Fractals

A permeability model for porous media considering the stress sensitivity is derived based on mechanics of materials and the fractal characteristics of solid cluster size distribution. The permeability of porous media considering the stress sensitivity is related to solid cluster fractal dimension, solid cluster fractal tortuosity dimension, solid cluster minimum diameter and solid cluster maximum diameter, Young's modulus, Poisson's ratio, as well as power index. Every parameter has clear physical meaning without the use of empirical constants. The model predictions of permeability show good agreement with those obtained by the available experimental expression. The proposed model may be conducible to a better understanding of the mechanism for flow in elastic porous media.

Even though many objects and phenomena of importance in geophysics have been shown to have fractal character, there are still many of them which show self-similar character and yet to be studied. The objective of the present work is to demonstrate that the fractal dimension of the boundary of a natural water body can be used to shed light on irregularity as well as other properties of a region. Owing to easy availability of satellite images and image processing softwares, this turns out to be a handy tool. In this study, we have analyzed several lakes in India mostly around the Western Ghats region. We find that the fractal dimension of their boundaries for the length scales between around 40 m to 2 km, in general, has broad variation from 1.2 to 1.6. But when they are grouped into three categories, viz., lakes along the ridge of Western Ghats, lakes in the planes and lakes in the mountain region, we find the first two groups to have a narrower distribution of dimensions.

In this paper, we use a time-frequency domain technique, namely, wavelet squared coherency, to examine the associations between the trading volumes of three agricultural futures and three different forms of these futures' daily closing prices, i.e. prices, returns and volatilities, over the past several years. These agricultural futures markets are selected from China as a typical case of the emerging countries, and from the US as a representative of the developed economies. We investigate correlations and lead–lag relationships between the trading volumes and the prices to detect the predictability and efficiency of these futures markets. The results suggest that the information contained in the trading volumes of the three agricultural futures markets in China can be applied to predict the prices or returns, while that in US has extremely weak predictive power for prices or returns. We also conduct the wavelet analysis on the relationships between the volumes and returns or volatilities to examine the existence of the two "stylized facts" proposed by Karpoff [J. M. Karpoff, The relation between price changes and trading volume: A survey, J. Financ. Quant. Anal.22(1) (1987) 109–126]. Different markets in the two countries perform differently in reproducing the two stylized facts. As the wavelet tools can decode nonlinear regularities and hidden patterns behind price–volume relationship in time-frequency space, different from the conventional econometric framework, this paper offers a new perspective into the market predictability and efficiency.

A nondistributive scator algebra in 1 + 2 dimensions is used to map the quadratic iteration. The hyperbolic numbers square bound set reveals a rich structure when taken into the three-dimensional (3D) hyperbolic scator space. Self-similar small copies of the larger set are obtained along the real axis. These self-similar sets are located at the same positions and have equivalent relative sizes as the small M-set copies found between the Myrberg-Feigenbaum (MF) point and -2 in the complex Mandelbrot set. Furthermore, these small copies are self similar 3D copies of the larger 3D bound set. The real roots of the respective polynomials exhibit basins of attraction in a 3D space. Slices of the 3D confined scator set, labeled (s;x,y), are shown at different planes to give an approximate idea of the 3D objects highly complicated boundary.

The oscillating multifractal formalism is a formula conjectured by Jaffard expected to yield the spectrum d(h, β) of oscillating singularity exponents from a scaling function ζ(p, s'), for p > 0 and s' ∈ ℝ, based on wavelet leaders of fractional primitives f_-s' of f. In this paper, using some results from Jaffard et al., we first show that ζ(p, s') can be extended on p ∈ ℝ to a function that is concave with respect to p ∈ ℝ and independent on orthonormal wavelet bases in the Schwartz class. We also establish its concavity with respect to s' when p > 0. Then, we prove that, under some assumptions, the extended scaling function ζ(p, s') is the Legendre transform of the wavelet leaders density of f_-s'. Finally, as an application, we study the validity of the extended oscillating multifractal formalism for random wavelet series (under the assumption of independence and laws depending only on the scale).

It is difficult to simulate the dynamical behavior of actual financial markets indexes effectively, especially when they have nonlinear characteristics. So it is significant to propose a mathematical model with these characteristics. In this paper, we investigate a generalized Weierstrass–Mandelbrot function (WMF) model with two nonlinear characteristics: fractal dimension D where 2 > D > 1.5 and Hurst exponent (H) where 1 > H > 0.5 firstly. And then we study the dynamical behavior of H for WMF as D and the spectrum of the time series γ change in three-dimensional space, respectively. Because WMF and the actual stock market indexes have two common features: fractal behavior using fractal dimension and long memory effect by Hurst exponent, we study the relationship between WMF and the actual stock market indexes. We choose a random value of γ and fixed value of D for WMF to simulate the S&P 500 indexes at different time ranges. As shown in the simulation results of three-dimensional space, we find that γ is important in WMF model and different γ may have the same effect for the nonlinearity of WMF. Then we calculate the skewness and kurtosis of actual Daily S&P 500 index in different time ranges which can be used to choose the value of γ. Based on these results, we choose appropriate γ, D and initial value into WMF to simulate Daily S&P 500 indexes. Using the fit line method in two-dimensional space for the simulated values, we find that the generalized WMF model is effective for simulating different actual stock market indexes in different time ranges. It may be useful for understanding the dynamical behavior of many different financial markets.

V-variable fractals, where V is a positive integer, are intuitively fractals with at most V different "forms" or "shapes" at all levels of magnification. In this paper, we describe how V-variable fractals can be used for the purpose of image compression.

The aim of this paper is to formulate the best version of the Fundamental theorem of Calculus for real functions on a fractal subset of the real line. In order to do that an integral of Henstock–Kurzweil type is introduced.

The difference between the inverse power function and the negative exponential function is significant. The former suggests a complex distribution, while the latter indicates a simple distribution. However, the association of the power-law distribution with the exponential distribution has been seldom researched. This paper is devoted to exploring the relationships between exponential laws and power laws from the angle of view of urban geography. Using mathematical derivation and numerical experiments, I reveal that a power-law distribution can be created through a semi-moving average process of an exponential distribution. For the distributions defined in a one-dimension space (e.g. Zipf's law), the power exponent is 1; while for those defined in a two-dimension space (e.g. Clark's law), the power exponent is 2. The findings of this study are as follows. First, the exponential distributions suggest a hidden scaling, but the scaling exponents suggest a Euclidean dimension. Second, special power-law distributions can be derived from exponential distributions, but they differ from the typical power-law distributions. Third, it is the real power-law distributions that can be related with fractal dimension. This study discloses an inherent link between simplicity and complexity. In practice, maybe the result presented in this paper can be employed to distinguish the real power laws from spurious power laws (e.g. the fake Zipf distribution).

Multifractal detrended fluctuation analysis (MF-DFA) method is applied to analyze the daily electric load time series. The results of the MF-DFA show that there are three crossover timescales at seven days, 15 days and 365 days approximately in the fluctuation function. Also we find that these fluctuations have multifractal nature with long range correlation behavior. The multifractal singularity spectrum of the daily electric load series has been fitted by the quadratic function model. Comparing the MF-DFA results of the original load series with those of shuffled and surrogate series, it concludes that the multifractal characteristics of the daily electric load time series is due to both broadness of the probability density function and long-range correlation, and the long-range correlation is dominant.

In this paper, we introduce a model of the double-weighted Koch networks based on actual road networks depending on the two weight factors w,r ∈ (0, 1]. The double weights represent the capacity-flowing weight and the cost-traveling weight, respectively. Denote by the capacity-flowing weight connecting the nodes i and j, and denote by the cost-traveling weight connecting the nodes i and j. Let be related to the weight factor w, and let be related to the weight factor r. This paper assumes that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the capacity-flowing weight of edge linking them. The weighted time for two adjacency nodes is the cost-traveling weight connecting the two nodes. We define the average weighted receiving time (AWRT) on the double-weighted Koch networks. The obtained result displays that in the large network, the AWRT grows as power-law function of the network order with the exponent, represented by θ(w,r) = ½ log₂(1 + 3wr). We show that the AWRT exhibits a sublinear or linear dependence on network order. Thus, the double-weighted Koch networks are more efficient than classic Koch networks in receiving information.

Gas diffusion behavior in fractal porous media is simulated by Monte Carlo technique in this work. Based on the fractal character of pore size in porous media, the probability model of the effective gas diffusion coefficient is derived. The proposed model of the effective gas diffusion coefficient is explicitly expressed as a function of structural parameters of porous media, such as porosity, pore size, the fractal dimensions for pore area and tortuosity. The effect of structural parameters of porous media has been studied in detail. The results show that the present results from the Monte Carlo simulations present a good agreement with those from the available analytical model and the available experimental data. The proposed Monte Carlo simulation technique may have the potential in predictions of other gas transport properties in fractal porous media.

Given an integer n ≥ 2 and an ordered pair (A, B) with A ⊂ {k₁α + k₂β : k₁ + k₂ ≤ n - 1 and k₁, k₂ ∈ ℕ ∪{0}} and B ⊂ {k₁α + k₂β : 2 ≤ k₁ + k₂ ≤ n and k₁, k₂ ∈ ℕ}, where . Let T ≔ T(A, B) be unique compact set of ℝ² satisfying the set equation: T = [(T+A)∪(B-T)]/n. In this paper, we show that such self-similar sets which are totally disconnected are determined to within Lipschitz equivalence by their Hausdorff dimension.

This paper formulates a fractal-tree network model to address the challenging problem of characterizing the hydraulic fracture network in unconventional reservoirs. It has been proved that the seepage flow in tight/shale oil reservoirs is much more complicated to the conventional formation. To further understand the flow mechanisms in such a complex system, a semi-analytical model considering "branch network fractures" was established stage by stage using point source method and superposition principle. Fractal method was employed to generate and represent induced fracture network around bi-wing fractures. In addition, based on the new established model and solution, deterministic fractal-tree-like fracture network patterns and heterogeneity were carefully investigated and compared with the simulation model. Results show that the fractal dimension for the fracture network has significant effect on the connectivity of the stimulated reservoir. The proposed fractal model may capture the characteristics of the heterogeneous complex fracture network and help in understanding the flow and transport mechanisms of multiple fractured horizontal wells.

This paper presents an investigation of the use of ϵ-hulls to characterize and classify the totally disconnected Sierpiński relatives. These fractals all have the same fractal dimension, and the same topologies, but the ϵ-hulls provide ways to distinguish between such relatives. Examples of specific relatives are presented to illustrate the theory.

The three-point sinuosity (TPS) method is proposed to calculate the fractal dimension of surface profile accurately. In this method, a new measure, TPS is defined to present the structural complexity of fractal curves, and has been proved to follow the power law. Thus, the fractal dimension can be calculated through the slope of the fitted line in the log–log plot. The Weierstrass–Mandelbrot (W–M) fractal curves, as well as the real surface profiles obtained by grinding, sand blasting and turning, are used to validate the effectiveness of the proposed method. The calculation values are compared to those obtained from root-mean-square (RMS) method, box-counting (BC) method and variation method. The results show that the TPS method has the widest scaling region, the least fit error and the highest accuracy among the methods examined, which demonstrates that the fractal characteristics of the fractal curves can be well revealed by the proposed method.

Multiphase flow in porous media is very important in various scientific and engineering fields. It has been shown that relative permeability plays an important role in determination of flow characteristics for multiphase flow. The accurate prediction of multiphase flow in porous media is hence highly important. In this work, a novel predictive model for relative permeability in porous media is developed based on the fractal theory. The predictions of two-phase relative permeability by the current mathematical models have been validated by comparing with available experimental data. The predictions by the proposed model show the same variation trend with the available experimental data and are in good agreement with the existing experiments. Every parameter in the proposed model has clear physical meaning. The proposed relative permeability is expressed as a function of the immobile liquid film thickness, pore structural parameters (pore fractal dimension D_f and tortuosity fractal dimension D_T) and fluid viscosity ratio. The effects of these parameters on relative permeability of porous media are discussed in detail.

Much is known in the analysis of a finitely ramified self-similar fractal when the fractal has a harmonic structure: a Dirichlet form which respects the self-similarity of a fractal. What is still an open question is when such a structure exists in general. In this paper, we introduce two fractals, the fractalina and the pillow, and compute their resistance scaling factor. This is the factor which dictates how the Dirichlet form scales with the self-similarity of the fractal. By knowing this factor one can compute the harmonic structure on the fractal. The fractalina has scaling factor , and the pillow fractal has scaling factor .

We obtain a special version of gluing lemma related to the theory of iterated function systems (IFS). As an application, we verify that a family of concrete n-dimensional self-affine tiles {T_{n, r} : 0 ≤ r < 3, n ≥ 3} are homeomorphic with the unit cube [0,1]ⁿ. The tiles T_{n, r} are "nontrivial" in the sense that each of them is neither a self-affine polytope nor the product of an interval with an (n - 1)-dimensional self-affine tile.

Currently European Union Allowances (EUA) and secondary Certified Emission Reduction (sCER) have become two dominant carbon trading assets for investors and their linkage attracts much attention from academia and practitioners in recent years. Under this circumstance, we use the empirical mode decomposition (EMD) approach to decompose the two carbon futures contract prices and discuss their correlation from the multi-frequency perspective. The empirical results indicate that, first, the EUA and sCER futures price movements can be divided into those triggered by the long-term, medium-term and short-term market impacts. Second, the price movements in the EUA and sCER futures markets are primarily caused by the long-term impact, while the short-term impact can only explain a small fraction. Finally, the long-term (short-term) effect on EUA prices is statistically uncorrelated with the short-term (long-term) effect of sCER prices, and there is a medium or strong lead-and-lag correlation between the EUA and sCER price components with the same time scales. These results may provide some important insights of price forecast and arbitraging activities for carbon futures market investors, analysts and regulators.

In this paper, we calculate the number of spanning trees on two families of generalized pseudo- fractal networks with two controllable parameters. The initial state is a complete graph with an arbitrary number of nodes as a generalization of a triangle. In the subsequent steps, each existing edge (newly produced edge) gives birth to finite new nodes. Using the electrically equivalent transformations, we obtain the changes of edge weights of corresponding equivalent networks and derive the relationships for enumerating spanning trees between original networks and transformed networks. Finally, we obtain closed-form formulas for the number of spanning trees, which is verified by numerical simulations.

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.