Fractals

The general transport equation of multiscaling disordered fractal media in three-dimensional case is derived from conservation of mass. Multiscaling fractional transport equation is obtained on the basis of discussing Brownian motion, fractional Brownian motion and standard diffusion equation of fractals, which is consistent with the result of literature.

The collective behavior of complex physical models has often been used to model complex social phenomena. Herein, we describe one such process using a fractal stochastic point process. A coarse-graining of the relative dispersion, the ratio of the standard deviation to the mean in a time-series, is shown to decrease as a modulated inverse power law in time. This is an empirically motivated (renormalization group) model of teen birth statistics and only indirectly addresses the social mechanisms that promote sexual activity, subsequent pregnancy and finally child birth. This prediction agrees with the statistics observed across 27 years of data in Texas, and among twelve different Texan regions with differing demographics for 16 years.

This paper introduces the concept of point-fractal to analyze the time-scale variability of rainfall data on the northern and southern regions of Taiwan. It is evident that scale invariance exists in time and clustering will decrease in accord with the increase of threshold. Under the variation of threshold, it can be verified that the maximum values of the homogenous scale-invariant interval are just the same. In addition, taking the probability-scale law on different levels of threshold, the relation can be established between the saturation scale (return period) and the threshold (design hydrologic variable). The methodology is different from the traditional one, i.e. frequency analysis method. When both methods are compared, it is found that the former does not require either the probability density function or the calculation of parameters. We conclude that this study provides a new alternative method in computation.

A modified type of iterated function system (IFS) has recently been shown to generate images qualitatively similar to "classical" chaotic attractors. Here, we use time-delay embedding reconstructions of time-series from this system to generate three-dimentional projections of IFS attractors. These reconstructions may be used to access the topological structure of the periodic orbits embedded within the attractor. This topological characterization suggests an approach by which a rigorous comparison of IFS attractors and classical chaotic systems may be attained.

From the reductionist perspective of molecular biology, proliferation or differentiation of eucaryotic cells is a well-defined temporal, spatial, and cell type-specific sequence of molecular cellular events. Some of those events, such as passing of the restriction point in the cell cycle, are of a stochastic nature. Results of this study indicate that, in spite of the intracellular stochasticity, cancer cells can form collective structures with fractal dimension and self-similarity. A transition from the monolayer culture to the aggregated colony facilitated interconnectedness between P19 cells, altered constitutive expression of randomly chosen retinoid-responsive genes, and increased fractal dimension of the entire population. Retinoid-induced emergence of neuron-like phenotype decreased fractal dimension significantly, slowing down dynamics of gene expression. Since the differentiated P19 cells retained both their cancer phenotype and a number of gene defects, we conclude that the appropriate dynamics of intracellular events is neccessary for the proper course of differentiation. Owing to self-similarity, dynamics of cellular expansion can be measured by a fractal dimension in a single cell or in the entire population.

The purpose of this paper is to explain the connection between fractional Brownian motion (fBm) and non-Fickian diffusive processes, and at the same time, highlight three engineering applications: two requiring self-affine fBm trace functions and the other requiring self-similar fBm spatial trajectories.

The fractal dimension of fracture surfaces obtained within brittle-ductile transition regime is evaluated at various observation scales. Fracture surfaces are generated by the tensile fracture test. The brittle-ductile transition is accomplished by using the round-notched bar specimens with various notch radii, which cause the variation in stress triaxiality. The specimens are manufactured from mild steel, steel and cast-iron bar. The fracture model is identified according to the observation through scanning electron micrographs.

The fractal dimension for ductile fracture surfaces is almost constant despite variations in observing scale and changes in stress triaxiality. Meanwhile, the fractal dimension on brittle fracture surfaces shows the different values for macroscopic and microscopic observing scales. This transition-like scale dependence of fractal dimension for brittle fracture surfaces is considered to reflect such a characteristic of the fracture i.e. its specific length in microscopic fracture mechanism. The existence of transition in fractal dimension with observing scale is considered to be an index, used to distinguish the ductile fracture surface from the brittle fracture one.

This paper aims to study the phenomenon of light transmission through an interferometer by considering the passage of photons through the plates as a phenomenon of fractal percolation.

In addition to the percolation of contacts, porous percolation, the percolation of points and the percolation of rebounds are also studied here (in the form described below).

The Double Trace Moment method for the estimation of parameters of universal multifractal measures is revisited. It is found that the method applies strictly to bare measures but is approximate for dressed measures, which are the only observable ones. The approximation stems from the fact that the double dressed moments used by the method do not scale as required to produce unbiased estimates of the parameters. Modified moments are proposed, which correctly scale for both bare and dressed measures. The original and modified methods are compared theoretically and numerically. Conditions under which the original method gives accurate and heavily biased estimates are identified.

In this paper, we consider the problem of the derivation of real functions defined on sets X⊂ℝ with respect to a measure μ on X such that 0<μ(X)<∞. In order to do so, we define the concept of derivation with respect to a measure μ: the μ-derivative. This concept is consistent and useful to distinguish among different increase kinds displayed by the functions defined on X. We study some properties and, when the functions are defined on a self-similar fractal with Hausdorff dimension s, a new derivation concept is considered, the Θ_s-derivative, which is easier to calculate than the previous one. Finally, we study the relation between the two derivation concepts and an algorithm for the calculation of the Θ_s-derivative.

This paper presents a framework for dissipative dynamical systems excited by external temporal inputs. We introduce a set {I_l} of temporal inputs with finite intervals. The set {I_l} defines two other sets of dynamical systems. The first is the set of continuous dynamical systems that are defined by a set {f_l} of vector fields on the hyper-cylindrical phase space ℳ. The second is the set of discrete dynamical systems that are defined by a set {g_l} of iterated functions on the global Poincaré section Σ. When the inputs are switched stochastically, a trajectory in the space ℳ converges to an attractive invariant set with fractal-like structure. We can analytically prove this result when all of the iterated functions satisfy a contraction property. Even without this property, we can numerically show that an attractive invariant set with fractal-like structure exists.