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Functions of Several Variables
Published in John Srdjan Petrovic, Advanced Calculus, 2020
10.7.24. * The purpose of this problem is to present the Cantor ternary set, and some of its properties. Let I0 = [0, 1]. If we remove the middle third (1/3, 2/3), we obtain the set I1=[0, 1/3] ∪ [2/3, 1]. Next, we remove the middle thirds from each of the two parts, and we obtain I2 = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1]. By removing the middle thirds from each of the four parts of I2 we obtain I3. Continuing in this fashion, we obtain a sequence In of sets. The Cantor set C is defined as C = ∩n=0∞In. Prove that C is compact.Prove that C is a perfect set (a closed set with no isolated points).Prove that C is nowhere dense (its closure has no interior points).
Force-System Resultants and Equilibrium
Published in Richard C. Dorf, The Engineering Handbook, 2018
A whirl of dust, raised by the wind, may be very thin, or it may so thick as to fill the whole space. The fractal dimension of a dust can assume any value between zero and three. It equals zero, when the dust is extremely sparse, and three when the dust fills the whole three dimensional space. A Cantor set is a dust type of set; it has topological dimension 0, but it has a positive Hausdorff dimension that could be as big as the dimension of the surrounding space. The graph of a real nowhere differentiable of one real variable may have any Hausdorff dimension between 1 and 2. The trajectory of a particle driven by a Brownian motion has Hausdorff dimension greater than 1. A continuous graph describing the performance of a stock can sometimes fluctuate so much, during periods of market volatility, as to fill whole solid areas on the computer screen. The Hausdorff dimension of this graph can assume any value between one and two.
Fractal Dimension
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Pablo M. Blanco, Sergio Madurga, Adriana Isvoran, Laura Pitulice, Francesc Mas
The Cantor set (C) was developed by Georg Cantor (a German mathematician) in 1877. This fractal set is generated slicing a unit segment in three equal parts and removing the central one. Then, the two ending parts are sliced as well in three equal parts, and the central part is removed, and this process is infinitely repeated (Figure 16.2). Using this rule, in the k iteration there are 2k segments of (1/3)k length (Figure 16.3).
Metrics with unit discs of arbitrary shape
Published in International Journal of Mathematical Education in Science and Technology, 2021
We can modify the general construction above to give a metric on the plane where all discs of radius less than 1 are fractals. Fractals are irregularly shaped sets which have fractional dimension (Barnsley, 1993). For instance, the Cantor set has topological dimension 0 but fractal dimension ln(2)/ln(3). To obtain such a metric, let f: be a function whose graph is a fractal (Hunt, 1998). One such function is Let E0 = {(x, f(x)): x } be the graph of f. For α , define Eα = E0 + (0, α) = {(x, f(x) + α): x }. The collection of fractals {Eα: α } forms a partition of the plane 2.
A hybrid computational method for local fractional dissipative and damped wave equations in fractal media
Published in Waves in Random and Complex Media, 2022
Ved Prakash Dubey, Jagdev Singh, Ahmed M. Alshehri, Sarvesh Dubey, Devendra Kumar
In this part, the computer simulations have been performed for the solutions of LFWEs acquired through LFNVIM. All the 3D plots in the simulation analysis have been formed through the MATLAB software for this purpose. The LFNVIM generates a solution in a closed form of the Mittag-Leffler function. The basic properties of Mittag-leffler function report that the obtained solutions in Figures 1–7 are nondifferentiable, and the wave morphology can be adjusted by proper pick of the parameters in the Mittag-Leffler function. The Cantor set is prepared by the unit interval by continuous deletion of middle thirds and each interval acquires length 1/3 that of the preceding stage. This construction process is repeated again and again by removal of the open middle thirds of each closed interval remaining. And these steps are continued in an indefinite way. Now, the fractal dimension of the Cantor set is computed by utilizing the self-similar feature. At the limiting stage , the Cantor set consists of infinitely many isolated points with topological dimension zero. At any stage , boxes of length are required to cover the set. It is noted that for the n-box is reduced to a closed interval. Hence, the fractal dimension of the Cantor set is calculated as follows: