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City is proved to be a scale-free phenomenon, and spatial autocorrelation is often employed to analyze spatial redundancy of cities. Unfortunately, spatial analysis results deviated practical requirement in many cases due to fractal nature…
A pair of fractal gravity models can be derived from the spatial interaction models based on entropy maximizing principle and allometric scaling law. The models can be expressed as dual form in mathematics and are important for analyzing…
In this paper we study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to…
Many datasets exhibit a well-defined structure that can be exploited to design faster search tools, but it is not always clear when such acceleration is possible. Here, we introduce a framework for similarity search based on characterizing…
The area-perimeter allometric scaling is a basic and important approach for researching fractal cities and has been studied for a long time. However, the boundary dimension of a city is always numerically overestimated by the traditional…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on…
In this article a definition of optimal cover for fractal structures is proposed. Expression for Minkowsky dimension is rewritten in terms of functional equation on areas of covers that constructed for different scales.Given the definition,…
The spatial pattern of urban-rural regional system is associated with the dynamic process of urbanization. How to characterize the urban-rural terrain using quantitative measurement is a difficult problem remaining to be solved. This paper…
We apply the Principle of Maximum Entropy to the study of a general class of deterministic fractal sets. The scaling laws peculiar to these objects are accounted for by means of a constraint concerning the average content of information in…
Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot},…
Quantitative analysis of the structure of star clusters is crucial for understanding their formation and evolution. In this article, we explore the application of fractal dimension analysis to study the evolution of star clusters, also…
Kolmogorov introduced a concept of Epsilon-entropy to analyze information in classical continuous system. The fractal dimension of geometrical sets was introduced by Mandelbrot as a new criterion to analyze the complexity of these sets. The…
We develop an axiomatic framework for fractal analysis and fractal number theory grounded in hierarchies of definability. Central to this approach is a sequence of formal systems F_n, each corresponding to a definability level S_n contained…
Fractal dimension is an effective scaling exponent of characterizing scale-free phenomena such as cities. Urban growth can be described with time series of fractal dimension of urban form. However, how to explain the factors behind fractal…
The methods of determining the fractal dimension and irregularity scale in simulated galaxy catalogs and the application of these methods to the data of the 2dF and 6dF catalogs are analyzed. Correlation methods are shown to be correctly…
In this paper, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to…
We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$, a fractal distribution…
We develop the hypothesis that the dynamics of a given system may lead to the activity being constricted to a subset of space, characterized by a fractal dimension smaller than the space dimension. We also address how the response function…
This work joins aspects of reservoir optimization, information-theoretic optimal encoding, and at its center fractal analysis. We build on the observation that, due to the recursive nature of recurrent neural networks, input sequences…