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The Bochner periodic relations
Oliva, Filip ; Souček, Vladimír (advisor) ; Lávička, Roman (referee)
The Bochner periodic relations describe behaviour of the Fourier transform on isotypic components of the action of the special orthogonal Lie group SOm on the Schwartz space. Fourier transform can be expressed as a suitable element of the double cover ˜︃SL2 of special Lie group SL2. That is possible due to the Howe duality describing the decomposition of Schwartz space to ˜︃SL2 × SOm-invariant isotypic components. 1
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Howe duality and invariant differential equations
Beďatš, Daniel ; Lávička, Roman (advisor) ; Souček, Vladimír (referee)
Separation of variables for scalar-valued polynomials in k variables of dimension n, which asks for any such polynomial to be decomposed into a combination of invariant and harmonic polynomials, is known to be unique in the semistable range n ≥ 2k − 1. In this thesis, we explore the problem in the non-stable range n < 2k − 1. We give a systematic description of the non-uniqueness of separation of variables when n = 2k − 2 and n = 2k − 3 in terms of generalized Verma modules. We prove that the condition n ≥ 2k − 1 is not only sufficient, but also necessary for uniqueness. The problem is illustrated by a few detailed low-dimensional examples. As a prerequisite to the topic, we present a summary of the theory of classical Howe dual pairs and a classification of irreducible representations of the complex orthogonal group. 1
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Orthogonal polynomials in hypercomplex analysis
Malý, Marek ; Lávička, Roman (advisor) ; Salač, Tomáš (referee)
In this thesis, we describe a construction of orthogonal basis of polynomial solutions to the Laplace and Dirac operators over the Euclidian space Rm . A necessary property is rotational invariance of these operators. Described construction gives us so-called Gelfand- Tsetlin basis, which is orthogonal with respect to any rotational invariant scalar product, e.g. with recpect to the L2 -scalar product on the unit ball. For this basis, we calculate the norms of their elements and we apply our findings for dimension 3. 1
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Spin groups in low dimension
Knesel, Jakub ; Šmíd, Dalibor (advisor) ; Lávička, Roman (referee)
The aim of his thesis is to construct matrix representations of the Lie groups Spin(n) = Spin(0, n, R) in dimensions from one to six. After we construct the double-cover of the group SO(3) using the group SU(2) in the first chapter, we will define the Clifford algebra, which we will use to construct the spin group in general. We will also describe how the spin group Spin(n) provides a double-cover of the group SO(n). Using this theory, we will then construct matrix representations of the Clifford algebra and the spin group Spin(n) in all the low dimensions listed above respectively. Apart from Clifford algebra, all arguments in this thesis will be based only on linear algebra and elementary group theory. 1
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The generalized Dolbeault complexes in Clifford analysis
Salač, Tomáš ; Souček, Vladimír (advisor) ; Lávička, Roman (referee) ; Slovák, Jan (referee)
In the thesis we study particular sequences of invariant differ- ential operators of first and second order which live on homogeneous spaces of a particular type of parabolic geometries. We show that they form a reso- lution of the kernel of the first operator and that they descend to resolutions of overdetermined, constant coefficient, first order systems of PDE's called the k-Dirac operators. This gives uniform description of resolutions of the k-Dirac operator studied in Clifford analysis. We give formula for second order operators which appear in the resolutions. 1
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Canonical bases for solutions of invariant differential equations
Jančík, Michael ; Lávička, Roman (advisor) ; Souček, Vladimír (referee)
Spherical harmonics and spherical monogenics are, respectively, polynomial solutions of Laplace and Dirac equations. In R3 these solutions form irreducible representations of Lie algebra sl(2, C). The main aim is to construct orthogonal bases of such spaces. The well-known procedures like Gram-Schmidt orthogonalization procedure is quite clumsy and tedious. We show how to construct orthogonal bases in an easier way using repre- sentation theory. For description of rotations in R3 and R4 we use quaternions. Finally, we express constructed bases in spherical coordinates 1
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South Bohemian sacral architecture of the late Gothic 1450 - 1550 in the Rosenberg domain
Lavička, Roman ; Kuthan, Jiří (advisor) ; Jarošová, Markéta (referee)
South Bohemian sacral architecture of the late Gothic1450 - 1550 in the Rosenberg domain Mgr. Roman Lavička A group of late-Gothic churches, which keeps attracting our attention, was built in the Rosenberg domain during the reign of the Jagiellon dynasty. Limited archival material, results of dendrochronologic analyses and preserved buildings suggest that it was mainly building maintenance that was in progress in the Rosenberg domain in the second half of the 15th century. In the early 1480s the building of the new pilgrimage church in Kájov (1471/1474- 1485) was coming to its end. And it was only then that big town parish churches in Trhové Sviny, Dolní Dvořiště, Hořice na Šumavě, Chvalšiny and Nové Hrady started being rebuild or constructed anew almost simultaneously. Around 1485 almost twenty religious buildings were being built in the Rosenberg domain. They reflect the increasing economic prosperity and self-confidence of the town and village municipalities, corporations as well as individuals manifesting their wealth, success and strength. From a formal and ideological point of view it is possible to trace down a group of related buildings directly in this initial phase. They were probably designed by a single architect who is referred to as Master of Hořice presbytery. However, around 1495 we can...
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