This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A). In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration. In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  −→ A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.

A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes

Made available in DSpace on 2015-05-15T11:46:20Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 685495 bytes, checksum: 924951307927847259c1bd0253812600 (MD5) Previous issue date: 2014-02-19 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The main goal of this work is to furnish a proof of the well-known Rentschler s Theorem, which describes the structure of the locally nilpotent derivations on the polynomial ring in two indeterminates (over a field of characteristic zero), up to conjugation by tame automorphisms. As a central application of this result, we prove Jung s Theorem, concerning the generators of the group of automorphisms in two variables. Finally, some examples are discussed, illustrating connections to other important topics. / O principal objetivo deste trabalho é fornecer uma demonstração do bem-conhecido Teorema de Rentschler, que descreve a estrutura das derivações localmente nilpotentes sobre o anel de polinômios em duas variáveis (sobre um corpo de característica zero), a menos de conjugação por automorfismos tame . Como aplicação central deste resultado, provamos o Teorema de Jung, sobre os geradores do grupo de automorfismos em duas variáveis. Finalmente, alguns exemplos são discutidos, ilustrando conexões com outros tópicos importantes.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This dissertation addresses the universe ofwatersheds as the basic unit of environmental studies and has the catchment area of the resorttowns, and Santa Luzia Indiaroba Itanhy, members of the lower basin of the river Piauí, serving the southern coastal state of Sergipe. The overall objective was to analyze the process and forms of use and occupation of land in pursuit ofsocio reveal order and disorder. The theoretical and methodological was delimited from the Geossistêmico Bertrand model that allowed an analysis articulated between the naturaland anthropogenic lead. Therefore, the integration of the geo-environmental conditions was constituted in analysis of spatial order, while the disorders have been targeted by demographics,the field of livestock, conflicts of shrimp farming, industry and trade, highlighting the environmental impacts of socio-economic activities asresponsible for the process dynamics environment. As a conclusion, the hydrodynamics pretérita was the decisive factor in shaping the coastal plain and the behavioral pattern of rainfall shows good surpluswater in late autumn and winter and moderatesummer disabilities. The parameters, indicators of pollution by organic origin, were found at levels impacting water supplyfor the population. In the evolution of urban and rural population, we found an increase of 0.9% compared to 10:52% of rural population growth. Shrimp farming is an expanding activity in the three counties and has reduced the quality of our riversand lakes by input of organic matter released, causing the eutrophication of waters. In the context of the basin's preeminent urban resort, based on land use, the high concentration and for being a city of the industrial economy, with strongdemand for coastal tourism. / A presente Dissertação aborda o universo das bacias hidrográficas como unidade básica de estudos ambientais e tem como área de abrangência os municípios de Estância, Indiaroba e Santa Luzia do Itanhy, integrantes da bacia inferior do rio Piauí, porção costeira ao sul do estado de Sergipe. O objetivo geral foi analisar o processo e formas de uso e ocupação do solo, na busca de revelar ordem e desordens socioambientais. O referencial teórico-metodológico foi delimitado a partir do modelo Geossistêmico de Bertrand que permitiu uma análise articulada entre o ambiente natural e as derivações antropogênicas. Portanto, a integração entre os condicionantes geoambientais se constituiu em análise da ordem espacial, enquanto as desordens foram focalizadas pelos aspectos demográficos, domínio da pecuária, conflitos da carcinicultura, indústria e comércio, evidenciando os impactos ambientais das atividades socioeconômicas como responsáveis pelo processo da dinâmica ambiental. A guisa de conclusão, a hidrodinâmica pretérita foi fator decisivo na formação da planície costeira e o padrão comportamental das chuvas apresentou bons excedentes hídricos no final do outono e no inverno e moderada deficiência de verão. Os parâmetros, indicadores de poluição por origem orgânica, foram encontrados em níveis impactantes para águas de abastecimento da população. Na evolução da população urbana e rural, constatou-se um crescimento de 10.52 % frente a 0.9 % de crescimento da população rural. A carcinicultura é uma atividade em expansão nos três municípios e vem reduzindo a qualidade das águas dos rios e lagoas pelo aporte de matéria orgânica lançada, ocasionando a eutrofização das águas. No contexto da bacia há preeminência urbana em Estância, assentada no uso da terra, pela grande concentração e por ser um município de economia industrial, com forte demanda do turismo costeiro.

This work deals with the indebtedness of consumers. It examines the causes of the debotrs themselves. Based on a secondary analysis of past research in the area of financial literacy, consumer behavior, statistical data, but also of qualitative research in the form of case study seeks to answer the question why the debtor becomes the debtor, the extent to which the indebtedness and indebtedness viewed as rational behavior but also if the debtor is accompanied by rational reasoning or irrational behavior. It concludes that the debtor can not be considered entirely rational. Applied policy then the issue of over-indebtedness problem takes into account the irrational assumptions by the debtor only marginally and does not reflect as one of its main causes.

This work deals with the indebtedness of consumers. It examines the causes of the debotrs themselves. Based on a secondary analysis of past research in the area of financial literacy, consumer behavior, statistical data, but also of qualitative research in the form of case study seeks to answer the question why the debtor becomes the debtor, the extent to which the indebtedness and indebtedness viewed as rational behavior but also if the debtor is accompanied by rational reasoning or irrational behavior. It concludes that the debtor can not be considered entirely rational. Applied policy then the issue of over-indebtedness problem takes into account the irrational assumptions by the debtor only marginally and does not reflect as one of its main causes.

This thesis focuses upon how to calculate local components of Weil differentials of an elliptic function field. Because Weil differentials constitute a one-dimension vector space then one Weil differential is fixed. An algorithm calculating a local component is developed for the fixed one. The first algorithm computes local components of places of degree one. It is based upon elementary properties of local components. The definition of the Weil differential does not say enough about why it is defined in this way and about why it is useful. Thus there is the relationship between the Weil differential and some objects from complex analysis like the Laurent series and the residue. It provides a better understanding of properties of the Weil differential. The result of this thesis are other two algorithms calculating local components of Weil differentials. The algorithms employ the residue. 1

Este trabalho descreve o formalismo das gramáticas livres de contexto adaptativas com verificação de aparência. Esses dispositivos gramaticais possuem como núcleo uma gramática livre de contexto subjacente e, como mecanismo de auto-modificação, uma ou várias funções adaptativas que determinam quais produções são aplicáveis em cada passo de uma derivação. A verificação de aparência se refere a uma forma especial de aplicar algumas produções, escolhidas pelo projetista da gramática, sem alterar a forma sentencial nessa aplicação. É provado que esse formalismo tem poder de máquina de Turing demonstrando, em forma construtiva, sua equivalência com quatro formalismos gramaticais baseados em gramáticas livres de contexto com mecanismos de controle, que tem esse poder. São desenvolvidos dois analisadorers para linguagens dependentes de contexto a partir de um desses outros quatro formalismos. Um deles, que é baseado em autômatos-pilha, opera em forma ascendente; o outro, baseado em autômatos finitos adaptativos, opera em forma descendente. / This work introduces and describes the formalism of the context-free adaptive grammar with appearance checking. Such gramatical devices have as its kernel a subjacent context-free grammar and, as mechanism of self-modification, one or several adaptive functions which determines the productions able to be applied at each step of a derivation. The appearance checking refers to a special way to apply some productions, choosen by the designer of the grammar, without changing the sentential form in this application. It is proved that this formalism has Turing Machine power, proving, by construction, its equivalence with four grammatical formalisms based on context-free grammars and with control mechanisms, with such power. Two parsers have been developed for context-dependent languages from one of these four formalisms. One of them is based on stack-automata, and operates in a bottom-up fashion.

Este trabalho descreve o formalismo das gramáticas livres de contexto adaptativas com verificação de aparência. Esses dispositivos gramaticais possuem como núcleo uma gramática livre de contexto subjacente e, como mecanismo de auto-modificação, uma ou várias funções adaptativas que determinam quais produções são aplicáveis em cada passo de uma derivação. A verificação de aparência se refere a uma forma especial de aplicar algumas produções, escolhidas pelo projetista da gramática, sem alterar a forma sentencial nessa aplicação. É provado que esse formalismo tem poder de máquina de Turing demonstrando, em forma construtiva, sua equivalência com quatro formalismos gramaticais baseados em gramáticas livres de contexto com mecanismos de controle, que tem esse poder. São desenvolvidos dois analisadorers para linguagens dependentes de contexto a partir de um desses outros quatro formalismos. Um deles, que é baseado em autômatos-pilha, opera em forma ascendente; o outro, baseado em autômatos finitos adaptativos, opera em forma descendente. / This work introduces and describes the formalism of the context-free adaptive grammar with appearance checking. Such gramatical devices have as its kernel a subjacent context-free grammar and, as mechanism of self-modification, one or several adaptive functions which determines the productions able to be applied at each step of a derivation. The appearance checking refers to a special way to apply some productions, choosen by the designer of the grammar, without changing the sentential form in this application. It is proved that this formalism has Turing Machine power, proving, by construction, its equivalence with four grammatical formalisms based on context-free grammars and with control mechanisms, with such power. Two parsers have been developed for context-dependent languages from one of these four formalisms. One of them is based on stack-automata, and operates in a bottom-up fashion.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-28T14:05:33Z No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2017-08-28T15:55:51Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) / Made available in DSpace on 2017-08-28T15:55:51Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 917638 bytes, checksum: 80f3b489c75001e1f83c7c60e54c0c74 (MD5) Previous issue date: 2015-07-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The maingoalofthisdissertationisthepresentationofconcepts,examplesand characterizations{bothclassicalandrecent{concerningtheimportantandin uential theory oftheso-called freedivisors in thestandardhomogeneouscase.Tothisend,we beginwithabasicstudyonderivationsandwefocusonthemoduledubbed tangential idealizer of agivenhomogeneouspolynomial,whichgeometricallycorrespondstothe moduleoflogarithmicvector eldsalongthegivenprojectivehypersurface(thedivisor is saidtobe free if suchmoduleisfreeoverthegradedpolynomialring).Wewillalso discuss, inparticular,resultsaboutfreedivisorsintheprojectiveplane. / O principalobjetivodestadisserta c~ao eaapresenta c~aodeconceitos,exemplose caracteriza c~oes{tantocl assicasquantorecentes{arespeitodaimportanteein uente teoria doschamados divisoreslivres no casohomog^eneopadr~ao.Paraesta nalidade, iniciamos comumestudob asicosobrederiva c~oesefocalizamosnom odulodenomi- nado idealizadortangencial de umdadopolin^omiohomog^eneo,oquegeometricamente correspondeaom odulodoscamposvetoriaislogar tmicosaolongodahipersuperf cie projetivadada(odivisor edito livre quando talm odulo elivresobreoanelgraduado de polin^omios).Tamb emdiscutiremos,emparticular,resultadossobredivisoreslivres no planoprojetivo.