Fractals

In his fascinating book Wonders of Numbers, Clifford Pickover introduces the Ana sequence and fractal, two self-referential constructions arising from the use of language. This paper answers Pickover's questions on the relative composition of sequence terms and the dimension of the fractal. In the process, it introduces a novel way of obtaining fractals from iterative set operations. Also, it presents a beautiful variant of the Ana constructions involving the golden ratio. In conclusion, it suggests ways of constructing similar fractals for the Morse-Thue and "Look and Say" sequences.

Of fundamental importance to wave propagation in a wide range of physical phenomena is the structural geometry of the supporting medium. Recently, there have been several investigations on wave propagation in fractal media. We present here a renormalization approach to the study of reaction-diffusion (RD) wave propagation on finitely ramified fractal structures. In particular we will study a Rinzel-Keller (RK) type model, supporting travelling waves on a Sierpinski gasket (SG), lattice.

The multifractal analysis of data on nuclear fragments obtained from ²⁸Si-AgBr interactions at 14.5 A GeV is performed using three different methods (the factorial moments, G-moments and Takagi moments). The generalized fractal dimensions D_q is determined from all these methods. Data reflects multifractal geometry for the nuclear fragments. From the knowledge of D_q, the multifractal specific heat is calculated for this data and also for ¹⁶O-AgBr interactions at 60 A GeV and ³²S-AgBr interactions at 200 A GeV.

By prescribing their code run behavior, we consider some subsets of Moran fractals. Fractal dimensions of these subsets are exactly obtained. Meanwhile, an interesting decomposition of Moran fractals is given.

A simple method is proposed to determine the surface fractal dimension of swelling clay minerals. When swelling clay mineral is immersed into the water solution, swelling deformation and swelling pressure will occur. The swelling deformation and swelling pressure are proportional to the amount of water volume absorbed by swelling clay minerals. The water volume absorbed by swelling clay minerals is related to its surface fractality. The correlation of the normalized water volume by the volume of swelling clay minerals (V_w/V_m) to vertical overburden pressure (p) is obtained as V_w/V_m = Kp^D_s-3, from which the surface fractal dimension of swelling clay minerals can be estimated. The surface fractal dimension (D_s) of Wyoming bentonite is 2.64, obtained from the swelling tests. The surface fractal dimension of Wyoming bentonite determined from the swelling tests nearly equals to that obtained from the nitrogen adsorption isotherms, and is larger than that measured using the mercury intrusion porosimetry.

An empirical study accomplished on various samples of marks shows that their distribution on mark levels has some universal features. Using (classical) examination items based on commentaries and debate, the distribution displays a Gaussian form with a variance that does not depend on the profile of classes and age level of pupils. In the case of examination based on items with multiple choices, the deviation from the Gaussian form becomes relevant and the distribution gets the shape of a stable Lévy distribution with a fat tail that can be fitted to a power-law. The difficulties related to the infinite variance, typical for this class of models, can be overcome using the Truncated Lévy Distribution, frequently met in physics of socio-economic systems.

We conducted a set of experiments of impact fragmentation of samples with voids, such as pumice stones and bricks. We discovered that the fragment size distribution follows a power law, but that the exponent of the distribution is different from that of the distribution by the fragmentation of a space-filling sample like a gypsum ball. The value of the exponent is about 0.9. And the value seems universal for samples with voids.

A method is developed for the automatic detection of the onset of scaling for long-range dependent (LRD) time series and other asymptotically scale-invariant processes. Based on wavelet techniques, it provides the lower cutoff scale for the regression that yields the scaling exponent. The method detects the onset of scaling through the dramatic improvement of a goodness-of-fit statistic taken as a function of this lower cutoff scale. It relies on qualitative features of the goodness-of-fit statistic and on features of the wavelet analysis. The method is easy to implement, appropriate for large data sets and highly robust. It is tested against 34 time series models and found to perform very well. Examples involving telecommunications data are presented.

The branching structure of neurones is thought to influence patterns of connectivity and how inputs are integrated within the arbor. Recent studies have revealed a remarkable degree of variation in the branching structure of pyramidal cells in the cerebral cortex of diurnal primates, suggesting regional specialization in neuronal function. Such specialization in pyramidal cell structure may be important for various aspects of visual function, such as object recognition and color processing. To better understand the functional role of regional variation in the pyramidal cell phenotype in visual processing, we determined the complexity of the dendritic branching pattern of pyramidal cells in visual cortex of the nocturnal New World owl monkey. We used the fractal dilation method to quantify the branching structure of pyramidal cells in the primary visual area (V1), the second visual area (V2) and the caudal and rostral subdivisions of inferotemporal cortex (ITc and ITr, respectively), which are often associated with color processing. We found that, as in diurnal monkeys, there was a trend for cells of increasing fractal dimension with progression through these cortical areas. The increasing complexity paralleled a trend for increasing symmetry. That we found a similar trend in both diurnal and nocturnal monkeys suggests that it was a feature of a common anthropoid ancestor.

We have investigated the effect of loss of cell polarity at the individual cell level on the global pattern formation of squamous cell carcinoma. Simple models of cancer growth are able to reproduce qualitatively the sequence of observed abnormalities of metaplasia, dysplasia and carcinoma in situ and invasive carcinoma. The models are based on loss of cell polarity alone by cancerous cells, coupled to an otherwise normal growth rate and epithelial behavior. The models show that, as the probability of a wrong plane of cell division is increased, a transition from normal, well stratified epithelium, to an invasive, fractal, dendritic pattern is observed. This transition shows a sequence of morphologies in the following order as a function of loss of polarity: first an apparently normal but already diseased tissue, then metaplasic followed by a dysplasic tissue, and eventually carcinoma first in situ, then invasive. The model also suggests a killing mechanism of most severe cancer clones towards weaker ones.