Fractals

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.

While frequency-based methods for the characterization of fractals are popular and effective in many applications, they have limitations when applied to irregularly shaped images, such as nuclear images. The irregularity renders texture characterization by frequency domain methods, based upon Fourier transform, problematic. To address this situation, this paper presents an algorithm based upon the signal analysis in the spatial domain. An autocovariance function can be estimated regardless of the shape and size of regions where the image is defined. As in the continuous fractional Brownian motion (FBM) that results from inputting white noise into a specific fractional integral system, a discrete FBM can be related to white noise by a specific fractional summation system (FSS) that is linear, causal and shift-invariant. Although the method of direct sampling is not valid for converting a continuous fractional integral to a discrete fractional summation, discrete fractional summations similar to the sampled system functions can be obtained through an iterative process. While the continuous system function of a fractional integral is linear in the frequency domain when plotted in log-log scales, unfortunately, it is not true for the comparable discrete system function. The discrete system function is actually approximately linear in the log-log scales over a very limited range. The slope of the straight line that approximates the function curve in the mean-square-error (MSE) sense in a specific time range provides a description of the autocovariance function that reveals the statistical relations among the local textures. Applications to characterization of ovary nuclear images in groups of normal, atypical and cancer cases are studied and presented.

The use of multifractals in heterogeneous chemistry has proven useful in the characterization of some complex phenomena. This paper proposes a multifractal analysis in order to study the distribution of adsorbed particles for a reaction of a monomer and a dimer that dissociates, both of which are adsorbed on a surface and react. For a range of a control parameter, there is a steady reactive window that changes when adsorbate-adsorbate lateral interaction is considered. τ(q) and f(α) for different values of control parameter and interactions are calculated, studying distribution of adsorbed particles in relation to multifractal properties.

One of the most recent results in empirical finance is the achievement of an accuracy gain for the volatility estimates from the use of high-frequency samples. In many applications, for instance, the observed intra-day data become the best informative source of estimation for the daily volatilites. This information set, or filtration in probability terms, can in theory be optimally exploited while adopting an increasingly finer sampling rate, a fact which finds justification when assuming a stochastic characterization of the problem in terms of so-called semimartingales. These instruments are very useful for representing asset price dynamics, and in recent proposals supply realized and integrated volatility measures. The former volatility is the empirical approximation of the latter, an average of high frequency returns observed within a certain time frame. Semimartingales become suitable model tools by allowing for the quadratic variation principle to hold. This in turn means that the convergence of the realized to the integrated volatility can be verified; conversely, from both theoretical and experimental standpoints, the cumulative squared high frequency returns represent consistent estimators of the integrated volatility measure. The goal of this work is twofold: first, to show with simulations the quality of the convergence for time-based estimators, compared to that obtained when time-scale coordinate wavelet tranforms are considered. Second, to verify that special families of wavelet decompose returns and allow for multi-scaling features to be revealed, together with the possible presence of underlying nonlinear dynamics of stock index return volatility.

Scaling in random time series is a novel way of gaining insight into mechanisms of complex systems. Scaling parameters characterize searches for the presence of "essential" uncertainty in outcomes of complex systems. The presence of this type of uncertainty in the emergency system (ES) can point to unusual ways of relieving the financial burden to the health system and improve the public health. The objective of this paper is to test for the presence of "essential" uncertainty in an outcome variable of the ES, i.e. emergency-ward length of stay (EWLS). An inverse power law (IPL) function fit to data can assess "essential" uncertainty. The study of EWLS has been undertaken in ten large Quebec hospitals. We find that all hospital EWLS are well fit by an IPL, as determined by an aggregated allometric relation. The presence of "essential" uncertainty is further confirmed by stratifying the EWLS according to proxies of disease complexity and severity at entry in emergency ward. Results point out that the various hospital dynamic systems embed "essential" uncertainty to various degrees. We conclude that intervention to reduce hospital health care costs must be centered on the interaction and feedback characterizing the ES processing of the patients.

We study the singularities of a temperature profile obtained by means of balloon measurements in the troposphere and lower stratosphere. The data give the evolution of the temperature as the altitude of the probe increases. We compare the scaling exponents deduced from the Wavelet Transform Modulus Maxima (WTMM) method and the structure function method. In the lower stratosphere, the variations in the multifractal properties with the altitude deduced from wavelets allow us to detect thin layers of about 200 m depth exhibiting atmospheric turbulence.

The propagation of a laser beam through turbulent media is modeled as a fractional Brownian motion (fBm). Time series corresponding to the center position of the laser spot (coordinates x and y) after traveling across air in turbulent motion, with different strength, are analyzed by the wavelet theory. Two quantifiers are calculated, the Hurst exponent, H, and the mean Normalized Total Wavelet Entropy, . It is verified that both quantifiers give complementary information about the turbulence.

White blood cells slowly roll along the walls of blood vessels, due to the coordinated formation and breakage of chemical selectin-carbohydrate bonds. Using detailed computer simulations of cells rolling on a selectin surface under flow, we show the time series of the cell translational velocity to be fractal in nature over time scales ranging from 2²–2¹¹ ms. A rescaled range analysis was performed to determine the Hurst exponent of the velocity time series, for simulations of cells rolling on either a uniform or punctate distribution of P-selectin molecules. The rolling behavior was found to exhibit two very distinct regimes, with a negative Hurst exponent ranging from -(1.2-0.6) over time scales of 2³-2⁷ ms, and a positive Hurst exponent of +0.47±0.03 over time scales of 2⁷-2¹¹ ms. The short-time Hurst exponent was found to be a strong function of the molecular distribution and also a function of average molecular density, while the long-time Hurst exponent was unchanged over all conditions studied. The implication is that the short-time adhesive behavior of cells interacting with a reactive surface is sensitive to the spatial arrangement of molecules, and the total number of molecules on the surface.

We introduce a new model of the generation of pre-seismic electromagnetic emissions, which explains the observed phenomenology in terms of its geometry and fractal electrodynamics. Accumulated evidence indicates that an earthquake can be viewed as a critical phenomenon culminating in a large event that corresponds to a type of critical point. The principle feature of criticality is the fractal organization in both space and time. Earthquakes display a complex spatio-temporal behavior: in addition to the regularity in the rate of occurrence (e.g. Gutenberg-Richter law, Omori law), the spatial distribution of epicenters is fractal and earthquakes occur on a fractal structure of faults. Thus, the hypothesis that the fault develops as a fractal is reasonable. A mounting body of laboratory evidence suggests that micro-fracturing of rocks are associated with the appearance of spontaneous charge production and transient electromagnetic emissions (EME). The emitting, diffusing and recombination charge accompanying the micro-fracturing, can act as current generated during the crack opening. In this view, an active crack or rupture, can be simulated by a "radiating element." The idea is that a fractal geo-antenna (FGA) can be formed as an array of line elements having a fractal distribution on the ground surface as the critical point is approached. We test this idea in terms of fractal electrodynamics: we argue that the precursory VLF-VHF EM signals associated with recent earthquakes in Greece are governed by characteristics (e.g. scaling laws, temporal evolution of the spectrum content, broad band spectrum region and accelerating emission rate) predicted by fractal electrodynamics.