Title: Symmetries of the CR sub-Laplacian Author: Zuzana Vlasáková Department: Charles University Institute of Mathematics Supervisor: Prof. RNDr. Vladimír Souček, DrSc. Author's e-mail address: zuzana.kasarova@email.cz Supervisor's e-mail address: soucek@karlin.mff.cuni.cz Abstract: The aim of this work is to characterize the vector space of symme- try operators of the CR sub-Laplacian. To do this, we define a CR structure on some distinguished submanifold of Cn+1 (it is in fact the big cell in the CR sphere) and write down the CR sub-Laplacian on it. We also define the symmetries of the CR sub-Laplacian in general and using the ambient con- struction, which we introduce in the sequel, we construct all of them. Keywords: CR geometry, CR sub-Laplacian, symmetries of differential op- erator. 1

1 Title Helical symmetry and the non-existence of asymptotically flat periodic solutions in general relativity Author Martin Scholtz Department Institute of theoretical physics Faculty of Mathematics and Physics Charles University in Prague Supervisor Prof. RNDr. Jiří Bičák, DrSc., dr. h.c. Abstract. No exact helically symmetric solution in general relativity is known today. There are reasons, however, to expect that such solutions, if they exist, cannot be asymptotically flat. In the thesis presented we investigate a more general question whether there exist periodic asymptotically flat solutions of Einstein's equations. We follow the work of Gibbons and Stewart [3] who have shown that there are no periodic vacuum asymptotically flat solutions an- alytic near null infinity I. We discuss necessary corrections of Gibbons and Stewart proof and generalize their results for the system of Einstein-Maxwell, Einstein-Klein-Gordon and Einstein-conformal-scalar field equations. Thus, we show that there are no asymptotically flat periodic space-times analytic near I if as the source of gravity we take electromagnetic, Klein-Gordon or conformally invariant scalar field. The auxilliary results consist of corresponding confor- mal field equations, the Bondi mass and the Bondi massloss formula for scalar fields. We also...

This diploma thesis is concerned with the Robert Mundell's theory of optimum currency area (OCA) and its relation to European monetary union (EMU) or (if you like) Eurozone. The OCA theory belongs to one of the most comprehensive approach to evaluate whether adoption of a common currency by a certain block of countries is beneficial of not. Thus this paper tries to apply OCA theory to the current Eurozone member countries and is focused especially on three main attributes -- the symmetry of economic shocks, the labor market flexibility and the economic integration of international trade between the member states of the euro area. Results of this analysis are then applied to the new member states and the immediate candidate EU member states -- especially to the Czech Republic. The work assesses the feasibility of the OCA theory for classification the candidate countries and their readiness for entering the Eurozone and finally offers alternative criteria by which the readiness for adoption of the Euro should be measured. The text is based on available theoretical and empirical sources supplemented by the author's analyses.

This thesis deals with comparison the results of the frequency modal analysis of blade package of steam turbine with stiffening or damping bounding elements. Computational model was created through the finite element method, using cyclical symmetry. The results were compared with experimental analysis and analysis in VIPACK programme. Thesis also contains creating the solid blade model, using the reverse engineering.

In this work we study properties of the higher-dimensional generally rotating black hole space-time so-called Kerr-NUT-(A)dS and the related spaces with the same explicit and hidden symetries as the Kerr-NUT-(A)dS spacetime. First, we search commuta- tivity conditions for classical (charged) observables and their operator analogues, then we investigate a fulfilment of these conditions in the metioned spaces. We calculate the curvature of these spaces and solve the charged Hamilton-Jacobi and Klein-Gordon equations by the separation of the variables for an electromagnetic field, which pre- serves integrability of motion of a charged particle and mutual commutativity of the corresponding operators.

In this thesis we focus on Grand Unified Theories based on the SO(10) gauge group and, in particular, on the viability of the minimal nonsupersymmetric SO(10) models. Technically, this amounts to a detailed assessment of their vac- uum stability and related (non-)tachyonicity of their scalar spectrum. It turns out that the one-loop scalar mass corrections are important for elimination of the tachyonic behaviour of certain pseudo-Goldstone bosons. In this work these issues are briefly reviewed and two distinct methods for the calculation of the critical radiative corrections are discussed. More specifically, besides the revision of the effective potential approach to the 45 ⊕ 16 Higgs model also the standard perturbative theory method is employed for this purpose. The latter approach is particularly suitable for the more realistic 45 ⊕ 126 Higgs model since it appears to be practically impossible to construct the corresponding effective potential in that case. Consequently, diagrammatic methods are used to calculate the SO(10)- invariant leading scalar quantum correction to the problematic pseudo-Goldstone masses in the 45 ⊕ 126 model. 1

The minimal renormalizable SO(10) Higgs model with the unified gauge sym- metry broken down by the adjoint representation 45S is known to suffer from tachyonic instabilities along all potentially realistic symmetry breaking chains. Few years ago, this issue has been identified as a mere relic of the tree-level cal- culations and the radiative corrections to the masses of the pair of the "most dangerous" pseudo-Goldstone scalars transforming as (8, 1, 0) and (1, 3, 0) with respect to the SU(3)c × SU(2)L × U(1)Y Standard model gauge group were com- puted. Remarkably enough, it turns out that in the minimal potentially realistic renormalizable realization of the model - consisting of 45S ⊕ 126S scalar and 45G gauge fields - there is third pseudo-Goldstone scalar, a full singlet with respect to the Standard model gauge group that, until recently, happened to escape the community's attention. In this thesis we computed the one-loop corrections to its mass employing two different methods within the effective potential approach. In both cases we reduced the complexity of our calculations by decomposition into the Standard model irreducible representations. In the end, we cross-checked the resulting formulae in four distinct symmetry breaking limits in which the pseudo- Goldstone mass was degenerated with those of...

This paper proposes robust algorithm for detection of traffic signs in well light conditional. The algorithm uses colour based segmentation method for finding red traffic signs. Fast radial symmetry method FRS is used for identification of constituent shapes. Traffic signs are divided into four classes on the basis of the method.

The master thesis deals with modal analysis of turbine wheel of aircraft engine. The first part is concerned with the modal analysis of the computational model of turbine wheel and separated turbine blade using the cyclic symmetry of the ANSYS software. This part of the thesis set the task of determining the natural frequency depending on the operating parameters of the motor. The second part of the thesis occupies with the experimental simulation of the task. The results of experimental simulation are verified and compared with the results from the computational modal analysis. The goal is to create a Campbell diagram and to determine the intervals of the critical revolution of the turbine wheel.

Václav Drozd Object oriented political theory? A contribution of object oriented philosophy for political theory Abstract (in English): This diploma thesis is concerned with the turn to materiality and object in contemporary philosophy and explores its impact on political theory. It focuses on conceptions trying to reformulate the relation between subject and object, culture and nature or human and inhuman entities - symetrical ontology of Bruno Latour, speculative realism and object-oriented ontology. The aim of the study is to identify the benefits of these aproaches for political theory. The first frame topic important for investigated theories is the relation of human and state towards complex technologies. The second general topic is the existence under conditions of anthropocene and climate change. Keywords: anthropocene, speculative realism, object, corelationism, actor, vibrant matter, technologies, symmetry, actor-network-theory, Latour