Fractals

We show that the restriction of an eigenfunction of the Laplacian on the Sierpinski Gasket (SG) to any segment inside the SG is monotone on finite pieces, i.e. there is a subdivision of the segment, such that the function is monotone on all subintervals.

Ballistic particles interacting with irregular surfaces are representative of many physical problems in the Knudsen diffusion regime. In this paper, the collisions of ballistic particles interacting with an irregular surface modeled by a quadratic Koch curve, are studied numerically. The q moments of the source spatial distribution of collision numbers μ(x) are characterized by a sequence of "collision exponent" τ(q). The measure μ(x) is found to be multifractal even when a random micro-roughness (or random re-emission) of the surface exists. The dimensions f(α), obtained by a Legendre transformation from τ(q), consist of two parabolas corresponding to a trinomial multifractal. This is demonstrated for a particular case by obtaining an exact f(α) for a multiplicative trinomial mass distribution. The trinomial nature of the multifractality is related to the type of surface macro-irregularity considered here and is independent of the micro-roughness of the surface which, however, influences the values of α_min and α_max. The information dimension D_I increases significantly with the micro-roughness of the surface. Interestingly, in contrast with this point of view, the surface seems to work uniformly. This corresponds to an absence of screening effects in Knudsen diffusion.

A new fractal technique is used to investigate the onset of chaos in nonlinear dynamical systems. A comparison is made between this fractal technique and the commonly used Lyapunov exponent method. Agreement between the results obtained by both methods indicates that this technique may be used in a manner analogous to the Lyapunov exponents to predict onset of chaos. It is found that the fractal technique is much easier to implement than the Lyapunov method and it requires much less computational time. This fractal technique can easily be adopted to investigate the onset of chaos in many nonlinear dynamical systems and can be used to analyze theoretical and experimental time series.

Fractal interpolation function (FIF) is continuous on its interval of definition. As a special kind of continuous function, FIFs' integrations on various scales and Fourier transform are studied in this paper. All of them can be expressed by the parameters of the corresponding iterative function systems.

Spanning a planar graph the way D. Hilbert's curve does has various image processing and industrial applications. Spanning a planar graph by two disjoint curves with fractal properties has even more scientific and industrial uses. For example, given two liquids and an active osmosis through membrane between them, we would like to both cool the liquids and to find a cost-effective structure for the osmosis to occur. Another equivalent problem is to expose two liquids to light that passes through a transparent slab as the osmosis between them occurs. Two disjoint curves can be the answer for the required structure. Differences of lengths between the curves can also be useful. A fractal structure is obvious in the lungs, where osmosis of oxygen is vital. Fractal structures are often found in organic osmotic processes in Nature. In this article, a method for spanning a planar graph by two disjoint curves will be presented.

This work develops a preliminary method for coding random self-similar patterns as a series of numbers and investigates the corresponding algorithm to calculate the topological distance between starting point and the link in the generated fractal pattern from the code series. With reference to the wide range of stochastic property in natural patterns, a process for generating fractal patterns with various generating probabilities of the pattern links denoted as separately random self-similar generation or separately random fractal is proposed. To assess the adaptability of the process, the coding method is applied to the generation of a random self-similar river network and the corresponding algorithm for calculating topological distance of the links is used to determine the width function of the pattern. The width function-based geomorphologic instantaneous unit hydrograph (WF-GIUH) model is then applied to estimate the runoff of the Po-bridge watershed in northern Taiwan. The results show that the separately random self-similar generating algorithm can be implemented successfully to calculate hydrologic responses.

Approximately 90% of achondroplasia (ACH) cases are the result of a de novo mutation, with no phenotypical markers for the unaffected ACH parents being known to date. Here, the hypothesis of the presence of an oral mucosal microvascular abnormality in ACH children and unaffected ACH parents was tested. Two-dimensional vascular network geometry was analyzed in 15 children with sporadic ACH, 30 unaffected parents of children with typical ACH phenotype, and 45 control subjects, using high-resolution photographs of the lower gingival and vestibular oral mucosa. The vascular networks of ACH patients and ACH parents exhibited higher D(1–46) (P ≤ 0.013) and D(1–15) (P ≤ 0.0032) fractal dimensions, higher minimum-path dimension (P ≤ 0.0013), higher relative Lempel-Ziv (L-Z) complexity (P < 0.0001), and lower vessel-free area size (P < 0.00001) than controls. A vessel-free area size ≤ 56,832 μm² and L-Z complexity > 0.62 showed 100%-sensitivity and 100%-specificity in identifying unaffected ACH parents. These findings indicate (1) how complexity measures can be used to discover biological differences not demonstrable with traditional measures, and (2) the presence of a previously unrecognized microvascular network abnormality in both ACH patients and unaffected parents of children with de novo mutation ACH.