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Fractal derivatives, fractional derivatives and $q$-deformed calculus [PDF]

Entropy 2023, 25(7), 1008, 2023
This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff's concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function,
arxiv   +1 more source

Fractal Analytical Approach of Urban Form Based on Spatial Correlation Function [PDF]

Chaos, Solitons & Fractals, 2013, 49: 47-60, 2012
Urban form has been empirically demonstrated to be of scaling invariance and can be described with fractal geometry. However, the rational range of fractal dimension value and the relationships between various fractal indicators of cities are not yet revealed in theory. By mathematical deduction and transformation (e.g.
arxiv   +1 more source

Fractal Modeling and Fractal Dimension Description of Urban Morphology [PDF]

Entropy, 2020,22: 961, 2018
The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization ...
arxiv   +1 more source

Scattering from surface fractals in terms of composing mass fractals [PDF]

J. Appl. Cryst. 50, 919-931 (2017), 2015
We argue that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of surface fractal is shown to be a sum of the amplitudes of composing mass fractals. Various approximations for the scattering intensity of surface fractal are considered.
arxiv   +1 more source

On Fractal Features and Fractal Linear Space About Fractal Continuous Functions [PDF]

Fractals, 29(1), 2021, 2021
This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) $BV_{I}$ all fractal continuous functions with bounded variation is fractal linear space; (2) ${}^{1}D_{I}$ all fractal continuous functions with Box dimension one is a fractal ...
arxiv  

The solutions to uncertainty problem of urban fractal dimension calculation [PDF]

Entropy, 2019, 21, 453, 2017
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential ...
arxiv   +1 more source

Solving and Applying Fractal Differential Equations: Exploring Fractal Calculus in Theory and Practice [PDF]

arXiv, 2023
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving $\alpha$-order differential equations.
arxiv  

Small-angle scattering from fat fractals [PDF]

Eur. Phys. J, B (2014) 87: 139, 2014
A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural ...
arxiv   +1 more source

Further analysis of Multivariate fractal functions [PDF]

arXiv, 2023
The aim of this paper is to characterize a fractal operator associated with multivariate fractal interpolation functions (FIFs) and study the several properties of this fractal operator. Further, with the help of this operator, we characterize a latest category of functions and study their approximation aspects.
arxiv  

Betweenness Centrality of Fractal and Non-Fractal Scale-Free Model Networks and Tests on Real Networks [PDF]

, 2007
We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models.
arxiv   +1 more source