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Wearable Compact Fractal Antennas for 5G and Medical Systems
Published in Albert Sabban, Wearable Systems and Antennas Technologies for 5G, IOT and Medical Systems, 2020
The term “fractal curve” was introduced by B. Mandelbrot [18, 19] to describe a family of geometrical objects that are not defined in standard Euclidean geometry. Fractals are geometric shapes that repeat themselves over a variety of scale sizes. One of the key properties of a fractal curve is self-similarity. A self-similar object is unchanged after increasing or shrinking its size. An example of a repetitive geometry is the Koch curve, presented in 1904 by Helge von Koch; it is shown in Figure 13.1b. Koch generated the geometry by using a segment of a straight line and raising an equilateral triangle over its middle third. Repeating the process of erecting equilateral triangles over the middle thirds of straight lines led to what is presented in Figure 13.2a. Iterating the process infinitely many times results in a “curve” of infinite length. This geometry is continuous everywhere but is nowhere differentiable. If the Koch process is applied to an equilateral triangle, then, after many iterations, the iterations converge to the Koch snowflake shown in Figure 13.2b. This process can be applied to several geometries as shown in Figure 13.3 and Figure 13.4. Many variations of these geometries are presented in several papers.
Vlasov equation, waves and dispersion relations in fractal dimensions: Landau damping and the toroidal ion temperature gradient instability problem
Published in Waves in Random and Complex Media, 2022
Rami Ahmad El-Nabulsi, Waranont Anukool
Fractals are crinkly objects characterized by their fractional dimension and contravene standard and conventional geometrical and physical measures, such as length and area. They are self-similar since they could be decomposed into subsets that can be linearly mapped into the complete shape. They are characterized by a fractal dimension which is applied for fractal curves that are not self -similar, but are diagonally self-affine. The notion of fractal dimension is also obtained by means of the mass in a sphere and using uniform boxes as well. It is defined by the box-counting dimension and consequently, a number of fractals objects are characterized by an intricate geometrical structure and cannot be characterized by a unique fractal dimension. In general, fractals have played a leading role in various fields of sciences and engineering such as biology [1], medicine [2], the science of materials [3–5], elasticity theory [6–10], quantum mechanics and molecular physics [11–16], astronomy [17,18], fluid mechanics [19–21], statistical physics [11,22–30], plasma physics and magnetohydrodynamics (MHD) [31–33], and many others. In general, the notion of fractal dimension was introduced in aggregation, gelation, random walks, fracture of cracked metals, diffusion-limited aggregates, etc. [34–36]. It is noteworthy that fractal dimension is also used to measure the complexity of spatiotemporal objects in geometrical frames [37] and a complete spectrum of fractal dimensions such as the Renyi dimensions is therefore required [38–40].
The fractal behaviour of gravity field elements: case study in Egypt
Published in Journal of Spatial Science, 2023
Fractal curves and surfaces are supposed to be self-similar or self-affine. This implies that a small fraction of any fractal feature should – at least statistically – resemble its shape as a whole. That is why such fractal features are considered scale-invariant (Pelletier and Turcotte 1996). Mathematically, fractals may be of a deterministic or a stochastic (or random) type (Falconer 1990). While both types can be used in simulation purposes, the empirical analyses of natural phenomena would obviously presuppose random fractals. Spatially, fractal features may obey either mono- or multi-fractal behaviour (Falconer 1990). This corresponds to a homogeneous or spatially varying fractal behaviour, respectively (Mandelbrot 1989, Peitgen et al. 2004).
Design of a novel 5G MIMO antenna with its DGP optimisation using PSOGSA
Published in International Journal of Electronics, 2022
Amandeep Kaur Sidhu, Jagtar Singh Sivia
The Iterative Function System (IFS) is used to achieve the above-mentioned geometry of hybrid fractal curve by smearing affinal transformations (A) to an elementary shape (P). Affine conversion contains scaling, transformation and rotation and can be represented as: