A type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimension can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connect between entropy and fractal dimension, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurement into fractal dimension for the spatial analysis of scale-free urban phenomena using ideas from scaling. Urban systems proved to be random prefractal and multifractals systems. The entropy of fractal cities bears two typical properties. One is the scale dependence. Entropy values of urban systems always depend on the scales of spatial measurement.The other is entropy conservation. Different fractal parts bear the same entropy value. Thus entropy cannot reflect the spatial heterogeneity of fractal cities in theory. If we convert the generalized entropy into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. The essence of scale dependence is the scaling in cities, and the spatial heterogeneity of cities can be characterized by multifractal scaling. This study may be helpful for the students to describe and understand spatial complexity of cities.

Abstract:Multifractal theory provides a new spatial analytical tool to describe urban form and growth, but many basic problems remain to be solved. Among various pending issues, the most significant one is how to obtain proper multifractal dimension spectrums. If an algorithm is improperly used, the parameter values will be abnormal. This paper is devoted to drawing a comparison between two OLS-based approaches for estimating urban multifractal parameters.Using observational data and empirical analysis, we will demonstrate how to utilize the double logarithmic linear regression to evaluate multifractal parameters. The OLS regression analysis has two different approaches. One is to fix the intercept to zero, and the other is not to fix it. The case studies show that the advisable method is to constrain the intercept to zero. The zero-intercept regression yields proper multifractal parameter spectrums within certain scale range of moment order, while the common regression results are not normal. In practice, the zero-intercept regression and the common regression can be combined to calculate multifractal parameters.Comparing the two sets of results, we can judge when and where multifractal urban structure appears. This research will inspire urban scientists to employ a proper technique to estimate multifractal dimensions for urban scientific studies.

The growth curves of fractal dimension of urban form take on squashing effect and can be described by sigmoid functions. The fractal dimension growth of urban form in western countries can be modeled by Boltzmann's equation and logistic function. However, these models cannot be well applied to the fractal dimension growth curve of Beijing city, the national capital of China. In this paper, the experimental method is employed to find parametric models for the growth curves of fractal dimension of Chinese urban form. By statistical analysis, numerical analysis, and comparative analysis, we find that the quadratic Boltzmann equation and quadratic logistic function can be used to characterize how the fractal dimension of the urban land-use pattern of Beijing increases in the course of time. The models are also suitable for many cities in the north of China.In order to convert the empirical models into theoretical models, we attempt to construct a model of spatial replacement dynamics of urban evolution, from which the logistic model of urban fractal dimension growth can be derived. The models can be utilized to predict the rate and upper limitation of Chinese urban growth. In particular, the models can be employed to reveal the similarities and differences between the fractal growth of Chinese cities and that of the cities in western countries.

The gravity model is one of important models of social physics and human geography, but several basic theoretical and methodological problems remain to be solved. In particular, it is hard to explain and evaluate the distance exponent using the ideas from Euclidean geometry. This paper is devoted to exploring the distance-decay parameter of the urban gravity model. Based on the concepts from fractal geometry, several fractal parameter relations can be derived from scaling laws of self-similar hierarchies of cities. Results show that the distance exponent is just a scaling exponent, which equals the average fractal dimension of the size measurements of the cities within a geographical region. The scaling exponent can be evaluated with the product of Zipf's exponent of size distributions and the fractal dimension of spatial distributions of geographical elements such as cities and towns. The new equations are applied to China's cities, and the empirical results accord with the theoretical expectations. The findings lend further support to the suggestion that the geographical gravity model is a fractal model, and its distance exponent is associated with a fractal dimension and Zipf's exponent. This work will help geographers understand the gravity model using fractal theory and estimate the distance exponent using fractal modeling.A set of gravity models have been applied to explain and predict various behaviors of spatial interactions in many social sciences. Among this family of models, the basic one is the gravity model of migration based on an inverse power-law distance-decay effect, which was proposed by analogy with Newton's law of gravitation. It can be used to describe the strength of interaction between two places (Haynes and Fotheringham, 1984;Liu and Sui et al, 2014;Rodrigue et al, 2009;Sen and Smith, 1995). According to the model, any two places attract each another by a force that is directly proportional to the product of their sizes and inversely proportional to the bth power of the distance between them, and b is the distance-decay exponent, which is often termed distance exponent for short (Haggett et al, 1977). The size of a place can be measured with appropriate variables such as population numbers, built-up area, and gross domestic product (GDP). Recently, the gravity model has been employed to study the attractive effect of various new-fashioned human and physical activities by means of modern technology (Balcan et al, 2009;Goh et al, 2012;Kang et al, 2012;Kang et al, 2013;Jung et al, 2008;Lee et al, 2014;Liang, 2009;Liu and Wang et al, 2014;Simini et al, 2012). The model is empirically effective for describing spatial interactions, but it is obstructed by two problems on the distance exponent, b. One is the dimensional problem, that is, it is impossible to interpret the distance exponent in light of Euclidean geometry (Haggett et al, 1977;Haynes, 1975). The other is the algorithmic problem, namely, it is hard to estimate the numerical value of the distance exponent (Mikkonen and Luoma, 1999).Th...

Antarctic krill oil (KO) prepared using supercritical carbon dioxide extraction and characterized using gas chromatography-mass spectrometry was used to investigate its preventive effect on ethanol-induced gastric tissue damage in a rat model in vivo. KO characterization showed that 74.96% of the unsaturated fatty acids consist of oleic acid, eicosapentaenoic acid (EPA), and docosahexaenoic acid (DHA). Rats pre-treated with KO (100, 200, and 500 mg/kg) showed mitigated oxidative stress through enhanced antioxidant enzyme superoxide dismutase (SOD) and reducing enzymes malondialdehyde (MDA) and myeloperoxidase (MPO) in gastric mucosal injury induced by ethanol. Additionally, the secretion of pro-inflammatory cytokines (TNF-α, IL-6, and IL-1β), the expression of the IκBα/NF-κB signaling pathway, and nitric oxide (NO) production was suppressed. The results also demonstrated a significant decrease in histological injury and hemorrhage scores in a dose-dependent manner in the KO range. Therefore, KO has potential as a food supplement to alleviate ethanol-induced acute gastric mucosal injury.