K. Gohara and J. Nishikawa (Japan)

We have presented a theory for dissipative continuous dynamical systems stochastically excited by external temporal inputs. The theory shows that the dynamics is characterized by a set G(C) of trajectories in hyper-cylindrical phase space, where C is a set of initial states on the Poincare section. Two sets, G(C) and C, are attractive and invariant fractal sets. In this paper, using a nonlinear Duffing equation, we show numerically that the correlation dimension of the set C is approximately inversely proportional to the time length of inputs while the dimension is independent of the input amplitude. These obtained results might be universal characteristics of dissipative continuous dynamical systems stochastically excited by temporal inputs.

T. Huillet (France)

Heavy-tailed random variables or sequences constitute a source of numerous investigations and questions at the moment, both from the theoretical point of view and from applied fields as diverse as Finance, Hydrology, Seismology, Geology and Telecommunications. In all these applications, the central problem is to evaluate the probability of an outstanding i.e. extreme event to occur as the damage involved could be huge. In this article, it is suggested that the problem (borrowed from Geology or Environment Sciences) of deciding whether a natural resource (say such as a particular ore or pollutant) is abundant or sparse in Nature is intimately related to these tails' questions: we indeed supply a natural connection between the rareness/abundance investigation of a natural resource and questions arising from the distinction between heavy or light tails of its ''value''. On this basis, we suggest that a ''good'' statistical model for ''rare ore'' could be an homographic-Weibull distribution for its grade. In sharp contrast to this model, ore whose ''value'' is Pareto or log-normal distributed should represent data where ore and body of ore are strongly mixed. The Homographic-Weibull model present two unknown scale and shape parameters. One possible way to address the estimation problem from data is briefly discussed, exploiting a logratio transformation which appears ''natural'' in this problem. This is a preliminary step before the enrichment process of rare ore can be posed on statistical grounds.

B. Roland (France)

The bocage landscape has evolved very severly during the last fifty years with the agriculture developement. The shapes of the hedgerow lattice have many consequences for exemple in hydrological or ecological functions of this landscape. So we attempt to characterize the network structure by means of fractal geometry. The box-counting method permits to get significantly different results to analyse this structure with fractal parameters (dimension, superior and lower cuts off). Combining them with other parameters, we project to model the functionning of this landscape.

T. Martyn (Poland)

In this paper a method for ray tracing affine IFS fractals is presented. Like the previous approaches, the ray-fractal intersection algorithm presented here uses the object instancing principles. However, in contrast to the former ones, our method allows ray-tracing fractals at pixel size accuracy in an extremely efficient way. Moreover, our approach makes it possible to solve the spatial aliasing problem in a trivial and efficient way by appropriate corrections of sizes of instancing volumes. To take advantage of coherency and decrease computational redundancy, which is caused by searching for an intersection with each ray separately, the idea of tracing ray bundles is introduced.

G. Burkle-Elizondo (Mexico) and R. D. Valdez-Cepeda (Mexico)

It has been demonstrated that scribers, artists, sculptors and architects used a geometric system in ancient civilizations. There appears such system includes basically golden rectangles distributed in a golden spiral fashion. In addition, it is clear that we do not know the sequence in which the lines or pictures were originally traced or drawn. By this way, the artistic and architectural works can be considered as static objects and so they may be characterized by an inherent dimension. Our findings indicate a characteristic higher fractal dimension value for different groups of Mesoamerican artistic and architectural works: tablets from Palenque and other sites, Maya stelas, Maya hieroglyphs, pyramids and temples, calendars and astronomic stones, codex pages, murals, great stone monuments, astronomic stones and ceramic pots. Results could be suggesting that Mesoamerican artists and architects used specific patterns and they preferred works with higher (1.91) box and information fractal dimensions.

R. Gonzalez-Cinca (Spain)

We have studied sidebranching induced by fluctuations in dendritic growth by means of a phase--field model. We have considered a region where the linear theories are not valid and we have computed the contour length and the area of the dendrite at different distances from the tip. The dependence of the ratio of both magnitudes with the undercooling shows a behaviour in agreement with previous experiments. The derived scaling relation implies that dendrites are self--similar in the considered region.

V. Bychkov (Sweden)

Experimental observations indicate that the hydrodynamic flame instability result in development of a fractal structure at a flame front. We develop the theory of flame dynamics and stability and find estimates for the fractal dimension of a flame front. The obtained theoretical results are in a good agreement with experimental measurements.

A. Bari (Italy), A. Martin (Spain), D. Barranco (Spain), J. L. Gonzalez-andujar (Spain), G. Ayad (Italy) and S. Padulosi (Italy)

Species vary in space and in time as a result of evolution. To understand these patterns and their trends field surveys and genetic observations are usually carried out using morphological and molecular descriptors. Studies on the scoring of both morphological traits and molecular markers show that the researcher influences the delimitation of descriptor states and that expert knowledge appears to be of questionable value in defining descriptor states. This has encouraged the use of computers to detect variation, which has implications for both data capture and data analysis. This paper reports the findings of a study of the use of fractals as a tool to capture and analyze biodiversity data through plant morphology. The results show that fractals capture complexity better and detect more variation than other descriptors. This study used olive (Olea europea L.) as a model species. Further studies will examine the use of fractals in combination with neural networks to conduct further studies at different biodiversity levels including spatial genetic variation patterns.

Z. R. Struzik and W. J. van Wijngaarden (The Netherlands)

We introduce a special purpose cumulative indicator, capturing in real time the cumulative deviation from the reference level of the exponent h (local roughness, Hoelder exponent) of the fetal heartbeat during labour. We verify that the indicator applied to the variability component of the heartbeat coincides with the fetal outcome as determined by blood samples. The variability component is obtained from running real time decomposition of fetal heartbeat into independent components using an adaptation of an oversampled Haar wavelet transform. The particular filters used and resolutions applied are motivated by obstetricial insight/practice. The methodology described has the potential for real-time monitoring of the fetus during labour and for the prediction of the fetal outcome, allerting the attending staff in the case of (threatening) hypoxia.

Tadahiro Fujimoto, Yoshio Ohno, Kazunobu Muraoka, Norishige Chiba (Japan)

In this paper, we propose one-to-one and onto mappings, which are referred to as "Iterated Shuffle Transformation (IST)", defined on uniquely addressed spaces (UA-spaces) and discuss their properties. Each of the mappings constructs a fractal-like repeated structure on the relationship between points on the domain and those on the range. The property of this structure is referred to as "local resemblance in space/scale direction", which is considered to be the unification of "locality in space direction" in Euclidean geometry and "self-similarity in scale direction" in fractal geometry. We first present the definition of IST on code spaces, and then generalize it onto UA-spaces. We show various types of IST working on continuous geometric spaces and IFS attractors taken as examples of UA-space.

Kenichi Kamijo and Masahide Yoneyama (Japan)

In order to investigate the intrinsic behavior of actual organisms, idealized artificial life (AL) ecological system has been structured on a computer. The population change of AL entities conforming to a propagating rule has been simulated on it. The fractal dimension in both the original chaotic population change and the population change converted by the generation overlapping, which means the so-called moving total conversion in the change process, can be well denoted by the polynomial of degree 6 with the environment parameter except near the several values. Also the relationship between the generation overlap number and the fractal dimension has been investigated in the population change of the AL entities. It has been shown that the fractal dimension will decrease in proportion to the increase of the generation overlap number. Then the fractal dimension in some cases of the environment parameter for this AL ecological system can be well denoted by the polynomial of degree 6 with the generation overlap number.

J. Duchesne, P. Raimbault and C. Fleurant (France)

This article aims at establishing a very general law of the plant organization. By introducing the notion of hydraulic lengths which are=20 considers as the coordinates of a symbolic space with n-dimensions, a reasoning of statistical physics, derived of Maxwell's method, and combinig with the fractal geometry ends up a law of hydraulics lengths distribution which could appear very general because it is the remarkable gamma law form

E. Faleiro, J.M.G. Gomez and A. Relano (Spain)

High energy interactions of gamma rays and protons and also helium, oxygen and iron nuclei with the Earth atmosphere have been simulated by means of the CORSIKA Monte Carlo code, and the secondary-particle density distributions in the resulting extensive air showers, at ground level, have been studied. It is shown that the fluctuations of the particle density distributions have features typical of a 1/f noise. The multifractal spectrum of the samples is obtained and its features are found to be characteristic of the primary particle type. Using this property, a separation method for extensive air showers according to their primary cosmic ray is presented. A cutting parameter related to the multifractal spectrum is calculated and the efficiency of the cutting procedure is evaluated.

I. Kolumban and A. Soos (Romania)

The most known fractals are invariant sets with respect to a system of contraction maps, especially the so called selfsimilar sets. Recently Hutchinson and Ruschendorf gave a simple proof for the existence and uniqueness of invariant fractal sets and fractal functions using probability metrics defined by expectation. In these works a finite first moment condition is essential. In this paper, using probabilistic metric spaces techniques, we can weak the first moment condition for existence and uniqueness of selfsimilar fractal functions.

Wei-Xing Zhou (USA) & Zun-Hong Yu

Negative, or latent, dimensions have always attracted a strong interest since their discovery. When randomness is introduced in multifractals, the sample-to-sample fluctuations of multifractal spectra emerge inevitably, which has motivated various studies in this field. In this work, we study a class of multinomial measures and argue the asymptotic behaviors of the multifractal function as q tends to infinity. The so-called latent dimensions condition (LDC) is presented which states that latent dimensions may be absent in discrete random multinomial measures. In order to clarify the discovery, several examples are illustrated.

Nick Laskin (Canada)

We have studied the general properties of the fractional Schrodinger equation. The parity conservation law for the fractional quantum mechanics has been found. A new equation for the probability current density has been established. We also discuss the relationships between fractional and standard Schrodinger equations.

A. N. Kravchenko (USA)

Estimation and quantification of yield spatial variability and evaluation of spatial aspects of yield affecting factors are important issues in precision agriculture. In this study, joint multifractal theory was applied to analyze variability of crop grain yields and relationships between the yields, terrain elevation, and soil electrical conductivity (EC). Corn and soybean yield data from 1996 to 1999 were collected from a 20 ha agricultural field in Illinois, USA, along with elevation and soil EC measurements. Joint multifractal theory allowed successful delineation of the ranges of elevation and EC values that were of particular influence on crop yields. It was found to be an efficient tool for analysis of the yield spatial variability and is recommended for studying the relationships between scaling properties of two and more variables.

J.-M. Aubry and S. Jaffard (France)

Random Wavelet Series were introduced by the authors as a=20 generalization of the Lacunary Wavelet Series of Jaffard. They form a fairly broad class of random processes, with multifractal properties. We give three applications of this construction. First, we can synthesize random functions with a given spectrum of singularities, which is not necessarily concave. Secondly, we derive a multifractal formalism (a way of computing numerically the spectrum of singularities of a function) with a domain of validity not limited to concave spectra. Finally, we show that a particular case of our process satisfies a generalized selfsimilarity relation proposed in the theory of fully developed turbulence.

N. G. Makarenko, L. M. Karimova (Kazakhstan) and M. M. Novak (UK)

The paper focuses on diagnosis of extended nonlinear dynamical systems arising in the global solar magnetic field evolution. The methods of mathematical morphology, namely, Minkowski functionals and dimension are applied to analyzing topology of magnetic field cross-sections (synoptic charts). Time series of Euler characteristics obtained from solar magnetic field charts is used to investigate the Solar activity by the embedding methods.

October 2001