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Popis a distribuce škod působených vytloukáním a vystrouháváním srnce obecného v lesních porostech
Růžička, Pavel
This bachelor thesis is about the description and distribution of damage caused by roe deer in the form of fraying and rubbing. The thesis has been divided into two parts, the first part deals with a general view of the issue and the second part is the actual research, which consisted of collecting data on fraying and rubbing in a pre-selected suitable location, including the following suggestions and recommendations for practice. The research was carried out using a system of strip transects and recording damage sites in their vicinity. The map outputs show the amount of damage, its location and the type of damage, these signs clearly indicate the locations and individual trees most frequently visited by roe deer. The analytical outputs mention the occurrence of the most frequently damaged trees and their morphometric data. The most frequently damaged tree species was the norway spruce. The resulting connection between 2 roebucks territory boundaries which were found, the damage caused, and human-made natural boundaries cannot be ignored.
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Permutation groups and card shuffling
Sekera, Vojtěch ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we solve an old problem, named after the magician and mathematician Alex Elmsley, of raising a card to the top of the deck using faro shuffles. Furthermore we examine the group structure generated by these shuffles on an arbitrarily large deck of cards. Upon generalizing the faro shuffle in the third chapter, we reach a promising conjecture about these faro shuffle permutation groups. 1
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Continued fractions in local fields
Červenková, Eliška ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
The theses concerns the topic of p-adic Ruban and Browkin continued frations and their properties. To begin with, the concept of p-adic numbers is introduced and the necessary theory is shown. Next, continued fractions are defined and their convergence in both real and p-adic numbers is analyzed. Following this, the theses examines Ruban continued fractions and presents an algorithm for determining whether the expansion is terminating, along with a derivation of the maximum number of algorithmic steps required. It also holds that if Ruban expansion is not terminating, then it is periodic. A detailed description of the periodicity, including its properties, is provided. Then the focus is shifted to Browkin continued fractions. It holds that every rational number has a finite Browkin continued fraction. This claim is subsequently proven. The theses concludes with examples that demonstrate the properties of both Ruban and Browkin continued fractions. 1
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Elementary theory for groups of linear fractional transformations
Tomášková, Sára ; Drápal, Aleš (advisor) ; Růžička, Pavel (referee)
The thesis focuses on the properties of general projective linear group PGL2(F) and its action on the projective line P1 (F), both for a finite and an infinite field F. Only the basic knowledge from the Bachelor studies is used to prove these properties. Sharp 3- transitivity of the said group is discussed. Then, we deal with the subgroups consisting of identity and all elements whose sets of fixed points coincide. Furthermore, we show under which conditions all these subgroups have the property that all their finite subgroups are cyclic. We deduce that for a finite field F, it holds that all of these groups are cyclic if and only if F is equal to Zp for a prime number p. The thesis then focuses on the action of PGL2(F) by conjugation on the set of these sungroups. Finally, it is shown that projective special linear group PSL2(F) is simple. 1
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Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and non-commutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1
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Ultrafilters and their monads
Hladil, Josef ; Slávik, Alexander (advisor) ; Růžička, Pavel (referee)
Generalising the notion of an ultrafilter to structured sets, we construct the ultrafilter monad in the categories of partially ordered sets and finitely colourable graphs. This is done similarly to codensity monads, knowing that the codensity monad of the inclusion of finite sets into sets is the ultrafilter monad. We derive an equivalent definition of an ultrafilter on an object applicable for general graphs, also giving rise to a monad. We show that ultrafilters on a poset can be completely characterised in terms of suprema or infima of directed subsets when the poset has only finite antichains. We attempt to classify algebras over the poset ultrafilter monad; our results completely classify the algebras with all antichains finite as posets with a particular compact Hausdorff topology. 1
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Functional encryption for quadratic functions
Sýkora, Josef ; Žemlička, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis we will be studying functional encryption for quadratic functions. We want to encrypt a message, in form of a vector, z to a plaintext ct and create a secret key skf for a quadratic function f, which will allow us to decrypt ct to f(z), while the ct and skf will not leak any information about z. We will introduce one concrete design. The aim of this thesis will be the preparation of necessary preliminaries, which will allow us to describe the design, and to verify correctness of the algorithms. We will describe Arithmetic Branching Programs. Such objects will help us represent function f. Furthermore, we will introduce Garbling and Partial Garbling schemes. Those will allow us to "randomize"a part of the algorithm. We will also specify an encryption algorithm for linear transformation and use it to describe the main encryption algorithm for quadratic functions. 1
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Rank Two Commutative Semifields
Tittl, Ondřej ; Göloglu, Faruk (advisor) ; Růžička, Pavel (referee)
In this thesis we will explain what are semifields and what interesting properties these algebraic objects possesses. In the first chapter we will go over some basics and preliminaries to understand what semifields are. In the second chapter we will prove some useful lemmata for either commutative and non-commutative case of semifields and provide some examples. At last we will try to do some research by ourselves, where we will try to find some examples of semifields. 1
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