Fractals

The Hausdorff dimension of Julia sets of expanding maps can be computed by the eigenvalue algorithm. In this work, an implementation of this algorithm for quadratic polynomial, that allows the calculation of the Hausdorff dimension of Julia sets for complex parameters, is done. In particular, the parameters in a neighborhood of the parabolic parameter $c=1/4$ are analyzed and a small oscillation in Hausdorff dimension is shown.

The ILF (Image Landscapes’ Fractal Dimension) method and $DF2d$ method obtained by a 2D generalization of Higuchi’s algorithm were applied to a set of 120 digital histological images of Anal Intraepithelial Neoplasia (AIN). The main goal of this research was to examine accuracy that means sensitivity and specificity of these methods and compare the applicability of both methods in the quantitative characterization and differentiation of clinical cases of AIN. Histological examination by an experienced pathologist revealed three grades of AIN tumors in the 120 histological slices: 36 of AIN1, 56 of AIN2 and 28 of AIN3. Statistical tests showed significant differences between calculated fractal dimension values in three datasets (AIN1, AIN2 and AIN3) using ILF and $DF2d$ methods at the level of significance of 0.05. Application of the ILF and $DF2d$ methods has an advantage when it comes to speed, accuracy, simplicity and time necessary for analysis. Both methods can be successfully applied for differentiation between AIN stages giving practically the same results. They can easily be adapted to other histological specimen.

Inversive geometry can be used to generate exactly self-similar space-filling sphere packings. We present a construction in two dimensions and generalize this construction to search for packings in higher dimensions. We discover 29 new three-dimensional topologies of which 10 are bearings, and 13 new four-dimensional topologies of which five are bearings. To characterize the packing topologies, we estimate numerically their fractal dimensions and we analyze their contact networks.

An innovative reconfiguration application is proposed to re-calculate the parameters of the Ferragina and Manzini exact search algorithm (or FM indexes), using a modular and efficient hardware implementation to accelerate alignment programs of short DNA sequence reads. Although these programs use multi-core execution strategies or multiple computers, they have become slow considering the very high speed at which the new massively parallel sequencing machines produce the reads to be aligned. Consequently, a search for different ways to accelerate the alignment is crucial. The proposed design runs with software functions in a hybrid system, and has the ability to align millions of reads to reference as large as the human genome. Tests on the M505k325t card show that a single alignment core can accelerate the computation by a factor close to $10.64\times$ in relation to BWA. Due to the minor consumption of area and power, multiple alignment cores can fill the Field Programmable Gate Array (FPGA) by multiplying the computation speed. With a multiple-core implementation, the processing speed of the design outperforms applications that are accelerated by GPUs and competes with similar FPGA proposals whose cost is much higher.

In this paper, we examine the number of geodesics between two points of the Sierpinski Gasket ($S$) via code representations of the points and as a main result we show that the maximum number of geodesics between different two points with respect to the intrinsic metric on $S$ is five.

This paper investigates the Hadamard fractional calculus of a fractal function. It is proved that there exists some linear relationship between the order of Hadamard fractional calculus and the fractal dimension of the Weierstrass function including Box dimension, $K$-dimension and Packing dimension. Furthermore, numerical result is given to show the linear relationship.

Let $((0,1],T)$ be the doubling map in the unit interval and $\phi$ be the Saint-Petersburg potential, defined by $\phi (x)=2^n$ if $x\in (2^{-n-1},2^{-n}]$ for all $n\ge 0$. We consider asymptotic properties of the Birkhoff sum $S_n(x)=\phi (x)+\cdots +\phi (T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{nlogn}{S}_n(x)$ converges to $\frac{1}{log2}$ in probability. We determine the Hausdorff dimension of the level set $\{x:{lim}_{n\to \infty }{S}_n(x)/n=\alpha \}(\alpha >0)$, as well as that of the set $\{x:{lim}_{n\to \infty }{S}_n(x)/\mathrm{\Psi }(n)=\alpha \}(\alpha >0)$, when $\mathrm{\Psi }(n)=nlogn$, $n^a$ or $2^{n^{\gamma }}$ for $a>1,\gamma >0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.

In this work, we study the eccentric distance sum of Vicsek networks. To obtain the eccentric distance sum of networks, we investigate the corresponding integral on self-similar measure for Vicsek fractals. We use the self-similarity of distance and measure to solve the integral.

Great efforts have been made to determine total suction using elaborate laboratory tests, because there is no a specific correlation relating the total suction to the water content of bentonites. However, elaborate laboratory tests are difficult and time consuming to perform on bentonites. Thus, a theoretical equation is a necessary choice to correlate the total suction to the water content. A simple method is proposed to calculate the total suction from the water content based on the surface fractal model for bentonites. The correlation relating the total suction to the water content is expressed by a power-law function with the exponent of $D_{\mathrm{\text{s}}}-3,D_{\mathrm{\text{s}}}$ is the surface fractal dimension of bentonites. The surface fractal dimension can be determined using the nitrogen adsorption isotherm tests. The total suction of bentonites calculated from the proposed method is in satisfactory agreement with the experimental data found in the literature. In addition, the correlation of the total suction to the water content also offers another method to determine the surface fractal dimension of bentonites.

A microseismic (MS) monitoring system in a mine can monitor the MS signals generated by coal rock rupture and blasting waves and can distinguish the two types of waves more clearly to monitor and analyze the rupture and evolution process of coal rock. According to the nonlinearity characteristics of the waveform, the fractal characteristics of a mine’s MS and blasting waves are analyzed by simple fractal and multifractal theory, and the simple fractal dimension $D$ and multifractal parameters are obtained, respectively. Results show that the simple fractal dimension $D$ reflects the complexity and frequency structure of the wave. The simple fractal dimension $D$ of a blasting wave is larger than that of a mine MS wave, which indicates that the blasting wave is relatively complex with higher frequency, while the mine MS wave is relatively simple with lower frequency. However, the simple fractal dimension $D$ can only describe the wave integrity features, not the local features. The multifractal parameters can describe the local characteristics of the wave more finely, and the multifractal spectrum describes the probability information of the singularity exponent $a$. The singularity exponential range and multifractal spectral width $\mathrm{\Delta }\alpha$ of the blasting wave are smaller than those of the mine MS wave, which indicates that the probability measure of distribution unevenness and the degree of partial parameter fluctuation of the blasting wave are more severe than those of the mine MS wave. Wave signal analysis based on simple fractal and multifractal methods can not only obtain the characteristics of the wave strength and spectral structure but also other important information, such as local singularity. Therefore, it is possible to more clearly and conspicuously identify mine MS and blasting waves, so that coal rock rupture can be monitored more accurately.

For an infinite sequence $t={({𝜀}_k)}_k$ of $1$ and $-1$ with probability $p$ and $1-p$, we mainly study the multifractal analysis of one-dimensional biased walks. Let $H(t)=-t{log}_{2}t-(1-t){log}_2(1-t)$ and $S_n(t)={\sum }_{k=1}^n{𝜀}_k$. The Hausdorff and packing dimensions of the sets $E_b=\{t\in {\{\pm 1\}}^{\mathbb{N}}:{lim}_{n\to \infty }\frac{1}{n}[S_n(t)-𝔼(S_n(t))]=b\}$ are ${dim}_H{E}_b={dim}_P{E}_b=H(\frac{2p+b}{2})$, which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934) 321–330] on random walk, saying that: For any $0<b<1$, the set $\{t\in {\{\pm 1\}}^{\mathbb{N}}:\underset{n\to \infty }{lim}\frac{S_n(t)}{n}=b\}$ has Hausdorff dimension $H(\frac{1+b}{2})$.

In order to research the properties of the fractional order calculus of broken line segments’ fractal interpolation function (FIF) generated by the linear iterated function system (IFS), the concepts of the Riemann–Liouville fractional order calculus and the method of the IFS are used to prove the properties of the fractional calculus of the broken line segments’ FIF generated by the linear IFS. There are two conclusions as follows. First, the fractional order integral of the broken line segments’ FIF formed by the linear IFS is continuous and first-order differentiable on the closed interval $[0,x_N]$. Second, the broken line segments’ FIF formed by the linear IFS exists with fractional order differential, but the differential function is not continuous.

The aim of this work is to quantitatively explore the texture evolution of amphibole aggregation and residual melt with pressure and temperature. The amphibole aggregation growth from a basaltic melt and the residual melt at high pressure (0.6–2.6GPa) and high temperature (860–970${}^{\circ }$C) exhibit statistical self-similarity which made us consider studying such characteristic by fractal analysis. The bi-phase box counting method was applied for fractal analysis of each product to identify the fractal phase and the fractal dimension was estimated. In the experimental products, the residual melt is identified as the fractal and amphibole as the Euclidean except for one experiment. The results show that the residual melt can be quantified by the fractal dimension $(D_B)$ within the range of 1.782–1.848. The temperature has a significant effect on the morphology of amphibole and the fractal dimension of the residual melt. The higher the crystallization temperature is, the more regular the amphibole grains are. At lower temperature (from 860${}^{\circ }$C to 915${}^{\circ }$C), the fractal dimension of the residual melt decreased with the increasing crystallization temperature, but at higher temperature (970${}^{\circ }$C), the fractal phase changed to amphibole and the fractal dimension of amphibole is 1.816. The pressure may be the dominant factor that controls the morphology of the mineral aggregation and the residual melt. The fractal dimension of melt decreased linearly with the increasing pressure and if the linear relationship between the fractal dimension and pressure can be further verified in the future, it can be used as a potential geological barometer.

Urban form can be described with fractal dimension, which is a measurement of space filling of urban evolution. However, how to model and understand the fractal dimension growth of urban morphology are still pending questions. This paper is devoted to the research on the fractal dimension curves of urban growth. The principle of squashing function and empirical evidences are employed to demonstrate the following inference: the fractal dimension time series of a city’s spatial form take on a sigmoid curve. Among various sigmoid functions, the logistic function is the most probable selection. The observational data of fractal dimension of different cities from different sources support this logic judgment. A further discovery is that the fractal dimension curves of cities in the developed countries differ from those in the developing countries. A generalized logistic function is thus proposed to model the fractal dimension curves of different types of cities. The general logistic models can be used to predict the missing values and estimate the growth rates of fractal dimension of city development. Moreover, these models can be utilized to analyze when and where there is a fractal of urban form.

For $\beta \in (0,\frac{1}{2})$, a middle-$(1-2\beta )$ Sierpinski carpet ${\mathrm{\Gamma }}_{\beta }$ is defined as the self-similar set generated by the iterated function system (IFS) ${\{F_i\}}_{i=1}^4$, where $F_i:{ℝ}^2\to {ℝ}^2$ is defined by

Understanding gas transport behavior though fractured porous media is essential in many fields including fiber science, energy science, soil science, environmental engineering, chemical engineering, etc. In this paper, a fractal model is developed to characterize gas diffusion through fractured porous media, where a bundle of fractal-like tree branching networks is used to represent the fracture system according to fractal scaling laws. The analytical expression for relative gas diffusion coefficient of fractured porous media is derived. The proposed fractal model has been validated by the available experimental data and empirical correlations. From the parametrical study, it can be seen that structural parameters of fractured porous media (for example porosity, the fractal dimension, the diameter ratio, the length ratio and the branching angle) have a significant effect on equivalent gas transport properties. Gas relative diffusion coefficient has a positive correlation with the porosity, the pore size fractal dimension, or the diameter ratio, whereas it has a negative correlation with the length ratio, the branching levels, or the branching angle. The proposed fractal model does not only shed light on gas transport physics of fractured porous media, but also reveals more mechanisms than experimental measurements.

This paper presents a preliminary work to evaluate the thermal resistance of a spherical medium. In the sphere, a three-dimensional pore fractal network combined with electrical double layers is constructed to represent the pore structure of medium. The construction and structural parameters of a network are illustrated in detail. The thermal resistance of a sphere is primarily associated with structural parameters of network, pore structure parameters of medium, thermal conductivities of solid phase and pore channels. In this work, the evolution of thermal resistance with the fractal network is analyzed comprehensively. Besides, the evolution mechanisms of thermal resistance with pore structure parameters of medium are revealed. The influences of types of porous materials on thermal resistance are systematically studied.

Jamin effect, which is a capillary pressure obstructing the drop/bubble flow through the narrow throat, has an important effect on the multiphase flow in the low permeability reservoir porous media. In this work, a novel model for the relative permeability with Jamin effect is developed to study the two-phase flow through porous media based on the fractal theory. The proposed relative permeability is expressed as a function of the applied pressure difference, shape parameters of the drop/bubble, the physical parameters of the wetting and nonwetting fluids, and microstructural parameters of porous media. Good agreement between model predictions and available experimental data is obtained, and the advantage of the present fractal model can be highlighted by comparisons with the empirical model predictions. Additionally, the influences of Jamin effect on the two-phase relative permeability are discussed comprehensively and in detail. The model reveals that the length ratios ($\xi$ and ${\eta }_x$) have significant effects on the relative permeabilities. It is also found that the nonwetting phase relative permeability strongly depends on the interfacial tension, applied pressure difference, viscosity ratio and porosity of porous media at the lower wetting phase saturation. Furthermore, the fractal model will shed light on the two-phase transport mechanism of the low permeability reservoir porous media.

We study the aggregation process on the geometric graph. The geometric graph is composed by sites randomly distributed in space and connected locally. Similar to the regular lattice, the network possesses local connection, but the randomness in the spatial distribution of sites is considered. We show that the correlations within the aggregate patterns fall off with distance with a fractional power law. The numerical simulation results indicate that the aggregate patterns on the geometric graph are fractal. The fractals are robust against the randomness in the structure. A remarkable new feature of the aggregate patterns due to the geometric graph is that the fractal dimension can be adjusted by changing the connection degree of the geometric graph.

In this paper, we construct a special network based on the construction of the Sierpinski carpet. Using the self-similarity and renewal theorem, we obtain the asymptotic formula for the average path length of our evolving network.

Analyzing the influence of external stimuli on human eye movements is an important challenge in vision research. In this paper, we investigate the plasticity of eye movements due to the applied auditory stimuli (music). For this purpose, we use fractal theory, which provides us with tools such as fractal dimension as an indicator of process complexity. This study, for the first time, reveals the correlation between fractal dynamics of eye movements and fractal dynamics of auditory stimuli. Based on the performed analysis, the fractal structure of the eye movements shifts toward the fractal structure of the applied auditory stimuli, where the greater variation in fractal dynamics of auditory stimuli causes greater variation in the fractal dynamics of eye movements. The observed behavior is explained through the nervous system. As a rehabilitation purpose, the employed methodology in this research can be investigated in case of patients with vision problems, where the applied music could potentially improve their vision.

The pivotal aim of this paper is to propose an efficient computational technique, namely, Elzaki fractional projected differential transform method (EFPDTM) to solve the system of linear and nonlinear fractional differential equations. In the EFPDTM process, we investigate the behavior of independent variables for convergent series solution in admissible range. The EFPDTM manipulates and controls the series solution, which rapidly converges to the exact solution in a large admissible domain in a very efficient way. The solution procedure and explanation show the flexible efficiency of the EFPDTM, compared to the other existing classical techniques for solving the system of linear and nonlinear fractional differential equations.

The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we present an iterative algorithm for generating the weighted Koch network. Then, we study the eigenvalues for the transition matrix of the weighted Koch networks for weight-dependent walk. We derive explicit expressions for all eigenvalues and their multiplicities. Afterwards, we apply the obtained eigenvalues to determine the eigentime identity, i.e. the sum of reciprocals of each nonzero eigenvalues of normalized Laplacian matrix for the weighted Koch networks. The highlights of this paper are computational methods as follows. Firstly, we obtain two factors from factorization of the characteristic equation of symmetric transition matrix by means of the operation of the block matrix. From the first factor, we can see that the symmetric transition matrix has at least $3\cdot 4^{g-1}$ eigenvalues of $-\frac{1}{2}$. Then we use the definition of eigenvalues and eigenvectors to calculate the other eigenvalues.

This study presented the analytical solutions for water–gas flow through three-dimensional (3D) fracture networks subjected to triaxial stresses. The relationship between fractal dimension for fracture aperture distribution subjected to triaxial stresses and that subjected to no stresses is established, and the analytical solutions for fractal dimensions for aperture distribution and the equivalent permeability of both fluid and gas phases were derived. The results show that the calculated relative permeability of water phase-saturation curves agree well with those reported in the literature, which indicates that the proposed solutions are validate. With the increment of normal stresses applied on the fracture surface, both the maximum aperture and minimum aperture decrease; however, their ratio increases first and then decreases. The fractal dimensions for fracture aperture distribution of water and gas phases with respect to saturation are axisymmetric along saturation $=0.5$. With the increase in saturation, the fractal dimension for fracture aperture distribution of water phase increases significantly when the saturation is less than 0.1, and then gently when the saturation is continuously increased by up to 1.0. The normal stress increased by two orders of magnitude for a larger normal stress (i.e. increased from $10^{-2}$MPa to 10MPa) corresponds to smaller variations in equivalent permeability of both water and gas phases for a smaller normal stress (i.e. increased from $10^{-4}$MPa to $10^{-2}$MPa).