Regular Fractals Research Articles

REMARKS ON THE LOCAL STRUCTURE OF REGULAR FRACTAL FUNCTIONS WITH FRACTAL DIMENSIONS

In this paper, we have explored the local structure and fractal characteristics of fractal functions with certain fractal dimensions. The conclusion that points with inconsistent oscillation amplitudes with the upper Box dimension of the corresponding fractal functions have been proved to be nowhere dense. This will play an important supporting role in exploring the fractal dimension estimation of the combination of fractal functions.

New Classes of Regular Symmetric Fractals

The paper introduces new fractal families many of which approach optimal information dimension for annular and checkerboard structures and include the Sierpinski carpet and the Menger sponge as special cases. The complementary mapping is defined, and a notation to represent the families is proposed. The new classes represent an enhanced set that goes beyond the recently published results on optimal information dimensionality and they can be expected to have applications in natural and engineered self-similar systems.

Normalizing and classifying shape indexes of cities by ideas from fractals

A standard scientific study comprises two processes: one is to describe how a system works, and the other is to understand why the system works in this way. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe the size and shape of cities. However, the comparability of a shape index is often influenced by the resolution of remote sensing images or digital maps because the calculated values depend on spatial measurement scales. This paper is devoted to exploring the scaling property in classical shape indexes with mathematical methods. Two typical regular fractals, Koch's island and Vicsek's figure, are employed to illustrate the property of scale dependence of many shape indexes. A set of formula are derived from the geometric measure relation to associate the fractal dimension for boundary lines with shape indexes. Based on the ideas of fractals, two main problems are solved in this work. First, several different correlated shape indexes such as circularity ratios and compactness ratios are normalized and unified by substituting Feret's diameter for the major axis of urban figure. Second, three levels of scaling in shape indexes are revealed, and shape indexes are classified into three groups. The significance of this study lies in two aspects. One is helpful for realizing the deep structure of urban form, and the other is useful for clarifying the spheres of application of different types of shape indexes.

Conductivity Estimates of Fractal Models of Geological Media

AbstractGeological media are inherently fractal over multiple length scales, exhibiting a wide range of fractal dimensions . We study the size dependent conductivity of models of geological media based on so called regular random fractals. Specifically, we consider generalizations of the classic Menger sponges where the impermeable cells can take random locations. This results, for a given random fractal generator and a given construction order, in multiple realizations of the target media, each one maintaining the same . We use direct numerical simulation to estimate the conductivity, over a number of patterns, with orders . The conductivity bounds and estimates for regular fractal media are, in general, not applicable to these random fractal models. The conductivity of a specific realization may differ from the ensemble average by a few percents, which suggests a lack of self‐averaging; however, the self‐averaging is regained if the size of the domain of interest is beyond the maximal fractality length scale. We propose a low computational cost approach for estimating the ensemble average conductivity of a given random fractal at a given order, which matches the order of magnitude predicted from more intensive calculations. Through tabulation of conductivities of generators of random fractals with small scaling factors and fitting models, we explain how these fractals can model geological media in which fractal dimensions and conductivities (or effective diffusion coefficients) are measured.

Numerical Analysis of the Flow Effect of the Menger-Type Artificial Reefs with Different Void Space Complexity Indices

Based on fractal theory, a regular fractal is used to construct symmetrical reef models (e.g., cube and triangle reef models) with different fractal levels (n = 1, 2, 3). Using the concept of fractal dimension, we can better understand the spatial effectiveness of artificial reefs. The void space complexity index is defined to quantify the complexity of the internal spatial distribution of artificial reefs models under different levels. The computational fluid dynamics (CFD) flow simulation approach was used to investigate the effects of void space complexity on the flow field performances of the symmetrical artificial reef models. The upwelling convection index (Hupwelling/HAR, Vupwelling/VAR), wake recirculating index (Lwake/LAR, Vwake/VAR) and non-dimensionalized velocity ratio range were used to evaluate the efficiency of the flow field effect inside or around artificial reefs. The surface area and spatial complexity index of artificial reefs increase with increasing fractal level. The numerical simulation data shows that the Menger-type artificial reef models with a higher spatial complexity index have better flow field performances in the upwelling and wake regions. Compared to the traditional artificial reef models, the upwelling convection index (Vupwelling/VAR) and recirculating index (Vwake/VAR) of n = 3 fractal cube artificial reef increase by 37.5% and 46.8%, respectively. The efficiency indices of the upwelling region and wake region around the fractal triangle artificial reef model are 2–3 times those of the fractal cube artificial reef model when the fractal level is 3.

Equivalent relation between normalized spatial entropy and fractal dimension

Fractal dimension is defined on the base of entropy, including macro state entropy and information entropy. The generalized correlation dimension of multifractals is based on Renyi entropy. However, the mathematical transform from entropy to fractal dimension is not clear in both theory and practice. This paper is devoted to revealing the new equivalence relation between spatial entropy and fractal dimension using the box-counting method. By means of a set of regular fractals, the numerical relationship between spatial entropy and fractal dimension is examined. The results show that the ratio of actual entropy (Mq) to the maximum entropy (Mmax) equals the ratio of actual dimension (Dq) to the maximum dimension (Dmax). The spatial entropy and fractal dimension of complex spatial systems can be converted into one another by using functional box-counting method. The theoretical inference is verified by observational data of urban form. A conclusion is that the normalized spatial entropy is equal to the normalized fractal dimension. Fractal dimensions proved to be the characteristic values of entropies. In empirical studies, if the linear size of spatial measurement is small enough, a normalized entropy value is infinitely approximate to the corresponding normalized fractal dimension value. Based on the theoretical result, new spatial indexes of urban space filling can be defined, and multifractal parameters can be generalized to describe both simple systems and complex systems.

Mortal random walks on a family of treelike regular fractals with a deep trap

Due to the ubiquitous occurrence of evanescence in many physical, chemical and biological scenarios, mortal random walks that incorporate evanescence explicitly have drawn more and more attention. It has been a hot topic to study mortal random walks on distinct network models. In this paper, we study mortal random walks on T fractal and a family of treelike regular fractals with a trap located at central node (i.e., innermost node). First, with self-similar setting composed of T fractal, initial position of the walker and location of trap, the total trapping probability of the mortal walker reduces to a function of walker’s single-step survival parameter [Formula: see text]. In more detail, the total trapping probability is expressed by the [Formula: see text]th iteration of map (scaling function) of [Formula: see text]. Based on the map, the analytical expression of total trapping probability’s dominant behavior, the mean time to trapping (MFPT) and temporal factor are obtained, which are related to random walk dimension. Last, we extend the analysis to a family of treelike regular fractals. On them, the total trapping probability is still expressed as the [Formula: see text]th iteration of the map scaling [Formula: see text]. Accordingly, dominant behavior of total trapping probability, MFPT and temporal factor are determined analytically. Both analytical results obtained on T fractal and more general treelike regular fractals show that the mean time to trapping and desired random walk dimension can be obtained by tuning the survival probability parameter [Formula: see text]. In summary, the work advances the understanding of mortal random walks on more general deterministic networks.

On Fractal Constructions on Curvilinear Three-Web

We consider configurations of lines of curvilinear three-web, that can be inscribed in a triangle formed by the lines of this web. In the case when the inscribed configuration is triangulating, it generates a fractal in each such triangle. This allows us to associate with smooth function of two variables a certain fractal that generalizes the well-known Sierpinski triangle. We introduce the concept of a regular fractal and prove that a regular fractal is obtained only for a regular three-web (generalization of the basic theorem on hexagonal three-webs). We also find the fractal dimensions of some regular fractals and formulate problems related to fractal dimension.

The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation.

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpiński resonators: Effect of fractal depth and irregularity.

Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind, each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.

Diffraction on random fractal structures

In this paper, the formation of diffraction patterns for regular and random fractals of Cantor is numerically studied. The calculation of diffraction on one-dimensional, two-dimensional and radial fractal structures is performed using a fractional Fourier transform. This propagation operator makes it possible to obtain a field distribution (with the accuracy of scale) in any paraxial region, both in the Fresnel diffraction zone and in the far field.

The fractal peculiarities of graphene and intercalated graphite

The fractal properties of the hexagonal structure of graphene and the graphite intercalation compounds (GIC) are considered.The fractal properties of the single graphite monolayer are described in terms of the compact, perfect and totally disconnected Cantor set. The hexagonal fractal is built using David's Star as the fractal generator. The Hausdorff-Besikovitch fractal dimension is calculated for boundary of the single layer graphene and the GIC graphite monolayer boundary. Also different computational ways are considered for the fractal dimension calculation of the single layer graphene boundary. Comparison is carried out for the graphene boundary fractal with other regular fractals. The fractal dimension of the graphene boundary is shown to be equal to the Koch curve fractal dimension D = ln4/ln3. Thereby, from this point of view the contour boundary of the single graphene layer and the GIC graphite monolayer can be treated as an ideal one-atom-thickness and one dimensional nanoconductor. Such approach may be useful in study of the graphene and graphene's nanostructures. It is shown that this methodology can be expanded for computation of the fractal boundary dimension to the total family 2D crystals of the Kepler-Shubnikov 63net, including hBN, silicene, germanene, etc.

On Makarov’s Principle in Conformal Mapping

We examine several characteristics of conformal maps that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, they agree for domains whose boundaries are regular fractals such as Julia sets or limit sets of quasi-Fuchsian groups. We give a new proof of these dynamical equalities. We also show that these characteristics have the same universal bounds and prove a central limit theorem for extremals. Our method is based on analyzing the local variance of dyadic martingales associated to Bloch functions.

A novel method to identify the scaling region of correlation dimension

A random fractal exhibits self-similarity over the scaling region, this is different from the regular fractal. The scaling region obtained by the proper method for the exact fractal dimension is very important. And the correlation dimension is one of the fractal dimensions which is used widely in many fields. Therefore, it is necessary and timely to identify the scaling region that plays a critical role in calculating the correlation dimension accurately in various chaotic systems. Visual identification is widely used to determine the scaling region as a quick and simple subjective method. However, this method may lead to an inaccurate result in Grassberger Procaccia algorithm. In order to reduce the error caused by human factors from computing the correlation dimension, a novel method of identifying the scaling region based on simulated annealing genetic fuzzy C-means clustering algorithm is proposed. This new method is based on the fluctuating characteristics that the second-order derivative of the curve within the scaling region is zero or nearly zero. Firstly, the second-order differential of the double logarithm correlation integral discrete data is calculated. Secondly, the simulated annealing genetic fuzzy C-means clustering method is used for dividing the data into three groups: positive fluctuation data, zero fluctuation data, and negative fluctuation data. The zero fluctuation data are selected to retain, the rest is excluded. Thirdly, the 3 σ criteria are used for excluding gross errors to retain those valid from the zero fluctuation data. Fourthly, the data of the consecutive nature point interval are chosen from the retained data. Finally, the regression analysis and statistical test are used to identify the scaling region from the data valid. In order to verify the effectiveness of the proposed method, it is used to simulate the Lorenz and Henon systems. The calculated results are in good agreement with the theoretical values. Experimental results show that the proposed new approach is easy to operate, more efficient and more accurate than the subjective recognition, K-means method, and 2-means method in identifying the scaling region.

Synthesis of Complex Networks Regular Fractals

The article is sacred to the questions of design of topology of complex networks with the use of new methods of fractal geometry. A complex network structure is inprocess synthesized, through replacement of tops of count by a countfuse, and athe simplest, including affine transformations. The distinguishing feature of prefractal counts is pos sibility of receipt of numeral descriptions of network, coming from the analysis of properties of his countsfuses. For the got prefractal count properties o f fractal and scale invariance are investigational. An example of calculation of fractal dimension of prefracta l count of complex network is made at the use of square and threecornered blocks for coverage of network. It is shown that for the effective analysis of properties selfsimilarity of prefractal counts it is necessary to use V� coverage Dependence of fractal dimension of prefra ctal count is investigational on the linear size of grate e � coverage with the purpose of determining the amount of factors, influencing on the system. They got results are actual at the decision of intricate problems.

Species spatial distribution analysis using nearest neighbor methods: aggregation and self‐similarity

Abstract:Spatial aggregation and self‐similarity are two important properties in species spatial distribution analysis and modeling. The aggregation parameter k in the negative binomial distribution model and fractal dimension are two widely used measures of spatial aggregation and self‐similarity, respectively. In this paper, we attempt to describe spatial aggregation and self‐similarity using nearest neighbor methods. Specifically, nearest neighbor methods are used to calculate k and box‐counting fractal dimension of species spatial distribution. First, five scaling patterns of k are identified for tree species in a tropical rainforest on Barro Colorado Island (BCI), Panama. Based on the scaling patterns and the means of the nth nearest neighbor distance (NND), the mean NND of higher ranks can be accurately predicted. Second, we describe how to use the theoretical probability distribution model of the nth NND for a homogeneous Poisson process on regular fractals to estimate the fractal dimension. The results indicate that the fractal dimensions estimated using the nearest neighbor method are consistent with those estimated using the scale–area method for 85 tree species on BCI (abundance ≥ 100 individuals and ≤ 5000 individuals). For other tree species, the breakdown of self‐similarity in estimates of fractal dimension causes these two methods to be inconsistent. The applicability of the nearest neighbor method is also discussed.

Fractals based on regular convex polytopes

In the present paper the author presented new deterministic fractals based on regular convex polytopes, discussed their construction and presented special cases of some non-regular polytope-based fractals. Firstly, the fractals based on the regular convex poly- chora were investigated. Their construction was analyzed using different approaches and their Hausdorff dimension was determined. Then, the high-dimensional regular convex polytopes-based fractals were investigated. It was shown, that there is a possibility to de- termine and generalize the Hausdorff dimension of these fractals to R d . Moreover, two classes of fractals based on d-dimensional hyperpyramids and hyperbipyramids were dis- cussed.

DYNAMICS ON MULTILAYERED HYPERBRANCHED FRACTALS THROUGH CONTINUOUS TIME RANDOM WALKS

We introduce a new method to generate three-dimensional structures, with mixed topologies. We focus on Multilayered Regular Hyperbranched Fractals (MRHF), three-dimensional networks constructed as a set of identical generalized Vicsek fractals, known as Regular Hyperbranched Fractals (RHF), layered on top of each other. Every node of any layer is directly connected only to copies of itself from nearest-neighbor layers. We found out that also for MRHF the eigenvalue spectrum of the connectivity matrix is determined through a semi-analytical method, which gives the opportunity to analyze very large structures. This fact allows us to study in detail the crossover effects of two basic topologies: linear, corresponding to the way we connect the layers and fractal due to the layers' topology. From the wealth of applications which depends on the eigenvalue spectrum we choose the return-to-the-origin probability. The results show the expected short-time and long-time behaviors. In the intermediate time domain we obtained two different power-law exponents: the first one is given by the combination linear-RHF, while the second one is peculiar either of a single RHF or of a single linear chain.

Monomer–dimer model on a scale-free small-world network

The explicit determination of the number of monomer–dimer arrangements on a network is a theoretical challenge, and exact solutions to monomer–dimer problem are available only for few limiting graphs with a single monomer on the boundary, e.g., rectangular lattice and quartic lattice; however, analytical research (even numerical result) for monomer–dimer problem on scale-free small-world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale-free and small-world structures. In this paper, we address the monomer–dimer problem defined on a scale-free small-world network and obtain the exact formula for the number of all possible monomer–dimer arrangements on the network, based on which we also determine the asymptotic growth constant of the number of monomer–dimer arrangements in the network. We show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two-dimensional lattice and Sierpinski fractal having the same average degree as the studied network, which indicates from another aspect that scale-free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power-law behavior.

Formation of nanoscale tungsten oxide structures and colouration characteristics

In this work, pH dependent evolution of tungsten oxide (WO3) nanostructures is being reported along with physical characteristics. The synthesis was carried out via an inexpensive solvothermal cum chemical reduction route, with sodium tungstate (Na2WO4) and cetyl trimethyl ammonium bromide (C19H42NBr) as main reactants. The X-ray diffraction, together with transmission electron microscopic studies have revealed formation of regular polyhedral nanocrystalline structures and fractals as one goes from higher pH (= 5·5) to lower pH (= 2) values. The average crystallite size, as calculated through Williamson–Hall plots, was varied within 2·8–6·8 nm for different pH samples. Fourier transform infrared spectroscopy reveals in-plane bending vibration δ (W–OH), observable at ∼1630 cm − 1 and strong stretching ν (W–O–W) located at ∼814 cm − 1. Raman spectroscopy has divulged WO3 Raman active optical phonon modes positioned at ∼717 and 805 cm − 1. The thermochromic and photochromic properties of the nanoscale WO3 sample prepared at pH = 5·5, are also highlighted.