Fractals

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator . We show that the deviation , adequately normalized, is asymptotically normal and specify the asymptotic variance.

A specific system of difference equations derived from chemical reactions discrete chaotic dynamics (CRDCD) was used for generating visual images. Examples of symmetrical images generated by the proposed method were used to illustrate the impact of different color palettes on the resulting images. The image quality could be improved by applying an interpolation scheme.

The Julia sets theory of generalized Newton's method is analyzed and the Julia sets of generalized Newton's method are constructed using the iteration method. From the research we find that: (1) the basins of attraction of the Julia sets of generalized Newton's method depend on the roots of the equation and their orders and also the existence of the extraneous fixed points; (2) the Steffensen method is an exception to the law given in (1); and (3) if the order of the root is decimal, then the different choice of the range of the principal value of the phase angle will cause a different evolvement of the Julia sets.

We offer a new approach to the fabrication of remarkable autostereoscopic double-layer computer generated images or autostereoscopic staircase fractal structures (ASFSs) based on staircase fractals and moiré methods. Besides, we have found that the unwanted moiré fringes or moiré noises in the ASFSs will markedly decrease if high frequency fractal noises are employed. These ASFSs were observed to be similar in some sense to Denishuk or reflection holograms. Analyzing the geometry and parallax of the circle of skew lines (CSL-parallax) at the presentation of stereo depth information in ASFSs, we hope to provide design guidelines for autostereoscopic digital printing and so on and so forth. A far more important example is a computer synthesis of three-dimensional graphic symbols, a standard laser printer and this CSL-parallax being employed.

In the context of current Narratology, a novel can be regarded as an information generator system. This information can be symbolically and numerically codified and, subsequently, studied by nonlinear characteristic geometrical methods. Using this approach, geometric structures underlying the narrative discourse become evident. In this work, using a particular novel as our experimental data source, a formal expression for a discrete dynamical system is deduced, which generates a representative orbit of the narrative discourse evolution. The fractal dimension of this orbit is calculated from the correlation dimension and its deterministic character is unambiguously proved by solving the associated embedding problem. Finally, we describe the general features that a novel must satisfy in order to apply the proposed procedure.

With the rapid increase in the use of computers and the Internet, the demand for higher transmission and better storage is increasing as well. This paper describes the different techniques for data (image-video) compression in general and, in particular, the new compression technique called fractal image compression. Fractal image compression is based on self-similarity, where one part of an image is similar to the other part of the same image. Low bit rate color image sequence coding is very important for video transmission and storage applications. The most significant aspect of this work is the development of color images using fractal-based color image compression, since little work has been done previously in this area. The results obtained show that the fractal-based compression works for the color images works as well as for the gray-scale images. Nevertheless, the encoding of the color images takes more time than the gray-scale images.

Color images are usually compressed in a luminance-chrominance coordinate space, with the compression performed independently for each coordinate by applying the monochrome image processing techniques. For image sequence compression, the design of an accurate and efficient algorithm for computing motion to exploit the temporal redundancy has been one of the most active research areas in computer vision and image compression. Pixel-based motion estimation algorithms address pixel correspondence directly by identifying a set of local features and computing a match between these features across the frames. These direct techniques share the common pitfall of high computation complexity resulting from the dense vector fields produced. For block matching motion estimation algorithms, the quad-tree data structure is frequently used in image coding to recursively decompose an image plane into four non-overlapping rectangular blocks.

Fractional Euler-Lagrange equations were investigated in the presence of the elements of Berezin algebra. The super fractional Hessian was defined and the fractional Hamiltonian of the super symmetric classical model was constructed.

An analytical expression for the fractal dimension for tortuous streamtubes in porous media is derived and found to be a function of porosity and microstructures of porous media. There is no empirical constant in the proposed fractal dimension. The present model for the fractal dimension is verified by a comparison with the available analogy model.

This paper deals with the target excitation dependence of the multifractal specific heat of pions produced in nucleus-nucleus collisions at 60 and 200 AGeV. Fluctuation pattern of pions is studied in two dimensions considering anisotropy of phase space. The parameter characterizing the anisotropy of phase space, Hurst exponent, is extracted by fitting one-dimensional factorial moment saturation curves. Values of intermittency exponent α_q and generalized dimension D_q are determined. The fluctuation pattern reveals multifractal behavior at all target excitations at both energies. The multifractal specific heat c is found to depend on target excitation and the dependence is found to be more pronounced at lower energy. The results are discussed in detail.

Multiscale sample entropy (MSE) of human electroencephalogram (EEG) data from patients under different pathological conditions of Alzheimer's disease (AD) was evaluated to measure the complexity of the signal. Quantifying the complexity level with respect to various temporal scales, MSE analysis provides a dynamical description of AD development. When compared to EEG data from normal subjects, EEG data from subjects with mild cognitive impairment (MCI) showed nearly the same complexity profile, but a scale discrepancy which may occur from a spectral abnormality. EEG data from severe AD patients showed a loss of complexity over the wide range of time scales, indicating a destruction of nonlinear structures in brain dynamics. We compare the MSE method and spectral analysis to propose that nonlinear dynamical approach combining a multiscale method is crucial for revealing AD mechanisms.

We revisit the relation between the von Koch curve and the Thue-Morse sequence given in a recent paper of Ma and Goldener by relating their study to papers written by Coquet and Dekking at the beginning of the 1980s. We also emphasize that more general links between fractal objects and automatic sequences can be found in the literature.