Frederic Heliodore

Optical wave diffraction on fractal objects

In the frame of a generalized functions analysis, it is shown that the light scattered by fractal objects has fractal properties. The difference between mass diffraction and surface diffraction is presented and illustrated through clear and easy experiments based on the fundamental difference between the concepts of dimension and codimension.

Introduction to wave phenomena and uncertainty in a fractal space—I

Abstract The purpose of this note is to introduce the principle of dynamics in a fractal space. Although a lot of work has been devoted to fractal geometry, the introduction of time with regard to a fractal space must not be undertaken without some restrictions. In a first step, our analysis restricts itself to the case of electromagnetic theory which will be described in terms of differential geometry and homologic algebra. By an appropriate extension, some efficient mathematical tools, like the fractional derivative operator, will be introduced to enlarge dynamics to ‘fractal dynamics’ or, more generally speaking, to ‘scale dynamics’. This analysis enlightens the opportunities opened by the fractional derivative in quantum mechanics and, at the same time, restrains the generalisation. It is shown that one of the authorised generalisations is related to the ‘uncertainty principle’ which can be considered as a ‘universal principle’ concerning the defect of accuracy of the analysis with respect to a physical phenomenon. In the frame of quantum mechanics, the approach puts the emphasis on some conclusions that deal with the use of such operators and some adequate extension.

Thermodynamical properties and low frequency relaxation of interfacial space charges in polyethylene and polyethylene blends

An effort was made to determine whether the intimate structure of a polymer blend is homogeneous or fractal with phase segregation. This investigation involved the analysis of dielectric relaxation in polyethylene-polycarbonate blends in terms of the model of dissipation in fractal media. The effect of room-temperature aging on internal boundaries in polyethylene and polyethylene-polycarbonate blends was investigated. The results obtained are correlated with the presence of interphases in the bulk, as revealed by thermodynamical analysis.<<ETX>>

About the Fractal Relationship Between Kohlrausch-Williams-Watts Decay, Cole-Cole and Davidson-Cole Relaxations

It is shown that the Kohlrausch-Williams-Watts time decay (anomalous exponential decay \(\phi \left( t \right) \approx \exp \left( { - {{\left( {t/\tau } \right)}^\alpha }} \right)\)) and the inverse Fourier Transform of a Davidson-Cole (DC) transfer function \(\left[ {Zcc{{\left( p \right)}_ \sim }1/{{\left[ {1 + p{\tau _u}} \right]}^\beta }} \right]\) may experimentally be fitted together. On the other hand it is shown, from network theory, that physical processes whose transfer functions are Davidson-Cole or Cole-Cole \(\left[ {Zcc{{\left( p \right)}_ \sim }1/1 + {{\left( {p{\tau _u}} \right)}^\beta }} \right]\) are under the control of a same real order differential. This analysis suggests a filiation between all these common relaxation modes and raises the question of the relationship between the spectral dimension and the fractal dimension.