Fractals

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector $F=(f_1,\dots ,f_L)$ of $L$ functions, with $L\ge 2$, we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function $f_l$ has exactly a given value ${\alpha }_l$ of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of $F$. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.

Natural images exhibit a high degree of complexity, randomness and irregularity in color and texture, however fractal can be an effective tool to describe various irregular phenomena in nature. Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately by means of experiments. In this paper, we proposed a fractal dimension estimation method for RGB color images. In the proposed method, we present a hyper-surface partition method which considers the hyper-surface as continuous and divide the image into nonoverlapped blocks. We also defined a counting method in color domain. To validate the proposed method, experiments were carried on two types of color images: synthesized fractal images and natural RGB color images. The experimental results demonstrate that the proposed method is effective and efficient. The behaviors of the proposed method on the rescale images are also shown in the paper. And it can be performed as a reliable FD estimation approach for the RGB color images.

In the present paper, we introduce a new set which defines a generalized self-similar set for contractive functions ${\{f_i\}}_{i=1}^N$ on the unit ball ${\mathbb{Z}}_p$ of $p$-adic numbers. This set is called unconventional limit set. We prove that the unconventional limit set is compact, perfect and uniformly disconnected. Moreover, we provide an example of two contractions for which the corresponding unconventional limiting set is quasi-symmetrically equivalent to the symbolic Cantor set.

Relationships between material hardness, turning parameters (spindle speed and feed rate) and surface parameters (surface roughness $R$a, fractal dimension $D$ and characteristic roughness ${\tau }^{\ast })$ are studied and modeled using response surface methodology (RSM). The experiments are carried out on a CNC lathe for six carbon steel material AISI 1010, AISI 1020, AISI 1030, AISI 1045, AISI 1050 and AISI 1060. The profile of turned surface and the surface roughness value are measured by a JB-5C profilometer. Based on the profile data, $D$ and ${\tau }^{\ast }$ are computed through the root-mean-square method. The analysis of variance (ANOVA) reveals that spindle speed is the most significant factors affecting $R$a, while material hardness is the most dominant parameter affecting ${\tau }^{\ast }$. Material hardness and spindle speed have the same influence on $D$. Feed rate has less effect on three surface parameters than spindle speed and material hardness. The second-order models of RSM are established for estimating $R$a, $D$ and ${\tau }^{\ast }$. The validity of the developed models is approximately 80%. The response surfaces show that a surface with small $R$a and large $D$ and ${\tau }^{\ast }$ can be obtained by selecting a high speed and a large hardness material. According to the established models, $R$a, $D$ and ${\tau }^{\ast }$ of six carbon steels surfaces can be predicted under cutting conditions studied in this paper. The results have an instructive meaning to estimate the surface quality before turning.

We present an efficient method of constructing hyperbolic patterns based on an asymmetric motif designed in the central hyperbolic polygon. Since there is no rotational symmetry in each hyperbolic polygon, a subset of the hyperbolic group elements has to be selected carefully so that the central hyperbolic polygon is transformed to the other polygons once and only once. An efficient labeling procedure is proved by considering the group presentation and can be easily implemented using the computer. Illustrative hyperbolic patterns are constructed from given asymmetric motifs for the symmetry group [$p$, $q$]${}^{+}$ which consists of all compositions of an even number of reflections.

We discuss an optimization problem on the maximal invariance of a class of multiple-valued iterative dynamics (MVID) models of discrete-time nonlinear control dynamical systems with singular disturbance. We study the inner and outer bounds of the maximal invariance, between which all noninterfering MVID models of nonlinear discrete-time control dynamical systems with singular disturbance reside. We also study the invariant fractal structure and an optimization of Lyapunov multipliers associated to it.

The dynamics seismic activity occurred in the Cocos Plate—Mexico is analyzed through the evolution of Hurst exponent and 3D fractal dimension, under the mathematical fractal structure based on seismic activity time series, taking into account the magnitude ($M$) as the main parameter to be estimated. The seismic activity time series and, annual intervals are considered first for finding the Hurst exponent of each year since 1988 (the year in which the database is consistent) until 2012, and then the following years are accumulated describing the cumulative Hurst exponent. The seismic activity description is based on Cocos Plate data information; during a period conform from 1 January 1988 to 31 December 2012. Analyses were performed following methods, mainly considering that the Hurst exponent analysis provides the ability to find the seismicity behavior time–space, described by parameters obtained under the fractal dimension and complex systems.

The multifractal analysis of one time series, e.g. crude oil, gold and exchange rate series, is often referred. In this paper, we apply the classical multifractal and mixed multifractal spectrum to study multifractal properties of crude oil, gold and exchange rate series and their inner relationships. The obtained results show that in general, the fractal dimension of gold and crude oil is larger than that of exchange rate (RMB against the US dollar), reflecting a fact that the price series in gold and crude oil are more heterogeneous. Their mixed multifractal spectra have a drift and the plot is not symmetric, so there is a low level of mixed multifractal between each pair of crude oil, gold and exchange rate series.

In this study, the influence of several fractal identifiers of granular materials on dynamic behavior of a flexible pavement structure as a particulate stratum is considered. Using experimental results and numerical methods as well, 15 different grain-shaped sands obtained from 5 different sources were analyzed as pavement base course materials. Image analyses were carried out by use of a stereomicroscope on 15 different samples to obtain quantitative particle shape information. Furthermore, triaxial compression tests were conducted to determine stress–strain and shear strength parameters of sands. Additionally, the dynamic response of the particulate media to standard traffic loads was computed using finite element modeling (FEM) technique. Using area-perimeter, line divider and box counting methods, over a hundred grains for each sand type were subjected to fractal analysis. Relationships among fractal dimension descriptors and dynamic strain levels were established for assessment of importance of shape descriptors of sands at various scales on the dynamic behavior. In this context, the advantage of fractal geometry concept to describe irregular and fractured shapes was used to characterize the sands used as base course materials. Results indicated that fractal identifiers can be preferred to analyze the effect of shape properties of sands on dynamic behavior of pavement base layers.

This paper analyzes the complexity of stock exchanges through fractal theory. Closing price indices of four stock exchanges with different industry sectors are selected. Degree of complexity is assessed through Higuchi’s fractal dimension. Various window sizes are considered in evaluating the fractal dimension. It is inferred that the data considered as a whole represents random walk for all the four indices. Analysis of financial data through windowing procedure exhibits multi-fractality. Attempts to apply moving averages to reduce noise in the data revealed lower estimates of fractal dimension, which was verified using fractional Brownian motion. A change in the normalization factor in Higuchi’s algorithm did improve the results. It is quintessential to focus on rural development to realize a standard and steady growth of economy. Tools must be devised to settle the issues in this regard. Micro level institutions are necessary for the economic growth of a country like India, which would induce a sporadic development in the present global economical scenario.

The evolutional scenario of the resource distribution in the fractal-cluster system which is identified as the “organism” has been suggested. We propose a model in which the resource redistribution dynamics in the closed system is determined with the ultrametric structure of the system’s space. Moreover, each cluster has its own characteristic time of a transfer to the equilibrium state which is determined with the ultrametric size of the cluster. The general equation which determines this dynamics has been written. For the determined type of the resource transitions among clusters, the solution of this equation has been numerically received. The problem of the parameter identification’s modelling for the real systems has been discussed.

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

The construction of the ternary Cantor set is generalized into the context of hyperbolic numbers. The partial order structure of hyperbolic numbers is revealed and the notion of hyperbolic interval is defined. This allows us to define a general framework of the fractal geometry on the hyperbolic plane. Three types of the hyperbolic analogues of the real Cantor set are identified. The complementary nature of the real Cantor dust and the real Sierpinski carpet on the hyperbolic plane are outlined. The relevance of these findings in the context of modern physics are briefly discussed.

A random agent-based financial model is developed and investigated by the finite-range multitype contact dynamic system, in an attempt to reproduce and study the dynamics of financial markets. And an analysis method of detecting duration and intensity relationship in volatility series is introduced, called the volatility duration analysis. Then the auto-correlation analysis suggests that there exists evident volatility clustering feature in absolute volatility durations for the simulation data and the real data. Besides, the Lempel–Ziv complexity analysis is applied to study the complexity of the returns, the corresponding absolute returns and the volatility duration returns, which can reflect the fluctuation behaviors, the volatility behaviors and the volatility duration behaviors. At last, the multifractal phenomena of volatility durations of returns are comparatively studied for Shanghai Composite Index and the proposed model by multifractal detrended fluctuation analysis.

The fracture-pore double porosity medium is one of the most common media in nature, for example, rock mass in strata. Fracture has a more significant effect on fluid flow than a pore in a fracture-pore double porosity medium. Hence, the fracture effect on percolation should be considered when studying the percolation phenomenon in porous media. In this paper, based on the fractal distribution law, three-dimensional (3D) fracture surfaces, and two-dimensional (2D) fracture traces in rock mass, the locations of fracture surfaces or traces are determined using a random function of uniform distribution. Pores are superimposed to build a fractal fracture-pore double medium. Numerical experiments were performed to show percolation phenomena in the fracture-pore double medium. The percolation threshold can be determined from three independent variables (porosity $n$, fracture fractal dimension $D$, and initial value of fracture number $N_0$). Once any two are determined, the percolation probability exists at a critical point with the remaining parameter changing. When the initial value of the fracture number is greater than zero, the percolation threshold in the fracture-pore medium is much smaller than that in a pore medium. When the fracture number equals zero, the fracture-pore medium degenerates to a pore medium, and both percolation thresholds are the same.

In this paper, we propose a new matching (called a conditional matching), where the condition refers to the matching of the new constructed network which includes all the nodes in the original network. We then enumerate the conditional matchings of the new network and prove that the number of conditional matchings is just the product of degree sequences of the original network. We choose two families of fractal networks to show our obtained results, including the pseudofractal network and Cayley tree. Finally, we calculate the entropy of the conditional matchings on the considered networks and see that the entropy of Cayley tree is smaller than that of the pseudofractal network.