Tadahiro Fujimoto and Norishige Chiba (Japan)

In this paper, we propose a new fractal deformation technique. An ``extended unit Iterated Shuffle Transformation (ext-unit-IST)'' is a mapping that changes the order of the places of a code on a code space. When it is applied on a geometric space, it constructs a fractal-like repeated structure, named ``local resemblance''. In our previously proposed fractal deformation technique, a geometric shape was deformed by applying an ext-unit-IST to displacement vectors (d-vectors) given on the shape. In the new technique proposed in this paper, the ext-unit-IST is applied to the increasing rates of the d-vectors. This allows the d-vectors to change widely without disturbing the shape and improves the deformation quality. Several examples demonstrate the performance of the newly proposed technique.

Natalia A. Bryksina and William M. Last (Canada)

The gray-scale intensity data of finely-laminated sediments from Lake Bainbridge Crater, Gal=E1pagos, were analyzed in terms of Hurst exponent a nd fractal dimension. The sediment record preserved in this volcanic maar provides a continuous history of past El Ni=F1o/Southern Oscillation (ENSO= ) events over the last 6200 years. Gray-scale intensity data from analyses o= f x- radiography images of a 4.1 m long core from this basin were divided into two parts: Data1 and Data2. Data1 corresponds to the more recent part of stratigraphic sequence, which was deposited since 3000 yr BP; Data2 is fro= m the earlier part of record between 6200 and 3000 yr BP. A persistent behav= ior with a Hurst exponent of H=3D0.88 and H=3D0.8 was found by power spectrum analysis method for Data1 and Data2, respectively. The width method of analysis shows that ENSO fluctuation before 3000 yr BP had a persistent behavior on a time-scale up to 26 years, while the more recent El Ni=F1o occurrences have a persistent behavior on a time-scale up to 6 years.

Tomek Martyn (Poland)

In this paper we address and propose a solution to the problem of numerical estimating the generalized q-dimensions of affine RIFS invariant measures. Unlike the commonly used chaos game approach, our method gives good results for the potentially whole range of q (including the problematic large negative q) in an efficient and robust manner. In this goal, we use a deterministic, Markov-operator-based algorithm of approximating the measure on a lattice. We show that the algorithm makes it possible to approximate the measure at any accuracy with respect to the Hutchinson metric. Then, we give the rigorous proof that our lattice approximation is ideally suit for computing the generalized dimensions by means of the well-known overlapping box approach. The results included and their comparison with those obtained from the chaos game confirm the strength of our approach when applied in practice.

N. Nikolaou, A. Kakos, and V. Drakopoulos (Greece)

A new algorithm, called herein the Plotkin power domain algorithm, is discussed; it generates black and white images coded by an iterated function system, a technique used in fractal image compression. A simple complexity analysis for the algorithm is also derived.

I.I. Salakhutdinova and A.A. Golovko (Russia)

In this study we used H(-filtergrams obtained with the chromospheric telescope of the Baikal observatory operated by the ISTP SB RAS. The telescope includes the "Halle" birefringent filter with the 0.05 nm bandpass, and the Princeton Instruments 2048x2048 pixel CCD-camera with 16-bit amplitude resolution. We investigated the active regions NOAA 9077 (July 2000), and NOAA 0050 (July 2002). Images were taken at 1-min intervals. The method of structural functions was applied to obtain a number of characteristics of the chromospheric emission intensity field in the H(-line in the aforementioned active regions. A calculation of the parameters of the structural functions was performed in terms of Kolmogorov's theory of turbulence. The study revealed intervals of linear dependence of the structural functions on the scales of the field under consideration where a cascade redistribution of energy is taking place from larger to smaller scales. It is found that as solar activity progresses, the active region undergoes an intense structuring. This is associated with the development of an abundant multifractal structure. Fractal characteristics of the structural functions in this case undergo stepwise changes that come into play shortly before a next flare from among a cascade of flares preceding the main strong flare. Therein lies a possibility of predicting the flaring process.

H. Katsuragi, D. Sugino, and H. Honjo (Japan)

We report the binomial multiplicative model for low impact energy fragmentation. Experiments were performed for low impact energy region, and it was found that the weighted mean mass is scaled by the pseudo control parameter multiplicity. We revealed that the power of this scaling has non-integer (fractal) value and has a multi-scaling property. This multi-scaling can be interpreted by a binomial multiplicative (simple biased cascade) model. Although the model cannot explain the power-law of fragment cumulative distribution in fully fragmented states, it can produce the multi-scaling exponents that agree with experimental results well. Other models for fragmentation phenomena were also analyzed and compared with it.

Miguel A. Pasquale (Argentina)

Data from the dynamic scaling analysis of the growth front of silver patterns electroformed in a quasi-2D cell under localized and non-localized random quenched noise are reported. The plating solution either embedded in filter paper (FP), or containing disordered glass beads (GB), or as agarose gels (AG) were utilized. The scaling exponents from the displacement of the driven interface are the followings: roughening exponent = 0.63 (0.05) and the growing exponent = 0.60 (0.05) for FP, irrespective of its pore size distribution; roughening exponent = 0.64 (0.05) and growing exponent = 0.58 (0.05) for GB; and roughening exponent = 1.25 (0.10) and growing exponent = 0.88 (0.15) for AG. Exponents for FP and GB fit the predictions of the directed percolation depinning (DPD) model for D = 1, whereas for AG they coincide with those calculated by Leschhorn from a lattice model of probabilistic cellular automata. The difference between exponents resulting from FP, GB, and AG can be attributed to a non-localized random pinning in AG, which introduces a size-dependence mobility of obstacles in the gelled medium.

B G Sidharth (India)

There are a number of empirical relations which suggest a Quantum Mechanical behaviour at different scales, that is with different,scaled Planck constants.We argue that these relations are not accidental,but rather can be explained in terms of Newtonian dynamics in the context of an underlying Stochastic process together with critical point phenomena and the related coarse graining of the Renormalizing Group.

Seiji Tokunaga and Hidetsugu Sakaguchi (Japan)

Doublon is one of the typical patterns found in crystal growth. It is a pair of symmetry broken fingers. In this paper, we obtain numerically parameter range of coexistence of doublon and dendrite structure with a phase-field model. We perform numerical simulations in a two-dimensional channel, setting small seed of crystal at left-bottom side of the channel as an initial condition. The oscillation of groove of doublon appears in some parameter range even though without perturbation. In other parameter range, both of dendrite and doublon make their appearance along same growth direction.

Nicoletta Sala (Switzerland)

Fractal, in mathematics, is a geometric shape that is complex and detailed in structure at any level of magnification. The word "fractal" was coined less than thirty years ago by one of history's most creative mathematicians, Benoit Mandelbrot, whose seminal work, The Fractal Geometry of Nature, first introduced and explained concepts underlying this new vision. Although prior mathematical thinkers like Georg Cantor (1845-1818), Felix Hausdorff (1868-1942), Gaston Julia (1893-1978), Helge von Koch (1870-1924), Giuseppe Peano (1858-1932), Lewis Richardson (1891-1953), Waclaw Sierpinski (1882-1969) and others had attained isolated insights of fractal understanding, such ideas were largely ignored until Mandelbrot's genius forged them at a single blow into a gorgeously coherent and fruitful discipline. Fractal geometry is applied in different field now: engineering, physics, chemistry, biology, and architecture. The aim of this paper is to introduce an approach where the arts are analyzed using a fractal point of view. Different fractal components will be analyzed in different cultures. For example, in Western, Japanese, and African cultures.

L. Pachepsky, M. Kaul, Ch. Walthall, J. Lyden, C. Daughtry (USA)

Contemporary agriculture uses crop simulation models for crop management and yield prediction. However, a model validation remains a permanent problem and interfaces for users contain only quantitative information. L-systems model coupled with a crop model could provide an additional visual validation of the latter and serve as an interactive and attractive for practical users interface. An open parametric L-systems model was developed for soybean based on the data for two cultivars growing in controlled climate chambers at three temperatures. Detailed morphological observations accompanied by frequently taken photographs were conducted from emergence to seed filling and were used to create the virtual plants. A functional model of vegetative development [1, 2] was parameterized for these two cultivars and linked with the visual model. The results showed that the virtual plants reproduced well the series of photographs of the real plants. Therefore, it can be useful for visual validating the phenological modules of crop models and as a part of user interface. The software L-Studio that was used for the visual modeling demonstrated extraordinary abilities. However, a serious obstacle for its extensive using in crop simulation is the absence of a manual understandable for a wide range of researchers working with crop models.

Owen Dafydd Jones (UK)

We describe a class of self-similar processes that can be used to fit self-similar data, and give a fast, efficient on-line algorithm for simulating them.

J. Alberto Betancourt-Mar and E. Jonathan Suarez-Dominguez (Mexico)

It was studied the morphological changes in the crystal growth of mercury (II) chloride on agar extended on glass microscope slides. The time required for the complete growth (which depends on the growth velocity) controls the morphology of the generated structures. Long growth times produce DLA-like structures; instead, short growth times produce compact structures, periodic crystallization, rings and spirals (the spirals were observed at temperatures of fifty or more Celsius degrees). Fractal dimension of DLA-like structures is in accordance with the value reported in literature.

Hongquan Sun (P. R. of China)

This paper consists of three parts. The first part describes a mathematical model of fractal interpolation surface on a rectangle field and the calculation formula of the fractal interpolation surfaces. The second part presents the study of attitude (the trend and the obliquity) of the fault by using multivariate statistics. The third part discusses the simulation of roughness of the fault surface dealing with the improved methods, the partition of local domains and the determination of vertical scaling factor, of fractal interpolation surfaces. At the same time, the fractal dimension of the interpolated fault surface is obtained. The theory and method discussed in this paper provides a new way for studying the influence of the roughness of the fault surface in mining engineering and civil engineering.

A. Bari and G. Ayad (Italy), A. Martin & J.L. Gonzalez-Andujar (Spain) and M. Nachit & I. Elouafi (Syria)

To identify the olive cultivars that use water more efficiently, the fractal geometry of root branching/architecture was examined in relation to the water use efficiency parameters. The aim is to test the extent to which such root branching characteristics can help in locating and selecting olive plants that are more efficient than others in using water. The results revealed that the olive cultivars with high fractal dimension values in terms of their early root branching were those that best regulate water flow through their stomata. We are currently investigating further details of root branching/architecture in relation to water use efficiency parameters and genomics, to shed light on the inheritance-and, in particular, the function-of root architecture in relation to water use efficiency.

October 2003