Fractals

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.

In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.

Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

Fractal Dimensions (FD) are one of the popular measures used for characterizing signals. They have been used as complexity measures of signals in various fields including speech and biomedical applications. However, proper interpretation of such analyses has not been thoroughly addressed. In this paper, we study the effect of various signal properties on FD and interpret results in terms of classical signal processing concepts such as amplitude, frequency, number of harmonics, noise power and signal bandwidth. We have used Higuchi's method for estimating FDs. This study may help in gaining a better understanding of the FD complexity measure itself, and for interpreting changing structural complexity of signals in terms of FD. Our results indicate that FD is a useful measure in quantifying structural changes in signal properties.

Fingerprints are one of the simplest and most reliable human biometric features for identification. Geometry of the fingerprint is fractal and we can classify a fingerprint database with fractal dimension, but one can't identify a fingerprint with fractal dimension uniquely. In this paper we present a new approach for identifying fingerprint uniquely; for this purpose a new fractal is initially made from a fingerprint by using Fractal theory and Chaos Game theory. While making the new fractal, five parameters that can be used in identification process can be achieved. Finally a fractal is made for each fingerprint, and then by analyzing the new fractal and parameters obtained by Chaos Game, fingerprint identification can be performed. We called this method Fingerprint Fractal Identification System (FFIS). The presented method besides having features of fractals such as stability against turning, magnifying, deleting a part of image, etc. also has a desirable speed.

Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk.

The water pH series from Dongting Lake Inlet and Outlet in China are analyzed by detrended fluctuation analysis (DFA), spectral analysis and multifractal methods. The results show that these pH series are characterized by long-term memory, 1/f noise and multifractal scaling, and these characteristics have obvious difference between the Lake Inlet and Outlet. The comparison results show that monofractal (DFA exponent) and multifractal (Δα, Δf, B) parameters can be quantitative dynamical indexes reflecting the capability of anti-acidification of Dongting Lake. Furthermore, we investigate the frequency-size distribution of pH series from Dongting Lake Inlet and Outlet. Our findings suggest that water pH is an example of a self-organized criticality (SOC) process. Based on concept of self-organized ctiticality, we analysis the cause that different scale-free power-law behavior between pH series from Dongting Lake Inlet and Outlet. This work can be helpful to improvement of modeling of lake water quality.

The complexity of the Portevin-Le Chatelier effect in Al-2.5%Mg polycrystalline samples subjected to uniaxial tensile tests is quantified. Multiscale entropy analysis is carried out on the stress time series data observed during jerky flow to quantify the complexity of the distinct spatiotemporal dynamical regimes. It is shown that for the static type C band, the entropy is very low for all the scales compared to the hopping type B and the propagating type A bands. The results are interpreted considering the time and length scales relevant to the effect.

The average distance between points of a fractal is proposed as a natural measure of the way in which the points of a fractal are distributed. The average distance between points of the Cantor Set is found to be , the average distance between points of the Cantor p-set is , and the average distance between points of the Fat Cantor p-set is . A general formula for computing the average distance between points of a self-similar set satisfying the open set condition is found.

In this paper, we mainly study the s-sets (regular 1-sets), which is the most important fractal in the study of fractal geometry. The regular 1-sets are subsets of countable collection of rectifiable curves. Also we define new real maps on the s-sets by using the methodology based on fractal interpolation functions. In addition, some results are applied to the waveform signals for interpolation.

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

We considered Q-state Potts model on Bethe lattice in presence of external magnetic field for Q < 2 by means of recursion relation technique. This allows to study the phase transition mechanism in terms of the obtained one dimensional rational mapping. The convergence of Feigenabaum α and δ exponents for the aforementioned mapping is investigated for the period doubling and three cyclic window. We regarded the Lyapunov exponent as an order parameter for the characterization of the model and discussed its dependence on temperature and magnetic field. Arnold tongues analogs with winding numbers w = 1/2, w = 2/4 and w = 1/3 (in the three cyclic window) are constructed for Q < 2. The critical temperatures of the model are discussed and their dependence on Q is investigated. We also proposed an approximate method for constructing Arnold tongues via Feigenbaum δ exponent.

Let be a root of the polynomial p(x) = x² + 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a₀ + a₁α + ⋯ + a_ℓα^ℓ with a_i ∈ {0, 1, 2, 3, 4}. The set of points with integer part 0 in this number system