption and sustainable mobility.

Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric Full-size image (<1 K)-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the Full-size image (<1 K)-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.This research was partially supported by the Spanish Ministerio de Economía y Competitividad under grand contract CTM2014-55014-C3-3-R, and by the Government of Catalonia under grand contract 2014-SGR-1471. The work of the last author was supported by the European Commission through the Marie Sklodowska-Curie Actions (HiPerMeGaFlows project).Peer ReviewedPostprint (published version