Fractals

Recently Watts and Strogatz emphasized the widespread relevance of 'small worlds' and studied numerically networks between complete regularity and complete randomness. In this letter, I derive simple analytical expressions which can reproduce the empirical observations. It is shown how a few random connections can turn a regular network into a 'small-world network' with a short global connection but persisting local clustering.

A simple first-order recurrence in a (max, +) dynamic system is numerically investigated and shown to exhibit statistical long-range dependence, characterized by slowly decaying aggregated variances and power-law evolutions of the autocorrelation and spectrum. We propose this model as a basis for a very parsimonious modeling of some long-range dependent processes such as data traffic.

The Fraunhofer region of one-dimensional fractal diffraction grating is studied when this grating is defined using a set of scaled periodical (and rectangular) functions. The fractal structure is obtained through a Cantor density function, which is defined from an incremental product of such periodical components. Each of these components can be filtered individually and so, the corresponding intensity pattern can be evaluated. A mathematical foundation for such decomposition is developed using the concept of characteristic (or indicator) function of a set.

Graphic results are shown and the partial self-similarity of lateral intensity distribution calculated in any transversal plane to the optical axis for a certain value of magnification. With these results, fractal, periodical and intrinsic periodical domains can be distinguished to characterize the intensity patterns.

The self-similarity of intensity distributions and the contrast of that function are calculated for different states of the Cantor transmittance.

This article aims to define the conditions required in order that the series given by the amplitude transmitted in a non-constant pulse in a Fabry-Perot interferometer is a fractal series. This means obtaining the appropriate expressions so that the amplitude transmitted by a beam whose rays change amplitude in time, that is, a beam whose energy varies while is incident on the interferometer produces fractal figures.

We present the formation of unique and striking dendritic "seahorse" patterns in the growth of fullerene-tetracyanoquinodimethane (C₆₀-TCNQ) or pune TCNQ thin films. The films were fabricated by an ionized-cluster-beam deposition method. Energetic neutral and charged clusters were deposited on amorphous carbon substrates. Transmission electron microscopy reveals that the elemental pattern is a "seahorse" — that is, an S-shaped form, with "fins" on the outer edges of the curved arms forming the S. Such forms possess an approximate symmetry under rotations by π, but strongly break two-dimensional inversion symmetry. A novel formation mechanism is proposed, involving the aggregatio of neutral and charged clusters, such that some electrostatic charge is trapped on each growing island. This charge gives rise to a long-range field which biases the growth in a nontrivial way. The broken symmetry arises from the strong amplification of noise in the diffusive aggregation process by the effects of the electro-static field — that is, the symmetry breaking is spontaneous. This picture is tested by applying a transverse electric field during growth: for sufficiently strong fields, the S-shaped "seahorses" lose their curvature, while retaining the feature of having two main arms. These results demonstrate the importance of electrostatic effects in the growth process, and are consistent with the growth mechanism described here.

This paper considers the frequency-size statistics of wars. Using several alternative measures of the intensity of a war in terms of battle deaths, we find a fractal (power-law) dependence of number on intensity. We show that the frequency-size dependence of forest fires is essentially identical to that of wars. The forest-fire model provides a basis for understanding the distribution of forest firest in terms of self-organized criticality. We extend the analogy to wars in terms of the initial ignition (outbreak of war) and its spread to a group of metastable countries.

The physics of terrestrial radioactive contamination is complex, primarily due to the mixtures of radionuclides, their migration, the effects of soil chemistry, and the prevalent climate. Consequently, application of classical techniques is inappropriate in modeling such an environment. The classification of territories contaminated with radioactive isotopes is vital in establishing a detoxification plan. Application of the fractal geometry provides a good insight and helps in such classification. Several techniques are implemented to determine geometric aspects of contaminated fields, and these show a way to differentiate between natural and man-made isotopes.

The self-similarity properties of the functions (closed relations) associated with one- and two-sided cellular automata are studied. It turns out that these functions are generated by sequential machines, and their graphs are fractal sets generated by hierarchical iterated function systems. The Hausdorff dimensions of the graphs is one for one-sided cellular automata and two for two-sided automata.

We use the Discrete Wavelet Transform in order to study the power spectrum of data obtained in magnetoencephalographical measurements. α-wave phenomenon is found to occur independently 1/f^β noise, which is present over almost all channels. the β value is close to 1.

Image analysis was carried out on images of laboratory-made synthetic sandstone, which is used in air injection/in situ combustion experiments, and statistically similar sandstone was computer modeled in two (and three) dimensions. The pore space of both the synthetic and the modeled sandstones was found to be fractal. The equations governing the flow through porous rocks were developed, approximating the in situ combustion process with an immiscible displacement of resident oil by injected gas maintained at a constant pressure. A successive over-relaxation (SOR) scheme was applied to solve these equations. Finally, the similarity of simulations of flow through the laboratory-made synthetic and the computer-modeled sand-stones was demonstrated.