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\begin{center}
{\Large Abstracts of Papers Accepted for Presentation at} \\
\vspace{2cm}
{\Large \bf Fractal 95} \\
{\large Fractals in the Natural and Applied Sciences \\
$3^{rd}$ International Working Conference \\
February 7 -- 10, 1995, Marseille, France} \\
\vspace{5cm}
Conference chair: Miroslav M. Novak
\end{center}
\newpage
\begin{center}
{\Large\bf Fractal Image Coding Using Luminance-Based Partitioning}\\
\vspace{1.5cm}
{\bf S. Andonova, D. Popovi\'c, J. Janneck, M. Kowalzik,} and
{\bf M. Behrmann}\\
Institute of Automation Technology \\
University of Bremen, FRG\\
{\small Tel: +49 421 218 35 80}\\
{\small Fax: +49 421 218 36 01}\\
{\small email: andonova@theo.physik.uni-bremen.de}
\end{center}
\vspace{1cm}
{\bf Abstract:}
In order to accelerate the procedure of capturing the similarity
of image data in fractal image coding,
a partitioning scheme, based on the spatial luminance distribution, is proposed.
In the first step the image is partitioned using quad-tree technique,
until the
pre-selected size of coding unit (i.e the {\it range block}) is met.
According to the {\it luminance distribution}
within the range blocks, the {\it midrange} and the {\it edge} blocks
are further classified as $Z$, $O$ or $Y$ blocks.
During the search for similar image parts,
only the {\it range} and {\it domain} blocks of same type (i.e. {\it shade}, {\it
midrange} or {\it edge}) and
of the same $Z$, $O$, $Y$ class are
considered. \\
The algorithm of
fractal image coding has been implemented for intraframe coding in a
predictive interframe video coding scheme.
It has been found that
the time required for fractal coding using luminance-based partitioning
is significantly shorter than the time required for other partitioning schemes,
that is of fundamental importance in video coding.
\vspace{1cm}
{\bf Keywords: }{\it fractals, fractal image coding, image compression,
image partitioning,
contractive image transformations}
\newpage
\begin{center}
{\Large\bf Scale Relativity:
Many-Particle Schr\"odinger Equation}\\
\vspace{1.5cm}
{\bf Laurent Nottale}\\
CNRS, DAEC, Observatoire de Paris-Meudon\\ F-92195, Meudon Cedex, France
\end{center}
\vspace{1cm}
{\bf Abstract:} After having stated the principle of scale relativity and
summarized the new scale-dependent mathematical tool that we use in
order to implement it, we first recall how giving up the hypothesis of
differentiability of space-time implies its fractal structure and allows
one to recover the one-particle quantum mechanics. Then we obtain
the Hamiltonian form of the Schroedinger equation from the same
grounds, and finally apply this result to the many-identical particle
problem.
\newpage
\begin{center}
{\Large\bf The Use of the Fractal Dimension in Images for Plant
Recognition}\\
\vspace{1.5cm}
{\bf Reyer Zwiggelaar} and {\bf Christine R. Bull}\\
Silsoe Research Institute,
Wrest Park\\ Silsoe, Bedford MK45 4HS, UK\\
{\small tel. +44 (0)525 860000 x2520\\
fax. +44 (0)525 861461\\
email zwiggelaar@afrc.ac.uk}
\end{center}
\vspace{1cm}
{\bf Abstract:}
Fractal information from images can be used as a tool for plant recognition.
Depending on the application plant images at various scales can be used. For
the distinction between certain weeds in row crops images with a low number
of plants (excluding total canopy cover) are used. To determine the fractal
Hausdorff dimension links with the Fourier transform are exploited. However,
scatter in the power spectrum is reduced by integration in the spatial
frequency domain over band-limited spatial filters. Using this approach the
Hausdorff dimension is determined for whole images and subsets of these images.
The usefulness of the resulting fractal dimension information as a tool for
image segmentation and crop versus weed recognition are discussed.
\vspace{1cm}
{\bf Keywords:} {\it Plant images, Hausdorff dimension,
spatial frequency integration, plant discrimination.}
\newpage
\begin{center}
{\Large\bf Tumour Shape and Local Connected Fractal Dimension Analysis in
Oral Cancer and Pre-cancer}\\
\vspace{1.5cm}
{\bf G. Landini} and {\bf J. W. Rippin}\\
Oral Pathology Unit, School of Dentistry, University of Birmingham\\
St. Chad's Queensway, Birmingham B4 6NN, UK
\end{center}
\vspace{1cm}
{\bf Abstract:}
The local complexity of the epithelial-connective tissue interface (ECTI)
in normal tissues, epithelial dysplasia and squamous cell carcinoma of
the floor of the mouth was investigated using the concept of local
connected dimension. It was found that the distribution of the local
dimensions of the ECTI alone can classify the three types of
histopathological diagnoses (normal, dysplasia and carcinoma) with
85% accuracy. The values of the local fractal dimension were also used
to produce a colour-coded dimensional image of the ECTI that permits the
automatic location of supposedly "higher risk" areas.
\newpage
\begin{center}
{\Large\bf The Solution of the Inverse Fractal Problem Applied to a
Self-affine Fractal with the Help of Wavelet Decomposition}\\
\vspace{1.5cm}
{\bf Z.R. Struzik, E.H. Dooijes} and {\bf F.C.A. Groen}\\
Department of Computer Science,
University of Amsterdam\\
Kruislaan 403, 1098 SJ Amsterdam,
The Netherlands\\
{\small tel: +31 20 5257522, +31 20 5257523 or +31 20 5257463\\
fax: +31 20 5257490\\
email: zbyszek@fwi.uva.nl, edoh@fwi.uva.nl, groen@fwi.uva.nl}
\end{center}
\vspace{1cm}
{\bf Abstract:}
We solve the inverse fractal problem for self-affine functions in
${\bf R}^2$ by means of recovering the maps with which the function was
created.
A suitable representation is used - wavelet maxima bifurcation
representation - derived from the continuous wavelet decomposition which
possesses translational and scale invariance.
It allows us to uncover the invariance with respect to position and
scale in the case of the self-affine fractal, and may also prove itself
useful in other applications involving analysis of scale dependent features.
Two algorithms are presented which give satisfactory results for the
self-affine fractal and which potentially can be applied to a
variety of fractal types in order to solve the related inverse fractal
problem.
\vspace{1cm}
{\bf Keywords:} {\it inverse fractal problem, wavelet transform,
wavelet maxima bifurcation, representation, self-affine fractals}
\newpage
\begin{center}
{\Large\bf Qualitative Analysis of Human Blood Based on Elements
of Fractals}\\
\vspace{1.5cm}
{\bf M.V. Kurik} and {\bf S.F. Lyuksyutov}\\
Institute of Physics, Ukrainian Academy of Sciences\\
Prospect Nauki 46, 252 650 Kiev, Ukraine\\
{\small email: vasnet@oqe.ip.kiev.ua and
ukrlib@gluk.apc.org}\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
It has been shown experimentally that healthy red blood cells can create fractal
clusters with the dimension 2,5 after the process of mophilisation of the blood.
It was found that the blood of some children suffer against blood diseases
acquired after Chernobyl accident has a fairly uniform distribution of the red
blood cells.
\newpage
\begin{center}
{\Large\bf Fractal Properties of Relaxation Clusters and Phase Transition in
a Stochastic Sandpile Automaton}\\
\vspace{1.5cm}
{\bf S.~L\"{u}beck}\\
Theor. Tieftemperaturphysik, Universit\"{a}t Duisburg\\
Lotharstr.1, 47048 Duisburg, Germany\\
{\small email: sven@hal6000.thp.Uni-Duisburg.de}\\
{\bf B.~Tadi\'c}\\
Jo\v{z}ef Stefan Institute, University of Ljubljana\\
P.O. Box 100, 61111-Ljubljana, Slovenia\\
and\\
{\bf K.~D.~Usadel}\\
Theor. Tieftemperaturphysik, Universit\"{a}t Duisburg\\
Lotharstr.1, 47048 Duisburg, Germany
\end{center}
\vspace{1cm}
{\bf Abstract:}
We study numerically the spatial properties of relaxation clusters in
a two dimensional sandpile automaton with dynamic rules depending
stochastically on a parameter $p$, which models the effects of static
friction.
In the limiting cases $p=1$ and $p=0$ the model reduces to the
critical height model and critical slope model, respectively.
At $p=p_c$, a continuous phase transition occurs to the state
characterized by a nonzero average slope.
Our analysis reveals that the loss of finite average slope at
the transition is accompanied by the loss of fractal properties of the
relaxation clusters.
\vspace{1cm}
{\bf Keywords:}{\it Self-organized criticality, sandpile automata,
relaxation cluster, fractal dimension}
\newpage
\begin{center}
{\Large\bf Absorption Spectra and Photomodification
of Silver Fractal Clusters}\\
\vspace{1.5cm}
{\bf Yu. E. Danilova} and {\bf V. P. Safonov}\\
Institute of Automation and Electometry, Russian
Academy of Sciences\\ Siberian Branch, Novosibirsk, 630090,
Russia\\
{\small email: safonov@iaie-phys.nsk.su\\
fax:~~~~(3832) 354-851\\
tel:~~(3832) 351-044}
\end{center}
\vspace{1cm}
{\bf Abstract:}
Absorption spectra shape of silver colloid aggregates are
studied in combination with electron microscopic investigation of
the samples. The experimental results, concerned with the selective
photomodification of silver fractal clusters are reported. A
correlation was made between the experimental results and the
scaling theory of the optical absorption of fractal clusters. It
was found that absorption spectrum shape of quasi-percolated
aggregates in a silver colloid corresponds to the power law.
The optical spectral dimension was found to be
$d_O\approx0.5\div0.7$ for different realizations of colloid. The results
of the spectral hole burning experiments are in a qualitative
agreement with the estimations, based on the concept of
localized collective dipolar states in a cluster.
\vspace{1cm}
{\bf Keywords:} {\it fractal clusters, absorption, photomodification, scaling}
\newpage
\begin{center}
{\Large\bf Fractal Growth of Bacterial Colonies}\\
\vspace{1.5cm}
{\bf M. C. Ruzicka, M. Fridrich}\\
Inst. of Chemical Process Fundamentals, Czech Academy of Sciences\\
Prague 16502, Czech Republic\\
and\\
{\bf M. Burkhard}\\
Inst. of Physiology, Czech Academy of Sciences\\
Prague 14220, Czech Republic\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Complex growth patterns of bacteria on solid agar were studied. The colonies
were found to be not self similar objects, although the contrary was believed.
Their two-fold
fractal structure was revealed. Two different mechanisms governing the
pattern formation below and above a certain critical scale were specified. A
scenario of a colony development was suggested.
\newpage
\begin{center}
{\Large\bf Stochastic L-System Applied to the Calculation of the Leaf Area
of a Shrubby Legume for Forage (Chamaecytisus ruthenicus, F. ex
Wol.)} \\
\vspace{1.5cm}
{\bf Ana M. Tarquis$^1$} and {\bf Fernando Gonzalez-Andres$^2$}\\
$^1$ Dept. of Applied Mathematics and
$^2$ Dept. of Plant Biology\\
E.T.S. Ingenieros Agronomos (U.P.M.)
Ciudad Universitaria s.n.\\
28040 Madrid - SPAIN\\
{\small email: C0245005@VEC.CCUPM.UPM.ES}
\end{center}
\vspace{1cm}
{\bf Abstract:}
A stochastic and deterministic L-system was developed to calculate
the total leaf area of a shrubby legume used for forage at the end
of each growing season. Several tables were defined to describe the
structure (topology) and the physiology of the plant for each
life cycle. In order to include the influence of the field
temperature, several parameters were related to it. Also a growth
function for the deterministic model was defined and applied for a
bracketed L-system.
\vspace{1cm}
{\bf Keywords:}{\it TL-systems, stochastic, branching pattern, growth
function}
\newpage
\begin{center}
{\Large\bf Some Remarks on Phase Transitions and Their Disappearance in the
Descriptions of Multifractal Sets}\\
\vspace{1.5cm}
{\bf Chi-Ging Lee, Hsen-Che Tseng}, and {\bf Hung-Jung Chen}\\
Department of Physics,National Chung-Hsing University\\
Taichung402, Taiwan, ROC
\end{center}
\vspace{1cm}
{\bf Abstract:}
For a family of skew Ulam maps,the cycle expansion theory guarantees more
accurate location of transition points.Using this approach,we have found that
the disappearance of phase transitions has an interesting scenario.
\vspace{1cm}
{\bf Keywords:} {\it cycle expansions,multifractal}
\newpage
\begin{center}
{\Large\bf Chemically-Controlled Reaction Kinetics on Fractals:
Application to Proton Exchange in Proteins}\\
\vspace{1.5cm}
{\bf T. Gregory Dewey}\\
Department of Chemistry, University of Denver\\
Denver, CO 80208, USA\\
{\small tel: (303)-871-3100\\
fax: (303)-871-2254\\
email: gdewey@cair.du.edu}
\end{center}
\vspace{1cm}
{\bf Abstract:}
The effect of fractal diffusion on chemically-
controlled reactions in solutions is considered. A general
mechanism is examined that consists of a two step process.
First the reactants diffuse together to form an "encounter
complex." This is followed by the collapse of the complex
to the final product. The first step is diffusion controlled
and the second step is chemically controlled. For reactions
on fractals the rate constants associated with the diffusive
process will scale with time as $t^{-h}$ where h is a constant
between 0 and 1. The chemical processes are assumed to have
time-independent rate constants. For reactions in which the
encounter complex achieves a steady state, the differential
equations governing the time course of the reaction can be
solved exactly. At short times, the concentration of the
reactants decays exponentially, reflecting the time constant
of the chemical processes. At longer times, the decreasing
diffusive rate constants result in the process being
diffusion controlled. A stretched exponential of the form,
$exp(-kt^{1-h})$, is observed. Approximate solutions for the
pre-steady
state behavior of the system are also determined. These
theoretical results are applied to the analysis of proton
exchange kinetics in proteins. Exchange kinetics are
modeled as a reaction in the boundary volume of a fractal.
\newpage
\begin{center}
{\Large\bf Variogram and Wavelet analysis
of a cloud field advected by a baroclinic flow}\\
\vspace{1.5cm}
{\bf C. Serio$^1$} and {\bf V. Tramutoli$^2$}\\
$^1$Dipartimento d'Ingegneria e Fisica dell'Ambiente, Universit\`a
della Basilicata\\
Potenza, Italy\\
$^2$Facolt\'a di Scienze, Universit\`a
della Basilicata\\ Potenza, Italy
\end{center}
\vspace{1cm}
{\bf Abstract:}
In this paper scaling laws in
a relatively homogeneous cloud system generated and advected by
a strong baroclinic instability have been investigated.
The observations consist
in an infrared satellite image with a spatial (horizontal)
resolution of about 1 km.
The presence of two sizeable and unmistakable scaling
regions,
one extending from 4 to 15 km and
characterized by a power law with an exponent close to 1,
the other stretching from 20 km up to 100 km
and characterized by a power law with exponent close to 1/3,
have been revealed by variogram analysis.
These two scaling laws are in agreement with the idea of
universality of the turbulent motion and also suggest
the presence of a self-similar structure. To explore this
possibility, wavelet transform analysis at different spatial scales
has been used. Our findings are that
self-similarity is present at the smallest
scales, but this universal
characteristic is masked by non-universal
effects which influence the homogeneity of the underlying
turbulent motion.
\newpage
\begin{center}
{\Large\bf Some Experimental Results in Viscous Fingering}\\
\vspace{1.5cm}
{\bf S. Obernauer, A. D'Onofrio} and {\bf M. Rosen}\\
Grupo de Medios Porosos, Facultad de Ingenier\'{\i}a,
Universidad de Buenos Aires\\
Paseo Col\'on 850, C.P 1063\\ Capital Federal, Argentina\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We show experimental results carried out to analyze viscous instabilities in
an isotropic porous medium consisting of glass spheres packings. In all cases
the sample was first saturated with a polymer dispersion (xanthan) at
different
concentration and then flooded with dyed water. A rich variety of patterns was
found. The results obtained have shown the crucial influence of the fluid
viscosity ratio in the resulting structure. For values larger than 100 a
fractal
behavior is observed. A modification to the density-density correlation
function
is also proposed to take into account the dependence of the fractal dimension
with the relative position on the structure. The fractal dimension was
measured
on bands along the structure and values different are found.
\newpage
\begin{center}
{\Large\bf Interface roughening in magnetic systems with quenched
disorder}\\
\vspace{1.5cm}
{\bf M. Jost} and {\bf K.D. Usadel}\\
Theoretische Tieftemperaturphysik, Universit\"{a}t Duisburg,\\
Lotharstr.1, 47048 Duisburg, Germany\\
{\small email: mjt@thp.Uni-Duisburg.DE and usadel@thp.Uni-Duisburg.DE}
\end{center}
\vspace{1cm}
{\bf Abstract:}
The kinetic roughening of an interface between spin-up and spin-down
domains
in a model with non-conserved scalar order parameter with quenched
disorder
is studied numerically within a discrete time dynamics at zero
temperature.
Starting from a flat interface in a two dimensional system the time
evolution
of the height correlation function and of the width of the interface
is
analyzed. It is found that the interface is rough on length scales
smaller
than a characteristic length $\Delta$ which depends on the degree of
disorder.
A novel scaling ansatz for this situation is proposed and the data are
analyzed
within this framework.
\vspace{1cm}
{\bf Keywords:}{\it interface growth, kinetic roughening, model A dynamics
}
\newpage
\begin{center}
{\Large\bf Fractal properties and scaling exponents of the
Barkhausen effect}\\
\vspace{1.5cm}
{\bf G. Durin} and {\bf A. Magni}\\
Istituto Elettrotecnico Nazionale Galileo Ferraris and GNSM-INFM\\
C.so Massimo d'Azeglio 42, I - 10125 Torino, Italy
\end{center}
\vspace{1cm}
{\bf Abstract:}
Using the results of a theory of the Barkhausen effect based on a
Langevin approach of the domain wall motion in a medium with brownian
properties, we calculate the scaling exponents of the durations and
sizes distributions of the Barkhausen jumps. The Barkhausen signal is
found to be related to a random Cantor dust having a Hausdorff
dimension which is a linear function of the applied field rate. Using
simple properties of fractal geometry, the distributions are easily
calculated. The predictions are then verified for a Si-Fe alloy. The
fractal dimension {Delta} of the signal is also studied using four
different methods, all giving {Delta} ~ 1.5. The relation between the
above results and the power spectra data are analyzed in the frame of
self organized-criticality.
\newpage
\begin{center}
{\Large\bf Temporal Scaling Regions and Capacity Dimensions for
Microearthquake Swarms in Greece}\\
\vspace{1.5cm}
{\bf Yebang Xu} and {\bf Paul W. Burton}\\
School of Environmental Sciences, University of East Anglia \\
Norwich NR4 7TJ England U.K.\\
{\small Tel:44-0603-56161\\
fax:44-0603-507719 \\
email: Y.B.XU@uea.ac.uk or P.BURTON@uea.ac.uk}
\end{center}
\vspace{1cm}
{\bf Abstract:}
The characteristics of temporal scaling ranges and capacity dimensions are
studied in detail for microearthquakes recorded over a two year period in
three regions in central Greece. The three regions are centred on Volos,
Pavliani and Patras respectively and have very different long-term seismicity
and seismotectonic characteristics. The scaling range ($2^{7} - 2^{12}$ s )
exists for both the Volos and Patras swarms and these have a tectonic
background of strong seismicity including earthquakes of magnitude
$Ms\geq 6.0$. This might be an important indicator that distinguishes them
from the Pavliani swarm which has no historical evidence of large earthquake
occurrence. Under the condition of an identical threshold magnitude, the
capacity dimension values of temporal distributions for both the Pavliani and
Patras swarms are higher than that for the Volos swarm. Such differences in
dimension value might relate to the local complexity of the seismotectonic
environmental and its characteristic seismicity.
\newpage
\begin{center}
{\Large\bf Object Identification in Greyscale Imagery Using Fractal Dimension}\\
\vspace{1.5cm}
{\bf Philip Beaver, Stephanie M. Quirk }\\
Department of Mathematics, United States Military Academy \\
West Point, NY, USA\\
and \\
{\bf Joseph P. Sattler}\\
Army Research Laboratory, Sensors Directorate\\
Adelphi, MD 20783, USA
\end{center}
\vspace{1cm}
{\bf Abstract:}
A fractal dimension operator which assigns a local fractal dimension to each
pixel in a greyscale image can be used to help identify objects from a number
of imaging sources, to include aerial photography, satellite imagery, radar,
and x-rays. In this paper, we identify a problem with the consistency of
such operators, demonstrate a shortcoming in some existing algorithms, and
propose a general method for computing local fractal dimension which can be
tailored to different image sources. We do this in the context of our
specific application: identification of man-made objects in images produced
by aerial synthetic aperture radar.
\vspace{1cm}
\newpage
\begin{center}
{\Large\bf Object Instancing Graphtals }\\
\vspace{1.5cm}
{\bf Ph\@. Bekaert} and {\bf Y. D. Willems }\\
Department of Computing Science,
Katholieke Universiteit Leuven \\
Celestijnenlaan 200A, 3001 Leuven, Belgium \\
{\small
email: Philippe.Bekaert@cs.kuleuven.ac.be }
\end{center}
\vspace{1cm}
{\bf Abstract:}
This paper describes an application of fractals in computer graphics.
It is well known that fractal theory leads to algorithms for rendering
many objects in nature, like trees, mountains and seashells.
It is however our experience that strictly self-similar figures are
of limited use for everyday rendering: one often wants to render
figures that exhibit {\em some} self-similarity, but are not fractals
strictly according to the definition.
It is shown in this paper how by extending the notion of ``object
instancing'', a well known notion in computer graphics, a formalism
for describing such figures is obtained. The main topic in this
article is however a raytracing algorithm that was developed for
rendering such figures. The advantages of our approach are its
flexibility and its easiness to understand and to use.
\vspace{1cm}
{\bf Keywords: }{\it fractals, formal languages, raytracing}
\newpage
\begin{center}
{\Large\bf Study of Fractal Properties in Lichtenberg Figures}\\
\vspace{1.5cm}
{\bf M. C. Lan\c{c}a, J. Domingues} and {\bf I. Franco}\\
F\'{\i}sica Aplicada,
Faculdade de Ci\^encias e Tecnologia, Universidade Nova de Lisboa \\
P-2825 Monte de Caparica, Portugal \\
{\small Tel: 351 1 295 44 64 \\
Fax: 351 1 295 44 61 \\
e-mail: carmo@sgaaf.fct.unl.pt}
\end{center}
\vspace{1cm}
{\bf Abstract:}
The fractal stochastic model for dielectric breakdown is used to
simulate Lichtenberg figures produced in a point to plane geometry. Four
different models for the growth probability of the discharge are applied.
Solving the Laplace equation is on the basis of all the models. For the
simulated figures the fractal dimension was calculated using the box
counting method. The influence of the several computation parameters in the
fractal dimension of the patterns was studied. Simultaneously an
experimental setup was used to obtain discharges of the same kind. The
experimental figures were recorded with a photographic camera and
digitalized. These images undergo an image treatment process in order to
estimate the fractal dimension. A method of counting the number of branches
is implemented for this purpose.
\vspace{1cm}
{\bf Keywords: }{\it dielectric breakdown, fractal dimension,
leader discharges, Lichtenberg figures, simulation }
\newpage
\begin{center}
{\Large\bf Tori--in--a--Torus Fractals }\\
\vspace{1.5cm}
{\bf A.~Kittel, J.~Parisi, }\\
Physical Institute, University of Bayreuth\\
D--95440 Bayreuth, Germany \\
{\bf M.~Klein, G. Baier, O.E.~R\"ossler,}\\
Institute for Physical and Theoretical Chemistry, University of
T\"ubingen\\
D--72076 T\"ubingen, Germany\\
and\\
{\bf J.~Peinke}\\
C.R.T.B.T.--C.N.R.S.\\
F--38042 Grenoble, France\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We present an algorithm of an iteration rule capable to generate
attractors with dragon--, snowflake--, sponge--, or Swiss-flag--like
cross--sections. The idea behind is the mapping of a torus into two (or
more) shrinked twisted tori located inside the previous one. Upon
intersecting the attractor, we simultaneously obtain both connected and
disconnected fractal structures.
\vspace{1cm}
{\bf Keywords: }{\it Fractals, Dynamical Systems, Algorithms}
\newpage
\begin{center}
{\Large\bf Fractal Parameters of Soil Pore Surface Area Under a
Developing Crop}\\
\vspace{1.5cm}
{\bf L.P.Korsunskaia}\\
Institute of Soil Science and Photosynthesis\\
Pushchino 142292, Russia
{\small Tel: (095) 923 35 58\\
Fax: (095) 924 04 93\\
e-mail: sotnikov@issp.serpukhov.su}\\
{\bf Ya. A. Pachepsky}\\
Duke University Phytotron, Durham, NC 27608, USA\\
{\small Tel: (301) 504 74 68\\
Fax: (301) 504 58 23\\
e-mail: ypachepsky@asrr.arsusda.gov}\\
and\\
{\bf M. Hajnos}\\
Institute of Agrophysics, Lublin 20280, Poland\\
{\small Tel. (081) 450-61\\
Fax (081) 450-67\\
e-mail: kozakedm@demeter.ipan.lublin.pl}\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Fractal parameters of soils become increasingly
important in understanding and quantifying of transport
and adsorption phenomena in soils. It is not known how
soil plant development may affect fractal characteristics
of soil pores. We estimated pore surface area fractal
parameters from mercury porosimetry data on gray forest
soil before and during crop development in samples both
containing and not containing soil carbohydrates known to
be important structure-forming agents. Two distinct
intervals with different fractal dimensions were found in
the range of pore radii from 4 nm to 1 mkm. This could be
attributed to differences in mineral composition of soil
particles of different sizes. The interval of the
smallest radii had the highest average fractal dimension
close to 3. Smaller surface area fractal dimensions
corresponding to low surface irregularity were found in
the next interval of radii. The plant development
affected neither fractal dimensions nor the cutoff values
of soil samples. The carbohydrate oxidation caused a
significant increase of the fractal dimension in the
interval of larger radii, but did not affect fractal
dimension in the interval of small radii. The cutoff
values decreased after carbohydrate oxidation.
\vspace{1cm}
{\bf Keywords: }{\it Soil pore surface, Fractal dimension,
Structure-forming agents, Temporal variations}
\newpage
\begin{center}
{\Large\bf Similarity in the Problem of Discrete
Contact between Fractal Surfaces }\\
\vspace{1.5cm}
{\bf Feodor M. Borodich }\\
Division of Applied Mechanics (NIOM),
Smolenskiy Blvd., 6/8-92, Moscow, 119034, Russia,\\
Department of Applied Mathematics,
Moscow Institute of Scientific
Instrumentation Engineering (MIP),\\
Stromynka 20, Moscow, 107076, Russia \\
{\small email: F.Borodich@damtp.cambridge.ac.uk}
\end{center}
\vspace{1cm}
{\bf Abstract:}
Contact problems for rough surfaces described by fractal
parametric-homogeneous functions are under consideration.
It is assumed that the contact is discrete. Material of contacting
surfaces may have both linear and non-linear stress--strain
relationships. The qualitative conclusions
on the character of changes of the contact region and approaching
of surfaces are described. The results are derived using a
similarity method, without solving the field equations.
\vspace{1cm}
{\bf Keywords: }{\it fractal surfaces, contact problem, similarity}
\newpage
\begin{center}
{\Large\bf A Graphic Tool to Determine the Period of an Antenna Component of
the Mandelbrot Set }\\
\vspace{1.5cm}
{\bf M. Romera, G. Pastor, J. Negrillo} and {\bf F. Montoya}\\
Inst. de Electr\'onica de Comunicaciones, C. S. I. C. \\
Serrano 144, 28006 Madrid, Spain \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Determining the period of a cardioid-like component of the real part
of the Mandelbrot set, the antenna, can be a tedious task. To avoid it, an easy
graphic tool to obtain the period is introduced. We point out that the main
filament
of every component of period n attracts the escape lines from * to n and does
not attract the other ones. Then, the period n of a component can directly be
determined by knowing the last escape line attracted by its main
filament.
\newpage
\begin{center}
{\Large\bf Fractal Geometry of Quantum Paths and the Fractal Wilson Loop }\\
\vspace{1.5cm}
{\bf H. Kr\"oger }\\
D\'epartement de Physique, Universit\'e Laval, \\
Qu\'ebec, Qu\'ebec G1K 7P4, Canada \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We suggest that the fractal nature of quantum mechanical paths
should be seen in non-local order parameters in gauge theory.
e.g., the Wilson loop, and hence should play a role for the question of
confinement.
We present results of numerical simulations on the lattice
for imaginary time quantum mechanics, measuring the length of paths
and its critical exponent (Hausdorff dimension). For local potentials we find
results compatible with $d_{H}=2$.
Secondly, we consider the length of a quark propagator
in lattice $QCD$. We present a definition, discuss its properties and present
some
numerical results for a simplified free propagator.
Finally, we consider a fractal Wilson loop $$.
We show for non-compact $SU(2)$ lattice gauge theory
in the next to leading order of strong coupling expansion that $$ obeys
an area law behavior and is gauge invariant.
\newpage
\begin{center}
{\Large\bf Analysis of an Image of the Human Brain obtained by
Positron Emission Tomography in Terms of Fractal Geometry }\\
\vspace{1.5cm}
{\bf Martin Obert$^1$, Ralf Bergmann$^1$, Hartmut Linemann$^2$} and
{\bf Peter Brust$^1$}\\
$^1$Forschungszentrum Rossendorf\\
Inst. f. Bioanorganische u. Radiopharmazeutische Chemie\\
POB 51 01 19, D 01314 Dresden, Germany\\
$^2$Technische Universitaet Dresden, Universitaetsklinikum Carl Gustav Carus\\
Fetscherstrasse 74, D 01307 Dresden, Germany\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We introduce the concept of fractal geometry to describe the irregular
spatial distribution of [18F]2-fluoro-2-deoxy-D-glucose (FDG) in a
section of a human brain. FDG is labeled with a positron emitting
isotope that is measured by Positron Emission Tomography (PET) after
intravenous injection.
At first, we construct binary subsets from the image data for different
ranges of radioactivity. Such subsets contain only spatial information,
which allows the determination of self-similar properties. For a subset
generated from low to high radioactivity we find a fractal dimension,
D, of 1.9. For a subset representing basically high radioactivity
we find a D of 1.2. This indicates a multifractal behavior of the set,
which will be analyzed in detail in a future study. Further, the data is
analyzed as a "landscape" where two dimensions are given by the spatial
coordinates of the cross section of the brain and the third dimension,
the height of the "landscape", is defined by the radioactivity at the
spatial position on the slice. The analysis of this self-affine set with
the mass radius relation gives a D of 2.3.
We show in this preliminary study that D can serve as a mathematically
well-defined measure to describe the global irregularity of a PET image.
We assume that D will be of clinical relevance.
\vspace{1cm}
{\bf Keywords: }{\it Fractals, Positron Emission Tomography, Brain Research}
\newpage
\begin{center}
{\Large\bf New Approach to Synthesis of Fractal Materials
With a Given Fractal Dimension }\\
\vspace{1.5cm}
{\bf Peter E.Strizhak }\\
L.V.Pisarzhevskii Institute of Physical Chemistry\\
National Academy of Sciences of Ukraine\\
pr. Nauki 31, Kiev, Ukraine, 252039 \\
{\small Tel: (044) 265-62-09 \\
Fax: (044) 265-62-16\\
email: anal@chem.univ.kiev.ua or ipcukr@sovamsu.sovusa.com}
\end{center}
\vspace{1cm}
{\bf Abstract:}
A method for the synthesis of fractal films of CuS is proposed.
An approach is based on a possibility of fractal films formation
which is induced by spatial patterns formed in self-organized
chemically reactive media. We used a pattern formation in
'oxygen' oscillating chemical reaction - oxidation of ascorbic
acid by air oxygen in the presence of hydrogen sulfide ions,
methylene blue, and copper(II) coordination compounds. A CuS
film formation is observed for this system owing to the slow
reaction between copper(II) coordination compound and hydrogen
sulfide ions. A fractal dimension of CuS films correlates with
the fractal dimension of chemical patterns. The value of
fractal dimension for CuS films is always higher then for
chemical patterns. This difference might indicate that the
chemical patterns only induce the creation of a fractal film
through the formation of the growth centers for CuS, i.e.
chemical patterns are responsible only for the birth of fractal
clusters. The growth of these clusters is provided by the
independent mechanism of the CuS slow precipitation from the
solution. The electrical resistivity of fractal films and the
sensitivity of copper(II) ions selective electrodes with fractal
CuS surface to the copper(II) ions were measured. Our
experiments indicate that the behavior of Cu(II)-selective
electrodes with fractal surface is independent on the value of
a fractal dimension. The difference exists only between
electrodes with D = 2 and D < 2.
\vspace{1cm}
{\bf Keywords: }{\it fractal dimension, fractal film, copper sulfide,
oscillating chemical reaction, patterns formation}
\newpage
\begin{center}
{\Large\bf $1/f^{\beta}$-Fluctuations in Bipolar Affective Illness }\\
\vspace{1.5cm}
{\bf Klaus-D. Kniffki, Christian Braun, Andreas Klusch} and
{\bf Phuoc Tran-Gia$^{*}$}\\
Physiologisches Institut and Institut f\"ur Informatik$^{*}$,\\
R\"ontgenring 9 and Am Hubland, Universit\"at W\"urzburg,
\\D-97070 W\"urzburg, Germany \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Temporal fluctuations which cannot be explained as consequences of statistically
independent random events are found in a variety of physical and biological
phenomena.
These fluctuations can be characterized by a power spectrum density $S(f)$
decaying as $1/f^{\beta}$ at low frequencies with an exponent $0.5\leq \beta\leq
1.5$. We present
a new approach to reveal $1/f^{\beta}$-fluctuations in manic and depressive
episodes
in bipolar affective illness using published data from patients for whom daily
records
were obtained applying a 7-point magnitude category scale.
This time series $\{R(t_i)\}$ was described as a point process by introducing
discriminating rating
levels $r$ and $s$ for the occurrence of $R(t_i)\geq r$ (`mania') and $R(t_i)\leq
s$
(`depression').
For $\beta<1$ a new method to estimate the low frequency part of $S(f)$ was
applied
using
counting statistics without applying Fast Fourier Transform. The method reliably
discriminates these types of fluctuations from a random point process with
$\beta=0.0$.
It is very tempting to speculate that the neuronal/humoral mechanisms at
various
levels of the nervous system underlying
the manic and depressive episodes in bipolar affective illness are expressions
of a self-organized critical state.
But the most important result of the present study is the finding of a scaling
region
$1d\leq\Delta t \leq 200d$ for the `manias' and `depressions' where $S(f)$ is
decaying as $1/f^{\beta}$
with $\beta\approx 0.8$. Therefore, based on the monitored ratings for a given
time
period
it should be possible to predict future episodes with a certain probability by
applying
methods of nonlinear time series analysis or modified feed-forward neural
networks
learning
with the back-propagation algorithm. This could result in an improvement of the
treatment of patients.
\newpage
\begin{center}
{\Large\bf Fractal Surfaces as Self Organization of Microfractures }\\
\vspace{1.5cm}
{\bf Vladimir P. Stoyan }\\
Department of Civil Engineering,
Kuban State Agrarian University \\
Stavropolskaya Street,133/1, box 61,
Krasnodar, 350040,Russia\\
{\small Tel: (007-861-2) 33-38-04 (home), 56-09-39 (University)\\
Fax: (007-861-2) 54-63-22 (c/o Igor Popov )\\
email: evg@kgu.kuban.su (c/o E.Glushkov)}
\end{center}
\vspace{1cm}
{\bf Abstract:}
Fracture mechanisms are formulated on the grounds of physical
concepts on crack propagation in solid structure. The fracture is
treated as loss of stability in stretched structure, for which the
fluctuations of state are inevitable and lead to creation of zones
with disturbed continuity. The fact that stability is lost implies
the fractal surfaces growth phenomenon should be treated as a self
- organization of microfractures in non-equilibrium system which can
not be considered in terms of infinitesimal variations. Cracks have
the typical zigzag fractal microconfiguration depending on the structure
of loading the domain.
\vspace{1cm}
{\bf Keywords: }{\it crack, propagation, microconfiguration, directions,
regularity}
\newpage
\begin{center}
{\Large\bf An Aggregation Model for Town Growth }\\
\vspace{1.5cm}
{\bf L. Benguigui }\\
Solid State Institute and Department of Physics, Technion \\
Israel Institute of Technology, Haifa 32000, Israel \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
A new aggregation model based on the Eden model and its properties is
presented. The local density $\rho(r)$ exhibits scaling properties and can be
written as $\rho(r)=N^{\alpha}p^{-b}f(r/N^{c}p^{d})$ when $N$ is the total number
of particles and $p$ an integer characteristic of the model. The exponents are
determined: $a \approx d = 0.15 +- + 0.02, b = 0.32 +- 0.01, c = 0.425 -+
0.005$. Application of the model to London and the Rhine valley towns is
discussed.
\newpage
\begin{center}
{\Large\bf The `Smoothing Dimension' -
a New Fractal Analysis Method }\\
\vspace{1.5cm}
{\bf Florin Munteanu, Cristian Suteanu, Cristian Ioana} and
{\bf Edmond Cretu }\\
Romanian Academy, Institute of Duodenums \\
19-21 Jean-Louis Calderon Strew.,
Bucharest-37, R 70201 Romania \\
{\small Tel: 40 1 778 58 79\\
Fax: 40 1 312 58 38} \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Signal analysis involved in geophysical research or in
studies regarding biological systems often imposes the
difficult task of significant signal classification, in order
to distinguish different types of behaviour (reflected by the
same parameter at different moments in time) or to help
appreciating the agreement between models and modelled
systems.
We introduce a new fractal analysis method, which
emphasizes the way in which the length of the graph associated
with the time series scales with the cut-off frequency of a
filter used to "smooth" the graph. The obtained "Smoothing
dimension" $D_L$ can be determined with a good accuracy and is
significant even for signals not having power-law type power
spectrum. $D_L$ proves to be well correlated with the Hurst
exponent $H$ for a certain class of signals recorded in
experimental research, enhancing the possibilities of signal
classification.
\vspace{1cm}
{\bf Keywords: }{\it fractal analysis, Hurst exponent, signal classification}
\newpage
\begin{center}
{\Large\bf A Classification of Writings Relying on Fractal Behaviour }\\
\vspace{1.5cm}
{\bf N. Vincent\footnote {to whom correspondence should be sent}} and
{\bf H. Emptoz }\\
INSA de LYON - LISPI - Bat. 403 \\
20, Bd. A. Einstein,
69621 Villeurbanne Cedex
France \\
{\small Tel: 72 43 83 13\\
email: vincent@insa.insa-lyon.fr}
\end{center}
\vspace{1cm}
{\bf Abstract:}
Since the interest for fractal geometry has risen, the
applications are getting more and
more numerous in many domains. The atheism is to show that the
concepts may be applied
in the domain of automated writing treatment too and that they can
bring some help, to a certain
extend, to ease the solution of many problems. After the fractal
behaviour of writing has been
shown and its stability too, either through centuries or, for a single
writer, all along his life time,
a fractal dimension is computed for many types of writings and a
comparison is achieved. Thus,
the authors come to a classification. Two classification grids show the
distribution of different
types of writings. This study forms an important step that enables a
better orientation of the
research works in the optical reading domain and specially in the
recognition task. Of course,
our hypothesis has to be validated by use of an even greater number
of writing samples.
\vspace{1cm}
{\bf Keywords: }{\it fractal dimension, writing classification, legibility degree}
\newpage
\begin{center}
{\Large\bf Multifractality of the Giant Electric Field Fluctuations
in Semicontinuous Films Close to the Percolation Threshold }\\
\vspace{1.5cm}
{\bf F. Brouers, S. Blacher }\\
Etude Physique des Mat\'eriaux et G\'enie Chimique \\
Universit\'e de Li\`ege B5, 4000 Li\`ege, Belgium \\
and \\
{ \bf A. K. Sarychev} \\
Centre for Applied Problems in Electrodynamics \\
Russian Academy of Sciences, 127412 Moscow, Russia \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We show that the intensities of the local electric field in metal-dielectric
semicontinuous films may exhibit giant fluctuations in the frequency
range dominated by cluster plasmon absorption when the dissipation in
metallic grains is low. It is found that these fluctuations are highly
correlated in space and that for length scales smaller than the
corresponding correlation length, the distribution of these local fields
is multifractal.
\vspace{1cm}
{\bf Keywords: }{\it Multifractality, percolation, thin films,
optical properties, real space renormalization}
\newpage
\begin{center}
{\Large\bf Computing Orthogonal Polynomials for Fractal Measures and
the Dynamics of Quasi-Periodic Tight-Binding Models }\\
\vspace{1.5cm}
{\bf G. Mantica }\\
Int. Centre for Dynamical Studies, Via Lucini 3\\
I-20100 Como, Italy\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Iterated Functions Systems are a well developed mathematical formalism and
a convenient computational tool to describe a large class of fractal measures.
Yet, the problem of determining the associated orthogonal polynomials has been
solved satisfactorily only recently; in this paper we present the main lines of
this solution and its first, immediate applications to numerical integration and
modelling of solid state mesoscopic systems and chains of oscillators.
\newpage
\begin{center}
{\Large\bf Far Infrared and Optical Absorption of Fractal and
Multiscaling Metallic Clusters }\\
\vspace{1.5cm}
{\bf F.Brouers$^{1,2}$, D.Rauw$^1$, J.P.Clerc$^2$ and G.Giraud$^2$ }\\
$^1$Etude Physique des Mat\'eriaux, Institut de Physique B5 \\
Universit\'e de Li\`ege, Li\`ege, Belgium\\
$^2$D\'epartement de Physique des Syst\`emes D\'esordonn\'es \\
Universit\'e de Provence,
Centre de St.~Jerome, Marseille, France\\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We show that the electrical deterministic fractal lattice (DFL) model and its
extension to describe multiscaling fractal objects provide a good physical and
theoretical understanding of the observed far-infrared (FIR) anomalous
absorption properties of fractal metallic clusters as well as the cluster plasmon
resonances discussed previously by means of numerical methods. Using the
analytical properties of the DFL and the Decorated-DFL (DDFL) models, it is
possible to calculate exactly the power law exponents of the enhancement
factors B, of the cross-over localization length $L_{\omega}$, as well as the
Cantor structure of the self-similar cluster plasmon resonance spectrum in terms
of the fractal characteristics of the aggregate.
\vspace{1cm}
{\bf Keywords: }{\it Infrared absorption, optical properties, fractal
cluster, cluster plasmon modes, multiscaling}
\newpage
\begin{center}
{\Large\bf Intermittency and Fractal Behaviour of Medium
Energy Particles at High Energy }\\
\vspace{1.5cm}
{\bf Neeti Parashar }\\
Department of Physics \& Astrophysics, University of Delhi \\
Delhi - 110007, India \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
The experimental data on nuclear emulsion at 800 GeV has been analyzed to
study intermittency and fractal characteristics of medium energy particles, the
so called "grey particles", in terms of factorial moments. The fractal dimensions
and phase structure function determined are found to be in agreement with the
theoretical predictions.
\vspace{1cm}
{\bf Keywords: }{\it Multiparticle production, Intermittency, Fractal dimension,
Phase \linebreak transition}
\newpage
\begin{center}
{\Large\bf A theory for the Morphology of Laplacian Nonlinear Growth
Processes via Statistics of Equivalent Many-body Systems }\\
\vspace{1.5cm}
{\bf Raphael Blumenfeld }\\
Center for Nonlinear studies and Theoretical Division MS B258 \\
Los Alamos National Laboratory Los Alamos NM 87545 USA \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
The nonlinear evolution of two dimensional interfaces in Laplacian fields is
discussed. By mapping the growing region conformally onto the unit disk the
problem is converted to the dynamics of a many-body system. This problem is
shown to be Hamiltonian, with the Hamiltonian being the complex electrostatic
potential. The Hamiltonian structure allows introduction of surface effects as an
external field. An extension to a continuous density of particles is presented.
The results are used to formulate a first-principles statistical theory for the
morphology of the interface using statistical mechanics for the many-body
system. The distribution of the curvature and the moments of the growth
probability along the interface are calculated exactly from the distribution of
the particles. In a specific approximation the distribution of the curvature is
shown to develop algebraic tails which points to the onset of fractality.
LA-UR-94-2077
\newpage
\begin{center}
{\Large\bf Transition Processes on Noise-Induced Fractal Sets }\\
\vspace{1.5cm}
{\bf Simon J. Fraser} and {\bf Raymond Kapral} \\
Chemical Physics Theory Group, Department of Chemistry \\
University of Toronto
Toronto, Ontario M5S 1A1, Canada \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Attractor structure and transition dynamics for dissipative, bistable oscillators
driven by a periodic-dichotomous-noise process are described. The oscillator
motion constitutes a smooth trajectory flow in phase space between
noise-switching events, which are Bernoulli trials on a two-branched quartic
potential. This continuous-time stochastic dynamics induces a (two-branched)
smooth stochastic map of the plane in discrete time. As the oscillator mass
decreases, the corresponding attractor of this planar map changes, becoming
fractal or quasi-atomic for intermediate and small mass. The transitions between
\`left\' and \`right\' species defined in configuration space are studied in
relation to the changes in attractor structure. The \`reaction\' dynamics can be
understood in terms of the attractor structure in the transition region and the
short and long time behaviour of the species autocorrelation function.
\vspace{1cm}
{\bf Keywords: }{\it Fractal attractors, noise-induced transitions}
\newpage
\begin{center}
{\Large\bf Application of the Lattice-Boltzmann Equation
to Transport and Hydrodynamic Phenomena in Fractal and Disordered Media }\\
\vspace{1.5cm}
{\bf Alessandra Adrover$^1$} and {\bf Massimiliano Giona$^2$ }\\
$^1$Dipartimento di Ingegneria Chimica, Universit\`{a} di Roma "La
Sapienza" \\ Via Eudossiana 18, 00184, Roma, Italy \\ $^2$Dipartimento
di Ingegneria Chimica, Universit\`{a} di Cagliari \\ Piazza d'Armi,
09123, Cagliari, Italy \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
Lattice Boltzmann
equation is a powerful tool for studying fluid dynamics and transport
phenomena in complex geometries. In this article we focus attention on
two classical hydrodynamic problems: creeping flow around an array of
fractal objects and permeability measurements in disordered
(percolating) systems.
\newpage
\begin{center}
{\Large\bf Adsorption Equilibria in Microporous
Materials: the role of Fractality and of Energetic Heterogeneity }\\
\vspace{1.5cm}
{\bf Massimiliano Giona$^1$, Manuela Giustiniani$^2$} and
{\bf Douglas K. Ludlow$^3$ }\\
$^1$Dipartimento di Ingegneria
Chimica, Universit\'{a} di Cagliari \\ Piazza d'Armi, 09123 Cagliari,
Italy \\ $^2$Dipartimento di Ingegneria Chimica, Universit\'{a} di
Roma "La Sapienza" \\ Via Eudossiana 18, 00184 Roma, Italy \\
$^3$Department of Chemical Engineering, University of North Dakota \\ Grand
Forks, ND 58202-7129, USA \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We develop a thermodynamic analysis of adsorption
equilibria in microporous materials, focusing on the relationship
between the adsorption isotherms and the structural (energetic,
geometric) parameters characterizing the \linebreak adsorbate/adsorbent system.
Amongst the proposed theoretical models that include fractal scaling,
Keller adsorption isotherms are discussed in detail. In this model,
the fractality of the adsorbent plays the role of a closure condition,
which reduces the number of independent parameters. Experimental
results are reported to validate this model in the case of single and
multicomponent equilibria. The problems related to thermodynamic
consistency and to the low pressure behaviour of adsorption isotherms
are also discussed.
\newpage
\begin{center}
{\Large\bf Scale Invariance in Long-term Time Series }\\
\vspace{1.5cm}
{\bf Marina Shabalova }\\
Royal Netherlands Meteorological Institute (KNMI) \\
P.O. Box 201, 3730 AE De Bilt, The Netherlands \\
{\small Tel: +31 30 206759\\
Fax: +31 30 210407\\
email: konnen@knmi.nl}
\end{center}
\vspace{1cm}
{\bf Abstract:}
To evaluate climate variability at time scales
100-1,000,000 years, the continuous records of
paleotemperatures from SPECMAP project, the extended Vostok
ice-core temperature and air composition records, as well as GRIP
oxygen isotope record were analyzed in terms of structure
functions. It is shown that the characteristic temperature and
gas concentration changes follow the scale invariant law up
to time scales of about 10,000 years (smaller than formerly
recognized); the parameters of structure functions depend upon
site location. Although different sites yield different slopes,
there is a clear evidence for a break in the scaling at time
scales of about 10,000--20,000 years at high latitudes and of
about 7,000 years in equatorial belt. At time scales larger than
the scaling limit and up to 700,000 years the characteristic
amplitudes of temperature variations are almost constant, with
superposed on it series of local maxima and minima that
corresponds to the Milankovitch periodicities.
The break of scaling signifies the appearance of the basic
periods in climate system. The time scale corresponding to the
scaling limit identifies the boundary between the random and
deterministic regimes and thereby is a fundamental climatic
parameter. The earlier break of scaling and the scaling exponent
lower than 0.5 in equatorial sites, as well as a small range of
characteristic temperature fluctuations in tropics supports the
existence of stabilizing mechanisms in tropical climate, the
phenomenon which is largely discussed. Several breaks in structure
function of the GRIP oxygen-isotope profile and a very low
exponent in scaling regime suggest a significant role of regional
processes in the climate fluctuations over Greenland, with
characteristic periods of a few thousand years.
\newpage
\begin{center}
{\Large\bf IFS based on 3D Self-similar Contraction Mapping and
Modeling Trees and Coral}\\
\vspace{1.5cm}
{\bf Masaaki Shimasaki}\\
Computer Center\\
{\bf Yasuhiro Matsusaka} and {\bf Susumu Sakamoto} \\
Dept. of Computer Science and Communication Engineering, \\
Kyushu University, Fukuoka, Japan \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
We present an Iterated Function System (IFS) based on
three-dimensional (3D) self-similar
contraction mapping
to model trees and corals. IFS
is the fundamental mathematical method to generate fractal patterns,
such as snowflake and ferns. Those patterns are essentially two-dimensional.
There are many references on using IFS for fractal generation,
but most are concerned with 2D IFS that are related to complex variable
functions, suitable for the generation of 2D fractals.
We present an IFS based on truly self-similar 3D contraction mapping
functions in order to model 3D tree patterns. The method has proven
particularly suited when modeling Leeuwenberg's trees and corals.
It can be represented in a very simple form and is suited to
high-speed
computations.
In order to generate realistic tree images, we also propose methods
to generate leaves of various patterns.
\vspace{1cm}
{\bf Keywords: }{\it IFS, 3D self-similar contraction mapping, fractal tree,
coral}
\newpage
\begin{center}
{\Large\bf The Temperature Dependence of Nanocluster
Interface Morphology: a Monte Carlo Study}\\
\vspace{1.5cm}
{\bf N.V. Dolgushev} and {\bf S.A. Suvorov }\\
St.Petersburg State Institute of Technology,
Dept. of High Temperature Materials \\
Moscowsky Av. 26, St.Petersburg 198013, Russia \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
The Monte--Carlo simulation of two--dimensional cluster within the
approach of lattice--gas model is carried out. In the context of the model
used the generation of a "hanging--over" type configurations was allowed and
as an opportunity of vacancies, isolated atoms and their aggregates formation
were considered. Over the temperature region of $0.30\div 0.77{k_bT}/{J}$
the linear dependence of the cluster interface fractal dimension on
temperature was obtained. It has been proposed that the transition of linear
$D_p$ dependence to non--linear, some bounded overhead function, at the
reduced temperature ${k_bT}/{J}$ more then 0.8 is caused by a phase
transition.
\newpage
\begin{center}
{\Large\bf Fractals in Biology }\\
\vspace{1.5cm}
{\bf Shlomo Havlin }\\
Department of Physics, Bar-Ilan University,
Ramat-Gan 52900, Israel \\
\end{center}
\vspace{1cm}
{\bf Abstract:}
In the last decade it was realized that some
biological systems have no characteristic length or time scale,
{\it i.e.\/}, they have fractal --- or, more generally, self-affine
properties$^1$. The fractal properties in different
biological systems, have quite different nature and origin. In
some cases, it is the geometrical shape of a biological object
itself that exhibits fractal features, while in other cases the
fractal properties are the time-dependent properties.
We review several biological systems which can be characterized by
fractal geometry.
In particular, we focus on the long-range correlations and the
linguistic features found in non-coding DNA sequences and
suggest their possible implications. We discuss the application
of fractal analysis to the dynamics of heartbeat regulations and the
recent finding that the normal heart is characterized by long-range
``anticorrelations'' which are absent in the diseased heart. We also
discuss a dynamical fractal model for spreading and migration of
populations and diseases. A major part of the presented studies was
done in collaboration with S.~V.~Buldyrev, A.~Bunde, A.~L.~Goldberger,
H.~Larralde, R.~N.~Mantegna, M.~Meyer, C.-K.~Peng, M.~Simons,
H.~E.~Stanley and P.~Trunfio.
\vspace{1cm}
$^1$see {\it e.g.\/}, S.~V.~Buldyrev, A.~L.~Goldberger,
S.~Havlin, C.-K.~Peng and H.~E.~Stanley: in
{\sl Fractals in Science\/}, eds. A.~Bunde and S.~Havlin
(Springer, Berlin 1994)
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Dr. Miroslav M. Novak Tel: +44-181-547-2000
School of Physics Fax: +44-181-547-7562
Kingston University +44-181-547-7419
Surrey KT1 2EE, England Internet: novak@kingston.ac.uk