Nessa dissertação exploramos os efeitos geométricos das Transformações de Möbius em C utilizando projeções na Esfera de Riemann. Como aplicação, apresentamos a ação de algumas transformações aplicadas em cônicas no plano. Uma atividade didática voltada aos alunos do Ensino Médio sobre Transformações de Möbius utilizando o Geogebra é apresentada. / In the course of this dissertation we explore the geometric effects of the Möbius Transforms in C using projections in the Riemann sphere. As an application, we present the action of some transformations applied on conics in the plane. A didactic activity aimed at high school students about Möbius Transformations using Geogebra is presented.

Nessa dissertação exploramos os efeitos geométricos das Transformações de Möbius em C utilizando projeções na Esfera de Riemann. Como aplicação, apresentamos a ação de algumas transformações aplicadas em cônicas no plano. Uma atividade didática voltada aos alunos do Ensino Médio sobre Transformações de Möbius utilizando o Geogebra é apresentada. / In the course of this dissertation we explore the geometric effects of the Möbius Transforms in C using projections in the Riemann sphere. As an application, we present the action of some transformations applied on conics in the plane. A didactic activity aimed at high school students about Möbius Transformations using Geogebra is presented.

Orientadores: Reginaldo Palazzo Jr., Mercio Botelho Faria / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-08T06:25:10Z (GMT). No. of bitstreams: 1 Vieira_VandenbergLopes_D.pdf: 990187 bytes, checksum: 2212b8074f5503f78aa813ce4422cc4b (MD5) Previous issue date: 2007 / Resumo: Dentro do contexto de projetar sistema de comunicação digital em espaços homogêneos, em particular, em espaços hiperbólicos, é necessário estabelecer um procedimento sistemático para construção de reticulados O, como elemento base para construção de constelações de sinais geometricamente uniformes. E através desse procedimento que identificamos as estruturas algébrica e geométrica além de construir códigos geometricamente uniformes em espaços homogêneos. Propomos, a partir desses reticulados, a construção de grupos fuchsianos aritméticos Tp obtidos de tesselações hiperbólicas {p; q}, derivados de álgebras de divisão dos quaternios A sobre corpos de números K. Generalizamos o processo de identificação desses grupos em ordens dos quatérnios (reticulados hiperbólicos), associadas às constelações de sinais geometricamente uniformes, provenientes de grupos discretos. Esse procedimento permite rotular os sinais das constelações construídas por elementos de uma estrutura algébrica / Abstract: Within the context of digital communications system in homogeneous space in particular, in hyperbolic spaces, it is necessary to establish systematic procedure for the construction of lattices O ; as the basic entity for construction of eometrically uniforms signal constellations. By this procedure we identify the algebraic and geometric structures to construct geometrically uniforms codes in homogeneous spaces. We propose, from lattices, the construction of arithmetic fuchsian groups ¡p obtained by hyperbolic tessellations {p; q}, derived from division quaternion algebras A over numbers fields K. We generalize the process of identification of these groups in quaternion orders (hyperbolic lattices), which are associated with geometrically uniforms signal constellations, proceeding from discrete groups. This procedure allows us to realize the labelling of the signals belonging to such constellations by elements of an algebraic structure / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

En 1949, C. Loewner a demontré dans un travail non publié l'inégalité systolique optimale du tore T reliant l'aire au carré de la systole. Par la systole on désigne la longueur du plus court lacet non contractile de T. De plus, l' égalité est atteinte si et seulement si le tore est plat hexagonal. Ce résultat a donné naissance à la géométrie systolique. Dans cette thèse, nous étudions des inégalités de type systolique portant sur les longueurs minimales de différentes courbes et pas seulement la systole.Dans un premier temps, nous démontrons trois inégalités géométriques optimales conformes sur la bouteille de Klein reliant l'aire au produit des longueurs des plus courts lacets noncontractiles dans des classes d'homotopie libres différentes. Pour chaque classe conforme, nous décrivons la métrique extrémale réalisant le cas d'égalité.Nous établissons ensuite des inégalités géométriques optimales sur le ruban deMobius muni d'une métrique de Finsler. Ces inégalités géométriques relient la systole et la hauteur du ruban de Mobius à son volume de Holmes-Thompson. Nous en déduisons une inégalité systolique optimale sur la bouteille de Klein munie d'une métrique de Finsler avec des symétries. Nous décrivons également une famille de métriques extrémales dans les deux cas.Dans le troisième travail, nous démontrons une inégalité systolique critique sur la surface de genre deux. Plus précisément, il est connu que la surface de genre deux admet une métrique Riemannienne plate à singularités coniques qui est extrémale parmi les métriques à courbure nonpositive pour l' inégalité systolique. Nous montrons que cette métrique est en fait critique pour des variations lentes de métriques, cette fois-ci sans hypothèse de courbure, pour un autre problème systolique portant sur les longueurs des plus courts lacets non contractiles dans certaines classes d'homotopie libres données. Ces classes d'homotopie correspondent aux lacets systoliques et deux-systoliques de la surface extrémale / In 1949, C. Loewner proved in an unpublished work that the two-torus T satisfies an optimal systolic inequality relating the area of the torus to the square of its systole. By a systole here we mean the smallest length of a noncontractible loop in T. Furthermore, the equality is attained if and only if the torus is flat hexagonal. This result led to whatwas called later systolic geometry. In this thesis, we study several systolic-like inequalities. These inequalities involve the minimal length of various curves and not merely the systole.First we obtain three optimal conformal geometric inequalities on Riemannian Klein bottles relating the area to the product of the lengths of the shortest noncontractible loops in different free homotopy classes. We describe the extremal metrics in each conformal class.Then we prove optimal systolic inequalities on Finsler Mobius bands relating the systoleand the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimalsystolic inequality for Finsler Klein bottles with symmetries. We describe extremal metric families in both cases.Finally, we prove a critical systolic inequality on genus two surface. More precisely, it is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved Riemannian metrics. We show that this piecewise flat metric is also critical for slow metric variations, this time without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes. The free homotopy classes considered correspond to those of the systolic loops and the second-systolic loops of the extremal surface

Orientador: Claudio Aguinaldo Buzzi / Banca: Juliano Gonçalves Oler / Banca: Jéfferson Luiz Rocha Bastos / Resumo: O objetivo deste trabalho é ampliar os conhecimentos sobre números complexos já adquiridos no ensino médio. Diversas formas de representação e propriedades operatórias são abordadas. Para este fim, primeiramente, os números complexos são definidos a partir do conceito de matrizes quadradas de ordem 2, e portanto, serão definidos como pares ordenados de números reais. Na sequência, a partir da apresentação geométrica dos conceitos e operações, é estudado o plano complexo estendido, as Transformações de Möbius e a Projeção Estereográfica / Abstract: The objective of this paper is to extend the concepts of complex numbers already acquired in high school. Many forms of representation and operative properties are used. For that, first, the complex numbers are defined from the concept of square matrices of order 2, and will therefore be defined as ordered pairs of real numbers. Following, from the geometric presentation of concepts and operations, it is studied the extended complex plane, the M¨obius Transformations and the Stereographic Projection / Mestre

Many modern materials and material systems are layered. The properties related to layers are connected to interactions between atomic layers. In the present thesis, we introduce the interlayer potential (ILP), a novel model potential which fully describes the interaction between layers. The ILPs are different from the usual interatomic potentials which present interaction between atoms. We use the Chen-Möbius inversion method to extract the ILPs from ab initio total energy calculations. The so obtained ILPs can be employed to investigate several physical parameters connected with the particular set of atomic layers, e.g. surface energy, stacking fault energy, elastic parameters, etc. The interactions between the face centered cubic (fcc) (111) planes are described by two different ILPs. Using two close-packed model structures, namely the ABC stacking along the fcc ⟨111⟩ direction and AB stacking along the hcp ⟨0001⟩ direction, we demonstrate how these two ILPs are obtained via the Chen-Möbius method. Density function theory (DFT) is employed to generate the ILPs and also to compute the equilibrium structural properties of elemental metals Al, Ni, Cu, Ag, Au and Pd as well as of Pd-Ag random solid solutions. With the so established ILPs, we adopt the supercell method and the axial interaction model to calculate the stacking fault energy along the fcc ⟨111⟩ direction, including the intrinsic stacking fault energy, extrinsic stacking fault energy and twin stacking fault energy as well as the interactions between the intrinsic stacking faults. We find that the data derived from ILPs are consistent with those obtained in direct ab initio calculations. Along the fcc ⟨111⟩ direction, we study the surface energy and surface relaxation using the ILPs. The phonon dispersions are also described. We conclude that the interlayer potentials based on the Chen-M¨obius inversion technique may provide a new way to investigate the properties related to layers in layered materials. / <p>QC 20121101</p>

A síndrome de Moebius (SM) é uma diplegia congênita rara caracterizada por paralisia total ou parcial do VI e VII nervos cranianos, que leva à ausência ou deficiência dos movimentos dos músculos envolvidos na mímica facial e ao estrabismo convergente. As características bucais descritas nesses indivíduos incluem o palato ogival, micrognatia, malformação de língua, filtro curto, falta de coaptação de lábios, e maior incidência de lesões de cárie. O objetivo deste estudo foi avaliar as características salivares quantitativas e qualitativas, incluindo o proteoma salivar, de indivíduos com SM, associá-las com a saúde bucal, e compará-las com as características salivares de um grupo controle, não afetado pela SM. Foram incluídos 15 indivíduos com SM e 15 controles. O comprometimento facial do individuos com a SM foi avaliada e graduada em scores 0,1 ou 2, uni ou bilateral, para os nervos II, III, IV, V, VI, VII e XI. Os pesquisadores determinaram o índice de cárie (ICDAS), de doença periodontal (PSR) e de placa (Silness Löe) nos dois grupos de estudo. Também realizaram coletas de saliva total não estimulada, estimulada e parotídea bilateral, sendo o fluxo salivar estabelecido em ml/min. A capacidade tampão foi avaliada na saliva total estimulada através da titulação de HCl 0,01N. A atividade de ?-amilase nas amostrasmfoi medida através da produção de maltose. Para a análise proteômica optou-se pela divisão das amostras de saliva de acordo com o fluxo em ml/min. Desta forma, para cada grupo, estudo e controle, os 4 tipos de saliva (estimulada, não estimulada, parotídea esquerda, parotídea direita) foram subdivididos de acordo com baixo fluxo (abaixo da média do grupo) ou alto fluxo (acima da média do grupo), resultando em 16 subgrupos. O proteoma foi obtido por duas metodologias distintas, a primeira a partir da cromatografia líquida espectrometria de massas e a segunda que utilizou a técnica de eletroforese em gel de poliacrilamida contendo dodecil sulfato de sódio (SDS-PAGE) associada à eletroforese em gel de poliacrilamida (native cationic). A ocorrência das lesões de cárie foi significativamente mais alta entre os participantes com SM (p>0,05) no corte 2, bem como a ocorrência de doença periodontal (p>0,05), quando comparado ao grupo controle. Não houve diferença no índice de placa entre os grupos. A análise proteômica mostrou diminuição de cistatinas B, S e SN nos indivíduos com SM. Não houve diferença no perfil proteico entre os grupos de baixo e de alto fluxo salivar, para indivíduos com SM e controle. Houve aumento na quantidade de amilase salivar em e de histatina 1,3 e 5 em indivíduos com SM. Concluímos que indivíduos com SM apresentam diminuição de fluxo salivar, de capacidade tampão e alterações proteicas que colocam esses indivíduos em situação de maior risco para cárie e para doença periodontal. / The Moebius syndrome (MS) is a rare congenital diplegia characterized by total or partial palsy of the VI and VII cranial nerves, leading to the absence or disability of the movements of facial expression muscles and to convergent strabismus. The oral features described in these individuals include high-arched palate, micrognathia, tongue malformation, short filter, lack of lips coaptation, and higher incidence of caries lesions. The aim of this study was to evaluate the quantitative and qualitative salivary characteristics, including the salivary proteome of individuals with MS, associate them with the oral health, and compare them to the salivary characteristics of a control group, unaffected by SM. We included 15 subjects with MS and 15 controls. The facial involvement of individuals with MS was evaluated and graded on scores 0, 1 or 2, uni or bilateral to the nerves II, III, IV, V, VI, VII and XI. The researchers established the caries (ICDAS), periodontal disease (PSR) and plate (Silness Löe) indexes in both groups. We also performed unstimulated, stimulated and bilateral parotid saliva collections, and salivary flow was calculated (ml / min). The buffer capacity was measured in stimulated saliva by titration of 0.01N HCl. The ?-amylase activity was determined by maltose production. For proteomic analysis it was decided to split the saliva samples in accordance with the flow in ml/min. Thus, for each group, study and control, the 4 types of saliva (stimulated, unstimulated, left parotid, right parotid) were subdivided according to low flow (below the group average) or high flow (above average group), resulting in 16 subgroups. The proteome was obtained by two different methodologies, the first was liquid chromatography mass spectrometry and the second was sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) associated with cationic PAGE. The occurrence of caries lesions, related to cut-off 2, as well as the occurrence of periodontal disease, was significantly higher (p> 0.05) in participants with MS when compared to the control group. There was no statistical difference in plaque index between groups. Proteomics analysis showed decrease of cystatin B, S and SN in individuals with MS. There was no difference in protein profile between the low and high salivary flow groups, for individuals with MS and control. There was an increase in the amylase amount and histatin 1, 3 and 5 in individuals with MS. We concluded that individuals with MS present decreased salivary flow, decreased buffer capacity and protein alterations that place these individuals at increased risk for caries and periodontal disease.

A síndrome de Moebius (SM) é uma diplegia congênita rara caracterizada por paralisia total ou parcial do VI e VII nervos cranianos, que leva à ausência ou deficiência dos movimentos dos músculos envolvidos na mímica facial e ao estrabismo convergente. As características bucais descritas nesses indivíduos incluem o palato ogival, micrognatia, malformação de língua, filtro curto, falta de coaptação de lábios, e maior incidência de lesões de cárie. O objetivo deste estudo foi avaliar as características salivares quantitativas e qualitativas, incluindo o proteoma salivar, de indivíduos com SM, associá-las com a saúde bucal, e compará-las com as características salivares de um grupo controle, não afetado pela SM. Foram incluídos 15 indivíduos com SM e 15 controles. O comprometimento facial do individuos com a SM foi avaliada e graduada em scores 0,1 ou 2, uni ou bilateral, para os nervos II, III, IV, V, VI, VII e XI. Os pesquisadores determinaram o índice de cárie (ICDAS), de doença periodontal (PSR) e de placa (Silness Löe) nos dois grupos de estudo. Também realizaram coletas de saliva total não estimulada, estimulada e parotídea bilateral, sendo o fluxo salivar estabelecido em ml/min. A capacidade tampão foi avaliada na saliva total estimulada através da titulação de HCl 0,01N. A atividade de ?-amilase nas amostrasmfoi medida através da produção de maltose. Para a análise proteômica optou-se pela divisão das amostras de saliva de acordo com o fluxo em ml/min. Desta forma, para cada grupo, estudo e controle, os 4 tipos de saliva (estimulada, não estimulada, parotídea esquerda, parotídea direita) foram subdivididos de acordo com baixo fluxo (abaixo da média do grupo) ou alto fluxo (acima da média do grupo), resultando em 16 subgrupos. O proteoma foi obtido por duas metodologias distintas, a primeira a partir da cromatografia líquida espectrometria de massas e a segunda que utilizou a técnica de eletroforese em gel de poliacrilamida contendo dodecil sulfato de sódio (SDS-PAGE) associada à eletroforese em gel de poliacrilamida (native cationic). A ocorrência das lesões de cárie foi significativamente mais alta entre os participantes com SM (p>0,05) no corte 2, bem como a ocorrência de doença periodontal (p>0,05), quando comparado ao grupo controle. Não houve diferença no índice de placa entre os grupos. A análise proteômica mostrou diminuição de cistatinas B, S e SN nos indivíduos com SM. Não houve diferença no perfil proteico entre os grupos de baixo e de alto fluxo salivar, para indivíduos com SM e controle. Houve aumento na quantidade de amilase salivar em e de histatina 1,3 e 5 em indivíduos com SM. Concluímos que indivíduos com SM apresentam diminuição de fluxo salivar, de capacidade tampão e alterações proteicas que colocam esses indivíduos em situação de maior risco para cárie e para doença periodontal. / The Moebius syndrome (MS) is a rare congenital diplegia characterized by total or partial palsy of the VI and VII cranial nerves, leading to the absence or disability of the movements of facial expression muscles and to convergent strabismus. The oral features described in these individuals include high-arched palate, micrognathia, tongue malformation, short filter, lack of lips coaptation, and higher incidence of caries lesions. The aim of this study was to evaluate the quantitative and qualitative salivary characteristics, including the salivary proteome of individuals with MS, associate them with the oral health, and compare them to the salivary characteristics of a control group, unaffected by SM. We included 15 subjects with MS and 15 controls. The facial involvement of individuals with MS was evaluated and graded on scores 0, 1 or 2, uni or bilateral to the nerves II, III, IV, V, VI, VII and XI. The researchers established the caries (ICDAS), periodontal disease (PSR) and plate (Silness Löe) indexes in both groups. We also performed unstimulated, stimulated and bilateral parotid saliva collections, and salivary flow was calculated (ml / min). The buffer capacity was measured in stimulated saliva by titration of 0.01N HCl. The ?-amylase activity was determined by maltose production. For proteomic analysis it was decided to split the saliva samples in accordance with the flow in ml/min. Thus, for each group, study and control, the 4 types of saliva (stimulated, unstimulated, left parotid, right parotid) were subdivided according to low flow (below the group average) or high flow (above average group), resulting in 16 subgroups. The proteome was obtained by two different methodologies, the first was liquid chromatography mass spectrometry and the second was sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) associated with cationic PAGE. The occurrence of caries lesions, related to cut-off 2, as well as the occurrence of periodontal disease, was significantly higher (p> 0.05) in participants with MS when compared to the control group. There was no statistical difference in plaque index between groups. Proteomics analysis showed decrease of cystatin B, S and SN in individuals with MS. There was no difference in protein profile between the low and high salivary flow groups, for individuals with MS and control. There was an increase in the amylase amount and histatin 1, 3 and 5 in individuals with MS. We concluded that individuals with MS present decreased salivary flow, decreased buffer capacity and protein alterations that place these individuals at increased risk for caries and periodontal disease.

„Science, the Never-Ending Quest.“ The diploma thesis deals with a proposal of science center aimed at promoting science and education in the field of nanotechnology. The aim of the project is to design a science center that will complement the existing building complex of CEITEC Nano BUT but also through its architectural expression represent the value of such institution and enable it to stand out among the existing buildings, allowing it to become the new center „brain“ of the complex. Nanotechnology represents a significant sphere of the scientific world that holds big potential for the future in various industries including architecture and building construction. The aim of the diploma thesis is thus not only to design new science center but also to raise awarness of nanotechnolgy. The design proposal is based on simple geometric shape - circle which is segmented into three parts. Logical shape of circle complements the already existing complex and becomes the new center point "brain" of the whole complex. The symbol of infinity that circle and Mobius strip represent is associated with the never ending scientific research as the science itself is at the end nothing but a "never ending quest".

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