Fractals

Through analysis of the morphology of dendritic arborisation of neurons from the substantia gelatinosa of dorsal horns from four different species, we have established that two types of cells (stalked and islet) are always present. The aim of the study was to perform the intra- and/or inter-species comparison of these two neuronal populations by fractal analysis, as well as to clarify the importance of the fractal dimension as an objective and usable morphological parameter. Fractal analysis was carried out adopting the box-counting method. We have shown that the mean fractal dimensions for the stalked cells are significantly different between species. The same is true for the mean fractal dimensions of the islet cells. Still, no significant differences were found for the fractal dimensions of the stalked and islet cells within a particular species. The human species has shown as the only exception where fractal dimensions of these two types of cells differ significantly. This study shows once more that the fractal dimension is a useful and sensitive morphological descriptor of neuronal structures and differences between them.

We fully describe the homeomorphism classes of self-contacting symmetric binary fractal trees, and provide the explicit homeomorphisms for the classes. In developing this theory, we introduce new notations, definitions and results regarding symmetric binary fractal trees. The set of tip points at maximal height for a class of trees forms a middle Cantor set. We study various aspects of the holes in self-contacting trees, and the hole locator pairs can be used to describe the homeomorphism classes.

The role of the geometry of prefractal interfaces in Laplacian transport is analyzed through its "harmonic geometrical spectrum." This spectrum summarizes the properties of the Dirichlet-to-Neumann operator associated with these geometries. Numerical analysis shows that very few eigenmodes contribute significantly to the macroscopic response of the system. The hierarchical spatial frequencies of these particular modes correspond to the characteristic length scales of the interface. From this result, a simplified analytical model of the response of self-similar interfaces is developed. This model reproduces the classical low and high frequency asymptotic limits and gives an approximate constant phase angle behavior for the intermediate frequency region. It also provides an analytical description for the crossovers between these regimes and for their dependency on the order of the prefractal interface. In this frame, it is shown that the properties of any generation prefractal can be deduced from the properties of the fractal generator, which are easy to reach numerically.

This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

A fractal permeability model which accounts for the capillary pressure is derived, and the permeability contribution from the capillary pressure is analyzed. The predicted permeability is compared with the available experimental data, and good agreement is found between the present model predictions and available experimental data. The results show that at low porosity (< 0.35), the capillary pressure has the significant influence on the total permeability, and the capillary pressure effect cannot be neglected. But at high porosity (> 0.35), the effect of capillary pressure on permeability is insignificant.

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.

In some applications, for instance, finance, biomechanics, turbulence or internet traffic, it is relevant to model data with a generalization of a fractional Brownian motion for which the Hurst parameter H is dependent on the frequency. In this contribution, we describe the multiscale fractional Brownian motions which present a parameter H as a piecewise constant function of the frequency. We provide the main properties of these processes: long-memory and smoothness of the paths. Then we propose a statistical method based on wavelet analysis to estimate the different parameters and prove a functional Central Limit Theorem satisfied by the empirical variance of the wavelet coefficients.

The Bourrée Part I from Johann Sebastian Bach's Cello Suite No. 3 provides a clear example of structural scaling. The recursive form of this structure can be visualized in the manner of a well known fractal construction — the Cantor set.

We study the game theory from tick data of the won-dollar and yen-dollar exchange rates in financial markets. The standard deviation, the global efficiency, and the autocorrelation for arbitrary strategies are shown to give rise to properties of new dynamics, and these statistical quantities are very similar with the case of the majority game. Our results presented will be compared with numerical findings for other game models.