Fractals

A drying droplet changes its morphological pattern depending upon complex pattern forming system. To control the distribution of solute particles in a droplet during drying is an important aspect in many scientific and industrial purposes. In this work, with the help of optical microscopy, we study characteristic patterns generated in dried drops of colloidal copper sulphate (CuSO${}_4\cdot$5H₂O) solution on surface of glass. At lower concentration of salt solution the growth pattern follows a monofractal structure whereas at higher concentration, the self-assembled pattern gradually gets disappeared. Calculating the fractal dimension (FD) of the generated patterns by box counting method with help of imageJ, it is observed that the patterns resemble DLA structure through a specific range of concentration of the salt solution.

We introduce a method to reduce to the real case the calculus of the box-counting dimension of subsets of the unit cube $I^n$, $n>1$. The procedure is based on the existence of special types of $\alpha$-dense curves (a generalization of the space-filling curves) in $I^n$ called $\delta$-uniform curves.

In this paper, we generalize Sierpinski carpet and Menger sponge in $n$-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet and Menger sponge. Exactly, Menger sponge in $4$-dimensional space could be drawn out clearly under an affine transformation. Furthermore, the method could be used to a much broader class in fractals.

The diagonal effect of orders is well documented in different markets, which states that the orders are more likely to be followed by the orders of the same aggressiveness and implies the presence of short-term correlations in order flows. Based on the order flow data of 43 Chinese stocks, we investigate if there are long-range correlations in the time series of order aggressiveness. The detrending moving average analysis shows that there are crossovers in the scaling behaviors of overall fluctuations and order aggressiveness exhibits linear long-term correlations. We design an objective procedure to determine the two Hurst indexes delimited by the crossover scale. We find no correlations in the short term and strong correlations in the long term for all stocks except for an outlier stock. The long-term correlation is found to depend on several firm specific characteristics. We also find that there are nonlinear long-term correlations in the order aggressiveness when we perform the multifractal detrending moving average analysis.

A fractal permeability model coupling non-flowing boundary-layer effect for tight oil reservoirs was proposed. Firstly, pore structures of tight formations were characterized with fractal theory. Then, with the empirical equation of boundary-layer thickness, Hagen–Poiseuille equation and fractal theory, a fractal torturous capillary tube model coupled with boundary-layer effect was developed, and verified with experimental data. Finally, the parameters influencing effective liquid permeability were quantitatively investigated. The research results show that effective liquid permeability of tight formations is not only decided by pore structures, but also affected by boundary-layer distributions, and effective liquid permeability is the function of fluid type, fluid viscosity, pressure gradient, fractal dimension, tortuosity fractal dimension, minimum pore radius and maximum pore radius. For the tight formations dominated with nanoscale pores, boundary-layer effect can significantly reduce effective liquid permeability, especially under low pressure gradient.

This paper provides evidence of fractal, multifractal and chaotic behaviors in urban crime by computing key statistical attributes over a long data register of criminal activity. Fractal and multifractal analyses based on power spectrum, Hurst exponent computation, hierarchical power law detection and multifractal spectrum are considered ways to characterize and quantify the footprint of complexity of criminal activity. Moreover, observed chaos analysis is considered a second step to pinpoint the nature of the underlying crime dynamics. This approach is carried out on a long database of burglary activity reported by 10 police districts of San Francisco city. In general, interarrival time processes of criminal activity in San Francisco exhibit fractal and multifractal patterns. The behavior of some of these processes is close to $1/f$ noise. Therefore, a characterization as deterministic, high-dimensional, chaotic phenomena is viable. Thus, the nature of crime dynamics can be studied from geometric and chaotic perspectives. Our findings support that crime dynamics may be understood from complex systems theories like self-organized criticality or highly optimized tolerance.

The average geodesic distance is concerned with complex networks. To obtain the limit of average geodesic distances on growing Sierpinski networks, we obtain the accurate value of integral in terms of average geodesic distance and self-similar measure on the Sierpinski gasket. To provide the value of integral, we find the phenomenon of finite pattern on integral inspired by the concept of finite type on self-similar sets with overlaps.

In order to explore the causes of acoustic emission (AE) signals during coal failure, the coal samples with original joints were subjected to uniaxial compression experiments, and the AE signals were monitored by AEwin Test for Express-8.0. Based on the multifractal theory, the multifractal characteristics of AE were analyzed. The results showed that the AE counts and accumulative counts change over time corresponded well with the load-time, which reflected the degree of crack evolution and loading. During the initial loading stage, the cracks expanded gradually along the trace of the original cracks, which could induce a few AE events, while with the increase of load, the cracks enlarged gradually and then joined together to form a macroscopic fracture, which would cause much more AE events within a larger value. Multifractal spectrum [$f(\alpha )-\alpha$] of AE was more concentrated in the right side, illustrating that the frequency of small signals was greater than that of the large signals in AE sequences, which revealed cracks expanding and microfracture events dominated during the loading process. The greater the multifractal spectrum width ($\mathrm{\Delta }\alpha )$ was, the larger the AE signals differences were, which reflected that AE varied more intensely. The more developed the original cracks, the more obvious the multifractal characteristics. This research revealed the causes and percentage of the AE events within small or large signals, which would help us to recognize crack evolution of coal and generation mechanism of AE.

In this paper, we discuss the construction of new Cantor like sets in the hyperbolic plane. Also, we study the arithmetic sum of two of these Cantor like sets, as well as of those previously introduced in the literature. An hyperbolization, in the sense of Gromov, of the commutative ring of hyperbolic numbers is also given. Finally, we present the construction of a Cantor-type set as hyperbolic boundary.

In the present paper, a one-dimensional continuous function of unbounded variation on the interval $[0,1]$ has been constructed. Box dimension of this function has been proved to be 1. Furthermore, Box dimension of its Riemann–Liouville fractional integral of any order has also been proved to be 1.

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

In this paper, we first study the first-order network coherence, characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, on the weighted Cayley networks with the weight factor. The analytical formula of the EMFPT is obtained by the definition of the EMFPT. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor. Then, we study eigentime identity quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on the weighted Cayley networks with the weight factor. We show that all their eigenvalues can be obtained by calculating the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity on the weighted Cayley networks obey two laws along with the range of the weight factor.

Piezoelectric ultrasonic transducers have the ability to act both as a receiver and a transmitter of ultrasound. Standard designs have a regular structure and therefore operate effectively over narrow bandwidths due to their single length scale. Naturally occurring transducers benefit from a wide range of length scales giving rise to increased bandwidths. It is therefore of interest to investigate structures which incorporate a range of length scales, such as fractals. This paper applies an adaptation of the Green function renormalization method to analyze the propagation of an ultrasonic wave in a series of pre-fractal structures. The structure being investigated here is the Sierpinski carpet. Novel expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the operating frequency are presented. Comparisons of metrics between three new designs alongside the standard design (Euclidean structure) and the previously investigated Sierpinski gasket device are performed. The results indicate a significant improvement in the reception sensitivity of the device, and improved bandwidth in both the receiving and transmitting responses.

This study investigates the roles of fracture roughness, normal stress and shear displacement on the fluid flow characteristics through three-dimensional (3D) self-affine fractal rock fractures, whose surfaces are generated using the modified successive random additions (SRA) algorithm. A series of numerical shear-flow tests under different normal stresses were conducted on rough rock fractures to calculate the evolutions of fracture aperture and permeability. The results show that the rough surfaces of fractal-based fractures can be described using the scaling parameter Hurst exponent $(H)$, in which $H=3-D_f$, where $D_f$ is the fractal dimension of 3D single fractures. The joint roughness coefficient (JRC) distribution of fracture profiles follows a Gauss function with a negative linear relationship between $H$ and average JRC. The frequency curves of aperture distributions change from sharp to flat with increasing shear displacement, indicating a more anisotropic and heterogeneous flow pattern. Both the mean aperture and permeability of fracture increase with the increment of surface roughness and decrement of normal stress. At the beginning of shear, the permeability increases remarkably and then gradually becomes steady. A predictive model of permeability using the mean mechanical aperture is proposed and the validity is verified by comparisons with the experimental results reported in literature. The proposed model provides a simple method to approximate permeability of fractal-based rough rock fractures during shear using fracture aperture distribution that can be easily obtained from digitized fracture surface information.