A difference field M , is a field with a distinguished endomorphism, is called a generic difference field if it is existentially closed among the models of the theory of difference fields. In the language Ld, by a theorem of Hrushovski, it is characterized by the following: M is an algebraically closed field, s is an automorphism of M, and if W and V are varieties defined over M such that W is a subset of VU s (V ) and the projection maps W to V and W to s(V ) are generically onto, then there is a tuple a in M such that (a, s ( a)) in W. This thesis is a survey on the theory of generic difference fields, called ACFA, which has been studied by Angus Macintyre, Van den Dries, Carol Wood, Ehud Hrushovski and Zoe Chatzidakis. ACFA is the model completion of the theory of algebraically closed difference fields. It is very close to having full quantifier elimination, but it doesn&#039 / t. We can eliminate quantifiers down to formulas with one quantifier and hence obtain the completions of ACFA. This entails the decidability of the theory ACFA as well as its extensions obtained by specifying the characteristic. The fixed field of s is a pseudo-finite field

Thesis (Ph. D.)--University of North Texas, May, 2008. / Title from title page display. Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

Thesis (MSc (Mathematical Sciences))-- University of Stellenbosch, 2005. / It is an open question whether or not one can always lift Galois extensions of smooth algebraic curves in characteristic p to Galois extensions of smooth relative curves in characteristic 0. In this thesis we study some of the available techniques and partial solutions to this problem. Our studies include the techniques of Oort, Sekiguchi and Suwa where the lifting problem is approached via a connection with lifting group schemes. We then move to the topic of singular liftings and for this we study the approach of Garuti. Thereafter, we move to the wild smooth setting again where we study the crucial local − global principle, and apply it by illustrating how Green and Matignon solved the p2-lifting problem.

H-removable sequences, for arbitrary H, under &Lambda^* construction are presented here. In the first part we investigate Neighborhood Distinct (ND) graphs and ask some natural questions concerning disconnected H and H complement. In the second part, we introduce property * and investigate graphs that satisfy property *. Consequently we find $H$-removable sequences for all graphs H with up to 6 vertices except for G60. G60 is the only graph with up to 6 vertices for which neither it nor its complement satisfies property *. The last part of our work focuses on good and bad copies of arbitrary graphs $H$ and how to interchange from one to the other. The number of ways to count all possible copies of H in H_{pn} ^ &Lambda^* is also presented via examples.

Orientador: Paulo Roberto Brumatti / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-14T19:30:30Z (GMT). No. of bitstreams: 1 Veloso_MarceloOliveira_D.pdf: 662198 bytes, checksum: f119f8026ebe09649fca4a175b7cec47 (MD5) Previous issue date: 2009 / Resumo: Este trabalho é dedicado ao estudo das derivações localmente nilpotentes de certas K-álgebras finitamente geradas, onde K é um corpo de característica zero. Estes domínios são generalizações de anéis bem conhecidos na literatura sendo um deles o anel de Fermat. Mais precisamente, caracterizamos o conjunto das derivações localmente nilpotentes destes domínios ou de um subconjunto deste conjunto. Também calculamos o ML invariante destes domínios e como aplicação direta destas informações encontramos um conjunto de geradores para o grupo dos automorfismos de um destes domínos. No caso do anel de Fermat mostramos que nem sempre temos um domíno rígido. Além disso, verificamos que a Conjectura de Nakai é verdadeira para o anel de Fermat. / Abstract: This work is dedicated to the study of locally nilpotent derivations of certain finitely generated K-algebras, where K is a field of zero characteristic. These domains are generalizations of the well-known rings in the literature. One of this is the Fermat ring. More precisely, we characterize the set of locally nilpotent derivations of these domains or some subsets of this set. We also calculate the ML invariant of these domains and as a direct application of these results we find a set of generators for the group of automorphisms of some of these domains. We show that the Fermat ring is not always a rigid domain. Furthermore, we prove that Nakai's conjecture is true for the ring Fermat. / Doutorado / Algebra Comutativa / Doutor em Matemática

>Magister Scientiae - MSc / The pursuit of graphs which are vertex-transitive and non-Cayley on groups has been ongoing for some time. There has long been evidence to suggest that such graphs are a very rarety in occurrence. Much success has been had in this regard with various approaches being used. The aim of this thesis is to find such a class of graphs. We will take an algebraic approach. We will define Cayley graphs on loops, these loops necessarily not being groups. Specifically, we will define meta-Cayley graphs, which are vertex-transitive by construction. The loops in question are defined as the semi-direct product of groups, one of the groups being Z₂ consistently, the other being in the class of dihedral groups. In order to prove non-Cayleyness on groups, we will need to fully determine the automorphism groups of these graphs. Determining the automorphism groups is at the crux of the matter. Once these groups are determined, we may then apply Sabidussi's theorem. The theorem states that a graph is Cayley on groups if and only if its automorphism group contains a subgroup which acts regularly on its vertex set. / Chemicals Industries Education and Training Authority (CHIETA)

Let G be a group. A weak Cayley table isomorphism $\phi$: G $\rightarrow$ G is a bijection satisfying two conditions: (i) $phi$ sends conjugacy classes to conjugacy classes; and (ii) $\phi$(g1)$\phi$(g2) is conjugate to $\phi$(g1g2) for all g1, g2 in G. The set of all such mappings forms a group W(G) under composition. We study W(G) for fifty-six of the two hundred nineteen three-dimensional crystallographic groups G as well as some other groups. These fifty-six groups are related to our previous work on wallpaper groups.

This work is concerned with computing the invariant lattices of purely non-symplectic automorphisms of special elliptic K3 surfaces. Brandhorst gave a collection of K3 surfaces admitting purely non-symplectic automorphisms that are uniquely determined up to isomorphism by certain invariants. For many of these surfaces, the automorphism is also unique or the automorphism group of the surface is finite and with a nice isomorphism class. Understanding the invariant lattices of these automorphisms and surfaces is interesting because of these uniqueness properties and because it is possible to give explicit generators for the Picard and invariant lattices. We use the methods given by Comparin, Priddis and Sarti to describe the Picard lattice in terms of certain special curves from the elliptic fibration of the surface. We use symmetries of the Picard lattice and fixed-point theory to compute the invariant lattices explicitly. This is done for all of Brandhorst's elliptic K3 surfaces having trivial Mordell-Weil group.

This licentiate thesis consists of two papers treating subjects in rational homotopy theory. In Paper I, we establish two formality conditions in characteristic zero. We prove that adg Lie algebra is formal if and only if its universal enveloping algebra is formal. Wealso prove that a commutative dg algebra is formal as a dg associative algebra if andonly if it is formal as a commutative dg algebra. We present some consequences ofthese theorems in rational homotopy theory. In Paper II, we construct a differential graded Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 2: Manuscript.</p>