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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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Ricci flow and geometric analysis on manifolds
Eliáš, Jakub ; Somberg, Petr (advisor) ; Salač, Tomáš (referee)
Title: Ricci flow and geometric analysis on manifolds Author: Jakub Eliáš Ústav: Matematický ústav UK Supervisor: doc. RNDr. Petr Somberg Ph.D., Matematický ústav UK Abstract: This thesis discusses basis aspects of the Ricci flow on manifolds with a view towards the ambient space construction. We start with the back- ground review of the Riemannian geometry and parabolic partial differential equations, and the Ricci flow problem on manifolds is established. Then we aim towards the formulation of the Ricci flow problem on ambient spaces and provide several basic examples. There are two main parts: the first consists of general theory needed to formulate our problem and strategy, while the second part consists of particular calculations associated with the Ricci flow problem. Keywords: Ricci flow, Ambient space, Ambient metric, Poincaré-Einstein metric. 1
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Generalized Cartan geometries and invariant differential operators
Salač, Tomáš ; Souček, Vladimír (advisor) ; Krýsl, Svatopluk (referee)
We are getting familiar with difficulties with invariance of differential operators in case of parabolic geometries and fully characterize first order invariant operators. We define, so called curved Casimir operator. It is generalization of Casimir operator from representation theory. We give a new prove of characterization of first order invariant operators. We investigate more thoroughly behavior of curved Casimir operator on section of tractor bandle in conformal case and give list of various apllications
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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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Vector fields on spheres
Strakoš, Filip ; Salač, Tomáš (advisor) ; Golovko, Roman (referee)
This thesis deals with partial results concerning the problem of existence of vector fields on spheres. The proof of the Hairy Ball Theorem is given using the tools of the the- ory of characteristic classes. Basic notions of algebraic topology are stated in order to define the Euler class. Its definition is followed by the computation of the Euler charac- teristic class for the tangent bundle of even-dimensional sphere. In the rest of the text, the method of construction of vector fields on spheres using the orthogonal multiplica- tion is explained and the Radon-Hurwitz-Eckmann Theorem is proved. A brief historical background of the existence of the finite-dimensional real division algebras is mentioned at the end.
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Symmetry and Separation in the case of Laplace operator in low dimensions
Hudeček, Štěpán ; Krýsl, Svatopluk (advisor) ; Salač, Tomáš (referee)
In this thesis we analyze symmetry operators for partial differential opera- tors, in particular for Laplace and Helmholtz operators in dimension two and three. In both cases an important object is the Lie algebra of the Euclidean group. Separated solutions for partial differential operators are defined and il- lustrated for both of the mentioned operators. Examples of coordinate systems are listed, in which the solution separates. 1
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Ricci flow and geometric analysis on manifolds
Eliáš, Jakub ; Somberg, Petr (advisor) ; Salač, Tomáš (referee)
Title: Ricci flow and geometric analysis on manifolds Author: Jakub Eliáš Ústav: Matematický ústav UK Supervisor: doc. RNDr. Petr Somberg Ph.D., Matematický ústav UK Abstract: This thesis discusses basis aspects of the Ricci flow on manifolds with a view towards the ambient space construction. We start with the back- ground review of the Riemannian geometry and parabolic partial differential equations, and the Ricci flow problem on manifolds is established. Then we aim towards the formulation of the Ricci flow problem on ambient spaces and provide several basic examples. There are two main parts: the first consists of general theory needed to formulate our problem and strategy, while the second part consists of particular calculations associated with the Ricci flow problem. Keywords: Ricci flow, Ambient space, Ambient metric, Poincaré-Einstein metric. 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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Generalized Cartan geometries and invariant differential operators
Salač, Tomáš ; Krýsl, Svatopluk (referee) ; Souček, Vladimír (advisor)
We are getting familiar with difficulties with invariance of differential operators in case of parabolic geometries and fully characterize first order invariant operators. We define, so called curved Casimir operator. It is generalization of Casimir operator from representation theory. We give a new prove of characterization of first order invariant operators. We investigate more thoroughly behavior of curved Casimir operator on section of tractor bandle in conformal case and give list of various apllications
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