Fractals

Multi-resolution representation has wide application demands in the field of Geographic Information System (GIS). The existing fractal interpolation methods can implement the reconstruction of natural linear features from low resolution to high resolution, but they are unable to quantify the relationship between the number of fractal iterations and the resolution, and the fractal interpolation results cannot maintain the geographical characteristics of natural linear features. This study presents a multi-resolution reconstruction method for natural linear features in GIS based on the restricted fractal interpolation. First, the geographical characteristics of natural linear features are extracted to restrict the parameters of fractal interpolation function and control the process of fractal interpolation. Second, a mathematical function is deduced for identifying the relationship between the number of fractal iterations and the resolution. The experiment results demonstrate that this study can reconstruct various preset high-resolution linear features from the low-resolution linear data while the geographical characteristics and random fractal characteristics of natural linear features are kept well.

In this paper, we present two deterministic weighted scale-free networks controlled by a weight parameter $r(0<r\le 1)$. One is fractal network, the other one is non-fractal network, while they have the same weight distribution when the parameter $r$ is identical. Based on their special network structure, we study random walks on network with a trap located at a fixed node. For each network, we calculate exact solutions for average trapping time (ATT). Analyzing and comparing the obtained solutions, we find that their ATT all grow asymptotically as a power-law function of network order (number of nodes) with the exponent $f(r)$ dependent on the weight parameter, but their exponent $f(r)$ are obviously different, one is an increasing function of $r$, while the other is opposite. Collectively, all the obtained results show that the efficiency of trapping on weighted Scale-free networks has close relation to the weight distribution, but there is no stable positive or negative correlation between the weight distribution and the trapping time on different networks. We hope these results given in this paper could help us get deeper understanding about the weight distribution on the property and dynamics of scale-free networks.

In this paper, we check for existence of multifractal in volatility of Moroccan family business stock returns and in volatility of Casablanca market index returns based on multifractal detrended fluctuation analysis (MF-DFA) technique. Empirical results show strong evidence of multifractal characteristics in volatility series of both family business stocks and market index. In addition, it is found that small variations in volatility of family business stocks are persistent, whilst small variations in volatility of market index are anti-persistent. However, large variations in family business volatility and market index volatility are both anti-persistent. Furthermore, multifractal spectral analysis based results show strong evidence that volatility in Moroccan family business companies exhibits more multifractality than volatility in the main stock market. These results may provide insightful information for risk managers concerned with family business stocks.

To ensure the safe operation of many safety critical structures such as nuclear plants, aircraft and oil pipelines, non-destructive imaging is employed using piezoelectric ultrasonic transducers. These sensors typically operate at a single frequency due to the restrictions imposed on their resonant behavior by the use of a single length scale in the design. To allow these transducers to transmit and receive more complex signals it would seem logical to use a range of length scales in the design so that a wide range of resonating frequencies will result. In this paper, we derive a mathematical model to predict the dynamics of an ultrasound transducer that achieves this range of length scales by adopting a fractal architecture. In fact, the device is modeled as a graph where the nodes represent segments of the piezoelectric and polymer materials. The electrical and mechanical fields that are contained within this graph are then expressed in terms of a finite element basis. The structure of the resulting discretized equations yields to a renormalization methodology which is used to derive expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities. A comparison with a standard design shows some benefits of these fractal designs.

A theoretical study is made to derive an energy distribution equation for the size reduction process from the fractal model for the particle comminution. Fractal model is employed as a valid measure of the self-similar size distribution of comminution daughter products. The tensile strength of particles varies with particle size in the manner of a power function law. The energy consumption for comminuting single particle is found to be proportional to the 5(D−3)/3rd order of the particle size, $D$ being the fractal dimension of particle comminution daughter. The Weibull statistics is applied to describe the relationship between the breakage probability and specific energy of particle comminution. A simple equation is derived for the breakage probability of particles in view of the dependence of fracture energy on particle size. The calculated exponents and Weibull coefficients are generally in conformity with published data for fracture of particles.

The (detrended cross-correlation analysis) DCCA cross-correlation coefficient was proposed to measure the level of long-range cross-correlations between two non-stationary time series on multiple time scales. It has been applied to diverse areas of interest, although many properties of this method are not clear. In this paper, we theoretically study several fundamental properties of the DCCA cross-correlation coefficient, which contributes to acquiring more statistical characteristics of this measure. We resort to a synthetic time series that is followed by the integration and the detrending procedures of the DCCA cross-correlation coefficient, which divide the steps to estimate the coefficient into two portions. The former portion, including the integration and the detrending, is proved to be a linear transformation. The second portion is devoted to measuring Pearson’s $r$ between two synthetic time series. We confirm that the DCCA cross-correlation coefficient is also a linear measure by definition. The simulations including the ARFIMA processes and the multifractal binomial measures are numerically analyzed, which confirm the theoretical analysis.

Electronic sensor devices in geophysical processes are required to measure and automate different tasks. Throughout history, people have created multiple type devices, but acoustics have an important application such as the content form description in deep wells, watersheds, lakes, caves, among others. The acoustic signal is capable of reflecting where other types of signals cannot operate, either by drawbacks or where fluid is displaced. A mathematical model is presented in this paper described in state space as a basic acoustic sensor description. The objective is to adjust the parameters allowing the acoustic device to describe a signal in its trajectory, representing in geophysical manner the cavity form. Therefore, the control is performed on the response of the acoustic sensor model, adjusted with a parameter estimation process. The simulation results counts convergence between the reference and identified signals.

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order $\nu$ of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order $\nu$ of a CHFIF defined on any interval $[a,b]$ is also a CHFIF.

In this paper, the definition of unbounded variation point is introduced firstly. Then, we construct a continuous function containing uncountable unbounded variation points on the basis of Devil stair function. Finally, a mass of meaningful properties have been proved, such as Box dimension of continuous function constructed in the paper is 1 and the continuous function has infinite length on [0,1].

Self-similarity is a common tendency in nature and physics. It is wide spread in geo-physical phenomena like diffusion and iteration. Physically, an object is self-similar if it is invariant under a set of scaling transformation. This paper gives a brief outline of the analytical and set theoretical properties of different types of weak self-similar sets. It is proved that weak sub self-similar sets are closed under finite union. Weak sub self-similar property of the topological boundary of a weak self-similar set is also discussed. The denseness of non-weak super self-similar sets in the set of all non-empty compact subsets of a separable complete metric space is established. It is proved that the power of weak self-similar sets are weak super self-similar in the product metric and weak self-similarity is preserved under isometry. A characterization of weak super self-similar sets using weak sub contractions is also presented. Exact weak sub and super self-similar sets are introduced in this paper and some necessary and sufficient conditions in terms of weak condensation IFS are presented. A condition for a set to be both exact weak super and sub self-similar is obtained and the denseness of exact weak super self similar sets in the set of all weak self-similar sets is discussed.

This study reveals the two-scale characteristics in prime number distribution. It is observed a sub-diffusion process of power law decay at the small scale of the natural number $1\le N\le 10^{35}$, but is found to obey the classical Brownian motion of an exponential decay at the large scale $N\ge 10^{220}$. Such two-scale mechanism gives rise to the multi-fractal scaling from the power law to the exponential law distributions in a transition region of the natural number $10^{35}\le N\le 10^{220}$. In the small range, the sub-diffusion of prime number distribution is well depicted by the fractional derivative equation model, and in the large scale, exponential decay distribution can accurately be described by a classical diffusion equation model. The Riemann diffusion equation proposed recently by the present authors can accurately model the prime distribution from small to moderate to large scales and is reduced to the fractional derivative sub-diffusion equation at small scale and the classical Brownian motion diffusion equation at large scale, respectively.

Fracture network and fractured porous media as well as their transport properties have received great attentions in many fields from engineering application and basic theoretical researches. Fracture will dynamically extend in length and aperture to form complex fracture network under some external conditions such as percussion drilling, wave propagation, desiccation and hydrofracturing. The complexity of fracture network can be well quantitatively characterized by fractal dimension. In this work, the dynamic characterization of fracture network extension in porous media under drying process is measured by the improved box-counting technique, and fractal dimensions of fracture network are respectively related to drying time, average aperture, moisture content and fracture porosity. The fractal dimension increases exponentially with drying time and average aperture, and decreases with moisture content in the form of power law. Specially, the fractal dimension is approximatively increased with porosity in the form of linearity in a narrow porosity range. The transport capacity of fracture network, described by seepage coefficient, is also related to the fractal dimension with drying time in the form of exponential function. The presented fractal analysis of fracture network could also shed light on the hydrofracturing application in subsurface unconventional oil and gas reservoirs.

Fractal interpolation filter is proposed for the first time in the literatures to transform original signals. Using the multifractal detrended fluctuation analysis (MFDFA), the authors investigate how the filter affects the multifractal scaling properties for both artificial and traffic signals. Specifically, the authors compare the multifractal scaling properties of signals before and after the transforms. It is shown that the fractal interpolation filter changes slightly the maximum value of the multifractal spectrum, while the values of spectrum width and maximum point of spectrum are much more affected by vertical scaling factor. The multifractal spectrum shrinks dramatically after the fractal interpolation filter. The fractal exponents in the signal change dramatically for the negative values of vertical scaling factor while remain stable otherwise. Thus, an appropriate vertical scaling factor can be found in order to minimize the effects of filter when one uses the fractal interpolation filter.

Automatic Face Recognition (FR) presents a challenging task in the field of pattern recognition and despite the huge research in the past several decades; it still remains an open research problem. This is primarily due to the variability in the facial images, such as non-uniform illuminations, low resolution, occlusion, and/or variation in poses. Due to its non-intrusive nature, the FR is an attractive biometric modality and has gained a lot of attention in the biometric research community. Driven by the enormous number of potential application domains, many algorithms have been proposed for the FR. This paper presents an overview of the state-of-the-art FR algorithms, focusing their performances on publicly available databases. We highlight the conditions of the image databases with regard to the recognition rate of each approach. This is useful as a quick research overview and for practitioners as well to choose an algorithm for their specified FR application. To provide a comprehensive survey, the paper divides the FR algorithms into three categories: (1) intensity-based, (2) video-based, and (3) 3D based FR algorithms. In each category, the most commonly used algorithms and their performance is reported on standard face databases and a brief critical discussion is carried out.