National Repository of Grey Literature 104 records found  1 - 10nextend  jump to record: Search took 0.00 seconds. 
The Gabriel-Roiter measure in representation theory
Krasula, Dominik ; Šťovíček, Jan (advisor) ; Trlifaj, Jan (referee)
The Gabriel-Roiter measure is a module-theoretic invariant, defined in 1972 by P. Gabriel. It is an order-preserving function that refines a composition length of a module by also taking lengths of indecomposable submodules into account. We calculate all Gabriel-Roiter measures for finite-length representa- tions of an orientation of a Dynkin graph D4 and an orientation of a Euclidean graph ˜A3. In 2007, H. Krause proposed a combinatorial definition of the Gabriel-Roiter measure based on other length functions instead of composition length. We study these alternative Gabriel-Roiter measures on thin representations of quiv- ers whose underlying graph is a tree. 1
Classification of finite dimensional modules over string algebras
Macháč, Ondřej ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis we classify indecomposable finite dimensional modules over string alge- bras. In the introductory part we define string algebras and string modules and band modules. In the third chapter we prove the classification theorem and define functors used. In the fourth and fifth chapter we verify assumptions on the functors regarding string modules or band modules, respectively. In the last chapter we show that these functors are sufficient and we finish proofs of all the remaining assumptions of the main theorem. At last we show examples of classification. 1
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
Representation theory of gentle algebras
Mlezivová, Anna ; Šťovíček, Jan (advisor) ; Chan, Aaron (referee)
The object of study of this thesis is a special class of quiver algebras called gentle algebras. To study modules over them, we can use a combinatorial or geometric view. Thanks to Theorem 6.1. in the article Chan and Demonet [2020], we can find the lattice of torsion classes of modules over gentle algebras using string combinatorics. In the thesis, we apply this theorem for a few examples. Especially we derive the lattice of torsion classes of Kronecker algebra, and we do the first steps for finding the lattice for Markov algebra. The emphasis is placed on understanding the relationship with the geometric view. 1
Hyperfields and their applications in tropical geometry or matroid theory
Andr, Břetislav ; Patáková, Zuzana (advisor) ; Šťovíček, Jan (referee)
Hyperfields are algebraic structures generalizing the concept of an algebraic field. In contrast to classical fields, summation in a hyperfield is multivalued, that is, the sum of two elements is not a single element, but a whole set of elements. Hyperfields find practical use in the theory of matroids and in tropical geometry, a variant of algebraic geometry. Matroid is an algebraic structure generalizing the concept of linear independence. There exist more types of matroids expanding the basic definition, e.g. oriented or valuated matroids. All of these definitions can be generalized to a single concept of an F-matroid, where F is a hyperfield. Tropical geometry is concerned with similar problems as algebraic geometry, only over the so-called tropical semifield. It finds many applications due to its combinatorial nature. Tropical geometry and algebraic geometry are closely tied by the so-called Litvinov-Maslov dequantization and hyperfields may be used to describe its generalized version. 1
Classification problems from linear algebra and representations of quivers
Borýsek, Martin ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
This thesis deals with the description of categories of finite-dimensional representati- ons of quivers. Its aim is to present a classification of indecomposable objects in this category for quivers whose underlying graph is Dynkin and to discuss the theory on the example of the so-called three-subspace problem. In the first chapter, the basic concepts of quiver representations are introduced. In the second chapter, the proof itself is de- monstrated using reflection functors and reflection transformations. Then, in the third chapter, this thesis deals with the basics for the theory of M. Auslander and I. Reiten. In the conclusion, the Auslander-Reiten quiver is discussed for the category of finite- dimensional representations of the above-mentioned problem of three subspaces. 1
The use of natural antibiotics and herbal medicine in the treatment of SIBO
Svitavská, Jana ; Vaníčková, Zdislava (advisor) ; Šťovíček, Jan (referee)
This bachelor thesis deals with the syndrome of small intestinal bacterial overgrowth (SIBO), a gastrointestinal disorder caused by increased total bacterial counts and abnormal microbiome composition in the small intestine. SIBO often manifests through non-specific dyspeptic symptoms, including gas bloating, abdominal pain, diarrhea, malabsorption and malnutrition. A non-invasive breath test is most commonly used to diagnose SIBO. Treatment of SIBO should include therapy for the underlying disease, bacterial eradication and nutritional support. The main aim of this bachelor thesis is to investigate the efficacy of herbal preparations and natural antibiotics in the treatment of SIBO and to highlight the potential usage of these antibacterial agents as a stand-alone or adjunctive therapy to conventional treatment. Another aim is to confirm the hypothesis that it is impossible to predict the diagnosis of SIBO based on gastrointestinal symptoms. Therefore, treating this disease empirically without confirmation by laboratory examination is not appropriate. The data was collected in the Gastroenterology Laboratory of the Institute of Medical Biochemistry and Laboratory Diagnostics, 1st Faculty of Medicine, Charles University in Prague and in VFN. The prevalence of gastrointestinal symptoms and other...
Decoding of Reed-Solomon Codes
Procházka, Dalibor ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
Reed-Solomon codes are a typical example of MDS codes, that are frequently used in practise. In this thesis, we go over three different algorithms of decoding these codes, including the initial view from the original article, as well as the modern approach of currently used algorithm and of another possible efficient algorithm. We compile various sources and unite them under the same notation. We describe in detail the theory be- hind each algorithm, show its correctness, discuss every algorithm's time complexity and demonstrate its steps on simple examples. 1
The Tate-Shafarevich group of an elliptic curve
Zvěřina, Adam ; Šťovíček, Jan (advisor) ; Příhoda, Pavel (referee)
This thesis deals with the Tate-Shafarevich group and its relation to rational points on the curve and its rank. We first define the notion of profinite groups and characterize them as Galois groups of field extensions. Then we define the Tate-Shafarevich group using Galois cohomology and explain its relation to the rational points on the curve. Finally, we formulate the Birch-Swinnerton-Dyer conjecture, which relates the rank of an elliptic curve and the order of its Tate-Shafarevich group. 1
Resolutions of singularities using blow-ups
Komora, Matúš ; Šťovíček, Jan (advisor) ; Hrbek, Michal (referee)
This bachelor's thesis aims to provide accessible treatment of the blow-up construction for algebraic varieties. The blow-up construction is a fundamen- tal technique in algebraic geometry that allows us to find a variety which has better properties than an original variety but is still equivalent to the original. This process can be used to resolve singularities. In the first two chapters, we begin by providing an introduction to the fundamental principles of alge- braic geometry, including the definitions of algebraic varieties but also basic topological concepts but also some construction such as Segre embedding and product of varieties. In the third chapter, we will introduce the concept of blow-ups and show the computation as on example. 1

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