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Fractal Analysis of Track Geometry
Nejezchlebová, Jitka ; Holcner, Petr (referee) ; Svoboda, Richard (advisor)
The master’s thesis deals with the fractal analysis of track geometry. The theoretical part is primarily focused on introducing the basics of fractal geometry. There is also concisely described the current methodology for evaluating track geometry. In the practical part is verified the accuracy of the methods for determining the fractal dimension of the curve. For different curves with the same standard deviation is calculated the fractal dimension to demonstrate the possible advantages of using this analysis. Further is researched the use of the fractal dimension for the analysis of track geometry data. All mathematical procedures are done using the MATLAB system.
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A guide to fractal geometry
Hajmová, Kateřina ; Pokorný, Dušan (advisor) ; Boček, Leo (referee)
This text is intended for the general public. The aim of this work is acquaint readers with foundations of the fractal geometry. The thesis explains important terminology, such as the coastline paradox or the fractal dimension. A great emphasis is placed on explaining the concept of the box-counting dimension. The thesis includes the construction methods of the L-systems, IFS, TEA and random fractals. In addition, it shows the use of fractal geometry in practice. The text is completed with illuminating figures drawn in most cases in Geogebra software and Wolfram Mathematica.
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Hausdirff metric and its application in fractals
Roháľ, Branislav Ján ; Hušek, Miroslav (advisor) ; Pyrih, Pavel (referee)
Title: Hausdorff metric and its application in fractals Author: Branislav Ján Roháľ Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Miroslav Hušek, DrSc., Department of Mathematical Analysis Abstract: In this thesis we focus on the themes naturally connected with the con- cept of a fractal. In the first part of the thesis we pay attention to Banach fixed point theorem and to the Hausdorff metric which are later used when studying self-similar sets. There are included parts on the Hausdorff, similarity, and box- counting dimension, too. In the second part of the thesis the new approaches to fractal dimension and some their properties are refered. We introduce generaliza- tion of this concept for any space admitting a fractal structure and for a distance space where also the "size" of sets on each level of fractal structure is considered. In the last chapter the contribution of new approache is demonstrated, - this enables defining the notion needed and counting fractal dimension where it was not possible under the classical approaches, too. Application to the domain of words and counting of dimensions of a language generated by a regular expresion are presented. Keywords: Hausdorff metric, Banach fixed point theorem, self-similar set, Hausdorff dimension, fractal dimension
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X-ray micro-tomography characterization of voids caused by three-point bending in selected alkali-activated aluminosilicate composite
Kumpová, Ivana ; Rozsypalová, I. ; Keršner, Z. ; Rovnaníková, P. ; Vopálenský, Michal
This paper deals with the pilot characterization of a special alkali-activated aluminosilicate composite composed of waste brick powder, brick rubble and a solution of potassium water glass. Fracture tests were conducted on the specimens via three-point bending and fracture parameters were evaluated. Selected specimen was investigated using micro-tomography to supplement the results with visual information about the inner structure of this newly designed material before and after the mechanical loading. Tomographic measurements and image processing were conducted for a qualitative and quantitative assessment of changes in the internal structure with an emphasis on the calculation of porosimetric parameters and visualization of the fracture surface. Fractal dimension of fracture surface was estimated.
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Determination of Roughness Factor and Fractal Dimension of Zirconium in its Native and Surface Modified State using Atomic Force Microscopy. Effect of the Hydrogen Evolution Reaction on the Surface Structure
Novák, M. ; Kocábová, Jana ; Kolivoška, Viliam ; Pospíšil, Lubomír ; Macák, J. ; Cichoň, Stanislav ; Cháb, Vladimír ; Hromadová, Magdaléna
Atomic force microscopy (AFM) was used to characterize surface morphology of pristine zirconium, Si modified and FeSi modified zirconium electrodes prior and after hydrogen evolution at potentials negative of the open circuit potential value. Two main characteristic parameters were obtained from the ex situ AFM height images, namely, the roughness factor and fractal dimension of the studied surface. The effect of hydrogen evolution reaction on the electrode surface morphology was discussed. Fractal dimension values were used successfully to explain the non ideality of the interfacial capacitance.
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A guide to fractal geometry
Hajmová, Kateřina ; Pokorný, Dušan (advisor) ; Boček, Leo (referee)
This text is intended for the general public. The aim of this work is acquaint readers with foundations of the fractal geometry. The thesis explains important terminology, such as the coastline paradox or the fractal dimension. A great emphasis is placed on explaining the concept of the box-counting dimension. The thesis includes the construction methods of the L-systems, IFS, TEA and random fractals. In addition, it shows the use of fractal geometry in practice. The text is completed with illuminating figures drawn in most cases in Geogebra software and Wolfram Mathematica.
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The fractal dimension and forecasting of financial time series
Kaplan, Robert ; Krištoufek, Ladislav (advisor) ; Džmuráňová, Hana (referee)
In this thesis, we strive to build on the fractal market hypothesis and to develop two methods which aim to reveal whether the fractal dimension, as a property of the short memory, can be applied for forecasting of financial time series. In the first one, we use ten world market indices and repeatedly estimate the fractal dimension by boxcount, Hall-Wood, and Genton estimators on fixed number of returns and make one step ahead forecasts by AR(1) and ARMA(1,1) models; then, we look whether forecast errors from realized returns are lower when the fractal dimension is estimated lower. The second method incorporates only the fractal dimension and studies, if the sign of return persists in next period more likely with lower fractal dimension. The results indicate that the short memory is truly present in the markets and the fractal dimension may be potentially useful for prediction and increased profit for investors. However, the significance of our results is not strong. We recommend more sophisticated methods and models for further research.
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