|
|
Moduly s minimální množinou generátorů
Hrbek, Michal ; Růžička, Pavel (advisor) ; Trlifaj, Jan (referee)
Title: Modules with a minimal generating set Author: Michal Hrbek Department: Department of Algebra Supervisor: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: By a minimal generating set of a module we mean a subset which generates the module but any of its proper subsets does not. If the module is not finitely generated, an existence of a minimal generating set is not guaranteed. We say that a module is weakly based if it has a minimal generating set. In the presented thesis, we provide a characterization of weakly based modules over Dedekind domains. As an application of this, we show that the class of weakly based modules is not closed under extensions and the complement of this class is not closed under finite direct sums. Also, we show an example of an abelian group which is weakly based if and only if CH holds. Then we treat rings such that all modules are weakly based. We prove that a Baer regular ring has this property if and only if it is semisimple, and we show that any ℵ0-noetherian commutative semiartinian ring has this property. Final part of the text concerns the problem of Nashier and Nichols - does any generating set of any module over a perfect ring contain a minimal generating set? Keywords: module, minimal generating set, weak basis 1
|
|
|
Algorithms in Representation Theory
Trunkát, Marek ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
This thesis deals with an implementation of algorithm for computation of generator of almost split sequences ending at an indecomposable nonprojective module of path algebra over finite quiver. Algorithm is implemented in algebra system GAP (Groups, Algorithms, Programming) with additional package QPA (Quivers and Path Algebras). Powered by TCPDF (www.tcpdf.org)
|
|
|
A polynomial algorithm for the binary PCP
Kuřinová, Petra ; Holub, Štěpán (advisor) ; Růžička, Pavel (referee)
The Post correspondence problem, introduced in 1946 by Emil Post, is an important example of undecidable problem. Therefore PCP figures in pro- ofs of some results in theory of formal languages, matrix theory and other. The decidability of the Post correspondence problem proved Ehrenfeucht, Karhumäki and Rozenberg in the 1980s and Halava, Harju and Hirvensalo in 2002 ended the proof. Eight years later was verified that the solution can be found even in polynomial time. The main goal of this diploma thesis is to describe this algorithm in detail and to implement it in a web application. The thesis also introduces basics of com- binatorics on words and some facts about PCP and produces some interesting examples of instances of PCP. Keywords: Post correspondence problem, generalized Post correspondence problem, binary PCP, polynomial algorithms on words, successors of morphisms 1
|
|
|
Anylysis of cryptographic functions in the PDF format
Štíchová, Kateřina ; Růžička, Pavel (advisor) ; Joščák, Daniel (referee)
The present work is focused to cryptographic functions in PDF format. The aim is to examine problems with long-term archiving of documents in PDF format. It cryptographic functions used in the documents are for long-term archi- ving problem. There is a lot of ways how to provide security in PDF documents. I conclude it should be provided a simple risks analysis before providing long-term archiving of PDF document. An organization also should take this analysis when creating its retention plan.
|
|
|
Banschewského funkce na komplementárních modulárních svazech
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: Banaschewski function on countable complemented modular lattices Author: Samuel Mokriš Department: Department of Algebra Supervisor of the bachelor thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: A Banaschewski function on a bounded lattice L is an antitone self-map on L that picks a complement for each element of L. On any at most countable complemented modular lattice L, there exists a Banaschewski function with a Boolean range M. Moreover, such M is a maximal Boolean sublattice of L and is uniquely determined up to isomorphism. In the thesis we give a negative answer to the related question whether all maximal Boolean sublattices in an arbitrary countable complemented modular lattice are isomorphic and whether every max- imal Boolean sublattice in an arbitrary countable complemented modular lattice L is the range of some Banaschewski function on L. We also generalize the coun- terexample to greater cardinalities; for a given infinite cardinal κ we construct a complemented modular lattice L of cardinality κ and maximal Boolean sublat- tices B and E of L such that B is not the range of any Banaschewski function on L, that there exists a Banaschewski function on L of range E, and that B is not isomorphic to E. Keywords: complemented modular lattice, Banaschewski function, von...
|
|
|
Security of an Electronic Voting System
Fritzová, Petra ; Růžička, Pavel (advisor) ; Hojsík, Michal (referee)
Electronic elections, also known as i-voting might help in removing the crisis in our democracy, which is reflected in non-cooperation in the opportunity of expressing their opinions during direct elections. A reasonable set up of information and communication technologies in technical and financial terms could help that elections would be attended by more voters. The implementation of electronic elections could achieve that the way of governance in the democratic republic will be truly represented by the view of the vast majority of people who are authorized to elect. The introduction of the i-voting system could be efficient from the financial point of view. This electoral process could reduce the risk of human error as well as the risk of manipulation of votes since most of the processes would be automated. Thesis proposes a definition of the basic requirements for an ideal i-voting system which compares the requirements for ensuring the safety of two previously proposed electronic electoral systems. Thanks to a deeper analysis of these two systems the thesis also describes the imperfection in safety and it raises the possibility of basic attacks on components and systems properties due to imperfections in security.
|
|
|
Security of an Electronic Voting System
Fritzová, Petra ; Růžička, Pavel (advisor) ; Joščák, Daniel (referee)
Electronic elections, also known as i-voting might help in removing the crisis in our democracy, which is reflected in non-cooperation in the opportunity of expressing their opinions during direct elections. Automation that is brought by i-voting process could mean reducing the risk of human error and also the risk of votes manipulation. On the other hand, the introduction of i-voting means the rise of new risks and vulnerabilities, which could attacker exploit to breach the security i-voting process. Thesis analyzes the Estonian electronic election system and system SERVE in terms of infrastructure and functionality of their components, identifies and suggests the basic possibilities of attacks on these systems. A special section is devoted to the electronic election system in Norway, where the cryptographic theory, which is the base for this system, was reviewed and the evidence that proofs the safety of this system was designed. Powered by TCPDF (www.tcpdf.org)
|
|
|
Constraint Satisfaction Problem and Universal Algebra
Kazda, Alexandr ; Barto, Libor (advisor) ; Růžička, Pavel (referee) ; Valeriote, Matt (referee)
The thesis consists of a collection of my contributions to universal algebra. Motivated by the Constraint Satisfaction Problem (CSP), we study the algebras of polymorphisms of relational structures. We begin by showing by an algebraic argument (and a bit of calculus) that random relational structures' CSP is almost always NP-complete. We then study digraphs with a Maltsev polymorphism, and conclude that such digraphs must also have a majority polymorphism. Next, we show how to use absorption tech- niques to prove that congruence modular reflexive digraphs must have an NU operation. We close our work by giving an algebraic proof of a result (first obtained by graph theorists) that 3-conservative relational structures with only unary and binary relations either define NP-complete CSP, or CSP for them can be solved by the local consistency algorithm. 1
|
|
|
Nestandardní analýza dynamických systémů
Slavík, Jakub ; Pražák, Dalibor (advisor) ; Růžička, Pavel (referee)
In the presented thesis, we study an application of nonstandard analysis to dynamical systems, in particular to ω-limit set, stability and global attractor. We recall the definition and properties of elementary embedding, in detail ex- plore the introduction of infinitesimals to the real line and study metric spaces using nonstandard methods, in particular continuity and compactness which are closely related to the theory of dynamical systems. Last we attend to dynamical systems and present nonstandard characterizations of some of its properties such as asymptotic compactness and dissipativity and using these characterizations we prove one of the basic results of this theory - existence of a global attractor. 1
|
|
| |
guest ::
RSS feed.