Fractals

In this paper, we employ the detrended cross-correlation analysis (DCCA) to investigate the cross-correlations between different stock markets. We report the results of cross-correlated behaviors in US, Chinese and European stock markets in period 1997–2012 by using DCCA method. The DCCA shows the cross-correlated behaviors of intra-regional and inter-regional stock markets in the short and long term which display the similarities and differences of cross-correlated behaviors simply and roughly and the persistence of cross-correlated behaviors of fluctuations. Then, because of the limitation and inapplicability of DCCA method, we propose multiscale detrended cross-correlation analysis (MSDCCA) method to avoid "a priori" selecting the ranges of scales over which two coefficients of the classical DCCA method are identified, and employ MSDCCA to reanalyze these cross-correlations to exhibit some important details such as the existence and position of minimum, maximum and bimodal distribution which are lost if the scale structure is described by two coefficients only and essential differences and similarities in the scale structures of cross-correlation of intra-regional and inter-regional markets. More statistical characteristics of cross-correlation obtained by MSDCCA method help us to understand how two different stock markets influence each other and to analyze the influence from thus two inter-regional markets on the cross-correlation in detail, thus we get a richer and more detailed knowledge of the complex evolutions of dynamics of the cross-correlations between stock markets. The application of MSDCCA is important to promote our understanding of the internal mechanisms and structures of financial markets and helps to forecast the stock indices based on our current results demonstrated the cross-correlations between stock indices. We also discuss the MSDCCA methods of secant rolling window with different sizes and, lastly, provide some relevant implications and issue.

In this study, we analyze some critical points of the application of the box-counting method to the evaluation of the fractal dimensions of the natural and built landscapes. A brief theoretical discussion of the eventual drawbacks of the method is supported by experimental results of two box-counting programs applied to classical fractals. The optimized version of the algorithm, based on the results of computations for the classical fractal images, is proposed and employed for the evaluation of the complexity level of the chosen historical buildings and surrounding environment in the well-known case of Amasya city. The hypothesis of the relationship between the visual complexity of built and natural settings is analyzed for the Amasya case and for two historical Brazilian cities.

In the paper, we generalized the idea of circle inversion to star-shaped sets and used the generalized inversion to replace the circle inversion transformation in the algorithm for the generation of the circle inversion fractals. In this way, we obtained the star-shaped set inversion fractals. The examples that we have presented show that we were able to obtain very diverse fractal patterns by using the proposed extension and that these patterns are different from those obtained with the circle inversion method. Moreover, because circles are star-shaped sets, the proposed generalization allows us to deform the circle inversion fractals in a very easy and intuitive way.

Since the existence of market memory could implicate the rejection of the efficient market hypothesis, the aim of this paper is to find any evidence that selected emergent capital markets (eight European and BRIC markets, namely Hungary, Romania, Estonia, Czech Republic, Brazil, Russia, India and China) evince long-range dependence or the random walk hypothesis. In this paper, the Hurst exponent as calculated by R/S fractal analysis and Detrended Fluctuation Analysis is our measure of long-range dependence in the series. The results reinforce our previous findings and suggest that if stock returns present long-range dependence, the random walk hypothesis is not valid anymore and neither is the market efficiency hypothesis.

In present times the security of image data is a major issue. So, we have proposed a novel technique for security of color image data by public-key cryptosystem or asymmetric cryptosystem. In this technique, we have developed security of color image data using RSA (Rivest-Shamir-Adleman) cryptosystem with two-dimensional discrete wavelet transform (2D-DWT). Earlier proposed schemes for security of color images designed on the basis of keys, but this approach provides security of color images with the help of keys and correct arrangement of RSA parameters. If the attacker knows about exact keys, but has no information of exact arrangement of RSA parameters, then the original information cannot be recovered from the encrypted data. Computer simulation based on standard example is critically examining the behavior of the proposed technique. Security analysis and a detailed comparison between earlier developed schemes for security of color images and proposed technique are also mentioned for the robustness of the cryptosystem.

Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal, which is defined as the closure of the orbit Φ^ω(*_Φ) = ⋃_n∈ωΦⁿ(*_Φ) of the set *_Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals and for a contracting compact-valued function Φ : X ⊸ X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

P. Jarratt has developed a family of fourth-order optimal methods. He suggested two members of the family. The dynamics of one of those was discussed previously. Here we show that the family can be written using a weight function and analyze all members of the family to find the best performer.

IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝ^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

The tortuosity is a very important parameter for description of fluid flow in porous media, and it has been shown that porous media in nature have the fractal characteristics. The Sierpinski carpet is an exactly self-similar fractal model, which is often used to simulate fractal porous media. In this work, the tortuosity of different generations of Sierpinski carpet is calculated and analyzed by the finite volume method. A simple linear relation between the generations and tortuosity in pore fractal model of porous media is obtained. The results are compared with the available conclusions and show a more realistic tortuosity predication for fluid flow in the two-dimensional pore fractal models of porous media.

This paper examines the pure-strategy subgame-perfect equilibrium payoffs in discounted supergames with perfect monitoring. It is shown that the equilibrium payoffs can be identified as sub-self-affine sets or graph-directed iterated function systems. We propose a method to estimate the Hausdorff dimension of the equilibrium payoffs and relate it to the equilibrium paths and their graph presentation.