Fractals

The Sierpinski tetrahedron is used to construct evolving networks, whose vertexes are all solid regular tetrahedra in the construction of the Sierpinski tetrahedron up to the stage $t$ and any two vertexes are neighbors if and only if the corresponding tetrahedra are in contact with each other on boundary. We show that such networks have the small-world and scale-free effects, but are not fractal scaling.

Nanopore structure and its multiscale feature significantly affect the shale-gas permeability. This paper employs fractal theory to build a shale-gas permeability model, particularly considering the effects of multiscale flow within a multiscale pore space. Contrary to previous studies which assume a bundle of capillary tubes with equal size, in this research, this model reflects various flow regimes that occur in multiscale pores and takes the measured pore-size distribution into account. The flow regime within different scales is individually determined by the Knudsen number. The gas permeability is an integral value of individual permeabilities contributed from pores of different scales. Through comparing the results of five shale samples, it is confirmed that the gas permeability varies with the pore-size distribution of the samples, even though their intrinsic permeabilities are the same. Due to consideration of multiscale flow, the change of gas permeability with pore pressure becomes more complex. Consequently, it is necessary to cover the effects of multiscale flow while determining shale-gas permeability.

In this paper, an analysis of temporal variation of wind speed and wind direction recorded at 10 min intervals are presented. The measurements were carried out at Hambanthota, a site located in the southern coastal belt of Sri Lanka which has a high potential for wind power generation. The multifractal detrended fluctuation analysis was used to analyze the temporal scaling properties of wind speeds and wind directions. The analysis was carried out for seasonal variation of wind speed and wind direction. It was observed that the scaling behavior of wind speed in Hambanthota is similar to the scaling behavior observed in previous studies which were carried out in other parts of the world. The seasonal wind and wind direction change exhibits different scaling behavior. No difference in scaling behavior was observed with heights. The degree of multifractality is high for wind direction when compared with wind speed for each season.

The spatial pattern of the urban–rural regional system is associated with the dynamic process of urbanization. How to characterize the urban–rural terrain using quantitative measurement is a difficult problem remaining to be solved. This paper is devoted to defining urban and rural regions using ideas from fractals. A basic postulate is that human geographical systems are of self-similar patterns correlated with recursive processes. Then multifractal geometry can be employed to describe or define the urban and rural terrain with the level of urbanization. A space-filling index of urban–rural region based on a generalized correlation dimension is presented to reflect the degree of geo-spatial utilization in terms of urbanism. The census data of America and China are used to show how to make empirical analyses of urban–rural multifractals. This work is a normative study rather than a positive study, and it proposes a new way of investigating urban and rural regional systems using fractal theory.

The experiments were performed by sliding a pin against a plate under lubricated sliding condition. The friction temperature signal was measured by a thermocouple through the wear life. The multifractal spectrum was utilized to study the dynamic evolution of the friction temperature. The width of the multifractal spectrum, $\mathrm{\Delta }\alpha$, and the difference between the fractal dimensions of the subset with the largest singularity exponent and that with the smallest singularity exponent, $\mathrm{\Delta }f(\alpha )$, are computed for qualitative analysis. The results demonstrate that the friction temperature signal possesses multifractal properties. In the running-in period, both $\mathrm{\Delta }\alpha$ and $\mathrm{\Delta }f(\alpha )$ decrease gradually, and the spectrum changes from a wide, hook-like shape to a narrow, bell-like shape. In the steady state, $\mathrm{\Delta }\alpha$ maintains at a small value, $\mathrm{\Delta }f(\alpha )$ fluctuates slightly around 0, and the narrow, bell-like shape of the spectrum persists. In the increasing friction, $\mathrm{\Delta }\alpha$ and $\mathrm{\Delta }f(\alpha )$ increase gradually, and the spectrum changes from the narrow, bell-like shape to the wide, hook-like shape. The results provided insight into the dynamic behavior of friction system.

Stimulated reservoir volume (SRV) with large fracture networks can be generated near hydraulic fractured vertical wells (HFVWs) in tight oil reservoirs. Statistics show that natural microfractures and fracture networks stimulated by SRV were self-similar in statistical sense. Currently, various analytical models have been presented to study pressure behaviors of HFVWs in tight oil reservoirs. However, most of the existing models did not take the distribution and self-similarity of fractures into consideration. To account for stimulated characteristic and self-similarity of fractures in tight oil reservoirs, a mixed-fractal flow model was presented. In this model, there are two distinct regions, stimulated region and unstimulated region. Dual-porosity model and single porosity model were used to model stimulated and unstimulated regions, respectively. Fractal geometry is employed to describe fractal permeability and porosity relationship (FPPR) in tight oil reservoirs. Solutions for the mixed-fractal flow model were derived in the Laplace domain and were validated among range of the reservoir parameters. The pressure transient behavior and production rate derivative were used to analyze flow regimes. The type curves show that the fluid flow in HFVWs can be divided into six main flow periods. Finally, effect of fractal parameters and SRV size on flow periods were also discussed. The results show that the SRV size and fractal parameters of fracture network have great effect on the former periods and fractal parameters of matrix mainly influence the later flow periods.

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of ${{𝒞}}^1$-rational fractal interpolation surfaces (FISs) on a rectangular grid in the present paper that contain scaling factors in both directions and two types of positive real parameters which are referred as shape parameters. The graphs of these ${{𝒞}}^1$-rational FISs are the attractors of suitable rational iterated function systems (IFSs) in ${ℝ}^3$ which use a collection of rational IFSs in the $x$-direction and $y$-direction and hence these FISs are self-referential in nature. Using upper bounds of the interpolation error of the $x$-direction and $y$-direction fractal interpolants along the grid lines, we study the convergence results of ${{𝒞}}^1$-rational FISs toward the original function. A numerical illustration is provided to explain the visual quality of our rational FISs. An extra feature of these fractal surface schemes is that it allows subsequent interactive alteration of the shape of the surfaces by changing the scaling factors and shape parameters.

In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc. 155 (2013) 73–86], Lau, Luo and Rao completely classified the topological structure of so called fractal square $F$ defined by $F=(F+{𝒟})/n$, where ${𝒟}⊊{\{0,1,\dots ,n-1\}}^2,n\ge 2$. In this paper, we further provide simple criteria for the $F$ to be totally disconnected, then we discuss the Lipschitz classification of $F$ in the case $n=3$, which is an attempt to consider non-totally disconnected sets.

The simplest infinite sequences that are not ultimately periodic are pure morphic sequences: fixed points of particular morphisms mapping single symbols to strings of symbols. A basic way to visualize a sequence is by a turtle curve: for every alphabet symbol fix an angle, and then consecutively for all sequence elements draw a unit segment and turn the drawing direction by the corresponding angle. This paper investigates turtle curves of pure morphic sequences. In particular, criteria are given for turtle curves being finite (consisting of finitely many segments), and for being fractal or self-similar: it contains an up-scaled copy of itself. Also space-filling turtle curves are considered, and a turtle curve that is dense in the plane. As a particular result we give an exact relationship between the Koch curve and a turtle curve for the Thue–Morse sequence, where until now for such a result only approximations were known.

Objectives: To test the capacity of the digital tool, fractal dimension (FD) analysis, in identifying subtle differences in bone pattern in patients with renal osteodystrophy (RO), correlated with the time of hemodialysis, in different regions of interest, delineated on panoramic and periapical radiographs. Study design: A total of 34 patients with chronic renal disease undergoing hemodialysis were submitted to panoramic and periapical radiographs. Different regions of interest were delineated on the mandibular body and ramus. FD was analyzed by means of the software program ImageJ and correlated with the time of hemodialysis. Results: The sample consisted of 34 subjects. The time of hemodialysis varied from 1 to 286 months. There was significant correlation between the time of hemodialysis and the FD values in the region delineated in the mandibular angle ($r=0.498;p=0.003$) and this was shown in the periapical radiographs as well ($r=-0.349;p=0.043$). Conclusions: FD analysis was a useful tool in detecting alterations caused by RO in bone pattern, in panoramic and periapical radiographs.

This study aimed to investigate the asymmetric structure between the carbon and energy markets from two aspects of different trends (up or down) and volatility-transmission direction using asymmetric detrended cross-correlation analysis (DCCA) cross-correlation coefficient test, multifractal asymmetric DCCA (MF-ADCCA) method, asymmetric volatility-constrained correlation metric and time rate of information-flow approach. We sampled 1283 observations from January 2008 to December 2012 among pairs of carbon and energy markets for analysis. Empirical results show that the (1) asymmetric characteristic from the cross-correlation between carbon and returns in the energy markets is significant, (2) asymmetric cross-correlation between carbon and energy market price returns is persistent and multifractral and (3) volatility of the base assets of energy market returns is more influential to the base asset of the carbon market than that of the energy market.

In this paper, a GIS-based method was developed to extract the real-time traffic information (RTTI) from the Google Maps system for city roads. The method can be used to quantify both congested and free-flow traffic conditions. The roadway length was defined as congested length (CL) and free-flow length (FFL). Chengdu, the capital of Sichuan Province in the southwest of China, was chosen as a case study site. The RTTI data were extracted from the Google real-time maps in May 12–17, 2013 and were used to derive the CL and FFL for the study areas. The Multifractal Detrended Fluctuation Analysis (MFDFA) was used to characterize the long-term correlations of CL and FFL time series and their corresponding multifractal properties. Analysis showed that CL and FFL had demonstrated time nonlinearity and long-term correlations and both characteristics differed significantly. A shuffling procedure and a phase randomization procedure were further integrated with multifractal detrending moving average (MFDMA) to identify the major sources of multifractality of these two time series. The results showed that a multifractal process analysis could be used to characterize complex traffic data. Traffic data collected and methods developed in this paper will help better understand the complex traffic systems.

Prediction of fractional flow in fractal porous medium is important for reservoir engineering and chemical engineering as well as hydrology. A physical conceptual fractional flow model of transient two-phase flow is developed in fractal porous medium based on the fractal characteristics of pore-size distribution and on the approximation that porous medium consist of a bundle of tortuous capillaries. The analytical expression for fractional flow for wetting phase is presented, and the proposed expression is the function of structural parameters (such as tortuosity fractal dimension, pore fractal dimension, maximum and minimum diameters of capillaries) and fluid properties (such as contact angle, viscosity and interfacial tension) in fractal porous medium. The sensitive parameters that influence fractional flow and its derivative are formulated, and their impacts on fractional flow are discussed.