|
|
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
|
|
|
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.
|
|
|
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
|
|
|
Tilting theory of commutative rings
Hrbek, Michal ; Trlifaj, Jan (advisor) ; Herbera Espinal, Dolors (referee) ; Šaroch, Jan (referee)
The thesis compiles my contributions to the tilting theory, mainly in the set- ting of a module category over a commutative ring. We give a classification of tilting classes over an arbitrary commutative ring in terms of data of geometrical flavor - certain filtrations of the Zariski spectrum. This extends and connects the results known previously for the noetherian case, and for Prüfer domains. Also, we show how the classes can be expressed using the local and Čech homology the- ory. For 1-tilting classes, we explicitly construct the associated tilting modules, generalizing constructions of Fuchs and Salce. Furthermore, over any commuta- tive ring we classify the silting classes and modules. Amongst other results, we exhibit new examples of cotilting classes, which are not dual to any tilting classes - a phenomenon specific to non-noetherian rings. 1
|
|
|
Structure of division rings
Reichel, Tomáš ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
This bachelor thesis deals with a theorem and its proof, which allows construction of division ring from cyclic field extension which satisfies certain conditions. The reader is expected to have basic knowledge of linear algebra, ring and module theory. For using this theorem the reader also needs some skills in counting Galois groups. In this work there are also included two basic examples of usage the theorem. During the proof we introduce a structure of tensor product and Brauer group. Powered by TCPDF (www.tcpdf.org)
|
|
|
Determining primeness of an ideal?
Stejskal, Adam ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
We present an algorithm for determining whether an ideal in a polynomial ring is prime or not. We use the Gröbner bases as a main tool for operations with ideals. We show an analogue of Buchberger's algorithm for computing a Gröbner basis for an ideal in polynomials over a ring, which not need to be a field. We also show a relation between prime ideals in polyno- mials over a ring R and prime ideals in polynomials over a quotient ring of R modulo a prime ideal. We are primarilly discussing the issues of theoretical corectness, but we also present the conditions of actual computability. 1
|
|
|
Witnessing of existential statements
Kolář, Jan ; Krajíček, Jan (advisor) ; Šaroch, Jan (referee)
The thesis formulates and proves a witnessing theorem for SPV -provable formulas in the form ∀x∃yA(x, y) where A corresponds to a polynomial time decidable relation. By SPV we understand an extension of the theory TPV (the universal theory of N in the language representing polynomial algorithms) by additional axioms ensuring the existence of a minimum of a linear ordering defined by a polynomial time decidable relation on an initial segment. As these additional axioms are not universal sentences, the theory SPV requires nontrivial use of witnessing Herbrand's and KPT theorems which have direct application only for universal theories. Based on the proven witnessing theorem, we derive a NP search problem characterizing complexity of finding y for a given x such that A(x, y). A part of the thesis is dedicated to arguments supporting the conjecture that SPV is strictly stronger than TPV . 1
|
|
|
Cryptosystems based on codes with rank metrics
Marko, Marek ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
The first part of this paper explains the uses of the element's rank and the metric induced by it in linear error-correcting codes over finite fields. Describing the effective decoding algorithm of rank-metric codes without the use of exhaustive search is essential. This algorithm is applied in cryptographic systems based on codes with rank metric pre- sented in the next chapter. Apart from the scheme of cryptosystem, we focus on the de- tailed illustration of a possible structural attack. Comprehension of the attack will be significant in order to show some methods how to withstand it. 1
|
|
|
Strongly compact cardinals and SCH
Narusevych, Mykyta ; Šaroch, Jan (advisor) ; Krajíček, Jan (referee)
The thesis is devoted to the cardinal arithmetic. The first step is to formulate the Singular Cardinals Hypothesis (SCH) which simplifies the cardinal exponentiation of sin- gular cardinal numbers. We then define stationary sets and closed and unbounded subsets of an ordinal number. The main goal is to prove the Silver's theorem and the corollary which states, that if SCH holds for all singular cardinals with countable cofinality, then it holds everywhere. In the last chapter we define strongly compact cardinal numbers and prove some of their properties. Finally, we prove the Solovay's theorem, which states that SCH holds everywhere above a strongly compact cardinal. 1
|
|
|
Rings with restricted minimum condition
Krasula, Dominik ; Žemlička, Jan (advisor) ; Šaroch, Jan (referee)
Ring is artintian if and only if all of its factors are artinian. We say that ring R satisfies the restricted minimum condition, if for every essenctial ideal, corresponding factor ring is artinian. We will call such ring RM ring for short. Similarly as the class of artinian rings, the class of RM rings is closed under fac- tors and finite direct products. In this thesis we prove that restricted minimum condition is satisfied in coordinate rings, ring (R × R)[x] and noetherian CDR domains. We investigate the relation between unique factorization domains and RM domains. In last chpater, we will focus are attention to polynomial rings, proving that if ring R[x] is RM then R is semisimple. Laurents polynomials over domain R are RM rings if and only if R is a field. 1
|
guest ::
