Fractals

A relationship between the Lipschitz-Holder exponent and the maximal possible order of derivative is used to look at fractal properties of functions representable by trigonometric series.

Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes non-trivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α. The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the point-process generation mechanism. In the context of point-process simulation, reduction of this discrepancy requires generating data sets with either a large number of points, or with low jitter in the generation of the points. In the context of fractal data analysis, the results presented here suggest caution when interpreting fractal exponents estimated from experimental data sets.

The dynamics of a free-falling body in complex materials such as a polymer fluid is phenomenologically modeled using a fractional generalization of the Riccati equation. The solution exhibits a rich behavior in its parametric dependence, and unlike normal free–fall there is no terminal velocity, instead a power–law increase in time is obtained. Within this approach the fractional order allows to tune the resulting equation.

This paper considers fractal trees with self-similar side branching. The Tokunaga classification system for side branching is introduced, along with the Tokunaga self-similarity condition. Area filling (D = 2) and volume filling (D = 3) deterministic fractal tree constructions are introduced both with and without side branching. Applications to diffusion limited aggregation (DLA), actual drainage networks, as well as biology are considered. It is suggested that the Tokunaga taxonomy may have wide applicability in nature.

Growth phenomena have been studied extensively in natural sciences. This interest has been renewed since the introduction of the fractal concept. In an attempt to understand the origin of irregular phenomena, several computer models and theoretical approaches have recently been developed. Studies using fractal analysis of the retinovasculature have proposed diffusion-limited aggregation (DLA) one of the most popular models to explain the formation of the retina. A deeper investigation of the physiological laws ruling the development of the retinovasculature has, however, revealed static and dynamic discrepancies with DLA, leading to rejection of the DLA model, and reopening the debate. In light of comparison of the two formation processes and of the absence of a DLA structural characteristic in retinovasculature, we discuss the validity of some previously proposed models.

In this paper, the methods of construction of a fractal surface are introduced, the principle of bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of an iterated function system of bivariate fractal interpolation functions is proved. Moreover, the theorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems, the fractal interpolated surfaces are created by using practical data.

We present the statistical properties of large scale galaxy distribution in the LEDA redshift database. This catalog contains more than 40,000 redshifts over all the sky. We find that LEDA, although seriously affected by incompleteness, shows quite stable statistical properties. In particular, we have considered the behavior of the two points correlation function and of the power spectrum of the density fluctuations, and we have done several tests to check whether the incompleteness of the catalog affect these statistical quantities. Our conclusion is that galaxy distribution in this catalog has fractal properties up to the distance ~ 200h^-1 Mpc with D = 2.1 ± 0.2, with no sign towards homogeneity: this result is statistically stable. Finally, we compare these results with those obtained in other redshift surveys, finding that this sample well reproduces the properties of galaxy distribution found in the different catalogs.

It is shown, using multifractal data on space-filling bearings, that there exists a multifractal crossover to an effective 'capacity' dimension in some multiplicative processes. This result is then used to interpret recent experimental data on multifractal structure of multiplicity distribution in electron-positron annihilation at high energies.

An iterative spectral method is presented for fractal characterization of irregularly shaped (i.e., nonrectangular) images such as microscopic images of cells. The algorithm extends the application of fractal characterization by frequency analysis to images with irregular supports by utilizing information in both the spatial and frequency domains. Experiments with synthetic fractal images demonstrate the efficacy of the proposed method.

Non-α/non-β cat retinal ganglion cell images were obtained from the published literature, and a homogeneous group of cells was chosen as a standard for each currently accepted cell type (γ, δ and ε). The NIH box-counting method was chosen to determine the fractal dimension (Df) of all cells. The 'standard' values allowed comparisons with other morphologically and physiologically non-α/non-β classified cell types in the literature. We suggest, based on fractal analysis of the dendritic trees, that the morphologically defined γ, δ, and ε cells are distinct types. The W-tonic and W-phasic cell types were further divided into 2 subcategories (W-tonic₁, W-tonic₂, W-phasic₁, W-phasic₂). The fractal dimension, of the ε cells being equivalent to the W-tonic₁ group and γ cell type equivalent to the W-phasic₁ group. Delta cells may be equivalent to either the W-tonic₂ or the W-phasic₂ group. We discuss the value of the fractal dimension as an added morphological parameter for future morphophysiological classification schemes of vertebrate retinal ganglion cells.

The function of many natural flow systems is to connect by a fluid flow a finite-size volume and one point. This paper outlines a strategy for constructing the architecture of the volume-to-point path such that the flow resistance is minimal (constructal theory¹). The given volume is viewed as an assembly of volume elements of various sizes. The main discovery is that the shape of each element can be optimized such that the elemental volume-to-point flow resistance is minimal. This optimization principle applies at every volume scale. The smallest volume element contains a fluid saturated porous medium with Darcy flow, which is collected by and channeled through a high permeability path (e.g., fissure) to one point on the element boundary. The geometric optimization is repeated for larger volume elements, which are constructs (assemblies) of optimized smaller volumes. The flow integrated over each new assembly is channeled through a high-permeability path to a point on the side of the assembly. One remarkable feature of the emerging minimal-resistance flow path is that the high-permeability channels of the various volume elements form a tree network which is completely deterministic. The interstices of the network are filled with low permeability porous medium. The method is extended to applications where the high-permeability paths are empty spaces (e.g., parallel-plate channels). It is shown that when the total void volume is constrained it can be distributed optimally among the volume elements to further decrease the overall flow resistance.

A statistical mechanical formalism is developed using the computer information concepts of algorithmic complexity and Kolmogorov universal probability. This formalism provides a thermodynamic description of microstates of a system. For an isolated classical system, the algorithmic complexity is equal to the thermodynamic entropy. This approach does not rely on probabilistic ensemble concepts and can be applied to non-ergodic systems. An H-function is developed from the Kolmogorov universal probability that satisfies the properties of an entropy function. Using this approach, the thermodynamics of irreversible growth processes far from equilibrium can be investigated. Entropy functions for fractal growth processes far from equilibrium are developed from the algorithmic complexity and are seen to be similar to Flory-type equilibrium functions. This development does not require energy minimization procedures associated with equilibrium arguments. Using this approach, constraints can be put on the types of mean field models that will yield fractal structures. The algorithmic complexity of a system can also be used to explore the role of fluctuations and criticality in highly ordered systems.