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An algorithmic approach to resolutions in representation theory
Ivánek, Adam ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we describe an algorithm and implement a construction of a projective resolution and minimal projective resolution in the representation the- ory of finite-dimensional algebras. In this thesis finite-dimensional algebras are KQ /I where KQ is a path algebra and I is an admissible ideal. To implement the algorithm we use the package QPA [9] for GAP [2]. We use the theory of Gröbners basis of KQ-modules and the theory described in article Minimal Pro- jective Resolutions written by Green, Solberg a Zacharia [5]. First step is find a direct sum such that i∈Tn fn∗ i KQ = i∈Tn−1 fn−1 i KQ ∩ i∈Tn−2 fn−2 i I. Next important step to construct the minimal projective resolution is separate nontri- vial K-linear combinations in i∈Tn−1 fn−1 i I + i∈Tn fn i J from fn∗ i . The Modules of the minimal projective elements are i∈Tn (fn i KQ)/(fn i I). 1
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Realization of refinement monoids
Jakubec, Tomáš ; Růžička, Pavel (advisor) ; Trlifaj, Jan (referee)
In this thesis are shown some results of realization of refinement monoid by Grothendieck monoids of von Neumann regular ring. The most important result of this thesis, recently published by P. Ara, J. Bosa and E. Pardo, claims that every finitely generated conical refinement monoid is realizable. We construct for each such monoid a von Neumann regular algebra such that monoid is realizable by this algebra. For this construction we use adaptable separated graphs and their relation to refinement monoids. 1
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Gauss'calculation of Ceres' orbit
Beďatš, Daniel ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
In 1801, a Sicilian astronomer Giuseppe Piazzi discovered a faint star-like object, which disappeared in the glare of the Sun after six weeks of observations. A year later, the object named Ceres was rediscovered thanks to calculations provided by Carl Friedrich Gauss. This thesis aims to present the full process of Gauss's contribution to the dis- covery of Ceres with focus on elementary and straightforward mathematical language. We introduce some basic notions and properties concerning elliptical orbits and develop a thorough account of Gauss's calculations. We also provide justification for multiple unexplained steps in the method. 1
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Integral attacks on block ciphers SIMON
Phamová, Kieu Trang ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
This thesis is focused on bit based division property using three subsets. Firstly, we introduce important definitions such as block cipher, integral attack and conventional division property. Then we define bit based division property using three subsets and prove its propagation rules: copy, AND-compression and XOR-compression. We often use these functions in cryptography, therefore in this thesis we will prove how bit based division property with three subsets propagates after the application of these functions.
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Quaternion algebras and units
Mišlanová, Kristína ; Kala, Vítězslav (advisor) ; Růžička, Pavel (referee)
The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially Lipschitz and Hurwitz order. The fourth chapter is dedicated to the relationship between unit quaternions and rotations in R3 , thanks to which we can characterize finite subgroups of H1 , or equivalently H× . This result will be used in the last chapter, where we are mainly focused on the problem of characterization of the group of units in orders in Hamiltonian quaternions. 1
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Permutation group algorithms
Plšková, Petra ; Bulín, Jakub (advisor) ; Růžička, Pavel (referee)
The Schreier Sims algorithm is a fundamental algorithm for permutation groups. Its purpose is to find a base and a strong generating set. Representation of a group using a base and a strong generating set is the core of many other algorithms. The aim of the thesis is to give the reader a detailed description of the algorithm. We present its pseudocode together with the pseudocode of its subroutines. We analyze the complexity with respect to the given pseudocode. Similarly, we describe an efficient, Monte Carlo version of the Schreier Sims algorithm whose time complexity is nearly linear. We give a detailed description of two improved algorithms for subroutines on which this version is based. We introduce the theoretical framework needed to support the correctness of both the deterministic and the probabilistic version of the algorithm. 1
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Sanitation installation and gas installation in the aparthotel
Růžička, Pavel ; Wierzbická, Helena (referee) ; Vrána, Jakub (advisor)
The master thesis focuses on the design of sanitary and gas installations in the particular aparthotel building in Pec pod Sněžkou. It is an aparthotel with two underground and max six above-ground floors incl. a gallery. The upper floors of the towers are used for accommodation. A parking lot and wellness area are situated on the first underground floor. On the second underground floor there is a technical background of the building, shops and a small restaurant for guests. All the roofs and terraces of the aparthotel are made to be green. The calculation part and the project contain the design of the sewerage, water supply, gas pipeline and their connection to the existing engineering networks. The theoretical part deals with increasing the pressure in the indoor plumbing, incl. design of equipment for increasing the pressure in the particular building.
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Production of check valve's part
Růžička, Pavel ; Řiháček, Jan (referee) ; Císařová, Michaela (advisor)
The bachelor's thesis deals with the production innovation of the return valve disc component used in pumping technology. It is a component made of sheet metal with 3 mm thickness, the quality of material is according to ČSN 11 320. Existing production using cutting technology in a process tool will be replaced by a modern, more accurate and especially more economical method by precise cutting with a pressure edge. Based on the literature study of precision cutting, assessment of production technology and processing of design solutions, the cutting tool was designed to be able to significantly speed up and reduce the cost of production of the component. It will be placed in a three-acting press from Feintool, namely XTF Speed with a nominal force of 1500 kN. In the final technical and economic evaluation, it was proved that the newly designed technology is able to bring a component of the same nature to the market, but at a significantly lower price.
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Lattice constructions and Priestley duality
Hartman, Juraj ; Růžička, Pavel (advisor) ; Tůma, Jiří (referee)
In this thesis after recalling some basic definitions and theorems in category theory, lattice theory and topology we first introduce the so called Stone duality of the category of boolean lattices and the category of boolean topological spaces. Then we introduce its generalization, the so called Priestley duality of the category of bounded distributive lattices and the category of total order disconnected topological spaces. Then we introduce the M3[.] lattice construction and prove that for every bounded distributive lattice L there is an isomorphism from the lattice M3[L] to the lattice of all continuous monotone maps from the Priestley space of L to the lattice M3 with discrete topology. Finally we introduce the so called boolean power, which we generalize to the so called priestley power and we prove that for every natural number n ≥ 3 and every bounded distributive lattice L there is an isomorphism from the lattice Mn to the priestley power of the lattice Mn by the lattice L. 1
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