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Construction of MDS matrices
Belza, Lukáš ; Žemlička, Jan (advisor) ; Šťovíček, Jan (referee)
This thesis focuses on Maximum Distance Separable (MDS) matrices over finite fields, with emphasis on circulant MDS matrices. At the beginning, concepts related to MDS codes and their characterization are introduced. This is directly followed by an introduction into circulant matrices and their relation to factor algebras of polynomials. In the second part, we shift our focus specifically on circulant MDS matrices. We start from the construction of such matrices in dimensions 3×3 and 4×4 and then proceed to a general construction of MDS matrices from Vandermond matrices.Finally, we find some restrictions on the existence of orthogonal circulant MDS matrices, namely that there are no such 2d × 2d matrices over any finite field of characteristic two. 1
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Complex algebraic curves
Zvěřina, Adam ; Šťovíček, Jan (advisor) ; Kazda, Alexandr (referee)
The thesis describes the relationship between algebraic curves and Riemann surfaces. We define Weierstrass ℘-function and prove some of its properties. We further prove that every complex algebraic curve can be regarded as a Riemann surface. Finally, we demonstrate that an elliptic curve can be parametrised with Weierstrass ℘-function. 1
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Applications of Groebner bases
Skalová, Marie ; Příhoda, Pavel (advisor) ; Šťovíček, Jan (referee)
Groebner bases are useful tool of algebraic geometry for geometry proving. In the thesis we are presenting an automatic geometric theorem proving method in two vari- ants. Firstly, a variant based on the book D. Cox, J. Little, D. O'Shea Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra and secondly a variant based on the book D. Stanovský, L. Barto, Počítačová algebra. We summarize theory, which is necessary for deduction of the method, then the- ory, which is necessary for definition of Groebner base and theorem about her properties. The thesis is including solved problems used for motivate several steps in method and solved exercises from already mentioned book by D. Cox, J. Little, D. O'Shea, some of them are solved by both variants. There is also own proof of decomposition of an affine variety in chapter 2. 1
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Basics on persistent homology
Novák, Jakub ; Šťovíček, Jan (advisor) ; Hrbek, Michal (referee)
Abstract:In this work, the reader is introduced to the theory of persistent ho- mology and its applications. In the first chapter we will show the basics of sim- plicial and singular homology and we will prove the basic relations, especially the independence of simplicial homological groups on the chosen △-complex and isomorphism between homological groups of homotopic spaces. In the second chapter, we explain the motivation behind persistent homology, describe its al- gebraic structure and how it can be visually represented. We describe and prove the corectness of the algorithm for its calculation. We then illustrate the theory on an example. 1
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Modules over string algebras
Löwit, Jakub ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
The aim of this thesis is to investigate the categories of modules over the so called string algebras. In particular, we try to understand the cotorsion pairs in these categories, which boils down to understanding the decompositions of extensions of such modules. For string algebras with some oriented tree for the underlying quiver, we describe some classes given by these cotorsion pairs in terms of purely combinatorial closure properties. For any string algebras, the combinatorics appears to be similar, althought more complicated.
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Counting the points on elliptic curves over finite fields
Eržiak, Igor ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
The goal of this thesis is to explain and implement Schoof's algorithm for counting points on elliptic curves over finite fields. We start by defining elliptic curve as a set of points satisfying certain equation and then proceeding to define an operation on this set. Theoretical background needed for the algorithm is presented in the second chapter. Finally, the Schoof's algorithm is introduced in the third chapter, supplemented by an implementation in SageMath open-source software.
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Elliptic curves over finite fields
Beran, Adam ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the theory of elliptic curves, with the main focus on elliptic curves over finite fields. We present basic theory, taking several technical aspects into consideration (singularity of the curve, effect of field characteristic on the form of the equation of elliptic curve). We algebraically deduce and formulate the group law, that is the definition of addition on a set of points on elliptic curve). We prove a known result saying that the set of points on elliptic curve under addition forms a group. We present an elementary proof, some of the calculations will be carried out in computer program Mathematica due to their complexity. Finally, we study endomorphisms of elliptic curves over finite fields (homomorphisms on the set of points on elliptic curve that are defined by rational functions). Using obtained results, we prove the Hasse's theorem, which provides an estimate of the order of the group of points on elliptic curve over finite field. 1
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Quotients in algebraic geometry
Kopřiva, Jakub ; Šťovíček, Jan (advisor) ; Příhoda, Pavel (referee)
This thesis is concerned with the existence of pushouts in two different settings of algebraic geometry. At first, we study the pushouts in the cat- egory of affine algebraic sets over an infinite field. We show that this can be regarded as an instance of much general problem whether the pullback of finitely generated algebras over a commutative Noetherian ring is finitely generated. We give a partial solution to this problem and study some ex- amples. Secondly, we examine the existence of pushouts in the category of schemes with an emphasis on diagrams of affine schemes. We use the methods of Ferrand [2003] and Schwede [2004] and generalise some of their results. We conclude by giving some examples and suggest another approach to the problem.
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Jones polynomial
Gajdová, Anna ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
The topic of this thesis is the Jones polynomial of a given knot and its com- putation. First we define the Jones polynomial in two ways: using skein relations and using the bracket polynomial and we prove that these definitions are equi- valent. Next we derive an algorithm for computation of the Jones polynomial based on its relation with the bracket polynomial. We prove that the time com- plexity of the algorithm is O 20.823n , where n denotes number of crossings in a link diagram. Lastly we present the results of running the algorithm and its variants on data. We test the algorithm among others on small table knots, bigger random knots and on torus knots. We estimate that the fastest vari- ant of the algorithm runs on random knots with the average time complexity O 20.487n+o(n) . 1
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Modules and localization
Lysoněk, Tomáš ; Trlifaj, Jan (advisor) ; Šťovíček, Jan (referee)
In the Thesis, we define the notion of a localization and explore its connections to properties of modules - local properties and AD-properties, especially the projectivity. We present a proof of the fact that projectivity is an AD propety of modules due to Raynaud and Gruson in the corrected version by Perry from 2010. The proof is presented in full detail and accompanied by examples, and by the role of the investigated notions in the context of algebraic geometry. 1
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