Abstract: This paper shows that the dendritic patterns formed by low-resistance channels in a river drainage basin are reproducible and can be deduced from a single principle that acts at every step in the development of the pattern: the constrained minimization of global resistance in area-to-point flow. The river basin is modeled as a two-dimensional territory with Darcy flow through a saturated heterogeneous porous medium with uniform flow addition per unit area. From one step to the next, small elements of the porous medium are dislodged and removed in ways that minimize the global flow resistance. The removed elements are replaced by channels with lower flow resistance. The channels form a dendritic pattern that is deterministic, not random. The finest details of this structure are sensitive to internal properties and external forcing, i.e. variations in the local properties of the flow medium, and the manner in which the total area-to-point flow rate varies as the structure develops. Remarkably insensitive to such effects are the basic type and rough size of the flow structure (channels versus no channels, dendrite, number of branches) and the minimized global resistance to flow.

Abstract: We propose a new way of evaluating the regularity of a graph of a function f. Our approach is based on measuring the growth rate of the lengths of less and less regularized versions of f. This leads to a new index, that we call regularization dimension, dim_R . We derive some analytical properties of dim_R and compare it with other fractional dimensions. A statistical estimator is derived, and numerical experiments are performed, which suggest that dim_R may be computed in a robust way. Finally, we apply the regularization dimension to the study of Ethernet traffic.

Abstract: A method to estimate the persistent behavior from a chaotic time series is proposed. Persistency means that here each value depends to some extent on the previous values and not only on the recent ones. The data were analyzed by means of Hurst's rescaled range method, i.e., R/S analysis (which was introduced by Mandelbrot and Wallis). The relation of the Hurst exponent to the self-affine and self-simialr fractal dimension is discussed.

Abstract: The large deviation multifractal spectrum gives important statistical informations on irregular measures. However it is difficult to estimate. In this paper, we propose two new definitions of the large deviation spectrum better adapted to the design of various estimators. They rely on the computation of the Lebesgue measure of the reunion of all intervals of same size whose coarse grain Holder exponent is equal to a Hoder exponent. In particular, we introduce the continuous large deviation spectrum for which we construct different estimators. We finally show some numerical results obtained on both deterministic and random synthetical signals.

Abstract: Importance of the influence of neighboring canopy gaps upon new gap creation has been clarified by the ecological study of a neotropical forest on Barro Colorado Island (BCI), Panama. A stochastic lattice model for the forest dynamics with interacting canopy gap expansion was introduced by Kubo et al. We give a theorem showing a condition that this model can be regarded as a stochastic Ising model, and that its stationary state is exactly given by a Gibbs state. Using this theorem, we obtain a Gibbs state which remarkably well approximates the real gap-size distribution in BCI.

Abstract: We calculate the box-counting dimension of a self-affine version of the Sierpinski triangle. This is done by investigating the singular values of the affine transformations. We also investigate multifractal features of self-affine measures supported by certain generalized Sierpinski triangles.

Abstract: Darwin's theory of evolution by natural selection revolutionized science in the nineteenth century. Not only did it provide a new paradigm for biology, the theory formed the basis for analogous interpretations of complex systems studied by other disciplines, such as sociology and psychology. With the subsequent linking of macroscopic phenomena to microscopic processes, the Darwinian interpretation was adopted to patterns observed in molecular evolution by assuming that natural selection operate fundamentally at the level of DNA. Thus, patterns of molecular evolution have important implications in many fields of science. Although the evolution rate of a given gene seems to be of approximately the same order of magnitude in all species, genes appear to differ in rate from one another by orders of magnitude, a fact which standard theory does not adequately explain. An understanding of the statistics of rates across different genes may shed light on this problem. The evolution rates of mammalian DNA, based on recent estimates of numbers of nonsynonymous substitutions in 49 genes of human, rodents, and artiodactyls, are studied. We find that the rate variations are better described by lognormal statistics, as would be the case for a multiplicative process, than by Gaussian statistics, which would correspond to a linear, additive process. Thus, we introduce a multiplicative evolution statistical hypothesis (MESH), in which the theoretical explanation of these statistics requires the evolution of different substitution rates in different genes to be a multiplicative process in that each rate results from the interaction of a number of interdependent contingency processes.

Abstract: An image texture measure based on the box counting algorithm is evaluated for its potential to characterize human trabecular bone structure in medical images. Although bone images lack the self-similarity of theoretical fractals, bone images are candidates for characterization using fractal analysis because of their highly complex structure. The fractal based measure, herein called the box counting dimension (BCD), is an effective dimension, and does not imply an underlying fractal geometry. The importance of resolution in quantifying bone characteristics using the BCD is addressed. The relationship of BCD to standard measures of trabecular bone structure is also analyzed. To evaluate the variability of the BCD with change in resolution, the BCD is determined for two sections from each of seven 3D X-ray Tomographic Microscopy (XTM) images of human radius bone specimens, while the resolution is varied using lowpass filtering. An automated method of choosing the range of scales for the fractal analysis curve regression is used. The relationship of BCD to trabecular bone width and spacing is analyzed both for the XTM images and for simulated images representing idealized structures. The range of BCD values is 1.21–1.54. Variation in BCD over a range of resolutions is found to be small compared to the variation in BCD between different bone specimens. Maximum change in BCD over a large range of resolutions (17.60–176 microns per pixel) is 0.08. BCD decreases as space between trabeculae increases. Fractal based texture measures may potentially allow clinical monitoring of changes in bone structure — for example, using Magnetic Resonance Imaging at 150–200 micron resolution.

Abstract: The Fraunhofer region of one-dimensional fractal diffraction grating is studied when this grating is defined using a set of scaled periodical (and rectangular) functions. The fractal structure is obtained through a Cantor density function, which is defined from an incremental product of such periodical components. Each of these components can be filtered individually and so, the corresponding intensity pattern can be evaluated. A mathematical foundation for such decomposition is developed using the concept of characteristic (or indicator) function of a set. Graphic results are shown and the partial self-similarity of lateral intensity distribution calculated in any transversal plane to the optical axis for a certain value of magnification. With these results, fractal, periodical and intrinsic periodical domains can be distinguished to characterize the intensity patterns. The self-similarity of intensity distributions and the contrast of that function are calculated for different states of the Cantor transmittance.

Abstract: The physics of terrestrial radioactive contamination is complex, primarily due to the mixtures of radionuclides, their migration, the effects of soil chemistry, and the prevalent climate. Consequently, application of classical techniques is inappropriate in modeling such an environment. The classification of territories contaminated with radioactive isotopes is vital in establishing a detoxification plan. Application of the fractal geometry provides a good insight and helps in such classification. Several techniques are implemented to determine geometric aspects of contaminated fields, and these show a way to differentiate between natural and man-made isotopes.

Abstract: In this paper we present a study of the variation of morphology of polyethylene oxide (PEO) films complexed with ammonium perchlorate (NH4ClO4) formed by evaporating a methanolic solution. Films with salt concentration x = 0 - 0.35 (x is the weight fraction of salt) were prepared for the morphology study. The wide varitey of structures obtained were systematically studied by photographs and polarizing microscopy. Some preliminary studies of x-ray diffraction, differential scanning calorimetry and variable temperature polarizing microscopy have also been done to identify the various phases present in the films.

Abstract: We plot the two-dimensional projections of the parameter spaces of the cubic mappings. The projection of the parameter points that have non-totally disconnected Julia sets can be seen as a combination of Mandelbrot-like sets. The regularities of these projections with respect to parameters are explained using elementary analysis.

Abstract: We propose the application of a recent vector space technique for the analysis of DNA sequences In this approach the similarity of two series of symbols is measured by the dot products of vectors corresponding to the given sequences In this paper we demonstrate that this measure, depending on the short-range properties of the sequences, may be used for identifying different elements of the genome

Abstract: The self-similarity properties of the functions (closed relations) associated with one- and two-sided cellular automata are studied. It turns out that these functions are generated by sequential machines, and their graphs are fractal sets generated by hierarchical iterated function systems. The Hausdorff dimensions of the graphs is one for one-sided cellular automata and two for two-sided automata.

Abstract: We present the formation of unique and striking dendritic "seahorse" patterns in the growth of fullerene-tetracyanoquinodimethane (C60-TCNQ) or pune TCNQ thin films. The films were fabricated by an ionized-cluster-beam deposition method. Energetic neutral and charged clusters were deposited on amorphous carbon substrates. Transmission electron microscopy reveals that the elemental pattern is a "seahorse" — that is, an S-shaped form, with "fins" on the outer edges of the curved arms forming the S. Such forms possess an approximate symmetry under rotations by π, but strongly break two-dimensional inversion symmetry. A novel formation mechanism is proposed, involving the aggregatio of neutral and charged clusters, such that some electrostatic charge is trapped on each growing island. This charge gives rise to a long-range field which biases the growth in a nontrivial way. The broken symmetry arises from the strong amplification of noise in the diffusive aggregation process by the effects of the electro-static field — that is, the symmetry breaking is spontaneous. This picture is tested by applying a transverse electric field during growth: for sufficiently strong fields, the S-shaped "seahorses" lose their curvature, while retaining the feature of having two main arms. These results demonstrate the importance of electrostatic effects in the growth process, and are consistent with the growth mechanism described here.

Abstract: The timescales which govern urban pollution processes are investigated by analyzing variance spectra and structure functions of observational time series. The range of analyzed scales stretches from one hour to several days. It is shown that characteristic fluctuations of CO, NOx (primary pollutants) and O3 (secondary pollutant) follow a scale invariant law up to timescales of about one day. Scaling exponents indicate the presence of stabilizing feedback mechanisms. Such a scale invariance is broken by the appearance of basic periods which, for primary pollutants, are expressions of traffic dynamics, whereas, for ozone, are closely linked to the diurnal and annual solar cycles.

Abstract: The magnetic field effect on the silver deposition pattern generated from Ag+/Cu redox reaction is simulated with the aid of a biased random walk model. In the model. In the model, one particle that represents an Ag+ ion is generated at the same time, and it walks randomly under the influence of a magnetic force. Comparing the simulated pattern with the experimental one, it is confirmed that the convection induced by the magnetic force contributes chiefly to the silver deposition pattern in a high gradient magnetic field.

Abstract: A simple first-order recurrence in a (max, +) dynamic system is numerically investigated and shown to exhibit statistical long-range dependence, characterized by slowly decaying aggregated variances and power-law evolutions of the autocorrelation and spectrum. We propose this model as a basis for a very parsimonious modeling of some long-range dependent processes such as data traffic.

Abstract: According to the standard diffusion equation, by introducing reasonably into an anomalous diffusion coefficient, the generalized diffusion equation, which describes anomalous diffusion on the percolating networks with a power-law distribution of waiting times, is derived in this paper. This solution of the generalized diffusion equation is obtained by using the method, which is used by Barta. The problems of anomalous diffusion on percolating networks with a power-law distribution of waiting times, which are not solved by Barta, are resolved.

Abstract: We use the Discrete Wavelet Transform in order to study the power spectrum of data obtained in magnetoencephalographical measurements. α-wave phenomenon is found to occur independently 1/fβ noise, which is present over almost all channels. the β value is close to 1.

Abstract: The fluctuations of air transmittency over small time scales (20–500 s), in the presence and the absence of fog, are analyzed from different points of view: Fourier and wavelet transforms, multiscaling exponents, the multifractal spectrum and structure functions. All results indicate a multifractal structure associated with intermittency, and whose characteristics do not seem to change with the level of visibility. The origin of this multifractality, whether statistical or dynamical, cannot be established with certainty: there are however indications in favor of the presence of chaos.

Abstract: Numerical and statistical methods are used to analyze and classify computer programs. Both computer source code and object files are examined. The results of the application of the Pearson and Spearman correlation methods to the source code are coupled with the random walk model applied to the binary code. One of the practical consequences of the analysis is the ability to quantify the degree of similarity between different computer programs and, hence, identify cases of plagiarism.

Abstract: Polyurethane foam has been examined extensively because of its use in many implantable devices. The influence of different environments on polymer structure has been investigated via a number of clinical studies, animal models, in vitro experiments and mathematical models to predict biomaterial behavior. In this work, we employed image analysis and fractal geometry in three models of polymer degradation. In the in vivo and in vitro degradation models, we observed structural collapse and thinning of the polymer backbone, which resulted in a decrease of fractal dimension. Also, an original graphic model was developed. This computer-simulated erosion model was generated by pixel withdrawal from images. A relative degree of erosion is attributed each time an erosion process is applied. Thus, these degrees of erosion can be associated with a general time-related operation. With the computer-simulated erosion model, an unambiguous relationship was demonstrated between the estimation of fractal dimension and the relative degree of erosion. Since such models are based on a loss of mass, the results of fractal dimension were plotted against image surface, allowing us to compare the three different sets of experiments. In all three models, a proportional relationship was found between fractal dimension and image surface. Indeed, the erosion process was well-characterized by decreased fractal dimension since it is a mass dimension. Finally, we showed that the combination of a graphic computer model and a box-counting algorithm is reliable for the quantification of surface erosion.

Abstract: Entropy spectrum of self-affine fractal interfaces created by contraction maps is investigated. Interfaces are created by a single (or multi-) generator(s) all of whose segments have the same anisotropy of scaling and different scaling factors. The whole interface is decomposed into many subsets and h (topological entropy), λ (decay exponent), DD (divider dimension) and DB (box dimension) of each subset are calculated. Entropy spectrum is obtained from the relation between h and λ. We find that H (Hurst or roughness exponent), DD and DB for only the remaning subset at n =∞ are related as and DB = 2 - H.

Abstract: We present an algorithm, an extension of Lavaurs' algorithm for z2+c, for determining the combinatorial arrangement of the hyperbolic components of the generalized Mandelbrot set for zn+c. Several external rays are plotted to support the algorithm.