Fractals

By using the frame representation method proposed by B. L. Hao et al. in 2000^1,2 and a method provided by P. Tino,^3,4 we will devise the generalized graphical representation for the strings defined on an alphabet Σ = {0, 1, …, n - 1} with cardinal n but all these graphs are still shown in a unit square, and generate its corresponding struture tables automatically. In particular, the words defined on Σ are mapped into n-adic numbers so that the index set Ind(B) can be computed. Thus, the fractals of the forbidden words can be naturally constructed. It has already been known that for biological sequences, the circular codes can be applied to some special bacteria and their forbidden words are the 5-necklaces. In this paper, we notice that the fractals of the forbidden words are particularly useful to represent the circular codes for some biological sequences.

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F^α-integral, where α is the dimension of F. A derivative along the fractal curve called F^α-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect.

Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F^α-integral and F^α-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F^α-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

In the present paper, the influence of the signal class (fBm/fGn) and the data length of time series on choosing the robust fractal analysis method have been studied. More than 1000 fBm/fGn generated time series in short, intermediate and long ranges have been analyzed using common fractal analysis methods. The chosen techniques were power spectral density, detrended fluctuation analysis, rescaled range analysis, box counting, average wavelet coefficients, and the variation method. Numerous graphs indicating the suitability of each method in terms of biases in calculating the fundamental fractal feature of time series, Hurst coefficient, were employed. The results strongly emphasized the crucial influence of the signal class as well as the data length when choosing the appropriate fractal analysis method. Furthermore, as a step forward, a study on the number of data points present in a classified class/length was performed. The effect of the number of data points could not be neglected either. Based on the results, a strategy flowchart for fractal analysis of time series has been proposed. Finally, as an empirical example, the monthly, weekly and daily scaled flow time series of Ghar-e-Aghaj River have been analyzed within the framework of the strategy flowchart.

This paper presents a comparative study of two families of curves in ℝⁿ. The first ones comprise self-similar bounded fractals obtained by contractive processes, and have a non-integer Hausdorff dimension. The second ones are unbounded, locally rectifiable, locally smooth, obtained by expansive processes, and characterized by a fractional dimension defined by M. Mendès France. We present a way to relate the two types of curves and their respective non-integer dimensions. Thus, to one fractal bounded curve we associate, at first, a finite range of Mendès France dimensions, identifying the minimal and the maximal ones. Later, we show that this discrete spectrum can be made continuous, allowing it to be compared with some other multifractal spectra encountered in the literature. We discuss the corresponding physical interpretations.

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

The Newton–Raphson method for solving nonlinear equations f(x) = 0 in ℝⁿ is discussed within the context of ordinary differential equations. This framework makes it possible to reformulate the scheme by means of an adaptive step size control procedure that aims at reducing the chaotic behavior of the original method without losing the quadratic convergence close to the roots. The performance of the modified scheme is illustrated with a few low-dimensional examples.

We propose an explicit way to generate a large class of Operator scaling Gaussian random fields (OSGRF). Such fields are anisotropic generalizations of self-similar fields. More specifically, we are able to construct any Gaussian field belonging to this class with given Hurst index and exponent. Our construction provides — for simulations of texture as well as for detection of anisotropies in an image — a large class of models with controlled anisotropic geometries and structures.

The complexity, entropy and other non-linear measures of the electroencephalogram (EEG), such as Higuchi fractal dimension (FD), have been recently proposed as the measures of anesthesia depth and sedation. We hypothesized that during unconciousness in rats induced by the general anesthetics with opposite mechanism of action, behaviorally and poligraphically controlled as appropriately achieved stable anesthesia, we can detect distinct inter-structure brain dynamic using mean FDs. We used the surrogate data test for nonlinearity in order to establish the existence of nonlinear dynamics, and to justify the use of FD as a nonlinear measure in the time series analysis. The surrogate data of predefined probability distribution and autocorrelation properties have been generated using the algorithm of statically transformed autoregressive process (STAP). FD then is applied to quantify EEG signal complexity at the cortical, hippocampal and pontine level during stable general anesthesia (ketamine/xylazine or nembutal anesthesia).

Our study showed for the first time that global neuronal inhibition caused by different mechanisms of anesthetic action induced distinct brain inter-structure complexity gradient in Sprague Dawley rats. EEG signal complexities were higher at cortical and hippocampal level in ketamine/xylazine vs. nembutal anesthesia, with the dominance of hippocampal complexity. In nembutal anesthesia the complexity dominance moved to pontine level, and ponto-hippocampo-cortical decreasing complexity gradient was established. This study has proved the Higuchi fractal dimension as a valuable tool for measuring the anesthesia induced inter-structure EEG complexity.

This paper presents the method of grading of cervical cancer images according to cell formation in the tissue. The variation of intensity and texture complexity of cancer cell images are calculated by fractal dimension methods. Box Counting Method (D_B) and HarFA Programme software are used to find out the dimension of the cells. Contact model and Epidemic Model are the two different models used here. The contact model shows the contact of the cells and the formation of the cancer in the part of the organ. The Epidemic model shows the growth of cancer. Lacunarity is formed between the cells. If the Lacunarity increases the dimension also increases. On seeing the growth of the cancer cell the pathologist can determine the grade of the cancer which in turn will help him to diagnose the cervical cancer and recommend appropriate treatment.