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Moufang plane and Spin groups
Stejskal, Dominik ; Krýsl, Svatopluk (advisor) ; Holíková, Marie (referee)
In this thesis we consider the action of the exceptional simple Lie group F4 on the so called (real) Moufang plane OP2 R. The goal of this thesis is to present a proof of the transitivity of this action, which is as complete as possible. We first define related concepts such as Clifford algebras, the groups Pin(r, s) and Spin(r, s) and the algebra of octonions O, and we prove their basic properties. The group F4 is defined as the automorphism group of the algebra J3(O) of hermitian octonionic matrices of order three. The Moufang plane is defined as a suitable subset of J3(O). In the group F4 we find isomorphic copies of the groups Spin(0, 8) and Spin(0, 9). By applying certain auxilliary results from the previous chapters we obtain the desired proof of the transitivity of the action of F4 on OP2 R. 1
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Symplectic Dirac operators on Gr2(C4)
Hudeček, Štěpán ; Krýsl, Svatopluk (advisor) ; Golovko, Roman (referee)
In this thesis we are presenting a construction of the symplectic Dirac operators as done by Katharina Habermann in 1995. We emphasize the differences with the classical Dirac operators. We are then computing the associated second order operator to the symplectic Dirac operators on the Kähler symmetric space Gr2(C4 ). We have also managed to find a way of inductive computing of its spectrum and we are presenting explicitly a part of the spectrum. 1
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New Integral Formulae in Hypercomplex Analysis
Sikora, Martin ; Souček, Vladimír (advisor) ; Krýsl, Svatopluk (referee) ; Vanžura, Jiří (referee)
Title: New Integral Formulae in Hypercomplex Analysis Author: Mgr. Martin Sikora Department: Mathematical Institute of Charles University Supervisor: prof. RNDr. Vladimír Souček, DrSc., MÚ UK Supervisor's e-mail address: soucek@karlin.mff.cuni.cz Abstract: The Dirac equation for Clifford algebra-valued functions on the even-dimensional Minkowski space can be understood as a hyperbolic sys- tem of partial differential equations. We show how to reconstruct the solution from initial data given on the upper sheet of the hyperboloid. In particular, we derive an integral formula which expresses the value of a function in a chosen point as an integral over a compact cycle given by the intersection of the null cone with the upper sheet of the hyperboloid in the Minkowski space. We also treat the ultra-hyperbolic case where the Dirac equation gives the ultra-hyperbolic system of partial differential equations. An analogue of the second order Cauchy formula is proved for (n − 1)-vector-valued holo- morphic functions. It reconstructs values inside a bounded domain in the 2n-dimensional complex space by integrating over the characteristic boun- dary of the domain. 1
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Calculus of variation in Physics and Geometry
Kuchařík, Jan ; Krýsl, Svatopluk (advisor) ; Scholtz, Martin (referee)
Název práce: Variační počet ve fyzice Autor: Jan Kuchařík Katedra / Ústav: Matematický ústav UK Vedoucí bakalářské práce: RNDr. Svatopluk Krýsl, Ph.D. Abstrakt: Ve své práci shrnuji některá základní použití variačního počtu v praktických aplikacích. Odvozuju zde nezbytný matematický aparát. Zavádím pojem matematického funkcionálu a jeho extremalizaci, odvozuji Euler-Lagrangeovu rovnici a její důsledek - Beltramiho identitu; dále se věnuji odvození metody řešení izoperimetrických úloh, která zobecňuje metodu Lagrangeových multiplikátorů. Ačkoliv se v práci vyskytují řešené úlohy nejrůznějšího typu, zaměřuju se na čtyři hlavní oblasti: Fermatův princip, Hamiltonův princip nejmenší akce, isoperimetrické úlohy a hledání geodetik. Title: Variational calculus in physics Author: Jan Kuchařík Department: Supervisor: RNDr. Svatopluk Krýsl, Ph.D. Abstract: In my research work, I try to collect some basic usage of variational calculus in practical applications. I derive all the necessary mathematical tools. I explain what is a fuctional and what it means to extremalize it, I derive Euler- Lagrange equation and its corollary - Beltrami identity. I also try to derive a method for solving isoperimetric problems which generalizes the one of the Lagrange multipliers. Although there is a variety of several different...
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(Conformal) Killing spinor valued forms on Riemannian manifolds
Zima, Petr ; Somberg, Petr (advisor) ; Krýsl, Svatopluk (referee)
The goal of the present thesis is to introduce on a Riemannian Spin- manifold a system of partial differential equations for spinor-valued differ- ential forms called Killing equations. We study basic properties of several types of Killing fields and relationships among them. We provide a simple construction of Killing spinor-valued forms from Killing spinors and Killing forms. We also review the construction of metric cone and discuss the re- lationship between Killing spinor-valued forms on the base manifold and parallel spinor-valued forms on the metric cone.
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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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