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Concept# Fractal dimension

Summary

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured.
It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.
The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure t

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SEM micrographs of the fracture surface for UO2 ceramic materials have been analysed. In this paper, we introduce some algorithms and develop a computer application based on the time-series method. Utilizing the embedding technique of phase space, the attractor is reconstructed. The fractal dimension, lacunarity, and autocorrelation dimension average value have been calculated.

The present work has been focused on the development of analytical and numerical techniques for the analysis of highly convoluted antennas and microwave devices including fractal shaped antennas. The accurate prediction of the frequency response of a high-iterated pre-fractal structure is frequently a very consuming task, in terms of computer resources. The techniques presented in this work try to ameliorate some of the bottlenecks in the solving process of fractal shaped or highly convoluted devices. First, in the frame of the Mixed Potential Integral Equation (MPIE) technique, a new set of basis functions for the discretization of the currents in the Method of Moments (MoM) solution is presented. The basis functions are defined over quadrangular domains and their aim is, on one hand, to allow a good representation of the current while preserving the main longitudinal direction existing in many practical surfaces, used as metalization in printed circuits, and on the other hand, to reduce the number of unknowns compared to a standard triangular mesh. The basis functions over quadrangular cells comprise as particular cases the classic rectangular and triangular rooftops. However, a new basis function over triangular domains is also included as a case derived from the general quadrangle, and has the particularity of being able to model the connection between two triangles with a common vertex, instead of the conventional attachment at the edge of the classic triangular pair. Second, different highly convoluted Euclidean structures have been studied in order to provide a benchmark for fractal devices performance. The considered structures are a meander line and a two-arm square spiral antenna. Both structures show miniaturization capabilities, the spiral being one of the outstanding shapes in terms of miniaturization keeping a reasonable frequency behavior. With the study of these two structures it has been shown that some properties, considered exclusive of the fractal shaped family, appear also in non-fractal shapes. Third, several analysis techniques based on a transmission line approach and specially suited to solve highly convoluted printed line devices have been developed. The aim is to have a simple and fast tool to allow a rapid analysis of complicated structures, providing reasonably accurate results taking into account the simplicity of the model. These methods have been applied to a set of fractal curves belonging to the family of the fractal tree. Some prototypes have been built in microstrip technology and measured to verify the validity of the method. Finally, in order to obtain a fast full-wave analysis of printed-line or wire-strip devices, a new technique is developed. The method takes advantage of the geometry of the structures presenting currents flowing mainly in the longitudinal direction. These 2D structures are then considered as 1D ones, thus, the cells are reduced to their axis or backbones. This approximation is valid for the limiting cases, namely, very narrow structures or interactions between far away cells. However, once the width compensation factor that is computed analytically is included, the approximation becomes valid for all the structures not having a width bigger than the standard mesh density limit in a 2D problem. Values of the error committed with respect to a classic 2D method are given, and the analysis of some line structures is performed, proving the validity of the proposed method.

Supramolecular hydrogels based on chitosan and monoaldehydes are biomaterials with high potential for a multitude of bioapplications. This is due to the proper choice of the monoaldehyde that can tune the hydrogel properties for specific practices. In this conceptual framework, the present paper deals with the investigation of a hydrogel as bioabsorbable wound dressing. To this aim, chitosan was cross-linked with 2-formylphenylboronic acid to yield a hydrogel with antimicrobial activity. FTIR, NMR, and POM procedures have characterized the hydrogel from a structural and supramolecular point of view. At the same time, its biocompatibility and antimicrobial properties were also determined in vitro. Furthermore, in order to assess the bioabsorbable character, its biodegradation was investigated in vitro in the presence of lysosome in media of different pH, mimicking the wound exudate at different stages of healing. The biodegradation was monitored by gravimetrical measurements, SEM microscopy and fractal analyses of the images. The fractal dimension values and the lacunarity of SEM pictures were accurately calculated. All these successful investigations led to the conclusion that the tested materials are at the expected high standards.

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