Fractals

In this paper, we present a construction of new nonlinear recurrent hidden variable fractal interpolation curves. In order to get new fractal curves, we use Rakotch’s fixed point theorem. We construct recurrent hidden variable iterated function systems with function vertical scaling factors to generate more flexible fractal interpolation curves. We also give an illustrative example to demonstrate the effectiveness of our results.

In this work, a novel fractal model for the laminar flow in roughened cylindrical microchannels is proposed. The average height of rough elements is derived using the fractal theory. The effects of relative roughness on the friction factor and the Poiseuille number are discussed. It is found that the Darcy friction factor and the Poiseuille number increase with the increase in the relative roughness in the cylindrical microchannel. Besides, it is observed that the Darcy friction factor decreases with the increase in the Reynolds number. Each parameter of the proposed model has a clear physical meaning. The present model can properly reveal some mechanisms that affect the laminar flow in roughened cylindrical microchannels. The present model improves the understanding of the physical mechanisms of fluid flows through roughened cylindrical microchannels. Our model predictions are compared with the existing experimental data, and good agreement can be found.

In this paper, we discuss a family of p.c.f. self-similar fractal networks which have reflection transformations. We obtain the average geodesic distance on the corresponding fractal in terms of finite pattern of integrals. With these results, we also obtain the asymptotic formula for average distances of the skeleton networks.

Surface finish is one of the most important issues that is discussed in machining of materials. In fact, reaching the required surface finish is the key scale in analysis of the quality of machined workpiece. Chip formation is an important factor that highly affects the surface finish of machined workpiece. In this research, we analyze the relationship between surface finish and chip formation by fractal analysis of their surfaces. Since both patterns have complex structures, we calculate their fractal dimension in case of different experiments in which machining parameters change. The result of analysis indicates that the variations of fractal dimension of machined surface show a reverse pattern compared to the fractal dimension of chip surface in case of variations of depth of cut and feed rate. However, in case of variations of spindle speed, the variations of fractal dimension of machined surface show a similar pattern with the fractal dimension of chip surface. Therefore, it can be said that the complexity of machined surface is linked to the complexity of chip surface. The method of analysis used in this research can be further applied to other manufacturing operations.

The coupling of heat transfer and water flow in rock fractures is a key issue to geothermal energy extraction. However, this coupling in a rough fracture has not been well studied so far. This paper will study this coupling in a rock fracture with different roughness. First, multi-scale and self-affine rough fracture are constructed through the Weierstrass–Mandelbrot function and embedded into a rock block. Its single scale trend line is also derived. Second, a roughness factor is proposed based on the standard deviation of each segment from the trend line and introduced into the governing equation of fracture flow. After coupling with heat transfer and matrix deformation, a thermal-hydro-mechanical (THM) coupling model is formulated for a rough fracture flow. Third, an analytical solution is derived through the Laplace transform and Stehfest method and used for the validation of this THM coupling model. Finally, the effects of fracture roughness and matrix deformation on heat transfer and fracture flow are numerically investigated. The simulation results indicate that the rock fracture with lower fractal dimension has higher heat transfer efficiency. The effect of fracture roughness on heat transfer is much greater than that of aperture variation induced by the shrinkage of rock matrix.

Tool wear is one of the unwanted phenomena in machining operations where tool has direct contact with the workpiece. Tool wear is an important issue in milling operation that is caused due to different parameters such as machine vibration. Tool wear shows complex structure, and machine vibration is a chaotic signal that also is complex. In this research, we analyze the correlation between tool wear and machine vibration using fractal theory. We run the experiments in which machining parameters, namely depth of cut, feed rate and spindle speed change, and accordingly analyze the variations of fractal dimension of tool wear versus the fractal dimension of machine vibration signal. Based on the obtained results, variations of complexity of tool wear are reversely correlated with the variations of complexity of vibration signal. Fractal analysis could potentially be applied to other machining operations in order to investigate the relation between tool wear and machine vibration.

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to $n$th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.

Cardiac tissue is characterized by structural and cellular heterogeneities that play an important role in the cardiac conduction system. Under persistent atrial fibrillation (persAF), electrical and structural remodeling occur simultaneously. The classical mathematical models of cardiac electrophysiological showed remarkable progress during recent years. Among those models, it is of relevance the standard diffusion mathematical equation, that considers the myocardium as a continuum. However, the modeling of structural properties and their influence on electrical propagation still reveal several limitations. In this paper, a model of cardiac electrical propagation is proposed based on complex order derivatives. By assuming that the myocardium has an underlying fractal process, the complex order dynamics emerges as an important modeling option. In this perspective, the real part of the order corresponds to the fractal dimension, while the imaginary part represents the log-periodic corrections of the fractal dimension. Indeed, the imaginary part in the derivative implies characteristic scales within the cardiac tissue. The analytical and numerical procedures for solving the related equation are presented. The sinus rhythm and persAF conditions are implemented using the Courtemanche formalism. The electrophysiological properties are measured and analyzed on different scales of observation. The results indicate that the complex order modulates the electrophysiology of the atrial system, through the variation of its real and imaginary parts. The combined effect of the two components yields a broad range of electrophysiological conditions. Therefore, the proposed model can be a useful tool for modeling electrical and structural properties during cardiac conduction.

We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpiński Gasket and its higher-dimensional variants ${\mathrm{\text{SG}}}_N$, $N>3$, proving results that generalize those of Teplyaev [Gradients on fractals, J. Funct. Anal. 174(1) (2000) 128–154]. When ${\mathrm{\text{SG}}}_N$ is equipped with the standard Dirichlet form and measure $\mu ,$ we show there is a full $\mu$-measure set on which continuity of the Laplacian implies existence of the gradient $\nabla u$, and that this set is not all of ${\mathrm{\text{SG}}}_N$. We also show there is a class of non-uniform measures on the usual Sierpiński Gasket with the property that continuity of the Laplacian implies the gradient exists and is continuous everywhere in sharp contrast to the case with the standard measure.

Magnetic resonance image (MRI) is an important tool to diagnose human diseases effectively. It is very important for research and clinical application to classify the normal and abnormal human brain MRI images automatically. In this paper, an accurate and efficient technique is proposed to extract features of MRIs and classify these images into normal and abnormal categories. We use two-dimensional multifractal detrended fluctuation analysis (2D MF-DFA) to obtain image features. These features are the local generalized Hurst exponents calculated by 2D MF-DFA. In this regard, the values of Hurst exponents are given as the training input vector and are taken to the classifiers. We use $k$-nearest neighbor ($k$-NN) and support vector machine (SVM) to classify a specific brain MRI as normal or glioma affected. For SVM, we apply the leave-one-out cross-validation method for experimental verification. The 2D MF-DFA-SVM system achieved accuracy, sensitivity, and specificity of $99.82\%$ ±$0.07$, $100\%$, and $99.81\%$ ±$0.09$, respectively. The 2D MF-DFA-$k$-NN system achieved accuracy, sensitivity, and specificity of $96\%$, $92.59\%$, and $100\%$, respectively. We find that when performing binary classification for brain MRIs, the SVM is superior to $k$-NN. In addition, our experimental results indicate that the proposed 2D MF-DFA-SVM achieved excellent outcomes compared to those of the previous works. The proposed system is a promising system to clinical use.

In this paper, we construct the Bäcklund transformations and the super-position formulas to the constant coefficients local fractional Riccati equation for the first time. Next, by means of the Bäcklund transformations and seed solutions which have been known in [X. J. Yang et al., Non-differentiable solutions for local fractional nonlinear Riccati differential equations, Fundam. Inform. 151(1–4) (2017) 409–417], we can get a class of exact solutions to the third-order modified KdV equation on the fractal set. These new type solutions can assist us to review different nonlinear phenomena better, which had been modeled via local fractional derivative.

In this paper, by using the theory of local fractional calculus and some techniques of real analysis, the structural characteristics of Hilbert-type local fractional integral inequalities with abstract homogeneous kernel are studied. At the same time, the necessary and sufficient conditions for these inequalities to take the best constant factor are discussed. As an application, some best constant factor inequalities with specific kernels are obtained.

Hydraulic properties of rock fractures are important issues for many geoengineering practices. Previous studies have revealed that natural rock fractures have variable apertures that could significantly influence the permeability of single rock fractures, yet the effect of aperture variation on the hydraulic properties of fractured rock masses has received little attention. The present study implemented a series of flow calculations on two kinds of 3D fractal-like tree network models constituted by fractures with heterogeneous aperture distributions and parallel plates with uniform apertures to gain insight into the effect of aperture variation on the hydraulic properties of fractured rock masses. The fluid flows through the two kinds of models are simulated using a developed numerical code, whose validity is verified by comparisons with the analytical solutions and previous experimental results. The effects of heterogeneous aperture distributions on the permeability of the models are illustrated by changing the geometrical parameters including the aperture ratio and mean aperture of the initial fracture branch. The results show that a significant channeling flow is observed in the 3D fractal-like tree network models constituted by fractures with heterogeneous aperture distributions. These preferential flow paths are mainly located in the large aperture channels, bypassing the contacts and/or low aperture barriers. The equivalent permeability of the 3D fractal-like tree network models increases with increasing fracture aperture ratio and/or increasing mean aperture of the initial fracture branch. The permeability ratio of the model of fractures with heterogeneous aperture distributions to the model with uniform mean apertures is always smaller than 1.0, and the ratio increases following a logarithmic relationship as the mean aperture of the initial fracture branch increases. A regression function is proposed to estimate the equivalent permeability of the model of fractures with heterogeneous aperture distributions. The deviations between the predicted permeability and the simulated/experimental results are less than half of the order of magnitude, which verifies that the proposed equation is capable of predicting the equivalent permeability of 3D fractal-like tree networks as a first-order estimation.

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.

Given an integer $n\ge 2$ and a digit set ${𝒟}\subset {\{0,1,\dots ,n-1\}}^2$, there is a self-similar set $K\subset {ℝ}^2$ satisfying the set equation $K=(K+{𝒟})/n$. This set $K$ is called a fractal square. By studying the line segments contained in $K$, we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

The understanding of the diffusion process and mechanisms of harmful species (e.g. chlorides) in porous cementitious materials is important to control and improve the material durability under harsh environments. In this paper, fractal analysis on the pore structure of porous cementitious materials was conducted and involved in a diffusion model. Macro material geometric parameters were considered in the model to avoid the difficulties in the measurements of microscopic pore parameters. The deformations of porous cementitious materials under the uniaxial elastic loads were considered to correct the diffusion model. The stress-affected diffusivity was displayed in an elegant expression involving some macro material parameters (e.g. total porosity, elastic modulus of solid skeleton, Poisson ratio). Results show that the effective diffusivity is greatly influenced by the porosity and stress ratio. The uniaxial elastic loads decrease the pore areas but increase the lengths of the pore channels for mass diffusion, which eventually causes the decrease of the effective diffusivity. The plots of the relative diffusivity against the stress ratio follow linear forms. The developed fractal diffusion model may help better understand the diffusion process in complex porous cementitious materials under elastic loads. Going beyond this, the fractal diffusion model may provide a new tool to predict the diffusivity of porous building materials under complex mechanical and environmental loads.

In this paper, we study the boundedness character and persistence, local and global behavior, and rate of convergence of positive solutions of following system of rational difference equations

Let $K_{\lambda }$ be the attractor of the following iterated function system(IFS):

The $q$-fractional differential equation usually describes the physics process imposed on the time scale set $T_q$. In this paper, we first propose a difference formula for discretizing the fractional $q$-derivative ${}^c{D}_q^{\alpha }x(t)$ on the time scale set $T_q$ with order $0<\alpha <1$ and scale index $0<q<1$. We establish a rigours truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial value problem of $q$-fractional differential equation: ${}^c{D}_q^{\alpha }x(t)=f(t,x(t))$ on the time scale set. We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.

Using multifractal detrended cross-correlation analysis (MF-DCCA) and nonlinear Granger causality test, this paper examines the return predictability of margin-trading activities. Results show that the predictive power of margin-trading activities on subsequent stock returns varies with respect to the different aspects of margin trading. In line with previous studies, we find no significant correlation between margin-buying amount and subsequent stock returns. However, the margin-covering amount is negatively associated with subsequent stock returns; and margin debt is positively associated with the future stock returns. In general, our findings suggest that margin traders may have no positive information when they conduct a margin-buying position, but may possess negative information when covering their positions.

This paper proposes a method to calculate the degree of fluctuation of the daily electrical load-curve using fractal dimension, which is a quantitative estimator of spatial complexity. The conventional methods for forecasting have not studied such a variable, being a new parameter that can be included to characterize the electrical load. The method of fractal dimension also allows us to propose a new numerical method to calculate the integral of a function, using the trapezoid rule, but splitting the curve with fractal segments, to discover other observations, which allows the elevation of new theoretical approaches. The results are compared with the other methods such as the conventional trapezoid rule and the box-counting. It is then a new contribution that expands the universal knowledge on the subject. The case study is the daily electrical load-curve, where the energy demanded corresponds to the area of the $R$ region bounded by the curve.

The rock cores of low permeability reservoirs have special pore structures, which are the essential factors to determine the seepage capacity and oil displacement efficiency and directly affect oil and gas reserves and oil well productivity. This paper studies 16 digital rock core samples. Based on the fractal theory for porous media, we carried out the fractal characterization of the pore structure of the samples by an approach combined with binarized CT images and fractal theory. The computing software based on the Box-counting method was used to measure and calculate the fractal dimension and porosity of the samples, and the calculated results were compared with the binarized CT data from the laboratory. It was found that among the tested samples, the results of 12 rock core samples were in good agreement with the experimental data, indicating that the fractal theory is effective in the measurement of fractal dimension and in the calculation of porosity of digital rock core samples. For the results with larger errors compared to laboratory data, we also analyzed and elaborated the reasons from the relevant binarized images and pore distribution images of samples. It is also found that the minimum pore size has a significant impact on the results when the fractal theory was applied to analyze the digital rock core samples. Finally, a standard or criterion is established whether the pore/particle size distribution in a digit rock core/porous medium is fractal or non-fractal.