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Norm-euclidean quadratic extensions of the field of rational numbers
Zemková, Kristýna ; Šaroch, Jan (advisor) ; Příhoda, Pavel (referee)
The goal of this work is to characterize all norm-euclidean quadratic ex- tensions of Q. The work treats completely the part of imaginary quadratic extensions. In the case of real quadratic extensions, we give a list of such dis- criminants D that the field Q( √ D) is norm-euclidean. Furthermore, we prove an estimate D < 214 for all norm-euclidean fields Q( √ D). Subsequently, the case D ≡ 1 (mod 4) is discussed in detail. For the case D ≡ 1 (mod 4) we mention references to the results of other authors. 1
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Kvazieuklidovské obory integrity
Čoupek, Pavel ; Šaroch, Jan (advisor) ; Glivický, Petr (referee)
In this thesis, we present an overview of some of the known facts about k-stage Euclidean and quasi-Euclidean rings and domains, certain generalisations of the concept of Euclidean ring, as well as some new results. Among the new results, the norm-free characterization of k-stage Euclidean rings based on a transfinite construction of k-stage Euclidean ring is fundamental and has many applications. Statements providing a way to construct new k-stage Euclidean rings from other k-stage Euclidean rings recieve special attention (with the integral domain case in mind). Also, we present an example of a 3-stage Euclidean integral domain which we believe is a good candidate for not being 2-stage Euclidean. 1
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Orthogonal bases and Jordan normal form
Kučera, Daniel ; Šaroch, Jan (advisor) ; Barto, Libor (referee)
There exists an ortonormal set of eigenvectors for a linear operator if and only if it commutes with its adjoint endomorphism. The aim of this thesis is to characterize endomorphisms for which there exists a matrix representation with respect to an orthogonal basis in Jordan form. We introduce the notion of unitarily jordanisable endomorphism to capture this property. The proof of the Spectral theorem as well as the existence and uniqueness of Jordan form can be found in the first two chapters. An interesting connection with bilinear forms arises in chapter three and is used to prove that any linear operator with single eigenvalue and the length of Jordan chains bounded by two is unitarily jordanisable. The last chapter is devoted to the discussion of uniqueness of othogonal polar basis for a bilinear form and an algorithm is introduced which can determine whether or not a linear operator is unitarily jordanisable. 1
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Structure of self-small groups and modules
Dvořák, Josef ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Title: Structure of self-small groups and modules Author: Josef Dvořák Department: Department of Algebra Supervisor: Mgr. Žemlička Jan, Ph.D. Supervisor's e-mail address: zemlicka@karlin.mff.cuni.cz Abstract: The thesis sums up the basic properties of self-small groups. Furthermore it thoroughly builds the theory od quotient categories by Serre classes, with focus on quotient category modulo the class B of boun- ded groups, which, as demonstrated, is equivalent to the quasicategory, i.e. category of abelian groups with Hom-sets being Q⊗Z HomA (A, B). This approach is developed into the theory of generalized quasi-categories. The dualities between quasi-caterogories od torsion-free and quotient-divisible categories of finite rank, resp. between categories of finite-rank self-small groups are studied and they are emloyed to the partial solution of Fuchs' problem no. 34. Keywords: self-small group, quotient divisible group, quasicategory, quo- tient category 1
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Semigroups of lattice points
Scholle, Marek ; Kepka, Tomáš (advisor) ; Šaroch, Jan (referee)
The thesis deals with subsemigroups of (Nm 0 , +), a special discussion is later devoted to the cases m = 1, m = 2 and m = 3. We prove that a subsemigroup of Nm 0 is finitely generated if and only if its generated cone is finitely generated (equivalently polyhedral) and we describe basic topological properties of such cones. We give a few examples illustrating that conditions sufficient for finite generation in N2 0 can not be easily trans- ferred to higher dimensions. We define the Hilbert basis and the related notion of Carathéodory's rank. Besides their basic properties we prove that Carathédory's rank of a subsemigroup of Nm 0 , m = 1, 2, 3, is less than or equal to m. A particular attention is devoted to the subsemigroups containing non-trivial subsemigroups of "subtractive" elements.
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Triangulation algorithm for non-linear equation systems
Väter, Ondřej ; Hojsík, Michal (advisor) ; Šaroch, Jan (referee)
The topic of this thesis is a triangulation algorithm and its use in cryptanalysis. First of all we will define a non-linear equation system on which we can apply triangulation algorithm and we will explain what its output is. Then we will demonstrate its application in cryptanalysis, more specificaly during the attack on the Rinjdael cifer. We will illustrate this attack by a search of collision for our hash function, created for this purpose in Davies-Mayer mode using Rijndael cipher This thesis also contains a practical part in which we will demonstrate the search of collision for our hash function mention before.
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Kombinatorika hashovacích funkcí
Sýkora, Jiří ; Holub, Štěpán (advisor) ; Šaroch, Jan (referee)
In this thesis, we study hash functions. We focus mainly on the famous Merkle-Damg˚ard construction and its generalisation. We show that even this generalised construction is not resistant to multicollision attacks. Combinatorics on words plays a fundamental role in the construction of our attack. We prove that regularities unavoidably appear in long words with bounded number of symbol occurences. We present our original results concerning regularities in long words. We lower some earlier published estimates, thus reducing the comlexity of the attack. Our results show that generalised iterated hash functions are interesting rather from the theoretical than practical point of view. 1
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Set-theoretic Methods in the Theory of Modules
Šaroch, Jan ; Trlifaj, Jan (advisor) ; Příhoda, Pavel (referee) ; Struengmann, Lutz (referee)
The thesis collects my actual contributions to the theory of cotorsion pairs, with closer attention paid to the application of set-theoretic methods in this area. It consists of an introduction and three papers with coauthors. The first two, already published, deal with the so-called Telescope Conjecture for Module Categories. We prove here, for instance, that a hereditary cotorsion pair (A, B) with the class B closed under direct limits is generated by a set of countably presented modules. Moreover, if the class A is closed under direct limits too, then the pair (A, B) is cogenerated by a set of indecomposable pure-injective modules. In the third paper, we deal with the cotorsion pairs which provide us with non-trivial examples of abstract elementary classes (in the sense of Shelah). Then we study the class D of all 1-projective modules, proving e.g. that-regardless of the ring-it always forms a Kaplansky class.
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Tiliting modules of finite type
Šaroch, Jan ; Žemlička, Jan (referee) ; Trlifaj, Jan (advisor)
The thesis studies properties of cotorsion pairs in the category of modules; we are mostly interested in conditions under which the given cotorsion pair is complete or, actually, of finite type. Methods of deconstruction of cotorsion pairs developed during our inquiry are eventually used to prove that every tilting module is of finite type. We show also a connection of presented problems with so-called telescope conjecture.
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Definable classes of modules and deconstruction of cotorsion pairs
Dohnal, Garik ; Šaroch, Jan (advisor) ; Šťovíček, Jan (referee)
The goal of this work was to prove the fact, that definable closure of any subclass of cotorsion modules closed under direct sums consists of $\Sigma$-cotorsion modules. The only known proof uses substantially the calculus of derived category, in this work we tried to prove the same, but only by means of a given category of all right $R$-modules and set-theoretic properties of partial orders indexing direct systems of $R$-modules. The main results of this work are proved under additional assumptions on the ring $R$, in particular $\vert R\vert\leq\aleph_{\omega}$ or $\text{dim}(R)<\aleph_{\omega}$. Attempts to give s proof in the same general situation, where the fact is known to hold, was not successful. Powered by TCPDF (www.tcpdf.org)
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