Fractals

In this paper we study a general separation property for subsystems G, whose attractor K_G is a sub-self-similar set. This is a generalization of the Lau-Ngai weak separation property for the bounded distortion case. For subsystems with positive Hausdorff measure in its similarity dimension, we characterize the subsets of K_G with positive measure where the separation property may fail. We exhibit two examples of fractal sets, one not satisfying the weak separation property and whose existence was questioned by Zerner, the other having positive Hausdorff measure in its dimension and with the separation property failing on a subset of positive measure.

Calculus on fractals, or F^α-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves F^α-integral and F^α-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The F^α-integral is suitable for integrating functions with fractal support of dimension α, while the F^α-derivative enables us to differentiate functions like the Cantor staircase. Several results in F^α-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of F^α-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour.

In this paper we show that, as operators, the F^α-integral and F^α-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes F^α-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes F^α-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and F^α-calculus.

This conjugacy is useful, among other things, to find solutions to F^α-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

Three-dimensional pore structure of four samples of geological rock was obtained using serial sectioning. Their fractal and flow properties have been successfully determined through image analysis. The measured 3D box counting fractal dimension of four geological rock samples are 2.59, 2.88, 2.88 and 2.29, which are not integer dimensions. This would imply that the structure of the four geological rock samples is fractal in nature. Porosity of the four rock samples are ranged from 0.18 to 0.31 and the specific surface area is ranged from 1.57 to 2.75 mm^-1. Tortuosity estimated from image analysis of the four samples is ranged from 1.11 to 5.83 and varied in each fluid flow direction. This would imply that the rock samples are not uniform in its pore structure. The fractal dimension as a representation of the isotropic level of the pore structure is not clearly clarified and it still needs further investigation.

Multifractal analyses of binary images of soil thin sections (STS) are widely used to characterize pore structure. However, no geometrical model is known to exist for a binary multifractal. Thus, the multifractality of binary images, and the accuracy of multifractal parameters estimated from them, need to be carefully evaluated. We captured 8-bit depth resolution digital grayscale images of three STS images with dimensions of 1024 × 1024 pixels and a pixel length of 1.9 μm. Random grayscale geometrical multifractal fields (GMF) with similar dimensions and known multifractal parameters were constructed using generators extracted from the STS images. The STS and GMF grayscale images were objectively thresholded to give six binary images. The method of moments was used to compute the log-transformed partition function, log(χ(q, δ)) versus log(δ) where δ is box size, for each grayscale image and its binary counterpart. Consistent linearity was observed in the resulting functions for the grayscale images, indicating, by definition, multifractal behavior. In contrast, the log(χ(q, δ)) versus log(δ) plots for the binary images exhibited a two-region response, with a flat plateau at small scales and linearity at larger scales, indicating they were not true multifractals. Generalized dimensions (D_q) computed from the linear portions of the binary log-transformed partition functions were significantly over estimated for q ≪ 0 and underestimated for q ≫ 0 relative to corresponding D_q values for the grayscale images. Based on these results we contend that binary images are not mathematical multifractals, and that generalized dimensions estimated from them cannot be used to quantify pore space geometry. Instead we encourage further exploration of the use of grayscale images for multifractal characterization of soil structure. This direct approach is theoretically sound and does not require any intermediate thresholding step, which is known to influence the results of multifractal analyses.

In this study, we illustrate the fractal nature of the wake shed by a periodically flapping filament. Such wake structure is a combination of primary vortex shedding resulting in the von Kármán vortex street, a series of concentrated vortex dipoles formed when the trailing edges of filaments reach their maximum amplitudes and small eddies form along the shear layer connected with the concentrated vortices due to the shear layer instability. The vortex dynamics of the flapping filament are visualized and imaged experimentally using a soap-film flow tunnel with a high-speed camera and a low pressure sodium lamp as a light source. The wake fractal geometry is measured using the standard box-counting method and it is shown that the fractal dimension of the soap pattern boundaries in the wake is D = 1.38 ± 0.05, which agrees well with those measured for fully developed turbulences and other shear flow phenomena. The invariant of the fractality in the wake induced by the flapping filament thus provides another illustration of the geometrical self-similarity and nonlinear dynamics of chaotic fluid flows.

Fractal growth patterns of polyaniline were developed during electropolymerization of aniline using the surfactants sodium dodecyl sulphate (NaDS) and NaDS containing cetyl trimethyl ammonium bromide (CTAB). Growth kinetics was studied and electric potential oscillations were monitored as a function of time. On addition of CTAB polymer growth was inhibited due to coordination of CTAB with the growing polyaniline chain. The average particle size of the polymer aggregate obtained from aniline-NaDS-H₂O system was ~150 nm as evident by Transmission Electron Microscopy (TEM) results. Polymer aggregates were characterized by electrical conductivity measurements, X-ray diffraction (XRD) and Thermogravimetric (TG) studies. An interaction between NaDS and aniline was observed in the absence of electric field as evident by (i) electrical conductivity of aqueous solution of NaDS in the absence and presence of aniline, and (ii) their crystallization patterns on microslides. A mechanism for the development of fractal patterns and electrical potential oscillations is proposed on the basis of diffusion limited aggregation process.

Multifractal detrended cross-correlation analysis (MF-DXA) has been developed to detect the long-range power-law cross-correlation of two simultaneous series. However, the synchronization of underlying data can not be guaranteed integrated by a variety of factors. We artificially imbed a time delay in considered series and study its influence on the multifractal cross-correlation analysis. Time delay is found to affect the multifractal characterization, where a larger time delay causes a weaker multifractality. We also propose an alternative modification on MF-DXA to make the process more robust. The logarithmic return and volatility of Chinese stock indices show cross-correlation scaling behavior and strong multifractality by MF-DXA as well as singularity spectrum analysis.

The non-homogeneity and complexity of micro network distribution of hydrogel matrices prepared with two scleroglucan biopolymers obtained with different fermentation times were analyzed using environmental scanning electron microscopy (ESEM) and dynamic rheology.

ESEM images were processed with the tools of multifractal theory using the box-counting method in order to obtain the gels multifractal spectra. Dynamic rheological measurements indicate that both polymeric networks correspond to physical gels that exhibit a solid like behavior. These results suggest the existence of a relationship between the fermentation time used in the polymer production, the degree of self-similarity and the rigidity of the scleroglucan gel structure.

This study discussed the application of a fractal interpolation method in satellite image data reconstruction. It used low-resolution images as the source data for fractal interpolation reconstruction. Using this approach, a high-resolution image can be reconstructed when there is only a low-resolution source image available.

The results showed that the high-resolution image data from fractal interpolation can effectively enhance the sharpness of the border contours. Implementing fractal interpolation on an insufficient image resolution image can avoid jagged edges and mosaic when enlarging the image, as well as improve the visibility of object features in the region of interest. The proposed approach can thus be a useful tool in land classification by satellite images.

A new fractal interpolation method called PPA (Pointed Point Algorithm) based on IFS is proposed to interpolate the self-affine signals with the expected interpolation error, solving the problem that the ordinary fractal interpolation can't get the value of any arbitrary point directly, which has not been found in the existing literatures. At the same time, a new method to calculate the vertical scaling factors is proposed based on the genetic algorithm, which works together with the PPA algorithm to get the better interpolation performance. Experiments on the theoretical data and real field seismic data show that the proposed interpolation schemes can not only get the expected point's value, but also get a great accuracy in reconstruction of the seismic profile, leading to a significant improvement over other trace interpolation methods.

In the plane IR², let A₀ be the unit interval on the x-axis, and let A₍₁₎ be the polygonal path with nodes (0, 0), , (½, 0), , (1, 0). Let S be the operator which, applied to a segment B₍₀₎ in IR², replaces it by a polygonal path B₍₁₎ = SB₍₀₎, a similar copy of A₍₁₎, but with the same endpoints as B₍₀₎. Denote by S⁽ⁿ⁾ the n-th iterate of S. The limit set (with respect to the Hausdorff metric) A_(∞) = lim_{n → ∞} S⁽ⁿ⁾A₍₀₎ is a space-filling curve which is the closure of its interior and the union of four half-size copies of itself, intersecting only in their boundaries. Although A_(∞) is of infinite connectivity, it is a tile tessellating the plane. It is related to the set of Eisenstein fractions and has a boundary of Hausdorff dimension

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.