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Twistor operator in symplectic spin geometry
Dostálová, Marie ; Krýsl, Svatopluk (advisor) ; Doubek, Martin (referee)
The topic of the diploma thesis is symplectic spinor geometry. Its re- search was started by D. Shale, B. Kostant and K. Habermann. We focus our attention to one of the so called symplectic twistor operators introduced by S. Kr'ysl. We investigate the action of this operator on real even dimensio- nal vector spaces considered as symplectic manifold, its invariance properties and regularity. We describe a part of the kernel of the symplectic twistor operator when acting on symplectic spinors on R2. The kernel forms a repre- sentation of the so called metaplectic group (double cover of the symplectic group). 1
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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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Integrability in Hamiltonian machanics
Kokoška, David ; Krýsl, Svatopluk (advisor) ; Švarc, Robert (referee)
Title: Integrability in Hamiltonian mechanics Author: David Kokoška Department: Mathematical Institute of Charles University Supervisor: doc. RNDr. Svatopluk Krýsl, Ph.D., Mathematical Institute of Char- les University Abstract: Hamiltonian mechanics can be formulated using symplectic manifolds and so called Hamiltonian systems. In the Theorem of Liouville-Arnold, conditi- ons are described, under which solutions of Hamilton equations stay on a torus of dimension equal to the dimension of the configuration space. Examples on application of the Liouville-Arnold theorem are contained. We study the pro- blem of motion in a gravitational central force field in the connection with the Runge-Lenz vector. Keywords: symplectic manifold, hamiltonian system, Liouville-Arnold theorem, Kepler's problem 1
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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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Space forms
Poppr, Marián ; Krýsl, Svatopluk (advisor) ; Lávička, Roman (referee)
Title: Space forms Author: Marián Poppr Institute: Mathematical Institute of Charles University Supervisor: doc. RNDr. Svatopluk Krýsl, Ph.D., Mathematical Institute of Char- les University Abstract: In the presented thesis, we focus on foundations of Riemannian geome- try. We are concerned with the problematics of the existence and uniqueness of metrics and connections on smooth manifolds. We explore the exponential map as a tool for a study of space forms - complete manifolds with constant secti- onal curvature. Using Jacobi fields we are going to prove the local case of the Killing-Hopf theorem, which describes isometries between space forms. Keywords: Riemannian manifolds, sectional curvature, Jacobi field, Killing-Hopf theorem. 1
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Semifields and planar functions
Hrubešová, Tereza ; Drápal, Aleš (advisor) ; Krýsl, Svatopluk (referee)
The aim of this diploma thesis is to introduce the topic of semifields and to explain its connection with planar functions. From its beginning the thesis leads to the formulation of relation between commutative se- mifields of odd order and planar Dembowski-Ostrom polynomials, which R. S. Coulter and M. Henderson introduce in their article from 2008. At the beginning of the thesis there is a short introduction to projective and affine planes. The thesis further describes coordinatization of projective plane by planar ternary ring. It also aims to investigate properties of ternary ring depending on the number of perspectivities in the projective plane. One of the chapters is dedicated to the isotopy of loops, which can be applied directly on the isotopy of semifields. The thesis mainly focuses on the proof of denoted correspondence between commutative semifields of odd order and planar Dembowski-Ostrom polynomials. Finally, several corrolaries of this relation and the isotopy of semifields are declared. 1
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Symplectic spin geometry
Holíková, Marie ; Krýsl, Svatopluk (advisor) ; Eelbode, David (referee) ; Souček, Vladimír (referee)
The symplectic Dirac and the symplectic twistor operators are sym- plectic analogues of classical Dirac and twistor operators appearing in spin- Riemannian geometry. Our work concerns basic aspects of these two ope- rators. Namely, we determine the solution space of the symplectic twistor operator on the symplectic vector space of dimension 2n. It turns out that the solution space is a symplectic counterpart of the orthogonal situation. Moreover, we demonstrate on the example of 2n-dimensional tori the effect of dependence of the solution spaces of the symplectic Dirac and the symplectic twistor operators on the choice of the metaplectic structure. We construct a symplectic generalization of classical theta functions for the symplectic Dirac operator as well. We study several basic aspects of the symplectic version of Clifford analysis associated to the symplectic Dirac operator. Focusing mostly on the symplectic vector space of the real dimension 2, this amounts to the study of first order symmetry operators of the symplectic Dirac ope- rator, symplectic Clifford-Fourier transform and the reproducing kernel for the symplectic Fischer product including the construction of bases for the symplectic monogenics of the symplectic Dirac operator in real dimension 2 and their extension to symplectic spaces...
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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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