Fractals

An experimental study of interactions between a high Reynolds number fluid flow and multi-scale, fractal, objects is performed. Studying such interactions is required to improve our current understanding of wind or ocean current effects on vegetation elements, which often display fractal-like branching geometries. The main objectives of the study are to investigate the effects of the range of scales (generation numbers) of the fractal object and of the incoming flow condition on the drag force and drag coefficient, and to observe flow features in the near wake region resulting from the interaction. In this study, Sierpinski carpets and triangles with the scale ratios of 1/3 and 1/2, respectively, are employed. The fractal dimensions of the Sierpinski carpet and triangle are D = 1.893 and 1.585, respectively. Each pre-fractal object is mounted on a load cell at the centerline in a wind tunnel. Two types of inflow conditions are considered: laminar flow and high-turbulence level, active-grid-generated, flow. As a first approximation, we find the drag coefficients are approximately constant of order unity, and do not depend upon generation number of the pre-fractal when defined using the actual frontal area that varies as function of generation number. Still, the drag coefficient of the Sierpinski carpet increases weakly with number of generations indicating that the drag force decreases less than the cross-sectional area. For the Sierpinski triangle a similar trend is observed at large scales. However, the drag coefficient displays a peak at the third generation and then shows a decreasing trend as smaller scales are included for higher generation cases. The drag coefficient for the turbulent flow is larger than that for the laminar flow for all the fractal generations observed. Flow features (mean velocity, mean vorticity, and turbulence root-mean-square distributions) are measured by using stereoscopic Particle Image Velocimetry to observe various scales of the motion in the near wake of the pre-fractal objects. Strong shear layers are formed behind the fractal objects depending on the hole locations of different generations, which results in the formation of various length scales of the dominant turbulence structures. The smaller scale wakes are found to merge behind the Sierpinski carpet, whereas they are merely damped behind the Sierpinski triangle.

The manure which we are using for plant growth does not help to increase the porosity of the soil. We use pyroligneous acid to treat the soil as softener, nutrient and porosity increaser. This helps to the soil to remove unwanted carbonates and separate the soil particles. In this way, we can increase 30% porosity of the soil. When we pour pyroligneous acid into soil, the soil increase the carbon content. We consider changes of pore space of a soil by image analysis and also assess the applicability of fractal scaling of pore surfaces. The calculated pore surface dimension showing that the surface exhibit fractal structure. Lacunarity shows the size distribution of holes. If Lacunarity increases, then the value become decreases. This study demonstrates that applicability of fractal scaling for evaluating pore surface of the soil images.

In the work, a simple and reliable algorithm is presented to calculate the fractal dimension of single pixel for the remote sensing images, and the fractal dimension values obtained by the algorithm proposed in this work have positive correlation with the complexity of surface features. On the basis of a scene of Landsat7 ETM+ (i.e., Enhanced Thematic Mapper Plus) data and the proposed algorithm, expert classification models and fractal technique were introduced to identify the ground objects in a Chinese subtropical hilly region, where surface features are very diverse and complex. In the work, the different land use/land cover types, especially the different vegetation categories were successfully identified using the ETM+ image, and this classification has an overall accuracy of 80.25% and a K_hat of 0.7738, which are higher than those of the traditional supervised classification.

Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

We prove that if a recurrent two-dimensional sequence with periodic initial conditions coincides in a sufficiently large starting square with a two-dimensional sequence produced by an expansive system of context-free substitutions, then they must coincide everywhere. We apply this result for some examples built up by homomorphisms of finite abelian p-groups, in particular for Pascal's Triangle modulo p^k, Pascal's Triangles modulo 2 with non-trivial periodic borders, and Sierpinski's Carpets with non-trivial periodic border. All these particular cases justify the conjecture that recurrent two-dimensional sequences generated by homomorphisms of finite abelian p-groups with periodic initial conditions can always be alternatively generated by expansive systems of context-free substitutions.

When probing the dynamical properties of complex systems, such as physical and physiological systems, the output signal may be not the expected one. It is often a linear or nonlinear filter (or a transformation) of the right one represented the properties we want to investigate. Besides, for a multiple-component system, it is necessary to consider the relations between different influence factors. Here, we investigate what effect kinds of linear and nonlinear filters have on the cross-correlation properties of monofractal series and binomial multifractal series relatively. We use the multifractal detrended cross-correlation analysis (MFDCCA) that has been known well for its accurate quantization of cross-correlations between two time series. We study the effect of five filters: (i) linear (y_i = ax_i + b); (ii) polynomial ; (iii) logarithmic (y_i = log(x_i + δ)); (iv) exponential (y_i = exp(ax_i + b)); and (v) power-law (y_i = (x_i + a)^b). We find that for both monofractal and multifractal signals, linear filters have no effect on the cross-correlation properties while the influence of polynomial, logarithmic and power-law filters mainly depends on (a) the strength of cross-correlations in the original series; (b) the parameter b of the polynomial filter; (c) the offset δ in the logarithmic filter; and (d) both the parameter a and b of the power-law filter. In addition, the parameter a and b of the exponential filter change the cross-correlation properties of monofractal signal, yet they have little influence on that of multifractal signal.

Two "pathological" properties of Minkowski content are that countable sets can have positive content (unlike Hausdorff measures) and the property of a set being Minkowski measurable is quite rare. In this paper, we explore both of these issues.

In particular, for each d ∈ (0,2) we give an explicit construction of a countable Minkowski measurable subset of ℝ² of Minkowski dimension d and arbitrary positive Minkowski content. We also indicate how this construction can be extended to ℝⁿ, to construct a countable subset with arbitrary positive Minkowski content of any dimension in (0, n). Furthermore, we give an example of a strictly increasing C¹ function which takes a Minkowski measurable subset of [0,1] onto a set which is not Minkowski measurable but of the same dimension.

Piezoelectric ultrasonic transducers typically employ composite structures to improve their transmission and reception sensitivities. The geometry of the composite is regular with one dominant length scale and, since these are resonant devices, this dictates the central operating frequency of the device. In order to construct a wide bandwidth device it would seem natural therefore to utilize resonators that span a range of length scales. In this article we consider such a device and build a theoretical model to predict its performance. A fractal medium is used as this contains a wide range of length scales and yields to a renormalization approach. The propagation of an ultrasonic wave in this heterogeneous medium is then analyzed and used to construct expressions for the electrical impedance, and the transmission and reception sensitivities of this device as a function of the driving frequency. The results presented show a marked increase in the reception sensitivity of the device.

This paper presents a study of the connectivity of the class of fractals known as the Sierpiński relatives. These fractals all have the same fractal dimension, but different topologies. Some are totally disconnected, some are disconnected but with paths, some are simply-connected, and some are multiply-connected. Conditions for these four cases are presented. Constructions of paths, including non-contractible closed paths in the case of multiply-connected relatives, are presented. Examples of specific relatives are provided to illustrate the four cases.