Fractals

Recent research indicates the presence of increased vascular density and irregularity on oral mucosal vascular networks in extracellular matrix (ECM)-related illness or conditions. Here, we estimated the frequency of occurrence of nodes of various degrees (K3, K4 and K5, where Kn designates a node with n connections) in patients with proven or suspected ECM-related conditions and in controls. Subjects with ECM-related conditions exhibited lower K3 and higher K4 frequency than controls (p < 0.0001) in their vascular networks. Inverse statistical correlations between the local fractal dimension and L-Z values and percentage of K3 (Pearson's r values range: -0.91 to -0.81; p values range: 0.0013 to < 0.0001), together with a positive relationship with K4 were observed (r values range: 0.81 to 0.86; p values range: 0.0015 to 0.0003). A positive correlation coefficient between D(1–46) and K5 frequency was also found (r = 0.6334, p = 0.027). K3 ≥ 52% or K4 <28% discriminated ECM patients from controls with 100% sensitivity (true positive cases to true positive + false negative ratio) and specificity (true negative cases to true negative + false positive ratio). These findings suggest that node degree distribution in oral vascular networks could be a helpful new marker of pathological conditions associated with proven or suspected ECM abnormalities.

Two methods for representing discrete image data on rectangular lattices using fractal surfaces are proposed. They offer the advantage of a more general fractal modeling compared to previous one-dimensional fractal interpolation techniques resulting in higher compression ratios. Theory, implementation and analytical study of the proposed methods are also presented.

In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.

We study crack patterns in laponite films of different thickness. The patterns show a self-similarity under coarse-graining, the fractal dimension determined by box-counting has a value around 1.66, independent of film thickness. The cracks on layers of different thickness show a remarkable scaling bahavior. We have measured the cumulative area covered by the cracks versus minimum crack-width resolved. Curves representing crack area for different thickness collapse onto a single curve, when the crack widths are scaled by the film thickness.

The time series data of the monthly rainfall records (for the time period 1871–2002) in All India and different regions of India are analyzed. It is found that the distributions of the rainfall intensity exhibit perfect power law behavior. The scaling analysis revealed two distinct scaling regions in the rainfall time series.

An agent-based model is applied for studying the group-forming behavior numerically. A fixed number of homogeneous agents are distributed on a two-dimensional square lattice system with L × L unit cells. The dynamics of the agents are described in terms of a potential and a friction factor, just like that of a weight on a slope. The potential is defined in an attempt to model the concept of the "biosocial attraction." Depending on the density of the population and the friction factor, four regimes of group-size distributions are identified. They are the fixed, the exponential, the power-law and the black-hole regimes. Especially, beyond the well-known exponential and power-law distributions, a black-hole regime is identified for the lowest values of the friction factor. In the black-hole regime, all the agents in the system are attracted to the only group remaining in the system. An example membership values for each of the four classes are suggested in terms of the two key parameters of the system, the population density and the friction factor. And a phase transition is observed between the exponential and the black-hole regimes. Also the results are discussed in relation to the city-size distributions in a country.

In this paper, scaling attractors of fractional differential systems are studied with the help of synchronization methods. The synchronization error systems between the drive and response systems are analyzed by using the theory of Laplace transform. An efficient computational scheme is developed. Both numerical simulations and computer graphics show that the developed techniques work well.

Periodicity is recurrent in nature and society. The problem of the periodicity is faced here with the support of fractal methodology. Some analytical tools for the understanding of the one-dimensional projections of periodic phenomena are proposed. The reconstruction of an unknown periodic sampled variable is approached, assuming a deterministic self-affine nature in its small oscillations. Fractal trigonometric polynomials are defined by means of suitable iterated function systems. These objects are fractal perturbations of the classical circular functions. The coefficients of the system enable the control and modification of the properties of the originals. Additionally, Fourier parameters and approximants for sampled signals are computed and the density of the fractal trigonometric polynomials in the most common spaces of periodic functions is proved.

We provide some special algorithms to construct the fractals which have not been constructed previously. Our generalized graphical representation can be applied to any alphabet with any cardinal number. It is noted that the Hao's frame representation and the Tino's generalization are the special cases of our result. Moreover, we will apply the recurrence formula to derive their characteristic equations and roots.