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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 < alpha < 2. After many (isotropic)
scatterings, particle trajectories x(n), n „ 0, are essentially
3-dimensional discrete-time Lévy-stable random walks (Levy-flights) of
index alpha, truncated at random times when a boundary is crossed.
Starting at x(0) = 0 with a step into the z > 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
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.
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.
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.
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.
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.
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.
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.
C. Ioana, F. Munteanu and C. Suteanu
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.
M. Bertolotti, P. Masciulli, F. Garzia and C. Sibilia
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.
H. Patzlaff, U. Behn and A. Lange
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.
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.
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.
H.-C. Tseng and H.-J. Chen
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.
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.
Y. Dang and L. H. Kauffman
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.
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:
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.
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.
A. N. Kravchenko and R. Zhang
R. Zhang
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.
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.
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.
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.
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.
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.
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. Tarquis, C. H. Díaz-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.
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.
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.
N. Vandewalle and M. Ausloos
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.
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.
Monitoring fire-induced landscape changes in Mediterranean regions
with a Fractal Algorithm.
Fractal Dimension of Fracture Surfaces in Ductile-Brittle Transition
Regime
Physical Simulation of High-Resolution Satellite Images for Fractal
Cloud Models
Fractal Characterization of the Surface of the Humin Fraction of Soil
Organic Matter
Fractal Properties of Soil Organic Matter
Lie Groups and Solution of Dynamical Problems on Fractal Lattices
Multifractal Decomposition for a Family of Overlapping Self-Similar
Measures
Smoothing Dimension Analysis - New Effective Tools in Fractal Signal Investigation
Scale Invariance in Transitional Pipe Flow
Optical Filtering Properties of a Self-Similar Multilayer Structure
Resources allocation and bird distribution in a sub-mountain ecotone
Analytical Explanation of a Phase Transition in the Multifractal
Measure Connected with a One-dimensional Random Field Ising Model
Scale Properties of the Diffraction Fields Induced by Pre-Fractal Random
Screens
Modification of Carbon Cluster Fractal Structure
Due to Capillary Forces
Cantor Staircases in Physics. A Connection with
Number Theory. Part 1.
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
Anomalous Diffusion and Chaotic Scattering in a Nonlocal Model
Julia and Mandelbrot-like Sets for Higher Order Koenig Rational
Iteration Functions
Hypercomplex Fractal Distance Estimation
Seasonal Changes in Fractal Landscape Surface Roughness Estimated From
Airborne Laser Altimetry Data
Scaling Analysis of the Critical Behavior of a Spin Chain Model
- a low-temperature region with a single fixed point attractor
- a high-temperature region with limit cycle attractor and an
unstable point.
Relation Between Fracture Toughness and Fractal Dimension of the Crack
Initiating Zone of Polycarbonate
Estimation of Soil Water Retention Function From Texture and Structure
Data: Fractal Approach
Scale-Dependant Soil Hydraulic Conductivity
Comparison between DC Voltage Reference Source Time Series
and Fractional Brownian Motion
Computer Aided Geometric Design with IFS Techniques
Superconducting Fractal Nb/Cu Multilayers
Fractal Characterization of Root System Architecture in Legume
Seedlings
Optical Properties of Self-affine Surfaces
Binary Expansions and Multifractals
Using the Interval Distribution of Level Sets Approach to
Determine the Phase Transition Point Based on a Time Sequence Data
A Deterministic L-System to Simulate Leaf Area Index in A Faba
Bean Canopy
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.
Spatial and Temporal Analysis of Brain Images Using Fractal Models
Aggregation and Trapping Phenomena along Growing Interfaces
Tree Formalism for Fractal Description
On the Relationship Between Mass and Fragmentation Fractal
Dimensions
30 December 1996