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The scaling exponent of a hierarchy of cities used to be regarded as a fractional dimension. The Pareto exponent was treated as the fractal dimension of size distribution of cities, while the Zipf exponent was considered to be the reciprocal of the fractal dimension. However, this viewpoint is not exact. In this paper, I will present a new interpretation of the scaling exponent of rank-size distributions. The ideas from fractal measure relation and the principle of dimension consistency are employed to explore the essence of Pareto's and Zipf's scaling exponents. The Pareto exponent proved to be a ratio of the fractal dimension of a network of cities to the average dimension of city population. Accordingly, the Zipf exponent is the reciprocal of this dimension ratio. On a digital map, the Pareto exponent can be defined by the scaling relation between a map scale and the corresponding number of cities based on this scale. The cities of the United States of America in 1900, 1940, 1960, and 1980 and Indian cities in 1981, 1991, and 2001 are utilized to illustrate the geographical spatial meaning of Pareto's exponent. The results suggest that the Pareto exponent of city-size distributions is a dimension ratio rather than a fractal dimension itself. This conclusion is revealing for scientists to understand Zipf's law on the rank-size pattern and the fractal structure of hierarchies of cities. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the effect of microwave absorbing material-based (MAM) substrate on the absorption properties of fractal frequency selective surface (FSS) absorbers was investigated in detail. The fractal shapes were proposed to construct the schematic diagrams of the FSS unit cell. The absorption properties of the absorber samples containing different substrates, including FR-4dielectric substrate and MAM-based substrate, were exhibited via the Naval Research Laboratory (NRL)-arch method, respectively. Then the power loss density distributions of the absorbers were illustrated by the finite integration technology (FIT) method to obtain an insight of the physical picture into the absorption properties. The results indicate that with the increase of the substrate thicknesses, the FR-4-based FSS absorbers could achieve the strongest absorbing performance at a certain thickness. However, employing the MAM-based substrate to replace the FR-4 substrate with the same substrate thickness enhances the absorbing properties greatly, including the peak values and the bandwidths. As a result, the way of introducing the MAM-based substrates instead of the traditional dielectric substrates provides a potential practical application in device miniaturization. [Copyright &y& Elsevier]

Abstract: This study applied logistic regression to model urban growth in the Atlanta Metropolitan Area of Georgia in a GIS environment and to discover the relationship between urban growth and the driving forces. Historical land use/cover data of Atlanta were extracted from the 1987 and 1997 Landsat TM images. Multi-resolution calibration of a series of logistic regression models was conducted from 50m to 300m at intervals of 25m. A fractal analysis pointed to 225m as the optimal resolution of modeling. The following two groups of factors were found to affect urban growth in different degrees as indicated by odd ratios: (1) population density, distances to nearest urban clusters, activity centers and roads, and high/low density urban uses (all with odds ratios<1); and (2) distance to the CBD, number of urban cells within a 7×7 cell window, bare land, crop/grass land, forest, and UTM northing coordinate (all with odds ratios>1). A map of urban growth probability was calculated and used to predict future urban patterns. Relative operating characteristic (ROC) value of 0.85 indicates that the probability map is valid. It was concluded that despite logistic regression’s lack of temporal dynamics, it was spatially explicit and suitable for multi-scale analysis, and most importantly, allowed much deeper understanding of the forces driving the growth and the formation of the urban spatial pattern. [Copyright &y& Elsevier]

Abstract Aside from deforestation and selective logging, mature tropical rain forests appear to be in a state of near-term flux. At plot scales over a 20-year period, forest inventory data (representing areas < 750 ha) have shown stands to be accumulating above-ground biomass pantropically while displaying signs of increasing turnover. To assess whether or not such disequilibrium is manifest at broader spatial scales (i.e. coarser grain sizes and larger extents), we performed a change detection analysis of landscape texture (i.e. the organization of reflective properties), in satellite images of closed-canopy tropical rain forests considered to be anthropically undisturbed. Here we show that fractal properties of pixel spectral values depicting low and high levels of photosynthetic activity underwent significant shifts from the 1970s to the 1980s. Following expectations for aggrading forests, canopy texture became more random throughout the tropics. Although subject to periodic disturbance events such as natural exogenic perturbations and/or synchronous die-off, which should produce no consistent trend, these forested landscapes across the globe exhibited similar dynamics at fine temporal (decadal) intervals. Such biophysical changes (representing areas > 1000 000 ha) directly affect atmospheric boundary layer conditions and could have implications with respect to biodiversity and carbon cycling in these systems. [ABSTRACT FROM AUTHOR]

The coal industry is dominated by the largest mines, with 1% of coal mines in the U.S. being responsible for 65% of the coal cumulatively produced. We show that this "heavy tail" can be well approximated by a power law, where mine-level cumulative production is inversely proportionate to distributional rank. Maximum likelihood and regression-based procedures estimate the counter-cumulative power-law parameter to be less than one, indicating there is no well-defined mean or variance for cumulative production. Goodness of fit tests indicate the power-law is a better fit to the data than other competing distributions, including the lognormal. • Mine-level cumulative coal production is power-law distributed. • Distribution indicates there is no well-defined mean or variance. • Distribution shows how heavily concentrated coal production is in the US. [ABSTRACT FROM AUTHOR]

Applications of the ideas gained from fractal theory to characterizerainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono-fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi-fractal approaches. The assumption behind multi-fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi-fractals. However, it has been observed recently that multi-fractal approaches mightprovide positive evidence of a multi-fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi-fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractalidentification methods indicate that the rainfall data exhibit multi-fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi-fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi-fractal approach for rainfallcha [ABSTRACT FROM AUTHOR]