Fractals

We consider a construction of recurrent fractal interpolation surfaces (RFISs) with function vertical scaling factors and estimation of their box-counting dimension. A RFIS is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given dataset on rectangular grids, we construct general RIFSs with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above-constructed RIFSs. Finally, we estimate lower and upper bounds for the box-counting dimension of the constructed RFISs.

A fractal tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the boundary dimension of $s_n$- and $u_n$-tilings. We first derive an explicit recursion formula for the boundary edges of $s_n$- and $u_n$-tilings. Then we present an analytical expression for their fractal boundary dimensions using matrix methods. Results indicate that, as $n$ increases, the boundaries of $s_n$- and $u_n$-tilings will degenerate into general Euclidean curves. The method proposed in this paper can be extended to compute the boundary dimensions of other kinds of fractal tilings.

Hurst exponent is an important measure of nonlinearity of dynamical time series. In this paper, using rescaled-range ($R$/$S$) analysis, multi-fractal detrended fluctuation analysis (MF-DFA) methods, the multiscale Hurst exponent (MHE) and the multiscale generalized Hurst exponent (MGHE) of coarse-grained silicon content ([Si]) time series in blast furnace (BF) hot metal were calculated. First, we collected these [Si] time series from No. 1 BF of Nanchang Iron and Steel Co. and No. 10 BF of Xinyu Iron and Steel Co. in Jiangxi Province, China. Then, we analyzed and compared the estimated Hurst exponents and the generalized Hurst exponent of these observed time series with some simulated time series. Our results show that the observed time series from these BFs have negative correlation with the Hurst exponent less than 0.5, the generalized Hurst exponent $H(q)$ is a nonlinear function of $q$, and such negative correlation and local various structure persist in their moving averages of the observed time series up to lag 5 or 10.

In this cryptosystem, we have presented a novel technique for security of video data by using matrix affine cipher (MAC) combined with two-dimensional discrete wavelet transform (2D-DWT). Existing schemes for security of video data provides only one layer of security, but the presented technique provides two layers of security for video data. In this cryptosystem, keys and arrangement of MAC parameters are imperative for decryption process. In this cryptosystem, if the attacker knows about all the exact keys, but has no information about the specific arrangement of MAC parameters, then the information of original video cannot be recovered from the encrypted video. Experimental results on standard examples support to the robustness and appropriateness of the presented cryptosystem of video encryption and decryption. The statistical analysis of the experimental results based on standard examples critically examine the behavior of the proposed technique. Comparison between existing schemes for security of video with the presented cryptosystem is also provided for the robustness of the proposed cryptosystem.

Let $S$ be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map

In order to research mechanics of rock fracture instant, it is one of methods that rock fracture mechanics are researched by rock fracture surfaces’ morphology. Some researched results which come from international and domestic researches in decades are described and summarized from mathematics in this paper. For example, fractal dimension method, “Small Island Method”, fractal interpolation method, Multi-fractal method, the accumulation power spectral density method. In addition, advantages and insufficiencies of every method are reviewed and commented. In the end, the future researched expectations are put forward from three aspects of rock fracture surfaces’ morphology.

Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

There are several cases when tortuous continuous lines have to be discretized in order to measure length and other related quantities. In this methodological paper, we are going to demonstrate the importance of the proper sequencing (ranking) of the discrete points and call attention to some typical errors caused by improper sequencing. For demonstration, the fractal dimensions of an animal track with well-known original sequencing and for some mixed sequencing are estimated and the extent of errors is shown.

For Gaussian Spin-Glasses in low dimensions, we introduce a simple Strong Disorder renormalization at zero temperature in order to construct ground states for Periodic and Anti-Periodic boundary conditions. The numerical study in dimensions $d=2$ (up to sizes $2048^2$) and $d=3$ (up to sizes $128^3$) yields that Domain Walls are fractal of dimensions $d_s(d=2)\simeq 1.27$ and $d_s(d=3)\simeq 2.55$, respectively.

In this paper, we introduce a new construction of the fractal interpolation surface (FIS) using an even more general iterated function systems (IFS) which can generate self-affine and non self-affine fractal surfaces. Here we present the general types of fractal surfaces that are based on nonlinear IFSs.

Based on cross-correlation algorithm, we analyze the correlation property of warehouse-out quantity of different warehouses, respectively, and different products of each warehouse. Our study identifies that significant cross-correlation relationship for warehouse-out quantity exists among different warehouses and different products of a warehouse. Further, we take multifractal detrended cross-correlation analysis for warehouse-out quantity among different warehouses and different products of a warehouse. The results show that for the warehouse-out behaviors of total amount, different warehouses and different products of a warehouse significantly follow multifractal property. Specifically for each warehouse, the coupling relationships of rebar and wire rod reveal long-term memory characteristics, no matter for large fluctuation or small one. The cross-correlation effect on long-range memory property among warehouses probably has less to do with product types,and the long-term memory of YZ warehouse is greater than others especially in total amount and wire rod product. Finally, we shuffle and surrogate data to explore the source of multifractal cross-correlation property in logistics system. Taking the total amount of warehouse-out quantity as example, we confirm that the fat-tail distribution of warehouse-out quantity sequences is the main factor for multifractal cross-correlation. Through comparing the performance of the multifractal detrended cross-correlation analysis (MF-DCCA), centered multifractal detrending moving average cross-correlation analysis (MF-X-DMA) algorithms, the forward and backward MF-X-DMA algorithms, we find that the forward and backward MF-X-DMA algorithms exhibit a better performance than the other ones.

The efficiency and accuracy of estimating the Hurst exponent have been two inevitable considerations. Recently, an efficient implementation of the maximum likelihood estimator (MLE) (simply called the fast MLE) for the Hurst exponent was proposed based on a combination of the Levinson algorithm and Cholesky decomposition, and furthermore the fast MLE has also considered all four possible cases, including known mean, unknown mean, known variance, and unknown variance. In this paper, four cases of an approximate MLE (AMLE) were obtained based on two approximations of the logarithmic determinant and the inverse of a covariance matrix. The computational cost of the AMLE is much lower than that of the MLE, but a little higher than that of the fast MLE. To raise the computational efficiency of the proposed AMLE, a required power spectral density (PSD) was indirectly calculated by interpolating two suitable PSDs chosen from a set of established PSDs. Experimental results show that the AMLE through interpolation (simply called the interpolating AMLE) can speed up computation. The computational speed of the interpolating AMLE is on average over 24 times quicker than that of the fast MLE while remaining the accuracy very close to that of the MLE or the fast MLE.

Given a continuous rectifiable function $f$, we present a simple Iterated Function System (IFS) with probabilities whose invariant measure is the normalized arclength measure on the graph of $f$.

In this paper, we introduce the $p$-circle inversion which generalizes the classical inversion with respect to a circle ($p=2$) and the taxicab inversion $(p=1)$. We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the $p$-circle inversion.

We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.