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Purpose: This paper aims to research the relationship between income gap (IG) and green economic growth based on the perspective of economic inequality. Design/methodology/approach: Based on the panel data of 283 prefecture-level cities in China from 2011 to 2020, this paper uses the super slack based model (Super-SBM) to measure the efficiency of regional green economic growth, and constructs a regression model of the IG and regional green economic growth. Findings: Firstly, the IG has an inhibitory effect on the growth of regional green economy (GE). Secondly, the relationship between the IG and regional green economic growth will be affected by the threshold value of income level. Thirdly, environmental regulation and government competition will increase the inhibitory effect of the IG. Originality/value: According to the characteristics of China's regional economy, the researchers construct an empirical model of the IG and green economic growth to study their relationship, the threshold effect of income level and the moderating effect of environmental regulation and local government competition. The research content, methods and conclusions of this paper provide new evidence for the sustainable growth of China's regional GE. [ABSTRACT FROM AUTHOR]

A pair of fractal gravity models can be derived from spatial interaction models based on the entropy maximizing principle and the allometric scaling law. The models can be expressed in dual form in mathematics and are important for analyzing and predicting spatial flows in networks of cities. However, the dual relationship of urban gravity parameters has remained an empirical relationship for a long time and lacks theoretical proof. This paper is devoted to proving the duality of fractal gravity models by means of ideas from scaling invariance. The results show that a fractal gravity model can be derived from its dual form. The observational data of interurban spatial flows of the Beijing-Tianjin-Hebei region of China are employed to make a case study, lending further support for the theoretical derivation. It appears that the duality of gravity models rests with the scaling symmetry of fractal structure. This work may be helpful for understanding the theoretical essence and application direction of spatial interaction modeling. • Fractal gravity models include inflow model and outflow model with dual relationships. • There is a dual relation between two scaling exponents of fractal gravity models. • The duality indicates commutative law of scaling exponents of fractal gravity models. [ABSTRACT FROM AUTHOR]

Abstract: This paper demonstrates self-affine fractal structure of city systems by means of theoretical and empirical analyses. A Cobb–Douglas-type function (C–D function) of city systems is derived from a general urban response equation, and the partial scaling exponent of the C–D function proved to be the fractal dimension reflecting the self-affine features of city systems. As a case, the self-affine fractal model is applied to the city of Zhengzhou, China, and the result is satisfying. A fractal parameter equation indicative of structural optimization conditions is then obtained from the C–D function. The equation suggests that priority should be given to the development of the urban element with a lower fractal dimension, or a higher partial scaling exponent, for utility maximization. Moreover, the fractal dimensions of different urban elements tend to become equivalent to each other in the long term. Accordingly, it is self-similar fractals rather than self-affine fractals that represent the optimal structure of city systems under ideal conditions. [Copyright &y& Elsevier]

Abstract: Cities as superorganisms confront disturbances from their metabolic processes, including large metabolic fluxes, low stocks of resources and products, and a low efficiency of the urban material metabolism. Based on ecological thermodynamics, an indicator system is established in this paper to evaluate the fluxes, stocks, and effiency of the urban material metabolism using emergy analysis. Also, a new model for the urban material metabolism is proposed to define the production possibility curve using a wealth index (WI) and an ecological effiency index (EEI). Then, six large Chinese cities including Beijing, Shanghai, Tianjin, Chongqing, Guangzhou, and Shenzhen are selected as typical cases to validate the proposed model. The results show that Shenzhen has the highest metabolic capacity, followed by Beijing, Shanghai, Guangzhou, Tianjin, and Chongqing. It is also reflected that different urban material metabolisms of the six cities are resulted from the varied regional metabolic capacities, thus providing insights into how cities improve their metabolic capacities. Meanwhile, Shenzhen has the highest WI and EEI, Chongqing has the lowest WI and EEI, and Beijing, Shanghai, Tianjin, and Guangzhou has low WI and high EEI. It is also shown that the sustainable operation of the urban material metabolism reflects the mutualism and symbiosis between socioeconomic development and ecological environment protection. [Copyright &y& Elsevier]

Abstract: This paper presents a computer-based analytical framework for the spatio-temporal evolution of urban systems using the ideas from the allometric growth associated with fractals. Both cities as systems and systems of cities follow the law of allometric growth, and the scaling factors of the allometric relations can compose the matrix equations as eigenfunctions. The fractal dimension arrays are just the eigenvectors of the scaling factor matrices while the numbers of variables are the greatest eigenvalues. The solutions of matrix equations can be employed to analyse city systems and evaluate relative levels of urban development. The method is applied to Hangzhou urban system of China. The results reveal clearly an urbanization process characterized as population concentration and an industrialization process characterized as industrial diffusion. The computation results are consistent with the reality, which indicate that the method is available for analyzing the spatio-temporal evolution of complex systems such as cities. [Copyright &y& Elsevier]