Fractals

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by . In the literature the best existing estimate is

In recent work, various fractal image coding methods are reported, which adopt the self-similarity of images to compress the size of images. However, till now, no solutions for the security of fractal encoded images have been provided. In this paper, a secure fractal image coding scheme is proposed and evaluated, which encrypts some of the fractal parameters during fractal encoding, and thus, produces the encrypted and encoded image. The encrypted image can only be recovered by the correct key. To maintain security and efficiency, only the suitable parameters are selected and encrypted through investigating the properties of various fractal parameters, including parameter space, parameter distribution and parameter sensitivity. The encryption process does not change the file format, keeps secure in perception, and costs little time or computational resources. These properties make it suitable for secure image encoding or transmission.

The sensitivity analysis for a class of hidden variable fractal interpolation functions (HVFIFs) and their moments is made in the work. Based on a vector valued iterated function system (IFS) determined, we introduce a perturbed IFS and investigate the relations between the two HVFIFs generated by the IFS determined and its perturbed IFS, respectively. An explicit expression for the difference between the two HVFIFs is presented, from which, we show that the HVFIFs are not sensitive to a small perturbation in IFSs. Furthermore, we compute the moment integrals of the HVFIFs and discuss the error of moments of the two HVFIFs. An upper estimate for the error is obtained.

In this paper we study a model of skin cancer (MM) in vitro, using geometry of fractals as the method of analysis. The fractal dimensions of moles (skin cancer cells) growth pattern have been measured by using the methods of Box-counting method (D_B) and Sausage method (D_S). The cell growth of this cancer can be modeled by Hidden Markov model (HMM) and percolation model which are depending upon the time complexity. From these models we can find the shape of the irregularity border by using the probability distribution of the cells. The variation in the irregular border of the skin cancer has been found out using ANOVA test and cell's compactness. The fractal approach led to very promising results which improved the determination and examination of the stage of skin cancer.

Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The two-dimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately D_f = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures.

By analyzing the phase diagram of Martin process on the cosine function, it is shown that with the change of system parameters the system will eventually converge to a chaotic attractor. The process is repeated and stable focus, period doubling bifurcation occurs during this process. Further computation gives the maximum Lyapunov exponent of the system and meanwhile, the bifurcation diagram is drawn. Thus it is proved from theory that the system exhibits strong chaotic properties.

We extend Kim's complex exponential function, come up with theory about the Julia set of Newton's transformation for general exponential equation, analyze the behavior of the roots of some complex exponential equation, and prove the symmetry, boundedness and embedding topology distribution structure of basins of attraction of the Julia set in theory.

Inspired by the study of 3D fractals based on quadratic polynomial maps,¹ this paper presents a novel approach for generating 3D Julia sets by utilizing a family of switching polynomial maps in order to further enrich the form of 3D fractals. Rotational symmetries in the structures of resulting Julia sets with different switching parameters are theoretically analyzed and proved. Experimental results obtained from various parameters and coefficients in the switching maps are provided and discussed in detail. It is hoped that the investigations conducted in this paper can bring on new perspectives for the generalization of 3D fractals.

In this paper, a class of generalized secant methods was investigated. The convergence of these methods was discussed, and their striking fractal patterns are generated to represent the chaotic behavior of these methods in the complex plane.

Stoev and Taqqu¹ presented a fast simulation method for the linear fractional stable motion (lfsm) process, using the fast Fourier transform. In this paper, we extend this method for fast simulation of the log-fractional stable motion (log-fsm) process. These two heavy-tailed processes, i.e. lfsm and log-fsm, can be used for modeling the modern telecommunication networks. In addition, a necessary and sufficient condition is given for the efficiency of this method. Finally, an upper bound for approximation error is presented.

The behavior of sandpile far from self-organized critical states (SOCS) is investigated in this paper. The results indicate that noise plays an important role in the critical phenomenon. The critical phenomena are detected in sandpile far from SOCS when external or internal noise is applied. In contrast to self-organized criticality, the exponent of avalanche distribution in sandpile far from SOCS increases with the distance of the sandpile from SOCS. At the same time, the exponent of avalanche distribution in a sandpile far from SOCS is consistent with some real systems such as earthquake.

We study the critical properties of the Bak–Sneppen coevolution model on scale-free networks by Monte Carlo method. We report the distribution of the avalanche size and fractal activity through the branching process. We observe that the critical fitness f_c(N) depends on the number of the node such as f_c(N) ~ 1/log(N) for both the scale-free network and the directed scale-free network. Near the critical fitness many physical quantities show power-law behaviors. The probability distribution P(s) of the avalanche size at the critical fitness shows a power-law like P(s) ~ s^-τ with τ = 1.53(5) regardless of the scale-free network and the directed scale free network. The probability distribution P_f(t) of the first return time also shows a power-law such as P_f(t) ~ t^-τ_f. The critical exponents τ_f are equivalent for both the scale-free network and the directed scale-free network. We obtain the critical exponents as τ_f = 1.776(5) at the scalinge regime. The directionality of the network does not change the universality on the network.