
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


Rational linear dependencies of periodic points of the logistic map
Mik, Matěj ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
Periodn points of a polynomial f are roots, and hence elements of the splitting field, of the polynomial fn (x) − x, where fn denotes the nth iterate of f. In the thesis, we will focus on describing rational linear dependencies of periodn points of the polynomial f(x) = 4x(1 − x), which defines the socalled logistic map. We will present a description of the dependencies for n = 1, . . . , 5 and a partial result for n = 6. We will be using computer calculated factorizations of polynomials over rational numbers and some finite field extensions. The factorizations will give us coordinates of the periodic points relative to some basis of their linear span, which will allow us to use a simple way of describing their dependencies. In the end of the thesis, we will put together an algorithm for describing the dependencies for a general n.


Sanitation installation and gas installation in a apartment building
Růžička, Pavel ; Vaščáková, Alena (referee) ; Vrána, Jakub (advisor)
The aim of the Bachelor thesis is a design of sanitary installations and gas installations in the apartment building. It is a fourfloor apartment building without a cellar. The apartments are located on all overground floors. The theoretical part is focused on infiltration of rainwater. The computation part and project includes a design of sanitary and storm sewers, water supply network, gas main a and their connection to existing utility lines.


Algorithms for factorization of integers of particular form
Lorenc, Filip ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
This bachelor thesis deals with three factorization algorithms  Pollard p1 method, Williams p+1 method and elliptic curve method ECM. This work aims to describe these algorithms theoretically and then compare them on real inputs. For each algorithm we describe its basic and extended version and then we derive their time complexity. In the first chapter we define Bpowersmooth and Bsmooth number and we state their approximation. The second, third and fourth chapter is about description of algorithms and in the last chapter we compare their effectiveness and performance. A part of the work contains basic theory about elliptic curves, which is necessary in ECM. There is also included a program containing all these algorithms.


Presentations of subgroups
Jakubec, Tomáš ; Růžička, Pavel (advisor) ; Kazda, Alexandr (referee)
This bachelor thesis shows, how to create the presentation of subgroup, if the presentation of group is known by ReidemeisterSchreier method. At first, a term presentation of group is defined and then the text shows, how to obtain the group, which is isomorphic to the original group, from this presentation and how the presentation can be changed, although the group, which is ob tained from the presentation, stays same. Then the text finds presentation of subgroup from the presentation of group, however this presentation cannot be in general used in practice. The obtained presentation of subgroup can be simplified by Reidemeister theorem, Schreier theorem and appropriate genera tors of subgroup. This thesis also contains solved examples of application of ReidemeisterSchreier method. The text is intended as an educational material for students of combinatorial group theory. 1


PILOT ANALYSIS OF CHEVRON NOTCH LIGAMENT AREA FOR APPLICATION\nON QUASIBRITTLE MATERIALS
Seitl, Stanislav ; Růžička, P. ; Miarka, P. ; Sobek, J.
Specimens for the bending tests with the chevron notch are standardized for the\nevaluation of the fracture toughness of various materials. The main advantage of this test\nsetup is that no sharp precrack has to be introduced, because a sharp crack is formed\nduring loading at the beginning of the test. Furthermore, no crack length measurement is\nrequired, and a stable crack growth can be reached due to geometry of the notch. In this\ncontribution a difference of the ligament area of the specimens with the straight through\nnotch and the chevron notch was investigated


Algorithms for the computation of Galois groups
Kubát, David ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
This thesis covers the topic of the computation of Galois groups over the rationals. Beginning with the classic algorithm by R. Stauduhar, we then review the theory necessary to explain the modular algorithm by K. Yokoyama. More precisely, we discuss the notion of the universal splitting ring of a polynomial. For a separable polynomial, we then study idempotents in the universal splitting ring. The modular algorithm involves computations in the ring of padic integers. Examples are given for polynomials of degree 3 and 4.


Discrete linear dynamical systems with control
Procházková, Zuzana ; Tůma, Jiří (advisor) ; Růžička, Pavel (referee)
Discrete linear dynamical systems with control Author: Zuzana Procházková Department: Department of Algebra Supervisor: doc. RNDr. Jiří Tůma, DrSc., Department of Algebra Abstract: In this thesis we describe elementary property of discrete linear dyna mical system. We define discrete linear dynamical system with control and its controllability and then we define descrete linear dynamical system with output and its observability. After that we show the duality of observability and con trollability with definition of dual system and its description. There are three problems solved in the last chapter. 1


Traditional working and quality of timber during repairs of timber structures
Kloiber, Michal ; Růžička, Petr
Extensive field experience with repairing timber structures shows two main limiting factors for the quality of work that are often circumvented or simplified. First, it is the quality of the timber for the repairs (replacements, additions) and the resulting problem when the recommended criteria for the selection of suitable material are not met, with regard to the longterm coexistence of the new timber with the original material. Second, a critical factor is the way in which the timber is worked  the preferred traditional working is for various reasons replaced by counterfeits. Due to the limited scope of this paper, we can only pay attention to the most blatant case, which is working of logs by hewing.


Plane geometry problems solved by algebraic geometry
Trummová, Ivana ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis I focus on a certain part of algebraic geometry which studies plane curves and their intersection points. The main part is a proof of Bézout's theorem and an overview of its corollaries, which have an interesting geometric visualization. The most important corollary is the proof of associativity of adding points on elliptic curves. This fact is widely used in modern cryptography. 21
