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Let F be a free group of finite rank. Given words u, v ∈ F, J.H.C. Whitehead solved the decision problem of finding an automorphism φ ∈ Aut(F), carrying u to v. He used topological methods to produce an algorithm. Higgins and Lyndon gave a very concise proof of the problem based on the works of Rapaport. We provide a detailed account of Higgins and Lyndon’s proof of the peak reduction lemma and the restricted version of Whitehead’s theorem, for cyclic words as well as for sets of cyclic words, with a full explanation of each step. Then, we give an inductive proof of Whitehead’s minimization theorem and describe Whitehead’s decision algorithm. Noticing that Higgins and Lyndon’s work is limited to the cyclic words, we extend their proofs to ordinary words and sets of ordinary words. In the last chapter, we mention an example given by Whitehead to show that the decision problem for finitely generated subgroups is more difficult and outline an approach due to Gersten to overcome this difficulty. We also give an extensive literature survey of Whitehead’s algorithm / Thesis / Master of Science (MSc)

K3 surfaces have garnered attention across various fields, from optics and dynamics to high energy physics, making them a subject of extensive study for many decades. Recent work by mathematicians, including Brandhorst [1], has focused on non-symplectic automorphisms, aiming to categorize K3 surfaces that admit such automorphisms. Brandhorst made a list of unique K3 surfaces with purely non-symplectic automorphisms and established specific criteria for a K3 surface to be isomorphic to one on his list. This thesis aims to provide an alternative representation of select K3 surfaces from Brandhorst's list. While Brandhorst predominantly characterizes these surfaces as elliptic K3 surfaces, we offer a description of these surfaces as hypersurfaces in weighted projective space. Our approach involves verifying the criteria established by Brandhorst, thereby establishing an isomorphism between the surfaces in question. Through this study, we contribute to the understanding of K3 surfaces and their automorphisms while also demonstrating the correspondence between different spaces and methodologies for analyzing K3 surfaces. This work lays the groundwork for further investigations into K3 surfaces with purely non-symplectic automorphisms, paving the way for deeper insights into their structural properties and geometric intricacies.

Made available in DSpace on 2016-03-22T17:26:43Z (GMT). No. of bitstreams: 1 Gesner.pdf: 787824 bytes, checksum: 39876589a5891eb7dea64c7a471b8166 (MD5) Previous issue date: 2010-12-21 / Traditional digital logic deals with variables assuming only two possible states: false and true. But for a large number of real world modeling, we want intermediary values. The concept of duality, stating that something can and must coexist with its opposite, makes the fuzzy logic seem natural, even inevitable. Thus, the fuzzy logic introduces the ability to infer conclusions and generates responses based on vague information, which is also ambiguous and qualitatively incomplete and inaccurate. In this context, the way of thinking of fuzzy-based systems is similar to humans, representing the expressions of natural language in a very simple and intuitive way, leading to the construction of systems easy to understand and to maintain. Other important area of research based on mathematical models for the treatment of uncertainty considers interval mathematics, which has been applied in the representation of inaccurate data. In interval mathematics, the principle of correctness is the assurance that in the computation of an algorithm, the interval output contains all possible outcomes corresponding to punctual data for an interval input. In addition, the optimality principle, determines that the interval output is the smallest possible one satisfying accuracy. Thus, the correctness is the minimum condition while the optimality is the ideal condition to be satisfied by interval computations. Based on these statements, intervals can be used to represent unknown values and to represent continuous values in scientific computing algorithms. The aims of interval valued fuzzy logic are to consider the interval fuzzy constructions as fuzzy constructors which are correct and to analyze criteria to ensure optimality. The extension of the interval connectives of fuzzy logic in this work is based on the canonical interval representation of real functions and in this case, restricted to the unit interval [0; 1] of the real line. Such representation always returns the smallest interval containing the image of the function. Considering concepts and foundations of both approaches, fuzzy logic and interval mathematics, this work studies the operators defined as coimplications. They are characterized as dual structure of the fuzzy implications. Moreover, it seeks to introduce the extension of the interval fuzzy coimplications, and analyses the satisfaction of properties similar to the respective classes of fuzzy coimplications. In particular, we show that valued interval fuzzy coimplications are representations of fuzzy coimplications satisfying these two principles. This Work also analise the dual structure of interval conjugate fuzzy implications, which are obtained from interval automorphisms / A l´ogica digital tradicional lida com vari´aveis assumindo apenas dois poss´ıveis estados: falso e verdadeiro. Mas para grande n´umero de modelagens do mundo real desejamos valores intermedi´arios. O conceito de dualidade, estabelecendo que algo pode e deve coexistir com o seu oposto, faz a l´ogica difusa parecer natural, at´e mesmo inevit´avel. Assim, a l´ogica fuzzy introduz a habilidade em inferir conclus oes e gerar respostas baseadas em informac¸ oes vagas, amb´ıguas e qualitativamente incompletas e imprecisas. Neste contexto, os sistemas de base fuzzy apresentam uma forma de raciocinar semelhante aos humanos, representando as express oes da linguagem natural de maneira muito simples e intuitiva, levando `a construc¸ ao de sistemas compreens´ıveis e de f´acil manutenc¸ ao. Outra importante ´area de pesquisa baseada em modelos matem´aticos para tratamento da incerteza considera a matem´atica intervalar, a qual vem sendo aplicada na representac¸ ao de dados inexatos. Em matem´atica intervalar, o princ´ıpio da corretude consiste na garantia de que, na computac¸ ao de um algoritmo, a sa´ıda intervalar cont´em todos os poss´ıveis resultados pontuais correspondentes aos dados pontuais referentes `a entrada intervalar. E, o princ´ıpio da optimalidade, determina que a sa´ıda intervalar seja a menor poss´ıvel satisfazendo a corretude. Assim, a corretude ´e a condic¸ ao m´ınima enquanto que a optimalidade ´e a condic¸ ao ideal a ser satisfeita por uma computac¸ ao intervalar. Com base nestes crit´erios, os intervalos podem ser aplicados para representar valores desconhecidos e para representar valores cont´ınuos em algoritmos da Computac¸ ao Cient´ıfica. O principal objetivo da l´ogica fuzzy valorada intervalarmente ´e considerar as construc¸ oes fuzzy intervalares como construc¸ oes fuzzy que s ao corretas e analisar crit´erios que garantam optimalidade. A extens ao intervalar dos conectivos da l´ogica fuzzy em estudo neste trabalho est´a baseada na representac¸ ao intervalar can onica de func¸ oes reais e, neste caso, restrita ao intervalo unit´ario [0; 1] da reta real, que sempre retorna o menor intervalo contendo a imagem da func¸ ao. Consideram-se conceitos e fundamentos de ambas abordagens, da l´ogica fuzzy e da matem´atica intervalar, para estudar os operadores definidos como coimplicac¸ oes, caracterizados como estrutura dual das implicac¸ oes fuzzy, buscando introduzir a extens ao intervalar das coimplicac¸ oes fuzzy, analisando a satisfac¸ ao de propriedades an´alogas `as respectivas classes de coimplicac¸ oes fuzzy valoradas pontualmente. Em particular, mostra-se que coimplicac¸ oes fuzzy valoradas intervalarmente s ao representac¸ oes de coimplicac¸ oes fuzzy satisfazendo estes dois princ´ıpios. O trabalho tamb´em contempla uma an´alise da estrutura dual das conjugadas de implicac¸ oes valoradas intervalarmente, as quais s ao obtidas por ac¸ ao de automorfismos intervalares

Cette thèse est consacrée à l'étude des différents problèmes liés aux structures des cubes , en théorie ergodique et en dynamique topologique. Elle est composée de six chapitres. La présentation générale nous permet de présenter certains résultats généraux en théorie ergodique et dynamique topologique. Ces résultats, qui sont associés d'une certaine façon aux structures des cubes, sont la motivation principale de cette thèse. Nous commençons par les structures de cube introduites en théorie ergodique par Host et Kra (2005) pour prouver la convergence dans $L^2 $ de moyennes ergodiques multiples. Ensuite, nous présentons la notion correspondante en dynamique topologique. Cette théorie, développée par Host, Kra et Maass (2010), offre des outils pour comprendre la structure topologique des systèmes dynamiques topologiques. En dernier lieu, nous présentons les principales implications et extensions dérivées de l'étude de ces structures. Ceci nous permet de motiver les nouveaux objets introduits dans la présente thèse, afin d'expliquer l'objet de notre contribution. Dans le Chapitre 1, nous nous attachons au contexte général en théorie ergodique et dynamique topologique, en mettant l'accent sur l'étude de certains facteurs spéciaux. Les Chapitres 2, 3, 4 et 5 nous permettent de développer les contributions de cette thèse. Chaque chapitre est consacré à un thème particulier et aux questions qui s'y rapportent, en théorie ergodique ou en dynamique topologique, et est associé à un article scientifique. Les structures de cube mentionnées plus haut sont toutes définies pour un espace muni d'une unique transformation. Dans le Chapitre 2, nous introduisons une nouvelle structure de cube liée à l'action de deux transformations S et T qui commutent sur un espace métrique compact X. Nous étudions les propriétés topologiques et dynamiques de cette structure et nous l'utilisons pour caractériser les systèmes qui sont des produits ou des facteurs de produits. Nous présentons également plusieurs applications, comme la construction des facteurs spéciaux. Le Chapitre 3 utilise la nouvelle structure de cube définie dans le Chapitre 2 dans une question de théorie ergodique mesurée. Nous montrons la convergence ponctuelle d'une moyenne cubique dans un système muni deux transformations qui commutent. Dans le Chapitre 4, nous étudions le semigroupe enveloppant d'une classe très importante des systèmes dynamiques, les nilsystèmes. Nous utilisons les structures des cubes pour montrer des liens entre propriétés algébriques du semigroupe enveloppant et les propriétés topologiques et dynamiques du système. En particulier, nous caractérisons les nilsystèmes d'ordre 2 par une propriété portant sur leur semigroupe enveloppant. Dans le Chapitre 5, nous étudions les groupes d'automorphismes des espaces symboliques unidimensionnels et bidimensionnels. Nous considérons en premier lieu des systèmes symboliques de faible complexité et utilisons des facteurs spéciaux, dont certains liés aux structures de cube, pour étudier le groupe de leurs automorphismes. Notre résultat principal indique que, pour un système minimal de complexité sous-linéaire, le groupe d'automorphismes est engendré par l'action du shift et un ensemble fini. Par ailleurs, en utilisant les facteurs associés aux structures de cube introduites dans le Chapitre 2, nous étudions le groupe d'automorphismes d'un système de pavages représentatif. La bibliographie, commune à l'ensemble de la thèse, se trouve en fin document / This thesis is devoted to the study of different problems in ergodic theory and topological dynamics related to og cube structures fg. It consists of six chapters. In the General Presentation we review some general results in ergodic theory and topological dynamics associated in some way to cubes structures which motivates this thesis. We start by the cube structures introduced in ergodic theory by Host and Kra (2005) to prove the convergence in $L^2$ of multiple ergodic averages. Then we present its extension to topological dynamics developed by Host, Kra and Maass (2010), which gives tools to understand the topological structure of topological dynamical systems. Finally we present the main implications and extensions derived of studying these structures, we motivate the new objects introduced in the thesis and sketch out our contributions. In Chapter 1 we give a general background in ergodic theory and topological dynamics given emphasis to the treatment of special factors. % We give basic definitions and describe special factors associated to a From Chapter 2 to Chapter 5 we develop the contributions of this thesis. Each one is devoted to a different topic and related questions, both in ergodic theory and topological dynamics. Each one is associated to a scientific article. In Chapter 2 we introduce a novel cube structure to study the actions of two commuting transformations $S$ and $T$ on a compact metric space $X$. In the same chapter we study the topological and dynamical properties of such structure and we use it to characterize products systems and their factors. We also provide some applications, like the construction of special factors. In the same topic, in Chapter 3 we use the new cube structure to prove the pointwise convergence of a cubic average in a system with two commuting transformations. In Chapter 4, we study the enveloping semigroup of a very important class of dynamical systems, the nilsystems. We use cube structures to show connexions between algebraic properties of the enveloping semigroup and the geometry and dynamics of the system. In particular, we characterize nilsystems of order 2 by its enveloping semigroup. In Chapter 5 we study automorphism groups of one-dimensional and two-dimensional symbolic spaces. First, we consider low complexity symbolic systems and use special factors, some related to the introduced cube structures, to study the group of automorphisms. Our main result states that for minimal systems with sublinear complexity such groups are spanned by the shift action and a finite set. Also, using factors associated to the cube structures introduced in Chapter 2 we study the automorphism group of a representative tiling system. The bibliography is defer to the end of this document

Neste trabalho, estudamos Loops de Código. Para este estudo, introduzimos os loops de código a partir de códigos pares e depois, provamos que loops de código de posto $n$ podem ser caracterizados como imagem homomórfica de certos loops de Moufang livres com n geradores. Além disso, introduzimos o conceito de vetores característicos associados a um loop de código. Com os resultados da teoria estudada, classificamos todos os loops de código de posto 3 e 4, encontramos todos os grupos de automorfismos externos destes loops e, finalmente, determinamos todas as suas respectivas representações básicas. / This work is about code loops. For this study, we introduce the code loops from even codes and then we prove that code loops of rank n can be characterized as a homomorphic image of a certain free Moufang loops with $n$ generators. Moreover, we introduce the concept of characteristic vectors associated with code loops. With the results of this theory, we classify all the code loops of rank 3 and 4, we find all the groups of outer automorphisms of these loops and finally we determine all their basic representations.

We investigate weak Cayley table isomorphisms, a generalization of group isomorphisms. Suppose G and H are groups. A bijective map phi : G to H is a weak Cayley table isomorphism if it satisfies two conditions:(1) If x is conjugate to y, then phi(x) is conjugate to phi(y); (2) For all x, y in G, phi(xy) is conjugate to phi(x)phi(y).If there exists a weak Cayley table isomorphism between two groups, then we say that the two groups have the same weak Cayley table.This dissertation has two main goals. First, we wish to find sufficient conditions under which two groups have the same weak Cayley table. We specifically study Frobenius groups and groups which satisfy the Camina pair condition. Second, we consider the group of all weak Cayley table isomorphisms between G and itself. We call this group the weak Cayley table group of G and denote it by W(G). Any automorphism of G is an element of W. The inverse map on G is also an element of W. We say that the weak Cayley table group is trivial if it is generated by the set of all automorphisms of G and the inverse map. Stephen Humphries proved that the symmetric groups S_n, the dihedral groups D_{2n} and the free groups F_n (n not equal to 3) all have trivial weak Cayley table groups. We will investigate the weak Cayley table groups of the alternating groups, certain types of Coxeter groups, the projective special linear groups and certain sporadic simple groups.

viii, 113 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation consists of four principal parts. In the first, we introduce the tracial quasi-Rokhlin property for an automorphism α of a C *-algebra A (which is not assumed to be simple or to contain any projections). We then prove that under suitable assumptions on the algebra A , the associated crossed product C *-algebra C *([Special characters omitted.] , A , α) is simple, and the restriction map between the tracial states of C *([Special characters omitted.] , A , α) and the α-invariant tracial states on A is bijective. In the second part, we introduce a comparison property for minimal dynamical systems (the dynamic comparison property) and demonstrate sufficient conditions on the dynamical system which ensure that it holds. The third part ties these concepts together by demonstrating that given a minimal dynamical system ( X, h ) and a suitable simple C *-algebra A , a large class of automorphisms β of the algebra C ( X, A ) have the tracial quasi-Rokhlin property, with the dynamic comparison property playing a key role. Finally, we study the structure of the crossed product C *-algebra B = C *([Special characters omitted.] , C ( X , A ), β) by introducing a subalgebra B { y } of B , which is shown to be large in a sense that allows properties B { y } of to pass to B . Several conjectures about the deeper structural properties of B { y } and B are stated and discussed. / Committee in charge: Christopher Phillips, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Huaxin Lin, Member, Mathematics; Marcin Bownik, Member, Mathematics; Van Kolpin, Outside Member, Economics

La thèse se compose de deux parties. La première partie est consacrée aux transformations des algèbres de dimension finie. Il est facile de voir que le groupe d'automorphismes d'une algèbre de dimension finie est un groupe algébrique affine. N.L. Gordeev et V.L. Popov ont démontré que n'importe quel groupe algébrique affine est isomorphe au groupe d'automorphismes de l'algèbre de dimension finie. Utilisant l'approche similaire nous démontrons que tout monoïde affine peut être obtenue comme un monoïde des endomorphismes d'une algèbre de dimension finie. Ensuite, nous étudions la solvabilité des groupes d'automorphismes d'algèbres commutatives de dimension finie. Nous introduisons un critère de leur solvabilité et l'appliquons aux intersections complètes et aux singularités isolées d'hypersurfaces. Nous étudions également les cas extrêmes du critère introduit. La deuxième partie de la thèse est consacrée à la transitivité infinie de groupes d'automorphismes spéciales de variétés affines et quasi-affines. Cette propriété est équivalente à la flexibilité pour les variétés affines. Tout d'abord, nous montrons l'équivalence entre la transitivité et la transitivité infinie des groupes d'automorphismes spéciaux sur un corps algébriquement clos de caractéristique arbitraire. Nous fournissons ensuite le critère de la flexibilité pour les cônes affines sur les variétés projectives et nous l'appliquons aux surfaces del Pezzo de degré 4 et 5. Enfin, nous étudions la flexibilité des torseurs universels sur les variétés couvertes par des espaces affines et fournissons une large gamme de familles de variétés flexibles. / The thesis consists of two parts. The first part is dedicated to transformations of finite-dimensional algebras. It is easy to see that the automorphism group of a finite-dimensional algebra is an affine algebraic group. N.L.~Gordeev and V.L.~Popov proved that any affine algebraic group is isomorphic to the automorphism group of some finite-dimensional algebra. We use a similar approach to prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional algebra. Next, we study the solvability of automorphism groups of commutative Artin algebras. We introduce a criterion of their solvability and apply it to complete intersections and to isolated hypersurface singularities. We also study extremal cases of the introduced criterion. The second part of the thesis is dedicated to the infinite transitivity of special automorphism groups of affine and quasiaffine varieties. This property is equivalent to the flexibility for affine varieties. Firstly, we prove the equivalence of transitivity and infinite transitivity of special automorphism groups over algebraically closed field of arbitrary characteristic. Then we provide the criterion of flexibility for affine cones over projective varieties and apply it to del Pezzo surfaces of degree 4 and 5. Finally, we study flexibility of universal torsors over varieties covered by affine spaces and provide a wide range of families of flexible varieties.

Made available in DSpace on 2015-03-26T13:45:37Z (GMT). No. of bitstreams: 1 texto completo.pdf: 458500 bytes, checksum: 8f45ac358c025eca77942539ace5f137 (MD5) Previous issue date: 2014-02-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The set of all automorphisms of a group G form a group denoted by Aut(G). In this work we study automorphisms of finite abelian groups, mainly following the approach by Christopher J. Hillar and Darren L. Rhea according to the paper Automorphisms of finite abelian Groups (American Mathematical Monthly 114 n. 10 (2007) 917-923). The main objective is to characterize the automorphism group Aut(G), where G is a finite abelian group and present a formula for the number of elements of Aut(G). The determination of this formula is done in two distinct ways: one from the calculation of the number of elements of the group Aut(G) viewed as the group of units of the endomorphisms ring End(G) and the other using certain characteristic subgroups of the group G. This latter method follows the development made by Heinrich Kuhn in his doctoral thesis. / O conjunto de todos os automorfismos de um grupo G forma um grupo denotado por Aut(G). Neste trabalho estudamos automorfismos de grupos abelianos finitos, seguindo principalmente a abordagem feita por Christopher J. Hillar e Darren L. Rhea no artigo Automorphisms of finite abelian Groups (American Mathematical Monthly 114 n. 10 (2007) 917-923). O objetivo principal ́e fazer uma caracterização do grupo de automorfismos Aut(G), onde G ́e um grupo abeliano finito e apresentar uma fórmula para o número de elementos de Aut(G). A determinação desta f ́ormula ́e feita de duas maneiras distintas: uma a partir do cálculo do número de elementos do grupo Aut(G) visto como grupo das unidades do anel de endomorfismos End(G) e a outra utilizando certos subgrupos característicos do grupo G. Esse último método segue o desenvolvimento feito por Heinrich Kuhn, em sua tese de doutorado.