Narrow Results

We perform a Lévy index analysis of relativistic shower particles produced in the interactions of ³²S nuclei at 200A GeV with nuclear emulsion using the results of factorial moment F_q in two-dimensional anisotropic (η - ϕ) space. We carry out the same investigation for target fragments produced in ³²S-AgBr interactions at 200A GeV energy for both events in emission angle space and azimuthal angle space. The analysis reveals that for pions the value of Lévy index μ is 1.491 ± 0.025 for ring-like events and μ ~ 2.004 ± 0.054 for jet-like events which indicates different degree of multifractality. In case of target fragments jet-like events show a value of μ ~ 1.871 ± 0.010 whereas ring-like events yields an unphysical value (μ > 2) the implication of which need to be explored.

We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly we consider the behavior of the spectrum as the number of branches increases.

This paper presents the method of grading of cervical cancer images according to cell formation in the tissue. The variation of intensity and texture complexity of cancer cell images are calculated by fractal dimension methods. Box Counting Method (D_B) and HarFA Programme software are used to find out the dimension of the cells. Contact model and Epidemic Model are the two different models used here. The contact model shows the contact of the cells and the formation of the cancer in the part of the organ. The Epidemic model shows the growth of cancer. Lacunarity is formed between the cells. If the Lacunarity increases the dimension also increases. On seeing the growth of the cancer cell the pathologist can determine the grade of the cancer which in turn will help him to diagnose the cervical cancer and recommend appropriate treatment.

By using the frame representation method proposed by B. L. Hao et al. in 2000^1,2 and a method provided by P. Tino,^3,4 we will devise the generalized graphical representation for the strings defined on an alphabet Σ = {0, 1, …, n - 1} with cardinal n but all these graphs are still shown in a unit square, and generate its corresponding struture tables automatically. In particular, the words defined on Σ are mapped into n-adic numbers so that the index set Ind(B) can be computed. Thus, the fractals of the forbidden words can be naturally constructed. It has already been known that for biological sequences, the circular codes can be applied to some special bacteria and their forbidden words are the 5-necklaces. In this paper, we notice that the fractals of the forbidden words are particularly useful to represent the circular codes for some biological sequences.

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

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.

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 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.

Telecommunication service providers, also called operators or carriers, must maintain 99.999% of their telecom network availability. To avoid or mitigate the effects of an outage in the network, carriers perform different activities to restore service. Spare parts management plays an important role in meeting the required service. Unfortunately, the closed-loop supply chain that supports the availability of spare parts experiences variability in its processes as a result of endogenous and exogenous variables that make matching the recovery process with the demand process difficult. Understanding the effect of the variables on the variability outcome and the emergence of the variability can help to guide the development of future mathematical models for the management of spare parts. In this article, a complex system approach to analyze the variability behavior is presented.

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.

We report on a fractal pattern formed in Rayleight-Taylor instability. The fractal dimension is 1.88 ± 0.06, which is constant in time t > t₁ = 70 s. The pattern area decreases according to exp(-t/τ), where t and τ are time and time constant 69.8 s, respectively. τ agrees with t₁. This result leads that it is important for the fractal formation that sufficient annihilation of the heavier solution at the surface.

Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function , and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

The Hurst exponent derived by the R/S analysis method of Shanghai stock market's logarithmic return series is about 0.6298. This shows that the Shanghai stock market exhibits fractal features, and a long memory cycle of about one-and-a-half years. With the reconstruction of phase space, the Shanghai Stock attractor dimension converges to 1.335, which means that the Shanghai stock market has chaotic features, and constructing a dynamic system of the Shanghai stock market needs at least two variables. The findings from the principal component analysis support the conclusion of the existence of chaotic features of the Shanghai stock market. The fractal and chaotic features of the Shanghai stock market reveal the nonlinear properties of the Chinese stock market, and the nonlinearity perspective will be more conducive to the formulation of countermeasures for the development of the Chinese stock market.

The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc. The new elements are non-differentiable and may serve as models for pseudo-random behaviour. The proposed fractal functions have good algebraic properties and good approximation properties as well. In the present paper it is proved that they constitute bases for the most important functional spaces as, for instance, the Lebesgue spaces (1 ≤ p < ∞), where I is a compact interval in the reals.

Natural data sets, such as precipitation records, often contain geometries that are too complex to model in their totality with classical stochastic methods. In the past years, we have developed a promising deterministic geometric procedure, the fractal-multifractal (FM) method, capable of generating patterns as projections that share textures and other fine details of individual data sets, in addition to the usual statistics of interest. In this paper, we formulate an extension of the FM method around the concept of "closing the loop" by linking ends of two fractal interpolating functions and then test it on four geometrically distinct rainfall data sets to show that this generalization can provide excellent results.

The fractal dimensions of real world objects are commonly investigated using digital images. Unfortunately, these images are unable to represent an infinitesimal range of scales. In addition, a proper evaluation of the applied methods that encompass the image processing techniques is often missing. Several mathematical well-defined fractals with theoretically known fractal dimensions, represented by digital images, were investigated in this work. The very popular Box counting method was compared to a new image pyramid approach as well as to the Minkowski dilation method. Effects from noise and altered aspect ratios were also considered. The new Pyramid method is quite identical to the Box counting method, but it is easier to implement. Additionally, the calculation times are much shorter and memory requirements are almost comparable.

This paper explores the three different methods to explicitly recognize the healthy and epileptic EEG signals: Modified, Improved, and Advanced forms of Generalized Fractal Dimensions (GFD). The newly proposed scheme is based on GFD and the discrete wavelet transform (DWT) for analyzing the EEG signals. First EEG signals are decomposed into approximation and detail coefficients using DWT and then GFD values of the original EEGs, approximation and detail coefficients are computed. Significant differences are observed among the GFD values of the healthy and epileptic EEGs allowing us to classify seizures with high accuracy. It is shown that the classification rate is very less accurate without DWT as a preprocessing step. The proposed idea is illustrated through the graphical and statistical tools. The EEG data is further tested for linearity by using normal probability plot and we proved that epileptic EEG had significant nonlinearity whereas healthy EEG distributed normally and similar to Gaussian linear process. Therefore, we conclude that the GFD and the wavelet decomposition through DWT are the strong indicators of the state of illness of epileptic patients.

The respiratory system of mammalians is made of two successive branched structures with different physiological functions. The upper structure, or bronchial tree, is a fluid transportation system made of approximately 15 generations of bifurcations leading to the order of 2¹⁵ = 30,000 bronchioles with a diameter of order 0.5 mm in the human lung.¹ The branching pattern continues up to generation 23 but the structure and function of each of the subsequent structures, called the acini, is different. Each acinus is made of a branched system of ducts surrounded by alveolae and play the role of a diffusion cell where oxygen and carbon dioxide are exchanged with blood across the alveolar membrane.² We show in this paper that the bronchial tree presents simultaneously several optimal properties of totally different nature. It is first energy efficient;^3-6 second, it is space filling;⁷ and third it is "rapid" as discussed here. It is this multi-optimality that is qualified here as magic. The multi-optimality physical characteristic suggests that, in the course of evolution, an organ selected against one criterion could have been later used for a totally different reason. For example, once energetic efficiency for the transport of a viscous fluid like blood has been selected, the same genetic material could have been used for its optimized rapidity. This would have allowed the emergence of mammalian respiration made of inspiration–expiration cycles. For this phenomenon to exist, the rapid character is essential, as fresh air has to reach the gas exchange organs, the pulmonary acini, before the start of expiration.

A 2D computer simulation method of random packings is applied to sets of particles generated by a self-similar uniparametric model for particle size distributions (PSDs) in granular media. The parameter p which controls the model is the proportion of mass of particles corresponding to the left half of the normalized size interval [0,1]. First the influence on the total porosity of the parameter p is analyzed and interpreted. It is shown that such parameter, and the fractal exponent of the associated power scaling, are efficient packing parameters, but this last one is not in the way predicted in a former published work addressing an analogous research in artificial granular materials. The total porosity reaches the minimum value for p = 0.6. Limited information on the pore size distribution is obtained from the packing simulations and by means of morphological analysis methods. Results show that the range of pore sizes increases for decreasing values of p showing also different shape in the volume pore size distribution. Further research including simulations with a greater number of particles and image resolution are required to obtain finer results on the hierarchical structure of pore space.

IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝ^d for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.