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Application of the Hermite basis for spectral analysis
Mihálik, Ondrej ; Tůma, Martin (referee) ; Jura, Pavel (advisor)
The work is concerned with an application of the Hermite functions in signal approximation. The purpose of the work is to show their properties in time and frequency domains, namely their orthogonality, Fourier transform, zeros and asymptotic behaviour as their order becomes high. The next subject of this work is the question of scaling these functions to minimize the square error of signal approximation. Several methods proposed by different authors are discussed. Finally these algorithms are tested by approximating simple signals so that their results can be compared.
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Application of Chebyshev Basis for Spectral Analysis
Ettl, Ondřej ; Jirgl, Miroslav (referee) ; Mihálik, Ondrej (advisor)
The work is focused on finding and verifying the basic properties of Chebyshev polynomials in Hilbert space. These include their generating functions, weight functions, orthogonality and recurrent relationships. Another goal was signal processing by Chebyshev’s transform and investigation of the resulting spectrum. Lastly the focus is shifted towards demostrain of two methods for modeling of frequency spectrum with help of Chebyshev polynomials.
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Multidimensional Liquid Phase Separations
Šesták, Jozef ; Čáslavský, Josef (referee) ; Pulkrabová, Jana (referee) ; Česla, Petr (referee) ; Kahle, Vladislav (advisor)
This dissertation is dedicated to the topic of multidimensional liquid phase separations. This separation techniques are developed for analysis of complex samples containing thermally labile, low volatile or high molecular weight components that can´t be analysed by two-dimensional (2D) gas chromatography. Concepts of peak capacity and orthogonality are explained and various methods of their determination are stated in theoretical part of dissertation. High performance column liquid chromatography (HPLC) and high performance capillary electrophoresis (HPCE) are suggested as the most suitable methods for automated multidimensional liquid phase separations on-line coupled to mass spectrometry. Configuration of simplified miniaturized liquid chromatograph is described in experimental part of this thesis. Original concept of the system has been extended by simple mobile phase gradient generation technique. Correct function was demonstrated on repeatable separation of alkylphenones, peptides, nitroaromatics, and nitroesters. This system has been utilized as a base for a couple of simple two-dimensional separation platforms for HILIC-MALDI-MS analysis of glycans, for separation of peptides based on off-line coupling of isoelectric focusing and capillary liquid chromatography, and finally for on-line IEC×RPLC, RPLC×RPLC, and HILIC×RPLC two-dimensional liquid chromatography. Correct operation of submitted platforms has been proved.
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Application of Chebyshev Basis for Spectral Analysis
Ettl, Ondřej ; Jirgl, Miroslav (referee) ; Mihálik, Ondrej (advisor)
The work is focused on finding and verifying the basic properties of Chebyshev polynomials in Hilbert space. These include their generating functions, weight functions, orthogonality and recurrent relationships. Another goal was signal processing by Chebyshev’s transform and investigation of the resulting spectrum. Lastly the focus is shifted towards demostrain of two methods for modeling of frequency spectrum with help of Chebyshev polynomials.
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Orthogonal bases and Jordan normal form
Kučera, Daniel ; Šaroch, Jan (advisor) ; Barto, Libor (referee)
There exists an ortonormal set of eigenvectors for a linear operator if and only if it commutes with its adjoint endomorphism. The aim of this thesis is to characterize endomorphisms for which there exists a matrix representation with respect to an orthogonal basis in Jordan form. We introduce the notion of unitarily jordanisable endomorphism to capture this property. The proof of the Spectral theorem as well as the existence and uniqueness of Jordan form can be found in the first two chapters. An interesting connection with bilinear forms arises in chapter three and is used to prove that any linear operator with single eigenvalue and the length of Jordan chains bounded by two is unitarily jordanisable. The last chapter is devoted to the discussion of uniqueness of othogonal polar basis for a bilinear form and an algorithm is introduced which can determine whether or not a linear operator is unitarily jordanisable. 1
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Orthogonal bases and Jordan normal form
Kučera, Daniel ; Šaroch, Jan (advisor) ; Barto, Libor (referee)
There exists an ortonormal set of eigenvectors for a linear operator if and only if it commutes with its adjoint endomorphism. The aim of this thesis is to characterize endomorphisms for which there exists a matrix representation with respect to an orthogonal basis in Jordan form. We introduce the notion of unitarily jordanisable endomorphism to capture this property. The proof of the Spectral theorem as well as the existence and uniqueness of Jordan form can be found in the first two chapters. An interesting connection with bilinear forms arises in chapter three and is used to prove that any linear operator with single eigenvalue and the length of Jordan chains bounded by two is unitarily jordanisable. The last chapter is devoted to the discussion of uniqueness of othogonal polar basis for a bilinear form and an algorithm is introduced which can determine whether or not a linear operator is unitarily jordanisable. 1
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Application of the Hermite basis for spectral analysis
Mihálik, Ondrej ; Tůma, Martin (referee) ; Jura, Pavel (advisor)
The work is concerned with an application of the Hermite functions in signal approximation. The purpose of the work is to show their properties in time and frequency domains, namely their orthogonality, Fourier transform, zeros and asymptotic behaviour as their order becomes high. The next subject of this work is the question of scaling these functions to minimize the square error of signal approximation. Several methods proposed by different authors are discussed. Finally these algorithms are tested by approximating simple signals so that their results can be compared.
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|
|
Multidimensional Liquid Phase Separations
Šesták, Jozef ; Čáslavský, Josef (referee) ; Pulkrabová, Jana (referee) ; Česla, Petr (referee) ; Kahle, Vladislav (advisor)
This dissertation is dedicated to the topic of multidimensional liquid phase separations. This separation techniques are developed for analysis of complex samples containing thermally labile, low volatile or high molecular weight components that can´t be analysed by two-dimensional (2D) gas chromatography. Concepts of peak capacity and orthogonality are explained and various methods of their determination are stated in theoretical part of dissertation. High performance column liquid chromatography (HPLC) and high performance capillary electrophoresis (HPCE) are suggested as the most suitable methods for automated multidimensional liquid phase separations on-line coupled to mass spectrometry. Configuration of simplified miniaturized liquid chromatograph is described in experimental part of this thesis. Original concept of the system has been extended by simple mobile phase gradient generation technique. Correct function was demonstrated on repeatable separation of alkylphenones, peptides, nitroaromatics, and nitroesters. This system has been utilized as a base for a couple of simple two-dimensional separation platforms for HILIC-MALDI-MS analysis of glycans, for separation of peptides based on off-line coupling of isoelectric focusing and capillary liquid chromatography, and finally for on-line IEC×RPLC, RPLC×RPLC, and HILIC×RPLC two-dimensional liquid chromatography. Correct operation of submitted platforms has been proved.
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