Abstracts/Titles of some of the accepted papers for Fractal 97:

Fractals, Memory and Levy Statistics in DNA Sequences

B. J. West, P. Allegrini, and P. Grigolini

The evolution of the probability density of a diffusing variable driven by a two-state process obeys an exact equation depending on the correlations function of the driver, that in the inverse-power-law case, leads to a Levy process. We impose such long-range correlation supplemented with a short-range randomization as a model of DNA sequences and find that the moments and the scaling exponents of our model are indistinguishable from those generated by a real DNA sequence. This makes it transparent how short-distance and long-distance regimes have different behaviors.

Crystal Pattern Formation Under Local Nonequilibrium Solidification

P.K. Galenko and M.D. Krivilyov

A model of local nonequilibrium solidification for solving the crystal pattern formation problem is developed. The main model equations include the mass conservation law, an evolution equation of a substance flux, and a phase interface motion equation. The numerical solution allows us to obtain a morphological spectrum of patterns, which are formed under different deviations from local equilibrium in the concentrational field and at the liquid-solid interface.

Fitting the Generic Multi-parameter Cross-over Model: Towards Realistic Scaling Estimates

Zbigniew R. Struzik, Edo H. Dooijes, Frans C.A. Groen

The primary concern of fractal metrology is providing a means of reliable estimation of scaling exponents such as fractal dimension, in order to prove the null hypothesis that a particular object can be regarded as fractal. In the context to be discussed in this contribution, the central question is what should be the minimum extent of the scaling range to give any meaning to the object's description as a fractal. Preceded by a short review of the motivations for the generic transition model, we present the straightforward extention of it to more free parameters. The price paid for fitting such a multi-parametric model is, however, not only computational expense but the danger of obtaining far from optimal fits. Deterministic cross-sections through the parameter space of the model are demonstrated to show insight into the sensitivity of the fitting procedure to parameter variations. Realistic confidence intervals obtained are demonstrated to allow for testing the fractality hypothesis on the base of globally uniform scaling (fractal dimension). This is demonstrated in both examples of non-linear fit to measurements on decreasingly lowered generation level deterministic pre-fractals and genuine human writing samples.

Fractal Characterization of Spatial and Temporal Variability in Site-Specific and Long-Term Studies

B. Eghball, R. B. Ferguson, G. E. Varvel, G. W. Hergert, and C. A. Gotway

Characterizing spatial and temporal variability is important in variable rate (VRAT) or long-term studies. This study was conducted to compare variability of soil nitrate and maize grain yield between nitrogen (N) treatments in a VRAT study and of crop yields in a long-term study. In the VRAT study, conventional uniform N application was compared with variable rate and variable rate minus 15% N. In the long-term experiment, continuous maize (Zea mays L.), soybean (Glycine max L.), and sorghum (Sorghum bicolor L.) were studied from 1975 to 1995. Semivariograms were estimated for soil nitrate and maize grain yield in 1995 in the VRAT and for grain yields of the crops in the long-term study. The slope of the regression line of log semivariogram vs. log lag (distance or year) was used to estimate fractal dimension [D = (4 - slope)/2], which is an indication of variability pattern. The intercepts of the log-log lines, which indicate extent of variability, were also compared between treatments. In the VRAT study, spatial variability of soil nitrate or grain yield was not significantly different among N treatments, even though VRAT N application resulted in a more homogeneous variability than uniform N application. Spatial variability of grain yield (D=1.71) was lower than soil nitrate (D=1.92). In the long-term study, maize had significantly less temporal yield variability than soybean or sorghum which were similar. Temporal variability was much more dominant than spatial variability in this study.

Fractal Patterns of Scalars Advected by Temporally Irregular Fluid Flows: The Random Map Approach

E. Ott and T. M. Antonsen

Temporally irregular, large spatial scale, fluid advection of passive tracers occurs in a wide variety of situations. Our main point is that these situations can be conveniently conceptualized as resulting from successive application of a sequence of random maps. This viewpoint is numerically convenient and also provides a useful theoretical framework for dynamical-systems-based analyses of the resulting fractal patterns. Examples of three different situations are discussed: (i) the fractal distribution of passive scalar gradients, (ii) the fractal pattern formed by a scum floating on the surface of a moving fluid, and (iii) the pattern of particles entrained as they flow past an obstacle in an open flow.

Dynamic Scaling of a Reaction Front in Porous Media

Jennifer M. Register and T. Gregory Dewey

Advances in the understanding of dynamic roughening at interfaces have had a significant impact on the analysis of fluid flow. To date, most efforts have focused on determining scaling exponents in simple experimental settings and modeling them with non-linear diffusion equations such as the Kardar-Parisi-Zhang (KPZ) equation. In the present work, flow of a fluid that is undergoing a chemical reaction is studied. The reaction front is created by flowing a reactant solution into a media that contains a second reactant species. The reaction of an acidic buffer solution with a basic adsorbent on porous media (alumina) is investigated. The diffusive flow of the chemical reaction front was visualized by absorbance changes of a pH indicator. The spatial and temporal scaling exponents of the leading edge, width and trailing edge of the reaction zone were determined. The system is modeled by including a chemical reaction term into the KPZ equation. The scaling behavior of the resulting coupled non-linear differential equations are investigated. The overall flow of the solution can be treated as a KPZ equation with quenched noise. However, the reaction width is dominated by temporal scaling and follows an Edwards-Wilkinson (EW) equation. This work provides a first step toward understanding chemical reactions in the complex settings that often occur in nature.

Multiscaling in Random Cluster Aggregates

V. Markel, V. M. Shalaev, E. Y. Poliakov, and T. F. George

Two-point density correlation functions are studied numerically on computer-generated three-dimensional lattice cluster-cluster aggregates with up to 20,000 particles. The ``pure'' aggregation algorithm is used, where subclusters of all possible sizes are allowed to collide. We find that large cluster-cluster aggregates demonstrate pronounced multiscaling, i.e., the power-law exponents in the pair-correlation function p(r) are not constants, but depend on r and the number of particles in a cluster. In particular, the fractal dimension determined from the slope of the two-point correlation function at small distances differs from that found from the dependence of the radius of gyration on the number of monomers (1.8 and 2.0, respectively, according to our data). We also consider different functional forms of p(r) and their general properties. We find that if the fractal dimension for the cluster-cluster aggregates can be defined as a continuous function D=D(r/R_g), where R_g is the radius of gyration, it must have a maximum at some value of r/R_g=x_m, where D(x_m)>2.

Efficient ray tracing of complex natural scenes

Christoph Traxler and Michael Gervautz

In this paper we present a method for the consistent modeling and efficient ray tracing of complex natural scenes. Both plants and terrain are modeled and represented in the same way to allow mutual influences of their appearance and interdependencies of their geometry. Plants are generated together with a fractal terrain, so that they directly grow on it. This allows an accurate calculation of reflections and the cast of shadows. The scenes are modeled with a special kind of PL-Systems and are represented by cyclic object-instancing graphs. This is a very compact representation for ray tracing, which avoids restrictions to the complexity of the scenes. To significantly increase the efficiency of ray tracing with this representation, an adaptation of conventional optimization techniques to cyclic graphs is necessary. In this paper we introduce methods for the calculation of a bounding box hierarchy and the use of a regular 3d-grid for cyclic graphs.

Scaling of Microcrack Patterns and Crack Propagation

F.M. Borodich

The similarity (scaling) properties of quasi-static propagation of a crack, when it is surrounded by a growing pattern of microcracks, are discussed. First we consider microcrack pattern growth using the fractal approach. We give some estimates for the total amount of absorbed energy of the pattern and also give a fractal interpretation of experimental results regarding the behaviour of fracture energy of concrete. Then a self-similar model of crack propagation based on the use of discrete group of coordinate dilations is presented. The model describes the case when a main crack extension is discontinous consisting of a sequence of finite growth steps (stick-slip regime). General expressions for changes of all parametric-quasi-homogeneous functions, giving the solutions to the problems during increasing of the external load are derived exactly without solving the field equations.

A study of the fractal structure of aggregates formed after heat-induced denaturation of beta-lactoglobulin.

Pierre Aymard, Dominique Durand, Taco Nicolai and Jean Christophe Gimel

Beta-lactoglobulin aggregates formed after heat-induced denaturation were studied by light and neutron scattering techniques over a range of wave vectors covering more than 4 decades. The results demonstrate that the aggregates have a fractal structure at larger length scales. The local structure of the aggregates depends on the pH and ionic strength of the solutions. The aggregates are polydisperse with a number distribution which has a power law dependence on the molar mass. The z-average structure factor is calculated by assuming streched exponential external cut-offs for the number distribution and the fractal structure. The calculated structure factors are in good agreement with the experimental data.

Smoothing Kinetics of Initially Fractal Grain Boundaries

P. Streitenberger, D. Foerster and P. Veit

The concept of fractal geometry is used to describe serrated and rugged grain boundaries in the pure Zinc and Titanium materials after deformation and heat treatment. The fractal dimension of the grain boundaries are determined by application of optical and scanning electron microscopy over a wide range of magnifications. Measurements of the coarsening kinetics of the initially fractal-like grain boundaries during isochronous as well as isothermal annealing are presented. The results of the annealing experiments can be explained by an analytic fractal coarsening model yielding the observed dependency of the time law of grain boundary smoothing on the initially fractal dimension of the grain boundaries. The results are supported by a Monte Carlo simulation of the smoothing process of single initially fractal grains.

Fractal Dimension of Simulated Traffic

J. Woinowski

Highway traffic can be simulated using cellular automata, as recently shown by M. Schreckenberg and others. This leads to space/time diagrams that have an obvious fractal outlook. One possibility to measure this fractal quality is the box dimension. This paper shows influences of traffic flow on the box dimension.

Levy Kinetics in Slab Geometry: Scaling of Transmission Probability

A. Davis and A. Marshak

We revisit a classic problem in kinetic theory, transport through a slab of thickness L, when particle free-path distributions have infinite variance but finite moments of order q 0 half-space, trajectories end in either transmission (z(n_T) >= L, n_T >= 0) or reflection (z(n_R) =< 0, n_R >= 1). The classic case is for exponentially distributed steps with mean-free-path l, leading to _e ~ (L/l)^2 and normalized flux (transmission probability) T_e(L) ~ (L/l)^-1. Numerical simulations of Levy/Gauss transport yield T_alpha(L) p.t. L^-alpha/2 whether l < infinity (1< alpha =< 2) or l = infinity (0< alpha =< 1). To derive this result from the standard scaling relation, p.t.n (recast here as _alpha p.t.L^alpha), zero-crossing events for the discrete-time Levy-stable process must be described by the correlation dimension D_2 = 1/2 (independent of alpha), rather than the better-known capacity (or Hausdorff) dimension D_f = max{1-(1/alpha), 0}. Applications to Earth's climatic equilibrium and cloud remote-sensing are discussed (including the effect of non-isotropic scattering described by kernels with forward peaks).

Monitoring fire-induced landscape changes in Mediterranean regions with a Fractal Algorithm.

Carlo Ricotta, Eric Olsen and Giancarlo Avena

In Mediterranean regions severe fires affect the landscape structure eliminating the spatial relationships among many preexisting patches with an intense homogenizing effect on the burned landscape. The dynamics of fire-induced landscape changes of a burned area can be monitored using remotely sensed data. The aim of this paper is to introduce a method to quantitatively estimate the fire-induced landscape changes in Mediterranean regions at the scale of the Landsat Thematic Mapper (TM) satellite based on a multitemporal analysis of the local variability of remotely sensed plant biomass data. The ability of the remotely sensed plant biomass texture data to enhance spatial variations in Mediterranean vegetation made it appropriate to the monitoring of fire-induced landscape changes at the scale of Landsat TM.

Fractal Dimension of Fracture Surfaces in Ductile-Brittle Transition Regime

Toshitaka Ikeshoji and Tadashi Shioya

The tensile fracture tests with notched round bar are conducted and the fractal dimension is calculated for the surface profile of the fracture surface. The specimens have various radii of notch and are made of 0.35% carbon steel and 0.55% one. The fractography shows the dimpled ductile fracture surface for both kinds of steel and brittle fracture surface with cleavage facet for 0.55% steel. The fractal dimension is about 1.4 for brittle fracture surface and about 1.3 for ductile one. The fractal dimension-absorbed energy plot shows the distinct two region corresponding to the brittle and ductile fracture.

Physical Simulation of High-Resolution Satellite Images for Fractal Cloud Models

A. Marshak, A. Davis, R. Cahalan, and W. Wiscombe

Based on fractal models for the horizontal distribution of cloud density, LANDSAT-type (i.e., 30 m resolution) radiance fields were simulated within the Nonlocal Independent Pixel Approximation (NIPA), an improved version of the Independent Pixel Approximation (IPA) widely used in remote-sensing. Scale-by-scale analyses (wavenumber spectra, structure functions, and singularity analysis) of liquid water variability inside stratus clouds indicate scale-invariance over three decades, from 9210 m to 9210 km. A simple two-parameter fractal cascade model reproduces the observed variability, thus capturing the rich turbulent structure in cloud density, hence optical thickness. IPA-based radiation fields of these models preserve scaling properties of fractal cloud models, at least for small moments; however LANDSAT cloud scenes show a characteristic scale (200-300 m) below which radiance fluctuations are much smaller. This is shown to be the effect of physical smoothing by horizontal photon transport. As a convolution of IPA field with gamma-type smoothing kernel, NIPA emulates this radiative smoothing and produces realistic Landsat-type images. Their statistical verisimilitude is checked with multifractal analyses. The simulations are graphically illustrated and compared with a real LANDSAT scene.

Fractal Characterization of the Surface of the Humin Fraction of Soil Organic Matter

Kalumbu Malekani, James A. Rice and Jar-Shyong Lin

Natural organic matter in soils interacts with surfaces of inorganic materials, primarily aluminosilicates, during the early stages of diagenesis to form an organo-mineral composite known as humin. Such composites typically represent >50 % of the organic carbon present in soils and sediments. Because of humin s insolubility it is recognized as the primary adsorbent of many anthropogenic organic compounds introduced into natural soil systems. Humin s insolubility and its high contaminant binding capacity have significant implications for the effective remediation of contaminated sites and the formulation and even application of various agrochemicals. Fractal analysis of small-angle X-ray scattering data was used to characterize the surface roughness of four humin samples following sequential removal of organic matter. The surface fractal dimensions were observed to decrease with the removal of organic matter which also resulted in a decrease in average surface pore size. These results suggest that the mineral components of humin have smooth surfaces over length scales of ~ 10-150 angstrons, and that it is the organic matter coatings which are responsible for their surface roughness. A physical model of humin/environment interface is proposed.

Fractal Properties of Soil Organic Matter

James A. Rice

Soil organic matter (SOM) is a heterogeneous assemblage of organic molecules that interact in a variety of ways with each other, with soil mineral surfaces and with soil mineral colloids. Because of SOM s heterogeneity it is very difficult to define its surface, or the surfaces of the composites produced by its interaction with soil minerals. Yet it is at these interfaces where chemical reactions that involve these materials are initiated. The physical heterogeneity of these surfaces can be quantified in either the solid-state or solution utilizing the fractal analysis of light and x-ray scattering data. Results will be presented which describe the fractal characterization of soil humic materials using x-ray scattering, dynamic light-scattering and static light-scattering experiments. Over the length scales studied, humic materials are surface fractals in the solid-state and mass fractals while in solution. These results will also show that in solution humic materials behave as either particles or polymers depending on the solution conditions. Applications of fractal analysis to the study of humic material aggregation and the study of organic coatings on mineral surfaces will be discussed.

Lie Groups and Solution of Dynamical Problems on Fractal Lattices

W. A. Schwalm, M. K. Schwalm and M. Giona

Lie theory is applied to recursion relations that arise from real space renormalization of dynamical problems on regular, finitely ramified fractals. The problems include diffusion, vibrations, the Schroedinger equation, spin waves, the linearized Landau-Ginzberg equation, etc. The recursion relations are systems of coupled, nonlinear difference equations. In many cases these discrete systems admit one or more continuous symmetries. A method of finding symmetry groups is introduced which depends on finding inverse images of the invariant sets of the recursion relations. In general, each such Lie group reduces the order of the recursion relations by one. The possibility of application to more general dynamical systems is considered.

Multifractal Decomposition for a Family of Overlapping Self-Similar Measures

Sze-Man Ngai

We discuss two methods for computing the multifractal dimension spectrum of a self-similar measure defined by a family of similitudes which does not satisfy the open set condition. Results are obtained by applying these methods to the measures defined by the well-known family of similitudes arising from the dilation equation in wavelet theory.

Smoothing Dimension Analysis - New Effective Tools in Fractal Signal Investigation

C. Ioana, F. Munteanu and C. Suteanu

Scale Invariance in Transitional Pipe Flow

F. Esposito, E. Nino and C. Serio

Using the tool of Extended Self-Similarity a hitherto undetected form of scaling for the structure functions of the velocity field has been revealedin transitional pipe flow. Velocity observations were recorded by a non-intrusive Laser Doppler Velocimetry system and cover the range of Reynolds numbers from 500 to 6000 (transition to turbulence and moderately developed turbulence). Although the nature of this form of scaling may be different from the celebrated Kolmogorovinertial-range scaling, it will be shown that the self-scaling properties of the structure functions are the same as those expected for fully developed turbulence. In addition, experimental evidence will be shown that the transition from puff to slug regime is characterized by a sort of bifractal state which is reminiscent of a second order phase transition.

Optical Filtering Properties of a Self-Similar Multilayer Structure

M. Bertolotti, P. Masciulli, F. Garzia and C. Sibilia

Resources allocation and bird distribution in a sub-mountain ecotone

Almo Farina

Environmental patternsproduced by non-uniform spatial and temporal distribution of resources is a central theme in ecology. At landscape scale where bounded patches with different characters are combined, the environment is perceived by organisms heterogeneous. The surfaces of contact between patches may be considered transitional habitats as well as ecotones. The distribution and abundance of birds in a montane ecotone (northern Italy), composed by scattered tree pastures, were utilized in a preliminary attempt to understand the spatial and temporal dynamic of these systems and to pattern resource allocation. The complexity of the patches of equal abundance (five categories: 1-5, 6-10, 11-15, 16-20, >20) produced by interpolating the spatial distribution of birds as well as the inter-seasonal overlap of these patches, have been measured regressing the log of perimeter with the log of area. Patches of low bird abundance, that probably indicate scarce resources, have higher fractal dimension (DAB). On the contrary high abundance patches have a lower DAB. During fall and winter period DAB decreases from the first to the third categories of abundance and increases again for higher abundance patches. This trend is less evident during the breeding season. The fractal dimension of the inter-seasonal abundance overlap decreases from the first abundance category (1.33) to the third category (1.26 ), then increases again reaching a peak in the fifth category (1.45). The highest value of DAB outside the breeding season should indicate a more fine-grained distribution of the resources but also a more similar feeding habitat of bird assemblage. During the breeding season bird distribution depends not only on food abundance but also on microsite availability where to locate the nests. Finally during the breeding season bird distribution is more difficult to be tracked because more diversified for the presence of migrant and permanent species and by a broader range of foraging guilds composed by frugivorous, granivorous and insectivorous species . From these preliminary results birds seem good indicators of distribution of resources and fractal analysis a promising tool to achieve this, especially outside the breeding season.

Analytical Explanation of a Phase Transition in the Multifractal Measure Connected with a One-dimensional Random Field Ising Model

H. Patzlaff, U. Behn and A. Lange

Scale Properties of the Diffraction Fields Induced by Pre-Fractal Random Screens

Dmitry A. Zimnyakov and Valery V. Tuchin

Correlation between Hausdorff dimension of the fractal amplitude and phase screens and scaling parameter of the far- and near-zone speckle intensity fluctuations is studied. Non-stationary speckle patterns are induced by the probe coherent beam diffraction on the moving screens; exponent of the structure function of the intensity oscillations is chosen as the scaling parameter. Relationships between this parameter and Hausdorff dimension of the studied objects are analyzed for different illumination and detection conditions. Some possible applications of the obtained results are discussed.

Modification of Carbon Cluster Fractal Structure Due to Capillary Forces

E. F. Mikhailov, S. S. Vlasenko, A. A. Kiselev, and T. I. Ryshkevich

The soot aerosol particles are known to have a porous low density structure that is effectively described as fractal. Therefore its physical and chemical properties exhibit strong correlation with the ambient factors, of which the water vapour presence in Earth's atmosphere is the most important. The laboratory study of soot properties on the base of fractal analysis combined with the control of chemical composition is presented in this report. The effect of the water vapour action on the fractal structure of the cluster was found to be in strong dependence on the cluster wettability, which in its own turn is highly sensible to the way of cluster origination. It was confirmed that layer of resins covering the surface of the natural soot particles is responsible for their liability to the vapour action. The special significance of charge localised on the branches of the cluster is demonstrated. On the base of suggested mechanism and experimental data the extent of fractal cluster restructuring due to the capillary condensation is evaluated.

Cantor Staircases in Physics. A Connection with Number Theory. Part 1.

M. Piacquadio and S. Grynberg

We analyze the Cantor staircase y=f(x), studied by Bruinsma and Bak, where x is the magnetic field H and y is the proportion of up-spins. We study the length of the stability stairsteps delta(H) arbitrarily close to a point (i',i) of the staircase, f(i')=i, an irrational number.
Any irrational number i in the interval (0, 1) can be expressed as an infinite continued fraction i=1/(a(1)+1/(a(2)+1/...)), a(i) natural numbers, which we denote (a(1),a(2),...,a(n),...), where (a(1),...,a(n))=Q(n)/P(n) is a rational number close to i. Irrational numbers are classified according to this closeness: it can happen, e.g., that the distance between i and Q(n)/P(n) diminishes as the inverse of the square of P(n), or as the cube, or as the k- power in general terms. We then say that i is in J(k), the k-Jarnik class of irrationals.
We found that the size of the stairsteps delta(H) arbitrarily near a point (i',i) of irrational height i depends on the J(k) to which i belongs. Concretely, the size of the largest delta(H) at epsilon- distance of (i',i), i in J(k), is a simple function of a power, whose base is the square root of epsilon, and whose exponent is the Hausdorff-dimension of the corresponding Jarnik class J(k) to which i belongs.
From this formula we deduce that the maximum value of the alpha- concentration of the Cantordust underlying the Cantor staircase is reached at the point i' such that f(i') is the golden mean (1,1,1,...). We recall that this is the case for the Cantordusts underlying the Cantor staircases y=f(x) studied by Procaccia, Halsey, and others, where x and y are time variables.

Phase Transitions Generated by Superposition of Multifractals with Different Supports in the Generalized Thermodynamic Formalism

H.-C. Tseng and H.-J. Chen

Anomalous Diffusion and Chaotic Scattering in a Nonlocal Model

S. Drozdz, J. Okolowicz, M. Ploszajczak, and T. Srokowski

Dynamics of fermions is studied in terms of the transport theory. It is shown that effects connected with antisymmetrization of the wave function increase chaoticity of motion by introducing an extra momentum dependence of the effective potential. Power spectral analysis indicates various types of anomalous diffusion which in presence of the nonlocality is, however, typically slower than for the corresponding local cases. At the same time the momentum dependence of the effective potential preserves the hyperbolic character of chaotic scattering.

Julia and Mandelbrot-like Sets for Higher Order Koenig Rational Iteration Functions

N. Argiropoulos, A. Boehm, and V. Drakopoulos

Koenig iteration functions K_sigma(z) are a generalization of Newton-Raphson's method, for which sigma=2. We give a simple algorithmic construction, to examine the orbits of all free critical points of the K_sigma(z) as applied to an one-parameter family of cubic polynomials and to examine the Julia sets of K_sigma(z) for increasing sigma, as applied to the cases f_n(z)=z^n-1, for n=2,3,4,..., with the help of microcomputer plots.

Hypercomplex Fractal Distance Estimation

Y. Dang and L. H. Kauffman

Seasonal Changes in Fractal Landscape Surface Roughness Estimated From Airborne Laser Altimetry Data

Yakov A. Pachepsky and Jerry C. Ritchie

Fractal geometry is a useful tool for the analysis of landscape data. Recently fractal scaling was applied to high resolution data from a profiling laser altimeter. Root-mean-square roughness (RMS) was scale-dependent and had more than one range of self-affine scaling. Distinctly different numbers of the self-affine scaling intervals, boundaries of intervals, and fractal dimensions over intervals were associated with different land covers. The objective of this work was to assess how persistent are these differences in scaling over a year. Data were collected at the USDA-ARS Jornada Experimental Range in New Mexico in May, September and February over grass-dominated, shrub-dominated, and a transitional area between shrub and grass-dominated sites along four transects at each site. A linearity measure was applied to find intervals of fractal scaling. The number and boundaries of fractal scaling intervals appeared to be persistent over the year. Grass and shrub transects had two and four linearity intervals, respectively. Transitional transects had a pattern of scaling that was intermediate between grass and shrub transects. The lowest fractal dimensions at small scales of 6-30 m corresponded to the maximum vegetation in September. In all three seasons, the ten meter scale was an appropriate one for discriminating between shrub and grass transsects by the fractal dimension of the linearity range that included this scale.

Scaling Analysis of the Critical Behavior of a Spin Chain Model

Nikita V. Dolgushev

Specific behavior of the heterogeneous systems caused by dividing interfaces has been attracting attention in recent years. It is well known that any crystal surface at any nonzero temperature has thermodynamically stable defects: steps, kinks, vacancies and adatoms. Such defects form non-planar crystal-surrounding interfaces. In this study we focus on the description of two-dimensional heterogeneous system interface by means of a spin chain model and discuss its cooperative behavior.

A spin chain model is considered for the description of one-dimensional objects such as the atomic steps on the crystal surface or the interface between 2D-phases. The real space renormalization group (RSRG) method was applied to analyze the spin chain critical behavior. It was found that the renormalization group's attractor undergoes a limit cycle bifurcation.

The bifurcation of the renormalization group mapping for pair correlation function divides the temperature interval into two regions:
- a low-temperature region with a single fixed point attractor
- a high-temperature region with limit cycle attractor and an unstable point.

As the renormalization group's attractor reflects the cooperative behavior of the spins in the chain, different types of attractors correspond to different spin chain organizations and, therefore, to different phases of the spin chain. Thus the bifurcation obtained can be interpreted in terms of a thermodynamic phase transition.

Relation Between Fracture Toughness and Fractal Dimension of the Crack Initiating Zone of Polycarbonate

Rachid Dekiouk, Frederic Bouyge, Zitoun Azari, and Guy Pluvinage

Any fracture surface can be considered as a fractal object and so caracterized by its fractal dimension. The shape of this surface is the direct consequence of the fracture mechanism which is related to the fracture toughness. Several relations between the fracture toughness and the fractal dimension have been proposed in the past but no local measurements of the fractal dimension have been done although the fracture initiation is purely a local process. We estimate the fracture toughness of polycarbonate by the means of the essential fracture work method and we determine the fractal dimension of the crack initiating zone by the means of image analisys based on Fourier transforms. We show that the fractal dimension increases when the fracture toughness decreases. We also notice two thresholds where the fractal dimension is quite steady: a first one which ranges from quasi-static strain rate to about 100/s and a second one for strain rates up to 150/s. This is explained by the differences in fracture mechanism involved as shown on polycarbonate constitutive law. Thus fractal dimension of fracture of polycarbonate seems to be related to fracture mechanism.

Estimation of Soil Water Retention Function From Texture and Structure Data: Fractal Approach

A. N. Kravchenko and R. Zhang

Scale-Dependant Soil Hydraulic Conductivity

R. Zhang

Comparison between DC Voltage Reference Source Time Series and Fractional Brownian Motion

Irena Nancovska, Anton Jeglic, Primoz Kranjec, Dusan Fefer

The time series under the examination are generated by precision voltage sources, in our case solid state voltage reference elements (VRE-s), of which a group DC voltage reference source (DCVRS) is composed. To characterize the DCVRS by the fact that the equations describing the system are not known, the rules governing the system must be found out. The measured time series which are mixtures of stochastic and deterministic components are characterized as deterministic chaos. Through this work we are trying to disentangle them (if possible) and to find out which of them is dominant. The phase space in which the full structure of the (possible) underlying attractor associated with the chaotic observations is unfolded is reconstructed by method of delays. The invariant properties of the dynamics such as correlation dimension and Lyapunov exponents are calculated to estimate the complexity of the underlying system. The white noise test performed on the measured signals showed the presence of colored noise. For purposes of comparison, we generate self-affine sequences that exhibit a power law spectrum of a form S(f) = f^(-a), 1 <= a <= 2, known as fractional Brownian motion (FBM). For some choices, the generated series have the same fractal properties as the measured signals. Beside this, simple tests differentiate between the measured series and FBM and confirm the thesis that some non-linear dynamical process influences the fluctuation of voltage.

Computer Aided Geometric Design with IFS Techniques

C. E. Zair and E. Tosan

Fractals represent powerful techniques for modeling complex irregular figures that cannot be described with classical geometry. However, they present many important restrictions in the case of shape modeling. Indeed, fractal figures generated with the aid of computers can not be manipulated efficiently. By analogy to free form techniques, we mean by manipulation the possibility to control the global shape of the fractal. The aim of our work concerns the definition of an IFS-based model which combines the advantages of fractals and free form techniques ones ( control using a set of control points). The use of IFS theory is motivated by the fact that subdivision techniques are common for both IFSs and free forms techniques. We are using this indication to demonstrate that subdivision algorithms for generating free form curves such as De Casteljau algorithm can be extended to IFS. We shall point out that a convenient choice of the complete metric space and the operators of the IFS allows the generation of fractal and smooth forms as IFS attractors.

Superconducting Fractal Nb/Cu Multilayers

A. Sidorenko, C. Surgers, T. Trappmann, and H. v. Lohneysen

Nb/Cu multilayers with several length scales (periodic, fractal, and "irregular fractal") have been prepared. The influence of the geometry on their superconducting properties is investigated by measurements of Tc and the temperature and angular dependence of the upper critical field Bc2. For low temperatures T << Tc, all samples show the characteristic behavior of two-dimensional superconductors independent of the stacking sequence, whereas for temperatures near Tc the type of layering determines the effective dimensionality, resulting in a "multi-crossover" behavior in fractal and irregular fractal samples.

Fractal Characterization of Root System Architecture in Legume Seedlings

J. Tatsumi and K. Takagai

Correlations between fractal dimension (D) and topology of root systems of legume plants grown in flat root boxes were studied. D increased rapidly throughout the experimental period (3 weeks) in the natural light conditions (control). High negative correlations were found between D and topological indices, a/E(a) and Pe/E(Pe), suggesting that increase in D was closely related to the alteration of root topology, from a simple branched herringbone type to a random branched type. This suggests that when roots develop under favorable conditions, D can be a good indicator for estimating the system size as well as the intricacy of root branching. Responses of roots to low light treatments revealed that low growth rate was closely associated with the gradual increase of D. In this situation D appeared to depend its change more upon the system size extension rather than the topological changes in root system architecture. We assume that D can be a useful tool for diagnosing root development.

Optical Properties of Self-affine Surfaces

V.M. Shalaev, E.Y. Poliakov, V.A. Markel, R. Botet, and E.B. Stechel

Optical properties of self-affine thin films are studied in the quasi-static approximation. The eigenmodes of a self-affine surface manifest strongly inhomogeneous spatial distributions and they are `sub-localized' on average. On a metal self-affine film, the intensities in areas of high local fields (`hot' zones) exceed the applied field intensity by approximately three orders of magnitude. The spatial locations of the `hot' zones are very strong functions of the frequency and polarization of the incident light. Surface-enhanced Raman scattering (SERS) from a self-affine surface is shown to be very large. A theory is developed expressing this SERS in terms of the eigenmodes of a self-affine surface; the theory successfully explains the observed SERS from cold-deposited thin films which are known to have a self-affine structure. `Hot' zones at the fundamental and Stokes frequencies are localized in nm-sized regions that can be spatially separated for the two waves. Nonlinear optical processes, such as second harmonic generation, also experience giant enhancements on a self-affine surface.

Binary Expansions and Multifractals

Manav Das

We study the multifractal decomposition of the closed unit interval, induced by the two maps that take the unit interval into its first and second halves. Let p, q be real numbers such that 0 < p < 1/2, q = 1-p. Assign probability p and symbol 0 to the first function (taking the interval into its first half), probability q and symbol 1 to the second function. Then we may obtain an invariant probability measure supported on the unit interval. For any point in the unit interval, and any open interval around this point, we may take the ratio of the logarithm of the measure of this interval to the logarithm of the length of the interval. If we take the limit as the length of the interval goes to zero, and if this limit exists, then we call this value the local dimension of the point under consideration, with respect to the probability measure constructed above. We characterize the set of points for which the local dimension exists, by connecting it explicitly with the frequency of occurrence of 0's and 1's in the dyadic expansion of a point in the unit interval. In particular, we show that the measure constructed above is supported on the set of these points.

Using the Interval Distribution of Level Sets Approach to Determine the Phase Transition Point Based on a Time Sequence Data

A.Yu. Tretyakov, H. Takayasu, and M. Takayasu

We show that the interval distribution of level-sets method can be used to determine the location of a critical point based on a time series data originating from an interacting particle system. The application of the method is demonstrated for the Contact Process in 2+1 dimensions and for a computer network following a deadlock prevention algorithm based on assigning to each packet a globally unique timestamp.

A Deterministic L-System to Simulate Leaf Area Index in A Faba Bean Canopy

A. Tarquis, C. H. Daz-Ambrona, and M. I. Mingues

In a closed canopy light interception depends on the incident solar radiation, the optical properties of the plant elements, the plant density and the canopy architecture. It is usually described by an adaptation of Beer's Law that assumes canopies to be homogeneus and continuous although during crop establishment or under water stress plants may be descretely distributed.
A deterministic Lindermayer system (L-system) was developed to simulate the growth of a legume (Vicia faba L. cv Alameda) and to calculate the leaf area index (LAI) during vegetative growth. This approach is based on fractal geometry and an individual plant is defined by growth rules derived from numerical files. A growth function based on L-system was constructed to describe the geometric structure (topology) and the physiological bechaviour of a faba bean plant during vegetative growth. The experimental data was collected on potted plants in a greenhouse during 1995. A function was then built in order to simulate LAI values considering a faba bean crop as a set of individual plants obtained with the L-system function.

Determination of Specific Surface of Soils with Nitrogen. The Fractal Dimension and Some Experimental Difficulties.

Roberto R. Filgueira(1), Guillermo O. Sarli, Agueda Piro, and Lidia L. Fournier

Specific surface of aggregates obtained from soil samples were investigated in order to estimate the fractal dimension using physical adsorption of nitrogen at low temperature (78 degree K). The resulting isotherms were interpreted with the model of Brunauer, Emmett and Teller (BET). The sand fraction was separated from the original sample and the remaining material processed to remove soluble salts, carbonates and organic matter. Aggregates of different size were separated by sieving and by sedimentation in water (Stokes). In a first set of experiments, the behaviour of the specific surface versus particle size (in the range 1 to 4000 microM), was investigated. It showed a curvilinear "anomalous" pattern and was not possible to reduce the results, in the whole range, to a straight line in a log-log representation. However, when only the range 1 to 14.5 microM was considered, it was possible to fit the data to a straight line in a log-log representation. From the slope, a fractal dimension D=2.79 was determined. In a second set of experiments, several samples were washed thoroughly to remove the clay fraction. The specific surface of particle size distribution in the range 1 to 32.5 microM was investigated. The plot of the specific surface values against particle size, in a log-log representation, could also be fitted to a straight line. From the slope a fractal dimension D=2.35 was determined. Different values for D were attributed to the effect of the clay content to the roughness of the aggregates. The fractal dimension of soil samples, could not be determined in a straightforward way by this method, because it depends both on the aggregate size and the aggregate composition.

Spatial and Temporal Analysis of Brain Images Using Fractal Models

V. Swarnakar, R. Acharya*, C. Sibata, K. Shin

During recent years it has been increasingly recognized that fractal models play an important role in the quantitative description of structural complexity of bio-morphological data. Despite the increasing use of fractal models for medical images, analysis of neuro-imaging data has scarcely been reported. In this article, a fractal model based image analysis framework is presented. This framework is applied towards observing temporal and spatial changes undergone by structures of interst within the brain. Fractal dimension is a basic parameter in any fractal model. Accuracy and robustness of the methods employed to estimate this parameter are crucial to the success of the fractal model. It is often desired, as is the case in this work, to observe variation in properties of structures that appear in small areas of the brain image. Most existing fractal dimension estimation methods fail when applied to such small image samples. A new method is presented here for fractal dimension estimation. Analysis of synthetic images, shows that this method is superior to the commonly employed box counting and power spectrum methods, in terms of accuracy and robustness. It also suggests that performance of the new method is not hindered when applied to small image samples. Temporal and spatial analysis results of MRI data from patients diagnosed with brain tumors, using the proposed framework, are presented. These preliminary results suggest that the proposed image analysis framework can be adequetely employed to observe structural changes undergone by the brain areas containing tumors.

Aggregation and Trapping Phenomena along Growing Interfaces

N. Vandewalle and M. Ausloos

Tree Formalism for Fractal Description

S. Duval and M. Tajine

We study a new notion of fractals based on the embedding of trees in a metric space (E, d). Actually, if R is a tree over an alphabet with arity (F, rho) then we associate to each symbol f of F a contracting mapping if rho(f)>0 and a compact of E otherwise. The embedding of R is obtained by the union of embeddings of its branches. To embed an infinite branch, we compose the mappings along it. We obtain, in this case, a point which belongs to E. To embed a finite branch, we successively apply the mappings along the branch to the leaf associated compact. The modeling of trees is done by using grammars of any order. The grammars of order 0, 1 and 2 generate respectively rational, algebraic, and functional fractals. More generally, a n-fractal is generated by a grammar of order n. This allows to classify the fractals according to the grammar order used. The rational fractals contain the IFS generated fractals. We can describe the high level operations most often found in object modeling. That way, given two fractals G and H, we model G u H, G x H, G + H and h(G) where x represents the cartesian product, + represents the sum of Minkowski and h is a homogeneous function in E. We are also able to describe the geometric simplicial complexes. The fractal geometry introduced here, is closely linked to discrete geometry in the sense that the model contains its own discretization. In fact, if a fractal G is the embedding of a tree R and R' is a tree obtained from R by truncation, then the embedding of R' is a discretization of G. It should be noticed that for the modeling of a segment, an adequate truncation permits to find the Bresenham's discretization. In the particular case of rational fractals, it is possible to effectively calculate the Hausdorff dimension. If G is a fractal obtained by the embedding of a rational tree R then the Hausdorff dimension of G is a function of the convergence radius of the generative series corresponding to R.

On the Relationship Between Mass and Fragmentation Fractal Dimensions

E. Perfect

Euclidean initiators have been used to model the fragmentation of solid materials. Fractal initiators may be more appropriate for predicting the fragmentation of porous media. I derived general expressions for the fragmentation of classical and fractal cubic initiators, assuming scale-invariant probabilities of failure, P. For classical cubes D=3+log(P)/log(b), where D is the fragmentation fractal dimension and b is the scaling factor. For fractal cubes D=d+log(P)/log(b), where d is the mass fractal dimension of the fractal cubic initiator. These expressions were tested for the fragmentation of soil aggregates by tillage. Aggregate mass- and fragment number-size distributions were determined for three tillage intensities replicated four times. Values for D (2.02- 2.55) were always less than those for d as predicted from theory. However, no correlation was found between D and d, possibly due to the narrow range in d (2.88-2.99) encountered for soil aggregates.

30 December 1996

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