The Lorentz group is a symmetry group on Minkowski space, and as such is central to studying the geometry of this and related spaces. The group therefore shows up also from physical considerations, such as trying to formulate quantum physics in anti-de Sitter space. In this thesis, the Lorentz group in 2+1 dimensions and its representations are investigated, and comparisons are made to the analogous rotation group. Firstly, all unitary irreducible representations are found and classified. Then, those representations are realised as the square-integrable, analytic functions on the unit circle and the unit disk, which turn out to correspond to the projective lightcone and the hyperbolic plane, respectively. Also, a way to realise a particular class of representations on 1+1-dimensional anti-de Sitter space is shown. / Lorentzgruppen är en symmetrigrupp på Minkowski-rum, och är således central för att studera geometrin i detta och relaterade rum. Gruppen dyker också därför upp från fysikaliska frågeställningar, såsom att försöka formulera kvantfysik i anti-de Sitter-rum. Denna uppsats undersöker Lorentzgruppen i 2+1 dimensioner och dess representationer, och jämför med den analoga rotationsgruppen. Först konstrueras och klassificeras alla unitära irreducibla representationer. Sedan realiseras dessa representationer som de analytiska funktioner på enhetscirkeln och enhetsskivan vars belopp i kvadrat är integrerbara. Det visar sig att denna cirkel respektive skiva svarar mot den projektiva ljuskonen respektive det hyperboliska planet. Dessutom visas att en särskild klass av representationer blir relevanta för att formulera kvantfysik i 1+1-dimensionellt anti-de Sitter-rum.

The design, analysis, construction and flight testing of a miniature Vertical Take-Off and Landing (VTOL) tail-sitter Unmanned Aerial Vehicle (UAV) prototype is presented in detail. Classic aircraft design methods were combined with numerical analysis to estimate the aircraft performance and flight characteristics. The numerical analysis employed a propeller blade-element theory coupled with momentum equations to predict the influence of a propeller slipstream on the freestream flow field, then the aircraft was analyzed using 3-D vortex lifting-line theory to model finite wings immersed in the flow field. Four prototypes were designed, built, and tested and the evolution of these prototypes is presented. The final prototype design is discussed in detail. A method for sizing control surfaces for a tail-sitter was defined. The final prototype successfully demonstrated controllability both in horizontal flight and vertical flight. Significant contributions included the development of a control system that was effective in hover as well as descending vertical flight, and the development of a strong but light weight airframe. The aircraft had a payload weight fraction of 14.5% and a maximum dimension of one meter, making it the smallest tail-sitter UAV to carry a useful payload. This project is expected to provide a knowledge base for the future design of small electric VTOL tail-sitter aircraft and to provide an airframe for future use in tail-sitter research.

Nesta tese, investigamos a geometria de curvas no 3-espaço e no 4-espaço de Minkowski usando a teoria de singularidades, mais especificamente, a teoria de contato. Para isto, estudamos as famílias de funções altura e de funções distância ao quadrado sobre as curvas. Os conjuntos discriminantes e conjuntos de bifurcação destas famílias são ferramentas essenciais para o desenvolvimento deste trabalho. Para curvas no 3-espaço de Minkowski, estudamos seus conjuntos focais e conjunto de bifurcação da família de funções distância ao quadrado sobre estas curvas para investigar o que acontece próximo de pontos tipo luz. Estudamos também os conjuntos focais e conjuntos de bifurcação esféricos de curvas nos espaços de Sitter do 3-espaço e do 4-espaço de Minkowski. Definimos imagens normal Darboux pseudo-esféricas de curvas sobre uma superfície tipo tempo no 3-espaço de Minkowski e estudamos as singularidades e propriedades geométricas destas imagens normal Darboux. Além disso, investigamos a relação da imagem normal Darboux de Sitter (hiperbólica) de uma curva tipo espaço em S21 com a superfície tipo luz ao longo desta curva tipo espaço. Definimos as superfícies horoesférica e dual hiperbólica de curvas tipo espaço no espaço de Sitter S31 e estudamos estas superfícies usando técnicas da teoria de singularidades. Damos uma relação entre estas superfícies do ponto de vista de dualidades Legendrianas. Finalmente, consideramos curvas sobre uma hipersuperfície tipo espaço no 4-espaço de Minkowski e definimos a superfície hiperbólica desta curva. Estudamos a geometria local da superfície hiperbólica e da curva hiperbólica, que é definida como sendo o local das singularidades da superfície hiperbólica. / We study in this thesis the geometry of curves in Minkowski 3-space and 4-space using singularity theory, more specifically, the contact theory. For this we study the families of height functions and of the distance square functions on the curves. The discriminant sets and bifurcation sets of these families are essential tools in our work. For curves in Minkowski 3-space, we study their focal sets and the bifurcation set of the family of the distance square functions on these curves in order to investigate what happens near the lightlike points. We also study the spherical focal sets and bifurcation sets of curves in the de Sitter space in Minkowski 3-space and 4-space. We define pseudo-spherical normal Darboux images of curves on a timelike surface in Minkowski 3-space and study the singularities and geometric properties of these normal Darboux images. Furthermore, we investigate the relation of the de Sitter (hyperbolic) normal Darboux image of a spacelike curve in S21 with the lightlike surface along this spacelike curve. We define the horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space S31 and study these surfaces using singularity theory technics. We give a relation between these surfaces from the view point of Legendrian dualities. Finally, we consider curves on a spacelike hypersurface in Minkowski 4-space and define the hyperbolic surface of this curve. We study the local geometry of the hyperbolic surface and of the hyperbolic curve that is defined as being the locus of singularities of the hyperbolic surface.

ABSTRACT CONSERVED CHARGES IN ASYMPTOTICALLY (ANTI)-DE SITTER SPACETIME G&Uuml / LL&Uuml / , iBRAHiM M.S., Department of Physics Supervisor: Assoc. Prof. Dr. Bayram Tekin August 2005, 77 pages. In this master&rsquo / s thesis, the Killing vectors are introduced and the Killing equation is derived. Also, some information is given about the cosmological constant. Then, the Abbott-Deser (AD) energy is reformulated by linearizing the Einstein equation with cosmological constant. From the linearized Einstein equation, Killing charges are derived by using the properties of Killing vectors. Using this formulation, energy is calculated for some specific cases by using the Schwarzschild-de Sitter metric. Last, the Einstein-Gauss-Bonnet model is studied. The equations of motion are calculated by solving the generic action at quadratic order. Following this, all energy calculations are renewed for this model. Some useful relations and calculations are shown in Appendix (A-B) parts. &Ouml / Z ASiMPTOTiK (ANTi)-DE SITTER UZAYZAMANINDA KORUNAN Y&Uuml / KLER G&Uuml / LL&Uuml / , iBRAHiM Y&uuml / ksek Lisans, Fizik B&ouml / l&uuml / m&uuml / Tez Y&ouml / neticisi: Assoc. Prof. Dr. Bayram Tekin Agustos 2005, 77 sayfa. Bu master &ccedil / aliSmasinda, Killing vekt&ouml / rler tanimlandi ve Killing denklemi &ccedil / ikarildi. Ayrica evrenbilimsel sabit, de-Sitter ve Anti-de Sitter uzaylari hakkinda bilgi verildi. Sonra, Abbott-Deser (AD) enerjisi, evrenbilimsel sabitli Einstein denklemi dogrusallaStirilarak yeniden form&uuml / le edildi. DogrusallaStirilmiS Einstein denkleminden, Killing vekt&ouml / rlerin &ouml / zellikleri kullanilarak Killing y&uuml / kleri (Deser-Tekin denklemi) &ccedil / ikarildi. Schwarzschild-de Sitter metrigi kullanilarak &ouml / zel durumlar i&ccedil / in enerji hesaplandi. Son olarak Einstein-Gauss-Bonnet (GB) modeli &ccedil / aliSildi. ikinci dereceden genel eylem &ccedil / &ouml / z&uuml / lerek hareket denklemleri hesaplandi. Bundan sonra, t&uuml / m enerji hesaplamalari bu model i&ccedil / in tekrarlandi. Bazi faydali hesaplamalar ek (A-B) kisimlarinda g&ouml / sterilmiStir.

Submitted by HENDRIK GERARD JOHAN JENNEN null (hjennen@ift.unesp.br) on 2016-02-23T14:54:31Z No. of bitstreams: 1 thesis_oneside.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-02-24T13:39:03Z (GMT) No. of bitstreams: 1 jannen_h_dr_ift.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) / Made available in DSpace on 2016-02-24T13:39:03Z (GMT). No. of bitstreams: 1 jannen_h_dr_ift.pdf: 1083742 bytes, checksum: eeb3f42f2937a777dba99b7615ef69c8 (MD5) Previous issue date: 2016-02-12 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Observações realizadas nas últimas três décadas confirmaram que o universo se encontra em um estado de expansão acelerada. Essa aceleração é atribuída à presença da chamada energia escura, cuja origem permanece desconhecida. A maneira mais simples de se modelar a energia escura consiste em introduzir uma constante cosmológica positiva nas equações de Einstein, cuja solução no vácuo é então dada pelo espaço de de Sitter. Isso, por sua vez, indica que a cinemática subjacente ao espaço-tempo deve ser aproximadamente governada pelo grupo de de Sitter SO(1,4), e não pelo grupo de Poincaré ISO(1,3). Nesta tese, adotamos tal argumento como base para a conjectura de que o grupo que governa a cinemática local é o grupo de de Sitter, com o desvio em relação ao grupo de Poincaré dependendo ponto-a-ponto do valor de um termo cosmológico variável. Com o propósito de desenvolver tal formalismo, estudamos a geometria de Cartan na qual o espaço modelo de Klein é, em cada ponto, um espaço de de Sitter com o conjunto de pseudo-raios definindo uma função não-constante do espaço-tempo. Encontramos que o tensor de torção nessa geometria adquire uma contribuição que não está presente no caso de uma constante cosmológica. Fazendo uso da teoria das realizações não-lineares, estendemos a classe de simetrias do grupo de Lorentz SO(1,3) para o grupo de de Sitter. Em seguida, verificamos que a estrutura da gravitação teleparalela--- uma teoria gravitacional equivalente à relatividade geral--- é uma geometria de Riemann-Cartan não linear. Inspirados nesse resultado, construímos uma generalização da gravitação teleparalela sobre uma geometria de de Sitter--Cartan com um termo cosmológico dado por uma função do espaço-tempo, a qual é consistente com uma cinemática localmente governada pelo grupo de de Sitter. A função cosmológica possui sua própria dinâmica e emerge naturalmente acoplada não-minimalmente ao campo gravitacional, analogamente ao que ocorre nos modelos telaparalelos de energia escura ou em teorias de gravitação escalares-tensoriais. Característica peculiar do modelo aqui desenvolvido, a função cosmológica fornece uma contribuição para o desvio geodésico de partículas adjacentes em queda livre. Embora tendo sua própria dinâmica, a energia escura manifesta-se como um efeito da cinemática local do espaço-tempo. / Observations during the last three decades have confirmed thatthe universe momentarily expands at an accelerated rate, which is assumed to be driven by dark energy whose origin remains unknown. The minimal manner of modelling dark energy is to include a positive cosmological constant in Einstein's equations, whose solution in vacuum is de Sitter space. This indicates that the large-scale kinematics of spacetime is approximated by the de Sitter group SO(1,4) rather than the Poincaré group ISO(1,3). In this thesis we take this consideration to heart and conjecture that the group governing the local kinematics of physics is the de Sitter group, so that the amount to which it is a deformation of the Poincaré group depends pointwise on the value of a nonconstant cosmological function. With the objective of constructing such a framework we study the Cartan geometry in which the model Klein space is at each point a de Sitter space for which the combined set of pseudoradii forms a nonconstant function on spacetime. We find that the torsion receives a contribution that is not present for a cosmological constant. Invoking the theory of nonlinear realizations we extend the class of symmetries from the Lorentz group SO(1,3) to the enclosing de Sitter group. Subsequently, we find that the geometric structure of teleparallel gravity--- a description for the gravitational interaction physically equivalent to general relativity--- is a nonlinear Riemann--Cartan geometry.This finally inspires us to build on top of a de Sitter--Cartan geometry with a cosmological function a generalization of teleparallel gravity that is consistent with a kinematics locally regulated by the de Sitter group. The cosmological function is given its own dynamics and naturally emerges nonminimally coupled to the gravitational field in a manner akin to teleparallel dark energy models or scalar-tensor theories in general relativity. New in the theory here presented, the cosmological function gives rise to a kinematic contribution in the deviation equation for the world lines of adjacent free-falling particles. While having its own dynamics, dark energy manifests itself in the local kinematics of spacetime.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The electromagnetic and gravitational quasinormal mode spectra of plane-symmetric anti-de Sitter black holes are investigated in the present work. According to the AdS/CFT correspondence, the black hole quasinormal frequencies correspond to the poles of R-current and stress-energy tensor correlation functions in the holographically dual field theory: the N = 8 super-Yang-Mills in 2 + 1 dimensions. In the present study, this duality is explored in order to choose the quasinormal mode boundary conditions and the gauge invariant variables governing the black hole perturbations. Among the main results, it is shown that zero wavenumber axial perturbations yield only small rotations on the system, while the polar perturbations lead to a change in the mass of the black hole and may also produce cylindrical gravitational waves. In relation to the quasinormal spectra, the long-distance and low-frequency limit of the dispersion relations present the hydrodynamical behavior that is characteristic of a conformally invariant theory, with diffusion, shear and sound-wave modes. In the electromagnetic perturbation sector, it also appears purely damped modes that tend to the bosonic Matsubara frequencies in the long-wavelength regime. / No presente trabalho, investigou-se o espectro de vibrações eletromagnéticas e gravitacionais de buracos negros anti-de Sitter com horizontes plano-simétricos. Segundo a correspond ência AdS/CFT, os modos quase-normais desses buracos negros estão associados aos pólos de funções de correlação de corrente-R e do tensor energia-momento na teoria de campos holograficamente dual: a super-Yang-Mills N = 8 em 2 + 1 dimensões. Neste estudo, a relação com a teoria de campos foi explorada ao se fixar as condições de contorno que definem os modos quase-normais, bem como na escolha das quantidades invariantes de calibre que governam as perturbações dos buracos negros anti-de Sitter. Para obter as equações de onda gravitacionais, foram utilizados formalismos baseados em variações da métrica e dos escalares de curvatura de Weyl. Entre outros resultados, mostrou-se que as perturbações axiais com número de onda nulo produzem somente pequenas rotações sobre o sistema, enquanto que as perturbações polares conduzem a mudanças na massa e também podem se propagar na forma de ondas gravitacionais cilíndricas. Em relação ao espectro quase-normal, no limite de baixas freqüências e grandes comprimentos de onda, algumas relações de dispersão apresentam o comportamento hidrodinâmico característico de uma teoria conformemente invariante, com o aparecimento de modos de difusão, cisalhamento e de onda sonora. Por fim, no setor eletromagnético das perturbações, surgem também modos puramente amortecidos que tendem às freqüências de Matsubara de um sistema bosônico no regime de grandes comprimentos de onda.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-19T13:02:13Z No. of bitstreams: 1 arquivototal.pdf: 1631315 bytes, checksum: 9ca68dbec4c6690247c6d2507322f997 (MD5) / Made available in DSpace on 2017-09-19T13:02:13Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1631315 bytes, checksum: 9ca68dbec4c6690247c6d2507322f997 (MD5) Previous issue date: 2014-07-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we investigate quantum properties associated with massive fields near a cosmic string within two distinct scenarios: a Minkowski flat background, and a maximally symmetric anti-de Sitter spacetime (AdS). Initially, we present a brief review of the formalism needed in our contribution. In Minkowski spacetime, we analyze the relativistic quantum dynamics of bosonic and fermionic charged particles in the presence of a magnetic field also considering the presence of scalar potentials. In order to develop this analysis, we assume that the magnetic field is parallel to the string axis, and the scalar potentials present a cylindrical symmetry with their center on the string. Two distinct configurations for the scalar potential are considered: (i) the potential proportional to the inverse of the polar distance, and (ii) the potential linearly proportional to this distance. The energy spectra are explicitly computed for different physical situations, and their dependence on the magnetic field strength, and scalar coupling constants are presented. In the context of a string in AdS, we investigate the quantum properties of a massive fermionic field. More specifically, we calculate the fermionic condensate (FC), (4RP), and the vacuum expectation value (VEV) of the energy-momentum tensor, (T"), induced by the presence of a cosmic string. In order to develop this analysis, we consider the MIT bag boundary condition on the boundary of AdS. The FC and the VEV of the energy-momentum tensor are decomposed into pure AdS and cosmic string induced parts. The string-induced part in the VEV of the energy-momentum tensor is diagonal, and the axial, and radial stresses are equal to the energy density. For points near the cosmic string, the effects of the curvature are sub-dominant and to leading order, the VEVs coincide with the corresponding VEVs for the cosmic string in the Minkowski bulk. At large proper distances from the string, the decay of the VEVs show a power-law dependence on the distance for both massless and massive fields, in contrast to the case of a Minkowski bulk where, for a massive field, the string-induced parts decay exponentially. / Nesta tese, investigamos aspectos quanticos associados a campos de materia na vizinhanca de uma corda cosmica em dois contextos distintos: a corda cosmica no espayo-tempo piano de Minkowski e a corda cosmica no espayo-tempo maximamente simetrico de anti-de Sitter (AdS). Inicialmente, fazemos uma breve revisao do formalismo utilizado nas nossas contribuicOes. No contexto da corda em Minkowski, analisamos a dinamica quantica relativistica de particulas carregadas, bosonicas e fermionicas, na pre­senca de um campo magnetico, considerando tambem a presenca de potenciais escalares. Para desenvol­vermos esta andlise, admitimos que o campo magnetico é uniforme e paralelo a corda e que os potenciais escalares apresentam simetria cilIndrica, cujos centros estao sobre a corda cosmica. Duas configuracOes distintas para o potencial escalar sao consideradas: (i) o potencial proporcional ao inverso da distancia polar e (ii) o potencial linearmente proporcional a esta distancia. Nesse sentido os espectros de ener­gia sao obtidos exatamentes em diferentes cendrios ffsicos e sua relayao com a intensidade do campo magnetico e com as constantes de acoplamento escalares é apresentada. No contexto da corda em AdS, analisamos o comportamento quantico de um campo fermi8nico massivo. Especificamente, investiga­mos o condensado fermionico, (4RP), e valor esperado no vacuo (VEV) do tensor energia-momento, (T"), induzidos pela presenca de uma corda cosmica. Para esta andlise, impusemos a condicao de contorno da sacola do MIT, na fronteira de AdS. 0 condensado fermionico e o VEV do tensor energia­momento sao decompostos, respectivamente, em duas contribuicOes: uma induzida pelo espayo de AdS e outra induzida pela presenca da corda. No que se refere ao VEV do tensor energia-momento, a parte induzida pela corda é diagonal e as press6es radial e axial sao iguais a densidade de energia. Para pontos proximos a corda, os efeitos da curvatura sao subdominantes e o VEV do tensor energia-momento coin­cide com o correspondente na geometria de Minkowski. ia para grandes distancias proprias, os VEVs decaem de acordo com uma lei de potencia, em contraste com o caso no espayo-tempo de Minkowski onde, para um campo massivo, o termo induzido pela corda decai exponencialmente.

Orientador: Prof. Dr. Vilson Tonin Zanchin / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017. / Nesta tese exploramos alguns aspectos de teorias de campo fortemente acoplados usando o respectivo dual gravitacional. Comecamos estudando perturbacoes gravitacionais de cordas negras em rotacao num espaco-tempo assintoticamente Anti-de Sitter (AdS) em quatro dimensoes. E apresentado uma analise completa do espectro dos modos quasinormais (QNMs), interpretacoes na teoria de campos dual, quando possivel, sao exploradas. No setor escalar das perturbacoes obtemos as relacoes de dispersao na aproximacao hidrodinamica seguindo tres abordagens: perturbacoes gravitacionais, flutuacoes hidrodinamicas de um fluido relativista em (2+1) dimensoes e uma analise de vetores de onda relativistas. Os resultados obtidos no limite hodrodinamico mostram efeitos relativistas como a contracao de Lorentz, dilatacao do tempo de amortecimento e o efeito Doppler da frequencia. Os resultados numericos tambem mostram um crossover do regime hidrodinamico para o relativista e do regime nao hidrodinamico para o relativista, o ultimo para o primeiro modo nao hidrodinamico. Adicionalmente reportamos o surgimento de uma nova classe de modos quasinormais. Na segunda parte exploramos modelos fenomenologicos para a cromodinamica quantica (QCD). Primeiramente, exploramos um modelo holografico para mesons vetoriais no contexto do modelo soft-wall, exploramos o derretimento desses estados de quasiparticula quando a temperatura e adicionado atraves de um buraco negro no fundo gravitational. Tambem calculamos os modos quasinormais neste modelo e as relacoes de dispersao neste modelo. Seguimos um procedimento semelhante para explorar o derretimendo dos mesons escalares. A diferencia em relacao aos mesons vetoriais e que pode ser introduzido um dilaton quartico no regime do ultravioleta (UV) para descrever o condensado de gluons, enquanto que o comportamento linear do espectro de massas e confinamento e garantido por um dilaton quadratico no regime do infravermelho (IR). Exploramos os efeitos da escala de energia associado com o condensado de gluons no espectro e derretimento dos mesons escalares. na ultima parte apresentando um modelo holografico para descrever o setor dos gluons do Lagrangiano de Yang-Mills implementado atraves do acoplamento do campo dilaton com a metrica. Neste contexto calculamos o espectro de massas do setor escalar e tensorial do campo dos gluons. Os resultados obtidos sao consistentes com os resultados da QCD na rede. Tambem calculamos o valor esperado do vacuo (VEV) e a densidade de energia do vacuo da QCD, ambos os resultados sao resoaveis comparados com os resultados disponiveis na literatura. No final, e proposto um novo dicionario relacionando a anomalia do traco de teorias 10 de campo conforme deformadas com a anomalia do traco da QCD. Como uma consequencia nao trivial deste dicionario observamos o surgimento de uma funcao-¿À semelhante ao resultado em QCD perturbativa a dois loops. / In this thesis we explore some aspects of strongly coupled field theories using the respective gravitational duals. We start studying gravitational perturbations of rotating black strings in asymptotically Anti-de Sitter (AdS) four-dimensional spacetime. A complete analysis of the spectrum of the quasinormal modes (QNMs) is presented, interpretations in the dual field theory, when possible, are explored. In the scalar sector of the perturbations we obtain the dispersion relations in the hydrodynamic approximation following three approaches: gravitational perturbations, hydrodynamic fluctuations of a relativistic fluid in (2+1) dimensions and an analysis of relativistic wave vectors. The results obtained in the hydrodynamic limit show relativistic effects like Lorentz contraction, damping time dilation and Doppler shift of the frequency. The numerical results also show a crossover from the hydrodynamicto- relativistic regimes and from the non-hydrodynamic-to-relativistic regimes, the last for the first non-hydrodynamic quasinormal mode (QNM). Additionally we report the emergence of a new class of QNMs. In the second part we explore phenomenological models for quantum chromodynamics (QCD). Firstly we explore a holographic model for vector mesons in the context of the softwall model. We explore the melting of these quasiparticle states when the temperature is added via a black hole on the gravitational background. We also calculate the quasinormal modes and the dispersion relations in this model. We follow a similar procedure to explore the melting of scalar mesons. The difference in relation to the vector mesons case is that a quartic dilaton in the ultraviolet (UV) regime can be introduced to describe the gluon condensate, while the linear behaviour of the mass spectrum and confinement is guaranteed by a quadratic dilaton field infrared (IR) regime. We explore the effects of the energy scale associated with the gluon condensate in the mass spectrum and melting of the scalar mesons. In the last part we present a holographic model to describe the gluon sector of the Yang- Mills Lagrangian implemented through the coupling of the dilaton field to the metric. In this context we calculate the mass spectrum of the scalar and tensor sectors of the gluons field. The results obtained are consistent with lattice QCD. We also calculate the vacuum expectation value (VEV) and the energy density of the QCD vacuum, both results are reasonable compared with results currently available in the literature. At the end a new dictionary is proposed, which relates the trace anomaly of deformed conformal field theories (CFTs) to the trace anomaly of QCD. As a non trivial consequence of this dictionary we observe the emergence of a â function similar to the two loops result in perturbative QCD.

Nesta tese, investigamos a geometria de curvas no 3-espaço e no 4-espaço de Minkowski usando a teoria de singularidades, mais especificamente, a teoria de contato. Para isto, estudamos as famílias de funções altura e de funções distância ao quadrado sobre as curvas. Os conjuntos discriminantes e conjuntos de bifurcação destas famílias são ferramentas essenciais para o desenvolvimento deste trabalho. Para curvas no 3-espaço de Minkowski, estudamos seus conjuntos focais e conjunto de bifurcação da família de funções distância ao quadrado sobre estas curvas para investigar o que acontece próximo de pontos tipo luz. Estudamos também os conjuntos focais e conjuntos de bifurcação esféricos de curvas nos espaços de Sitter do 3-espaço e do 4-espaço de Minkowski. Definimos imagens normal Darboux pseudo-esféricas de curvas sobre uma superfície tipo tempo no 3-espaço de Minkowski e estudamos as singularidades e propriedades geométricas destas imagens normal Darboux. Além disso, investigamos a relação da imagem normal Darboux de Sitter (hiperbólica) de uma curva tipo espaço em S21 com a superfície tipo luz ao longo desta curva tipo espaço. Definimos as superfícies horoesférica e dual hiperbólica de curvas tipo espaço no espaço de Sitter S31 e estudamos estas superfícies usando técnicas da teoria de singularidades. Damos uma relação entre estas superfícies do ponto de vista de dualidades Legendrianas. Finalmente, consideramos curvas sobre uma hipersuperfície tipo espaço no 4-espaço de Minkowski e definimos a superfície hiperbólica desta curva. Estudamos a geometria local da superfície hiperbólica e da curva hiperbólica, que é definida como sendo o local das singularidades da superfície hiperbólica. / We study in this thesis the geometry of curves in Minkowski 3-space and 4-space using singularity theory, more specifically, the contact theory. For this we study the families of height functions and of the distance square functions on the curves. The discriminant sets and bifurcation sets of these families are essential tools in our work. For curves in Minkowski 3-space, we study their focal sets and the bifurcation set of the family of the distance square functions on these curves in order to investigate what happens near the lightlike points. We also study the spherical focal sets and bifurcation sets of curves in the de Sitter space in Minkowski 3-space and 4-space. We define pseudo-spherical normal Darboux images of curves on a timelike surface in Minkowski 3-space and study the singularities and geometric properties of these normal Darboux images. Furthermore, we investigate the relation of the de Sitter (hyperbolic) normal Darboux image of a spacelike curve in S21 with the lightlike surface along this spacelike curve. We define the horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space S31 and study these surfaces using singularity theory technics. We give a relation between these surfaces from the view point of Legendrian dualities. Finally, we consider curves on a spacelike hypersurface in Minkowski 4-space and define the hyperbolic surface of this curve. We study the local geometry of the hyperbolic surface and of the hyperbolic curve that is defined as being the locus of singularities of the hyperbolic surface.

[pt] Assim como qualquer item, uma construção possui uma vida útil que considera o seu nascimento como momento que ela é concebida em projeto, e sua morte como sendo sua demolição final. Durante sua vida uma construção deve passar por manutenções (preventivas, adaptativas e corretivas) que permitem o prolongamento do seu uso, mantendo o seu nível de desempenho dentro do aceitável. Muitas vezes as ações corretivas se dão em períodos curtos de tempo, intervalos abaixo do esperado por seus usuários ou administradores. Obviamente toda intervenção trás consigo um custo financeiro, que cresce dependendo do momento em que ela ocorra dentro da vida da construção. Muito além do custo financeiro, toda atividade causa impacto no meio ambiente, gerando assim um custo ambiental. Determinar o preço ambiental do refazimento de uma obra, em um curto espaço de tempo (menor que o esperado) ilustra o peso deste custo, muitas vezes invisível ou negligenciado, é necessário. A Análise do refazimento de uma obra devido a falhas construtivas que trouxeram uma drástica queda no desempenho no uso do empreendimento demonstrou que o custo ambiental é proporcionalmente muito maior que o custo financeiro esperado. Sabendo que o custo financeiro de correções construtivas ao longo da vida do imóvel cresce em uma progressão geométrica de base 5, permite comparar o quanto o custo ambiental pode desequilibrar a sustentabilidade. Analisando que a intervenção de uma área de 4.200 m quadrados utilizou uma área ambiental de 1.360.000 m quadrados, faz com que os sinais de alerta se acendam demonstrando que a correção de um erro construtivo é muito maior para o meio ambiente que o custo financeiro envolvido. / [en] The world population has been growing at a dizzying rate in recent centuries. And this accelerated population gain brings with it numerous consequences, among them, the need to produce more food, housing and infrastructure. This all leads us to consume more and more natural resources and also increases the generation of waste and waste. The so-called carrying capacity of the planet (condition of sustaining a population), has not evolved in the last centuries in the same index of population growth, that is, humanity is consuming natural resources and generating waste at a speed higher than that which the planet is capable of. produce and absorb. To continue supporting the growing population of the planet, it is necessary to experiment with new technologies, methodologies and processes so that this growth is supported by the tripod of sustainability. The term sustainable development has the most common, and accepted, meaning that points to a tripod of economic growth, environmental preservation and social development. Civil construction is an essential economic sector in the development of any country and society, being responsible for a large fraction of the quality of life of human beings, since they alter the natural environment for better use of space. Understanding the environmental cost of correcting a construction failure is the objective of this work. There are countless studies that point to the financial cost of the so-called rework, but few look at this phenomenon under the environmental lens. The entire life cycle of an enterprise, from its design to its ruin, through its construction and use, causes environmental marks. To correct flaws in works already completed, or in use, there is a need to consume new materials, involving an entire production chain and generating new waste. To produce a certain input that will be used in the correction of a pathology, the following are required: consumption of raw materials to conceive it, energy consumption to manufacture it, waste to produce it, expenses with transportation to take it from the factory to the point of use. All of these steps in the process consume environmental resources. At the other end of the error correction, for the pathology to be eliminated, it must be removed from the site (demolition of a crooked wall, for example) using energy and producing residues from this removal. This waste will be transported to a suitable disposal site, that is, using more energy in this process. In addition, it is still necessary to transform a harmful waste into something less aggressive to the environment. Given the above, the purpose of this research is to understand the size of the impact that a constructive failure can cause to the environment depending on its severity and the moment it is detected. In order to carry out this work, a project was followed up with a short time of use, but which needed major interventions due to the flaws found. With the analysis of the presented pathologies it was possible to measure how much they weighed, and will weigh, to the environment. In addition, analyzing the origin and the correction method implemented will allow to index each of the flaws found environmentally, measuring how much the planet s carrying capacity could have been preserved had these defects not occurred. The useful life of a building can be understood as the time interval from its birth, marked by its design concept, until its death with its demolition and / or disuse. Project useful life (VUP) must be defined by the developer and the project designer. VUP, despite being a temporal measure, has an economic character, being defined as the best relation between global cost versus time to enjoy the good. Preventive maintenance takes place constantly and aims to increase the life of the project, whereas corrective maintenance must occur in a timely manner and correcting failures in points that are already performing below the desired level. Adaptive maintenance has the objective of adjusting the enterprise to receive new technologies, new equipment and to comply with the new legislation The economic character of the useful life of a good is characterized by its global cost, which must be defined as the sum of the cost of acquisition, or construction, of the good and the cost of maintenance throughout its life. The total cost of a construction during its life includes the costs of planning, design, construction, operation, maintenance and demolition. These construction costs represent between 15 percent and 20 percent of the total cost; 80 percent of the amount is spent on operation and maintenance and only 2 percent to 5 percent of the amount is spent on planning and design (conceptual and detailed). The total cost of a construction during its life includes the costs of planning, design, construction, operation, maintenance and demolition. These construction costs represent between 15 percent and 20 percent of the total cost; 80 percent of the amount is spent on operation and maintenance and only 2 percent to 5 percent of the amount is spent on planning and design (conceptual and detailed). The useful life of a building, for example, goes through the useful life of its components such as its foundations, superstructures, hydro-sanitary installations, electrical installations, facades, internal cladding, paintings and waterproofing. Studies show that corrective maintenance costs up to five times more than preventive maintenance. Corrective maintenance is often required in shorter time cycles than initially imagined (and desired) by those responsible for the enterprise. Currently, numerous failures in new construction (or with little use) are verified, such as buildings, bridges, roads, streets and public supply networks, which range from faults of all kinds, from simple to catastrophic. The service life can be extended with preventive, corrective and adaptive maintenance interventions. The extension of useful life is directly impacted on the overall cost of construction. The lowest global cost system is usually not the lowest initial cost nor the longest lasting. Seeking to optimize the cost-benefit ratio is the best option for society. The useful life of a building must be supported by the tripod of socio-environmental importance, cost of implementation and cost of maintenance over the years. When investors seek to save money by building buildings with low quality standards, and with low maintenance ease, they increase the cost of future maintenance. At the other end of the real estate market, users do not carry out preventive maintenance because they consider its cost to be high, often allowing certain components of the project to come close to the level of unacceptable performance and only then carry out the maintenance that has now become corrective, costing financially more than the preventive maintenance previously denied. The Sitter rule, or Law of 5, determines that the relative cost of an intervention grows in a geometric progression of ratio 5 over time in the project and its maintenance. The sooner a problem is perceived, the lower its cost. Sustainability, despite not having a unanimous definition, is a concept that must integrate aspects of social-ecological dimensions, economic factors, and the short, medium and long term advantages. Putting together all the concepts expressed by several authors, sustainability can be defined as the attempt to achieve economic and social growth while preserving the finite resources of the environment. For more than 40 years, humanity s demand for nature has exceeded the planet s replacement capacity. Currently 1.5 Earth planets would be needed to provide the ecological services that were used in the 1980s. Trees are cut faster than they can ripen, more fish are caught than the oceans can replenish and more carbon is emitted than forests and oceans can absorb. The carrying capacity of the planet has been compromised in a way never before experienced by humanity, to meet the current lifestyle of the population. Consumerism is seen as a behavior that leads to an increase in production and, consequently, to economic progress, but this equation is limited by resources that cannot sustain unlimited growth. Finite spaces cannot absorb waste that grows indefinitely. The carrying capacity of a system is obviously influenced by factors such as average income, material expectations and level of technology, that is, energy and material efficiency. There are few systems of indicators that analyze sustainable development in a generic way. The most commonly used indicators globally are as follows: (1) Sustainability Panel, (2) Sustainability Barometer and (3) Ecological Footprint. The indicator called Ecological Footprint has the advantage of being easily visualized, since the Ecological Footprint represents the ecological space necessary to sustain a given system, or community. It is a simple tool that counts the flows of matter and energy that enter and leave an economic system, converting them into areas of land, or water, necessary to sustain such a system. The Ecological Footprint is a method that transforms the consumption of raw materials and the assimilation of waste from an economic system, or from a human population, into an area corresponding to productive land or water. Using this method, it is possible to calculate the area of the ecosystem needed to ensure the eternal survival of a given population or system. Once this equivalent area of the ecosystem has been determined, it is possible to visualize how much it appropriates the carrying capacity of the planet as a whole. In fact, the size of the Footprint can change depending on the new technologies developed, which can be more or less resource-consuming and wastegenerating. The calculation method for measuring the Ecological Footprint, although easily intuitive, is difficult to carry out with regard to data collection.