Fractals

Let $({ℳ},c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in ({ℳ},c_k,n_k)$ is $[0,1]$. Let $A,B$ be two nonempty sets in $ℝ$. Suppose that $f$ is a continuous function defined on an open set $U\subset {ℝ}^2$. Denote the continuous image of $f$ by

In this paper, we study the doubling properties of self-affine measures supported on a class of self-affine sets named Gatzouras–Lalley sets. First, we show that every self-affine measure is doubling if the Gatzouras–Lalley sets satisfy “strong separation” in both horizontal and vertical directions. Then we provide a sufficient condition for self-affine measures supported on Gatzouras–Lalley sets to be doubling if the sets do not satisfy the “separation condition” in the horizontal direction. Finally, we investigate measures without “vertical separation”, and give sufficient conditions for them to be doubling.

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

Previous studies ignore the evolutions of pore microstructure parameters (pore diameter fractal dimension $D_{\lambda }$ and tortuosity fractal dimension $D_{\tau p}$) but these evolutions may significantly impact the gas transport during gas extraction. In order to investigate these evolutions of fractal dimension properties during gas extraction, following four aspects are studied. Firstly, surface diffusion in adsorbed multilayer is modeled for fractal shale matrix. Our new matrix permeability model considers the slip flow, Knudsen diffusion and surface diffusion. This model is verified by experimental data. Secondly, a new fracture permeability model is proposed based on fractal theory and the coupling of viscous flow and Knudsen diffusion. Thirdly, the multilayer adsorption and these permeability models are introduced into the equations of gas flow and reservoir deformation. Finally, sensitivity analysis is performed to determine the key factors on fractal dimension evolution. The results show that the multilayer adsorption can accurately describe the adsorption properties of real shale reservoir. Shale reservoir deformation and gas desorption govern the evolutions of fractal dimensions. The multilayer adsorption and adsorbed gas porosity ${\varphi }_a$ play an important role in the evolutions of fractal dimensions during gas extraction. The monolayer saturated adsorption volume $V_m$ is the most sensitive parameter affecting the evolution of fractal dimensions. Therefore, the effects of gas adsorption on the evolution of fractal dimensions cannot be neglected in shale reservoirs.

The dynamic evolutions of friction force and worn surface profile were qualitatively analyzed by phase trajectory and recurrence plots and quantitatively characterized by fractal dimension and percent determinism. The results show that phase trajectories first shrink to a small volume, then stabilize at a minimum volume, finally expand to a large volume in the wear process. The white areas on the recurrence plots increase with the wear time. The fractal dimension first increases, then stabilizes at a high value, and finally decreases rapidly. The percent determinism first decreases, then fluctuates in a certain range, and finally increases. It demonstrates that friction force and worn surface topography derived from one tribology system evolve in a similar but not exactly the same way. They have the consistent evolution law in the wear process. Specially, friction force is much more sensitive to the variation of wear states than the worn surface.

We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.

A model for predicting the penetration depth and penetration time of cementing fluid through the wellbore into weakly consolidated formations based on the analytical fractal permeability model for power-law fluids considering the particle size of fluid and irreducible water is proposed and verified in this paper. The penetration distance and utilization dosage are related to the properties of cementing fluid and weakly consolidated formations. All of the parameters in the model have clear physical meaning, and no empirical parameter is used in this model. The results show that the cementing fluid tends to migrate to a long distance under a high pressure differential and low flow rate. The maximum penetration time and maximum penetration volume of cementing fluid increases with the increasing of grouting pressure and decreases with the increase in the volume flow rate. Therefore, throughout the cementing operation, the parameters should be adjusted based on the real situation.

Classical Brownian motion (BM) techniques for statistical monitoring of clinical trials have been widely used. The conditional power (CP) and $\alpha$-spending function-based boundary crossing probabilities are popular procedures for statistical hypothesis testing under the assumption of BM. However, in some clinical trials, the assumptions of BM may not be fully met for the design and data analysis. Therefore, a more general class of stochastic processes, fractional Brownian motion (FBM), was proposed in the literature to model the test statistics derived from interim analysis of clinical trials. To investigate the properties of FBM, in this paper, we simulated a wide range of FBM data, e.g. $H=0.5$ (BM) versus $0.5<H<1$ (FBM), with treatment effects versus without treatment effects. Then the performance of CP-based interim analysis was compared by assuming that the data follow BM or FBM. Our simulation study suggested that CP under the FBM assumptions was generally higher than that under the BM assumptions when $H>0.5$ and also matched well with the empirical results.

Wang et al. established successfully a variational principle in a fractal space by the semi-inverse method. This paper argues that in the fractal space, time should be also considered as a fractal, and Wang’s variational principle is extended to fractal space and fractal time.

The computational complexity of the Hosoya index of a given graph is NP-Complete. Let $RT(G)$ be the graph constructed from $R(G)$ by a triangle instead of all vertices of the initial graph $G$. In this paper, we characterize the Hosoya index of the graph $RT(G)$. To our surprise, it shows that the Hosoya index of $RT(G)$ is thoroughly given by the order and degrees of all the vertices of the initial graph $G$.

Let ${\mu }_{\{p_n,d_n\}}$ be a Cantor–Moran measure given by the infinite convolution of finite measures with equal probability

This paper introduces a modified DLA model, based on a fractional diffusion mechanism, as a novel approach to modeling fractal growth. The specific memory performance of fractional operators can be reflected macroscopically in aggregated patterns eventually. The influence of the model’s order on the structure and behavior of its pattern is further quantitatively described by anisotropy index and fractal dimension. Some simulations are provided to illustrate the correctness and effectiveness of the main results.

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates $q$ connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when $q$ is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks.

For continued fraction dynamical system $([0,1),T)$, we give a classification of the underlying space $[0,1)$ according to the orbit of a given point $x_0\in [0,1)$. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of $x_0$ is one and the Hausdorff dimension of the asymptotic set is either zero or $\frac{1}{2}$ according to $x_0$ is rational or not.

Moroccan family business companies are becoming an attractive investment in Casablanca stock exchange. In this study, presence of fractal and chaos in Moroccan family business stock returns is examined. In particular, return series are analyzed by stationary wavelet transform to decompose original series into long- and short-term variations. Then, detrended fluctuation analysis and artificial neural networks are used to estimate, respectively Hurst exponent and largest Lyapunov exponent. Results from fractal analysis show that short variations in returns are anti-persistent, whilst long variations are persistent. Besides, results from chaos tests reveal that both short and long variations in returns are chaotic. Such conclusions should be taken into account in active portfolio optimization and management.

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct $\alpha$-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the $\alpha$-fractal rational quartic spline when the original function is in ${{𝒞}}^4(I)$. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the $\alpha$-fractal rational quartic spline to ${{𝒞}}^2$. The elements of the iterated function system are identified befittingly so that the class of $\alpha$-fractal function $Q^{\alpha }$ incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ $Q$. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

Effect of nanoscale pore size distribution (PSD) on shale gas production is one of the challenges to be addressed by the industry. An improved approach to study multi-scale real gas transport in fractal shale rocks is proposed to bridge nanoscale PSD and gas filed production. This approach is well validated with field tests. Results indicate the gas production is underestimated without considering a nanoscale PSD. A PSD with a larger fractal dimension in pore size and variance yields a higher fraction of large pores; this leads to a better gas transport capacity; this is owing to a higher free gas transport ratio. A PSD with a smaller fractal dimension yields a lower cumulative gas production; this is because a smaller fractal dimension results in the reduction of gas transport efficiency. With an increase in the fractal dimension in pore size and variance, an apparent permeability-shifting effect is less obvious, and the sensitivity of this effect to a nanoscale PSD is also impaired. Higher fractal dimensions and variances result in higher cumulative gas production and a lower sensitivity of gas production to a nanoscale PSD, which is due to a better gas transport efficiency. The shale apparent permeability-shifting effect to nanoscale is more sensitive to a nanoscale PSD under a higher initial reservoir pressure, which makes gas production more sensitive to a nanoscale PSD. The findings of this study can help to better understand the influence of a nanoscale PSD on gas flow capacity and gas production.