Related papers: Box-Counting Dimension Using Triangles
A fractal is in essence a hierarchy with cascade structure, which can be described with a set of exponential functions. From these exponential functions, a set of power laws indicative of scaling can be derived. Hierarchy structure and…
A multifractal analysis is performed on a three-dimensional grayscale image associated with a complex system. First, a procedure for generating 3D synthetic images (2D image stacks) of a complex structure exhibiting multifractal behaviour…
Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of…
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…
In this study, we present a method to measure changes over time of fractal dimension. We confirmed that our method can calculate the fractal dimension with the same precision as conventional methods, and tracking performance of our method…
Abstract Self-similar, fractal nature of turbulence is discussed in the context of two dimensional turbulence, by considering the fractal structure of the wave-number domain using spirals. In loose analogy with phyllotaxis in plants, each…
We develop an algorithm to construct new self-similar space-filling packings of spheres. Each topologically different configuration is characterized by its own fractal dimension. We also find the first bi-cromatic packing known up to now.
A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$…
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,…
We introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to…
A method is described for calculating the approximate fractal dimension from a set of N values y sampled from a waveform between time zero and t. The waveform was subjected to a double linear transformation that maps it into a unit square.
This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential theoretic methods are used to produce dimension bounds for images of sets under H\"older maps and…
Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions…
In this article, we present a novel box-covering algorithm for analyzing the fractal properties of complex networks. Unlike traditional algorithms that impose a predetermined box size, our approach assigns nodes to boxes identified by their…
We introduce two frameworks in order to deal with fractal and multi-fractal analysis for subset sum problems where some embedding into the $1$-dimensional Euclidean space plays an important role. As one of these frameworks, the notion of…
Recently the concept of self similarity in the structure of the proton at small x has been introduced. We estimate the fractal dimension of proton in analogy with classical monofractals.
Some aspects of Cauchy integrals on sets with dimension larger than 1 are briefly discussed.
A random sequential box-covering algorithm recently introduced to measure the fractal dimension in scale-free networks is investigated. The algorithm contains Monte Carlo sequential steps of choosing the position of the center of each box,…
The local structure of a fractal set is described by its dimension $D$, which is the exponent of a power-law relating the mass ${\cal N}$ in a ball to its radius $\epsilon$: ${\cal N}\sim \epsilon^D$. It is desirable to characterise the…
It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable…