National Repository of Grey Literature 49 records found  1 - 10nextend  jump to record: Search took 0.00 seconds. 
pp-elimination of quantifiers in module theories
Novák, Jindřich ; Šaroch, Jan (advisor) ; Ježil, Ondřej (referee)
The aim of this thesis is to prove the Baur-Monk Theorem and thereby show complete module-theories admit an elimination of quantifiers down to (Boolean combinations of) existential formulae. To achieve this, following a brief introduction in Chapter 1, the reader is fami- liarised in Chapter 2 with the notion of a positive-primitive formula in the lan- guage of right R-modules, and its close relationship with commutative groups, their cosets, and lattices. Chapter 3 first lays the technical groundwork for the proof of the Baur-Monk Theorem, presented in Section 3.3, in its opening two subsections which contain the needed combinatorial and group-theoretical results, namely the Neumann Lemma and a variation on the Inclusion-Exclusion Principle. Chapter 4 concludes the mathematical work contained herein with a brief over- view of some immediate corollaries of the the Baur-Monk Theorem and earlier results.
Kan extensions and adjoint functors
Otrubů, Mavis ; Šaroch, Jan (advisor) ; Žemlička, Jan (referee)
This thesis is devoted to Kan extensions. First, we provide needed definitions and prove a theorem which gives us an existence condition for a Kan extensions. The proof of this theorem also contains a guide to constructing Kan extensions. The main goal is to present a result which puts Kan extensions and adjoint functors in relation. We also connect this theorem to global Kan extensions. We apply these abstract results in the last chapter, where we formulate and solve a particular problem concerning adjoint functors between the categories of G-sets. 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
Inverse limits in module categories
Menčík, Matouš ; Trlifaj, Jan (advisor) ; Šaroch, Jan (referee)
For a class of modules C, we study the class lim ←− C of modules that can be obtained as inverse limits of modules from C. In particular, we investigate how additional properties of the class C are reflected by properties of the class lim ←− C. We also address the question of whether for a given module M, every inverse limit of products of M is an inverse limit of finite products of M. We provide examples of modules for which the answer is positive, negative, and for which there is a reason to believe that it depends on additional set-theoretic assumptions. 1
Module approximations and direct limits
Matoušek, Cyril ; Šaroch, Jan (advisor) ; Šťovíček, Jan (referee)
This master's thesis deals with questions about the existence of module appro- ximations, namely C-precovers and C-covers for a given class C of R-modules, and studies the relations of these approximations with direct limits. Thanks to a the- orem due to Enochs, we know that every R-module has a C-cover if the pre- covering class C is closed under direct limits, although the validity of the con- verse implication remains an open problem known as Enochs' conjecture. In this setting, we show that any module M with perfect decomposition satisfies that the class Add(M) is precovering and closed under direct limits; hence also cove- ring. Furthermore, we prove Enochs' conjecture for Add(M) if M is small, e.g. < ℵω-generated. Specifically, if M is small and Add(M) covering, then M has a perfect decomposition.
Models of bounded arithmetic
Narusevych, Mykyta ; Krajíček, Jan (advisor) ; Šaroch, Jan (referee)
We study mutual relations of various versions of the pigeonhole principle over bounded arithmetic theory T1 2 (R). The main two variants are the ordinary ontoPHPn+1 n (R), formalizing that R is not the graph of a bijective function from [n + 1] into [n], and its weak version, WPHP2m m (S), formalizing that S is not the graph of an injective function from [2m] into [m]. It is known that: T2 2 (R) proves WPHP2m m (R) (with m universally quantified) and, in fact, also WPHP2m m (S) for all relations S that are □p 1(R) (= polynomial-time definable from R) (J. Paris, A. J. Wilkie and A. R. Woods (1988)), neither I∃1(R) nor T1 2 (R) prove WPHP2m m (R), (J. Paris and A. J. Wilkie (1985) and J. Krajicek (1992)), full T2(R) does not prove ontoPHPn+1 n (R), (M. Ajtai (1988), J. Krajicek, P. Pudlak and A. R. Woods (1991), T. Pitassi, P. Beame and R. Impagliazzo (1993)). We generalize the method of J. Paris and A. J. Wilkie (1985) to show that theory T1 2 (R) plus ∀m WPHP2m m (□p 1(R)) does not prove ontoPHPn+1 n (R). This does follow from results mentioned above but such a combined proof involves complex probabilistic combinatorics which is difficult to modify for other principles. On the other hand our method is more elementary and thus better amenable to other principles. We demonstrate this by proving a...
Pseudofinite structures and limits
Ježil, Ondřej ; Krajíček, Jan (advisor) ; Šaroch, Jan (referee)
For a class of graph instances of a computational problem we define a limit object, relative to some computationally restricted class of functions. The key method here is forcing with random variables where the sample set is taken as instances of some nonstandard size. We study the general theory of these limits, called in the thesis wide limits, and their connection to classical problems such as finding a large clique or with the combinatorial problems associated with the classes of total NP search problems PPA and PPAD. Our main results are several 0-1 laws associated with these limits and existence of a significantly large clique of the wide limit of all graph consisting of one large clique. 1
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.
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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3 Šaroch, Jaroslav
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