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Of fundamental importance to wave propagation in a wide range of physical phenomena is the structural geometry of the supporting medium. Recently, there have been several investigations on wave propagation in fractal media. We present here a renormalization approach to the study of reaction-diffusion (RD) wave propagation on finitely ramified fractal structures. In particular we will study a Rinzel-Keller (RK) type model, supporting travelling waves on a Sierpinski gasket (SG), lattice.

A simple method is proposed to determine the surface fractal dimension of swelling clay minerals. When swelling clay mineral is immersed into the water solution, swelling deformation and swelling pressure will occur. The swelling deformation and swelling pressure are proportional to the amount of water volume absorbed by swelling clay minerals. The water volume absorbed by swelling clay minerals is related to its surface fractality. The correlation of the normalized water volume by the volume of swelling clay minerals (V_w/V_m) to vertical overburden pressure (p) is obtained as V_w/V_m = Kp^D_s-3, from which the surface fractal dimension of swelling clay minerals can be estimated. The surface fractal dimension (D_s) of Wyoming bentonite is 2.64, obtained from the swelling tests. The surface fractal dimension of Wyoming bentonite determined from the swelling tests nearly equals to that obtained from the nitrogen adsorption isotherms, and is larger than that measured using the mercury intrusion porosimetry.

During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal trees are examined by numerical simulations. Schemes for such generalizations of these geometries to vary their fractal dimensions are presented. These variations are found to have a direct influence on the primary resonant frequency, the input resistance at this resonance, and ratios resonant frequencies of these antennas. It is expected that these findings would further enhance the popularity of the study of fractals.

Database Tomography (DT) is a textual database analysis system consisting of two major components: (1) algorithms for extracting multi-word phrase frequencies and phrase proximities (physical closeness of the multi-word technical phrases) from any type of large textual database, to augment (2) interpretative capabilities of the expert human analyst. DT was used to obtain technical intelligence from a Fractals database derived from the Science Citation Index/Social Science Citation Index (SCI). Phrase frequency analysis by the technical domain experts provided the pervasive technical themes of the Fractals database, and the phrase proximity analysis provided the relationships among the pervasive technical themes. Bibliometric analysis of the Fractals literature supplemented the DT results with author/journal/institution publication and citation data.

Different forms of diffusion equations on fractals proposed in the literature are reviewed and critically discussed. Variants of the known fractional diffusion equations are suggested here and worked out analytically. On the basis of these results we conclude that the quest: "what is the form of the diffusion equation on fractals," is still open, but we are possibly close to obtaining a satisfactory answer.

Infantile hypertrophic pyloric stenosis (IHPS) is a common surgical condition of unknown etiology. The oral mucosal vascular networks of IHPS patients (n=25) and their unaffected parents showed lower blood vessel-free areas, as well as higher box-counting dimensions at two box size scales [D(1–46), D(1–15)], and relative Lempel-Ziv values (P<0.000001), as compared to those of gender- and age-matched controls. These findings may provide a useful phenotypical marker for identifying couples potentially at risk for the birth of an affected infant, while supporting the importance of a genetic component in this condition.

An analysis of multifractal characteristics of the potential time series of electrochemical noise generated during surface corrosion is performed. The generalized Hurst exponent, called H_q, is computed for potential fluctuations induced for steel corroding by a sulphate reducing bacterial consortia. The results indicate evidence of multifractal behavior in the sense that the qth-order (roughness) Hurst exponent H_q varies with changes in q. This multifractal structure is corroborated by means of microscopy inspection of the corroded surface.

We present a method of construction of vector valued bivariate fractal interpolation functions on random grids in ℝ². Examples and applications are also included.

We characterize the path length set of asymmetric binary fractal trees in terms of the scaling ratios, r and ℓ. We show that if r + ℓ < 1, then the path length set is a Cantor set, and if r + ℓ ≥ 1, then the path length set is an interval.

Hierarchically modular systems show a sequence of scale separations in some functionality or property in addition to their hierarchical topology. Starting from regular, deterministic objects like the Vicsek snowflake or the deterministic scale free network by Ravasz et al., we first characterize the hierarchical modularity by the periodicity of some properties on a logarithmic scale indicating separation of scales. Then we introduce randomness by keeping the scale freeness and other important characteristics of the objects and monitor the changes in the modularity. In the presented examples, a sufficient amount of randomness destroys hierarchical modularity. Our findings suggest that the experimentally observed hierarchical modularity in systems with algebraically decaying clustering coefficients indicates a limited level of randomness.

A family of the spread harmonic measures is naturally generated by partially reflected Brownian motion. Their relation to the mixed boundary value problem makes them important to characterize the transfer capacity of irregular interfaces in Laplacian transport processes. This family presents a continuous transition between the harmonic measure (Dirichlet condition) and the Hausdorff measure (Neumann condition). It is found that the scaling properties of the spread harmonic measures on prefractal boundaries are characterized by a set of multifractal exponent functions depending on the only scaling parameter. A conjectural extension of the spread harmonic measures to fractal boundaries is proposed. The developed concepts are applied to give a new explanation of the anomalous constant phase angle frequency behavior in electrochemistry.

Recent research indicates the presence of increased vascular density and irregularity on oral mucosal vascular networks in extracellular matrix (ECM)-related illness or conditions. Here, we estimated the frequency of occurrence of nodes of various degrees (K3, K4 and K5, where Kn designates a node with n connections) in patients with proven or suspected ECM-related conditions and in controls. Subjects with ECM-related conditions exhibited lower K3 and higher K4 frequency than controls (p < 0.0001) in their vascular networks. Inverse statistical correlations between the local fractal dimension and L-Z values and percentage of K3 (Pearson's r values range: -0.91 to -0.81; p values range: 0.0013 to < 0.0001), together with a positive relationship with K4 were observed (r values range: 0.81 to 0.86; p values range: 0.0015 to 0.0003). A positive correlation coefficient between D(1–46) and K5 frequency was also found (r = 0.6334, p = 0.027). K3 ≥ 52% or K4 <28% discriminated ECM patients from controls with 100% sensitivity (true positive cases to true positive + false negative ratio) and specificity (true negative cases to true negative + false positive ratio). These findings suggest that node degree distribution in oral vascular networks could be a helpful new marker of pathological conditions associated with proven or suspected ECM abnormalities.

In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.

The origin of fractal patterns is a fundamental problem in many areas of science. In ecological systems, fractal patterns show up in many subtle ways and have been interpreted as emergent phenomena related to some universal principles of complex systems. Recently, Lévy-type processes have been pointed out as relevant in large-scale animal movements. The existence of Lévy probability distributions in the behavior of relevant variables of movement, introduces new potential diffusive properties and optimization mechanisms in animal foraging processes. In particular, it has been shown that Lévy processes can optimize the success of random encounters in a wide range of search scenarios, representing robust solutions to the general search problem. These results set the scene for an evolutionary explanation for the widespread observed scale-invariant properties of animal movements. Here, it is suggested that scale-free reorientations of the movement could be the basis for a stochastic organization of the search whenever strongly reduced perceptual capacities come into play. Such a proposal represents two new evolutionary insights. First, adaptive mechanisms are explicitly proposed to work on the basis of stochastic laws. And second, though acting at the individual-level, these adaptive mechanisms could have straightforward effects at higher levels of ecosystem organization and dynamics (e.g. macroscopic diffusive properties of motion, population-level encounter rates). Thus, I suggest that for the case of animal movement, fractality may not be representing an emergent property but instead adaptive random search strategies. So far, in the context of animal movement, scale-invariance, intermittence, and chance have been studied in isolation but not synthesized into a coherent ecological and evolutionary framework. Further research is needed to track the possible evolutionary footprint of Lévy processes in animal movement.

We give a characterization of the convex hull of self-similar sets in ℝ³ which extends the results of Panzone¹ in ℝ². As an application, we show a convex set which cannot correspond to the convex hull of a self-similar set.

We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper-triangular matrices. Using methods from linear algebra, we obtain explicit formulae for the dimensions that are valid in many cases.

The Horton laws of stream numbers and magnitudes are proved in the limit of large network order for the broad class of Tokunaga model of river networks. Tokunaga model is built on the assumption of mean self-similarity in the side tributary structure, and an additional assumption of Tokunaga self-similarity, which is supported by data from real networks. Tokunaga model is gaining increasing recognition in the recent literature, because data supports several predictions of the model.

In this paper we study a model of skin cancer (MM) in vitro, using geometry of fractals as the method of analysis. The fractal dimensions of moles (skin cancer cells) growth pattern have been measured by using the methods of Box-counting method (D_B) and Sausage method (D_S). The cell growth of this cancer can be modeled by Hidden Markov model (HMM) and percolation model which are depending upon the time complexity. From these models we can find the shape of the irregularity border by using the probability distribution of the cells. The variation in the irregular border of the skin cancer has been found out using ANOVA test and cell's compactness. The fractal approach led to very promising results which improved the determination and examination of the stage of skin cancer.

Let X = {X(t, x), t ∈ ℝ, x ∈ ℝ^d} be a Lévy-based spatial-temporal random field proposed by Barndorff–Nielsen and Schmiegel¹ for dynamic modeling of turbulence. We describe some fractal geometry for this field, with a view toward a proper non-Gaussian aspect of Mandelbrot's paper.² Recent progress on multifractal scalings of the stationary exponential processes is also reported, and is toward the intermittency fields proposed in Barndorff-Nielsen and Schmiegel.¹

Intensive research on fractals began around 1980 and many new discoveries have been made. However, the connection between fractals, tilings and reptiles has not been thoroughly explored. This paper shows that a method, similar to that used to construct irregular tilings in ℜ² can be employed to construct fractal tilings. Five main methods, including methods in Escher style paintings and the Conway criterion are used to create the fractal tilings. Also an algorithm is presented to generate fractal reptiles. These methods provide a more geometric way to understand fractal tilings and fractal reptiles and complements iteration methods.