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Determination of a rectangular plate deformation
Mácha, Tomáš ; Profant, Tomáš (referee) ; Hrstka, Miroslav (advisor)
The aim of the thesis is to derive the analytical equations for calculating the deformation of a rectangular plate using Kirchhoff and Mindlin theory and their subsequent application. The first section provides some basic relations in the theory of elasticity, on which this derivation is based. Both of the theories are solved using Navier method of Fourier series. By the following the results of these two methods can be compared. The comparison is carried out using Python where the derived relations with given input parameters are used. This is how the magnitude of deflection over the entire surface of the plate is obtained. Historical part includes a description of the gradual development from the second half of the 18th century, up to the invention of the finite element method. Finally, these theories have been applied to some specific examples and the results have been analyzed and consecutively compared with results from ANSYS. The limitations and suitability of these theories are then outlined.
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Orthogonal bases and their application in signal processing
Kárský, Vilém ; Tůma, Martin (referee) ; Jura, Pavel (advisor)
This work is concentrates on finding basic properties of some orthogonal polynomials like a definition, weight function, orthogonality interval, recurrence relations, number of zeros and diferential eguations which they were suited on. Subsequently were founded formulas for calculating coefficients of the generalized Fourir series and I concentrate on calculating optimal free parameters on this orthogonal polynomials. In the end of this work are calculated and displayed spectrums of some functions in the bases of individual polynomials and was calculated and displayed aproximation error.
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Fourier transform of periodic structures
Zajíc, Tomáš ; Zahradník, Miloš (advisor) ; Krýsl, Svatopluk (referee)
Mathematical description of Fourier transform of the periodic structure. We introduce the concept of the Fourier series and we investigate the Dirichlet kernel. We also introduce the concept of distributions, the Fourier transform and convolution. Using this we discover the properties of the Dirac's delta, the Dirac comb and then we define the periodic structure. In conclusion, we mention the dual lattice. The thesis is designed to contain physical notes. Some of proofs are formal.
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Laplace equation in fractional Sobolev spaces
Bartoš, Ondřej ; Bárta, Tomáš (advisor) ; Vybíral, Jan (referee)
The goal of this thesis is to study Laplace's equation on a unit disc. The given function values on a unit circle can be interpreted as a 2π-periodic function and the solution can be derived using Fourier method. We introduce general integer Sobolev spaces and their alternatives useful for describing functions on a unit disc and a unit circle. Using elementary methods, we show how they are related to each other. The same results are shown for fractional Sobolev spaces. The main result is that functions from some Sobolev space on a unit disc that solve Laplace's equation correspond to functions from a one half lower Sobolev space on a unit circle. These results can be used to show for a function from some Sobolev space on a unit circle in how strong norm the solution of Laplace's equation converges to the given function. 1
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Dot product - definition and applications
Weissgráb, Lukáš ; Halas, Zdeněk (advisor) ; Rmoutil, Martin (referee)
This bachelor thesis presents various introductions of the dot product at several levels of difficulty. First part of the thesis deals with introduction of the dot product elementary, therefore only from knowledge of high school. The more advanced parts of this thesis are devoted to the introduction of the dot product as a bilinear form and focus on properties of this form. The final chapters are devoted to the Fourier series and the first fundamental form. All theorically explained pieces of knowledge are illustred with examples from mathematics and physics.
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Nerovnosti Friedrichsova a Poincarého typu a jejich výpočet
MOSKOVKA, Alexej
This thesis deals with the theory of Friedrichs' and Poincaré inequalities and their constants. They are important in mathematical analysis, functional analysis and theory of partial differential equations. The key property of them is the boundness of L-norms of functions by L-norms of gradients of functions. Constants can be derived analytically for simple geometries or approximated numerically. We provide an explicit derivation for an interval, a rectangle and a rectangular cuboid. We also perform a numerical computation for the interval and the rectangle as well as for an annulus, for which constants are unknown.
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Fourier transform of periodic structures
Zajíc, Tomáš ; Zahradník, Miloš (advisor) ; Krýsl, Svatopluk (referee)
Mathematical description of Fourier transform of the periodic structure. We introduce the concept of the Fourier series and we investigate the Dirichlet kernel. We also introduce the concept of distributions, the Fourier transform and convolution. Using this we discover the properties of the Dirac's delta, the Dirac comb and then we define the periodic structure. In conclusion, we mention the dual lattice. The thesis is designed to contain physical notes. Some of proofs are formal.
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Laplace equation in fractional Sobolev spaces
Bartoš, Ondřej ; Bárta, Tomáš (advisor) ; Vybíral, Jan (referee)
The goal of this thesis is to study Laplace's equation on a unit disc. The given function values on a unit circle can be interpreted as a 2π-periodic function and the solution can be derived using Fourier method. We introduce general integer Sobolev spaces and their alternatives useful for describing functions on a unit disc and a unit circle. Using elementary methods, we show how they are related to each other. The same results are shown for fractional Sobolev spaces. The main result is that functions from some Sobolev space on a unit disc that solve Laplace's equation correspond to functions from a one half lower Sobolev space on a unit circle. These results can be used to show for a function from some Sobolev space on a unit circle in how strong norm the solution of Laplace's equation converges to the given function. 1
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Convergence of Fourier series in Lp spaces
Michálek, Martin ; Zelený, Miroslav (advisor) ; Spurný, Jiří (referee)
The main question of this thesis is whether the partial sums of Fourier series converge in some sense to the function from which the series was derived. In our case we will analyze the convergence of Fourier series of Lebesgue integrable functions and the convergence will be meant in the sense of Lp spaces for p ∈ [1, ∞). The case p = 2 could be concluded from properties of orthogonal basis in Hilbert spaces. Our intention is to analyze the problem especially for the other p ∈ [1, ∞). Therefore we need to use some results from the theory of Banach (particularly Lp ) spaces.
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Orthogonal bases and their application in signal processing
Kárský, Vilém ; Tůma, Martin (referee) ; Jura, Pavel (advisor)
This work is concentrates on finding basic properties of some orthogonal polynomials like a definition, weight function, orthogonality interval, recurrence relations, number of zeros and diferential eguations which they were suited on. Subsequently were founded formulas for calculating coefficients of the generalized Fourir series and I concentrate on calculating optimal free parameters on this orthogonal polynomials. In the end of this work are calculated and displayed spectrums of some functions in the bases of individual polynomials and was calculated and displayed aproximation error.
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