Fractals

Telecommunication service providers, also called operators or carriers, must maintain 99.999% of their telecom network availability. To avoid or mitigate the effects of an outage in the network, carriers perform different activities to restore service. Spare parts management plays an important role in meeting the required service. Unfortunately, the closed-loop supply chain that supports the availability of spare parts experiences variability in its processes as a result of endogenous and exogenous variables that make matching the recovery process with the demand process difficult. Understanding the effect of the variables on the variability outcome and the emergence of the variability can help to guide the development of future mathematical models for the management of spare parts. In this article, a complex system approach to analyze the variability behavior is presented.

The Sudbury neutrino observatory (SNO) detects ⁸B solar neutrino fluxes from both the D₂O and Salt detector. In the present analysis we have taken into consideration the flux data from 2nd November, 1999 to 27th May, 2001 from the D₂O detector and that from 26th July, 2001 to 28th August, 2003 from the Salt detector. We have applied Delay Vector Variance analysis, 0-1 test, correlation dimension analysis, largest Lyapunov exponent method, recurrence plot and recurrence quantification analysis to explore the complexity and chaosity in these two time series. Present study reveals deterministic chaotic behaviour of these two signals which in turn suggests that long-term forecasting is not possible for these two signals but short-term forecasting can be made provided the model for the process dynamics is known to us.

Radon anomaly acts as a precursor to seismic events. Hence the fluctuation pattern of time series of radon bears a lot of significance. Multifractal detrended fluctuation analysis (MFDFA) has been applied successfully in past to various geo-electrical signals. The time series of earthquake related radon signal of duration three years (November 2005–November 2008) obtained from our experiment performed at Kolkata, India was analyzed in the framework of MFDFA. The study revealed that the fluctuation of radon concentration is multifractal in nature.

The vulnerabilities of the selective encryption scheme for fractal image coding proposed by Lian et al.¹ are identified. By comparing multiple cipher-images of the same plain-image encrypted with different keys, the positions of unencrypted parameters in each encoded block are located. This allows the adversary to recover the encrypted depth of the quadtree by observing the length of each matched domain block. With this depth information and the unencrypted parameters, the adversary is able to reconstruct an intelligent image. Experimental results show that some standard test images can be successfully decoded and recognized by replacing the encrypted contrast scaling factor and brightness offset with specific values. Some remedial approaches are suggested to enhance the security of the scheme.

Functions that are invariant with respect to the tetrahedral and cubic symmetries are determined. These invariant mappings are applied to serve as the density functions for automatic generation of the colorful images with such symmetries in three dimensional space from a dynamical system's point of view.

The Sierpiński relatives form a class of fractals that all have the same fractal dimension, but different topologies. This class includes the well-known Sierpiński gasket. Some relatives are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. This paper presents examples of relatives for which binary Cantor sets are relevant for the connectivity. These Cantor sets are variations of the usual middle thirds Cantor set, and their binary descriptions greatly aid in the determination of the connectivity of the corresponding relatives.

Computing the Hausdorff measure of C × C, where C is the classical ternary Cantor set, is a long standing difficult problem. It is well-known that for a self-similar set, calculating the Hausdorff measure is equivalent to determining its optimal sets.

This paper studies optimal sets of C × C: their diameters, measures, symmetries and the shapes. For this purpose, we introduce several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameter of the optimal set B is between 1.2993 and 1.3082. Two symmetry properties of B are proved. Finally, we show that the shape of B is very close to a disk. We conjecture that an optimal set might be a disk.

The scattering of particles in fractal superlattices has been analyzed by means of the transfer matrix method. The fractal superlattice consists of alternating layers of semiconductor materials following the rule of a Cantor set. This problem can be represented by a model of quantum particles scattering in piecewise constant potential wells. Fractal properties of the reflection coefficient versus the particle energy curves are examined comparatively to the curves when using the corresponding periodic potentials. The degree of self-similarity of the output variables has been quantified by means of the correlation function.

We report on a fractal pattern formed in Rayleight-Taylor instability. The fractal dimension is 1.88 ± 0.06, which is constant in time t > t₁ = 70 s. The pattern area decreases according to exp(-t/τ), where t and τ are time and time constant 69.8 s, respectively. τ agrees with t₁. This result leads that it is important for the fractal formation that sufficient annihilation of the heavier solution at the surface.

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has that Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.