Esta dissertação traz ideias para a inserção de novos conteúdos na matemática escolar. Ela trata da exploração de geometrias não-euclidianas, através de dois ambientes de geometria dinâmica, o "Spherical Easel” e o "Disco de Poincaré". O primeiro é um software livre e o segundo foi desenvolvido utilizando-se o recurso de macro-construção do GeoGebra. Na concepção das atividades tratamos as idéias que correspondem ao mundo não-euclidiano fazendo comparações com aquelas que fazem parte da geometria euclidiana e para cada atividade há um comentário que explica a sua intenção de aprendizagem. É a partir de considerações teóricas sobre a natureza da geometria e sua evolução histórica, bem como sobre o processo de aprendizagem da geometria, que é feita a apresentação da proposta. / This dissertation brings ideas to the inclusion of new contents in school mathematics. They are related to the exploitation of non-Euclidean geometries through two dynamic geometry environments, the "Spherical Easel" and the "Poincaré Disk". The first one is a free software and the second one was developed using the GeoGebra macro-construction. In the design of the activities the approach of ideas that correspond to non-euclidian worlds was made through comparison with the euclidian world and for each activity there is a comment that explain its learning objective. The proposal is supported by theoretical considerations about the nature of geometry and its historical evolution, as well as about the geometry learning process.

Orientador: Ermínia de Lourdes Campello Fanti / Banca: Denise de Mattos / Banca: Maria Gorete Carreira Andrade / Resumo: O obejtivo principal deste trabalho é estudar as obstruções "sing" que desempenham papel importante nas demonstrações de certos resultados sobre decomposição de grupos que satisfazem certas hipóteses de dualidade apresentados em [16] e [17], em particular, sobre decomposição de um grupo G adapatada a uma família S de subgrupos de G com (G,S) um par de dualidade de Poincaré. Alguns invariantes cohomológicos e certos resultados envolvendo tais invariantes, decomposição de grupos e/ou grupos e pares de dualidade são também apresentados. / Abstract: The main objective of this work to study the obstructions "sing" which play an important role in demonstrating certain results on the splittings of groups that satisfy certain hypotheses of duality presented in [16] and [17], in particular, the decomposition of a group G adapted to a family S of subgroups of G with (G,S) a Poincaré duality pair. Some cohomological invariants and certain results involving such invariants, a splittings of groups and/or groups and pairs of duality are also presented. / Mestre

Esta dissertação traz ideias para a inserção de novos conteúdos na matemática escolar. Ela trata da exploração de geometrias não-euclidianas, através de dois ambientes de geometria dinâmica, o "Spherical Easel” e o "Disco de Poincaré". O primeiro é um software livre e o segundo foi desenvolvido utilizando-se o recurso de macro-construção do GeoGebra. Na concepção das atividades tratamos as idéias que correspondem ao mundo não-euclidiano fazendo comparações com aquelas que fazem parte da geometria euclidiana e para cada atividade há um comentário que explica a sua intenção de aprendizagem. É a partir de considerações teóricas sobre a natureza da geometria e sua evolução histórica, bem como sobre o processo de aprendizagem da geometria, que é feita a apresentação da proposta. / This dissertation brings ideas to the inclusion of new contents in school mathematics. They are related to the exploitation of non-Euclidean geometries through two dynamic geometry environments, the "Spherical Easel" and the "Poincaré Disk". The first one is a free software and the second one was developed using the GeoGebra macro-construction. In the design of the activities the approach of ideas that correspond to non-euclidian worlds was made through comparison with the euclidian world and for each activity there is a comment that explain its learning objective. The proposal is supported by theoretical considerations about the nature of geometry and its historical evolution, as well as about the geometry learning process.

Esta dissertação traz ideias para a inserção de novos conteúdos na matemática escolar. Ela trata da exploração de geometrias não-euclidianas, através de dois ambientes de geometria dinâmica, o "Spherical Easel” e o "Disco de Poincaré". O primeiro é um software livre e o segundo foi desenvolvido utilizando-se o recurso de macro-construção do GeoGebra. Na concepção das atividades tratamos as idéias que correspondem ao mundo não-euclidiano fazendo comparações com aquelas que fazem parte da geometria euclidiana e para cada atividade há um comentário que explica a sua intenção de aprendizagem. É a partir de considerações teóricas sobre a natureza da geometria e sua evolução histórica, bem como sobre o processo de aprendizagem da geometria, que é feita a apresentação da proposta. / This dissertation brings ideas to the inclusion of new contents in school mathematics. They are related to the exploitation of non-Euclidean geometries through two dynamic geometry environments, the "Spherical Easel" and the "Poincaré Disk". The first one is a free software and the second one was developed using the GeoGebra macro-construction. In the design of the activities the approach of ideas that correspond to non-euclidian worlds was made through comparison with the euclidian world and for each activity there is a comment that explain its learning objective. The proposal is supported by theoretical considerations about the nature of geometry and its historical evolution, as well as about the geometry learning process.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T15:40:44Z No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T18:55:21Z (GMT) No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) / Made available in DSpace on 2017-05-29T18:55:21Z (GMT). No. of bitstreams: 1 arturafonsoguedesrossini.pdf: 414354 bytes, checksum: 7de0172af679cf245fd9b870000235c4 (MD5) Previous issue date: 2011-12-15 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Uma folheação algébrica do plano projetivo sobre um corpo k pode ser dada tanto por um campo de vetores como por uma 1-forma em P2, já que dimensão um e codimensão um são a mesma noção visto que a dimensão de P2 é igual a 2. Então surge uma pergunta natural: Como se relacionam os campos vetoriais e as 1-formas em P2? Veremos que uma 1-forma ω e um campo de vetores X definem a mesma folheação do plano projetivo quando ω(p)(X(p)) = 0 para todo ponto p ∈P2. Uma segunda questão é a existência de curvas algébricas invariantes por uma folheação de P2. Originalmente, Poincaré formulou o seguinte problema: É possível limitar o grau de uma curva algébrica invariante por um campo de vetores em termos do grau do campo de vetores? A resposta para este problema é negativa, como podemos ver no Exemplo 3.18. Entretanto adicionando-se algumas hipóteses sobre tal curva invariante este problema pode possuir resposta positiva. No caso em que tal curva invariante é suave, mostra-se que o grau da curva é no máximo igual ao grau do campo vetorial mais um. Se uma curva invariante não for suave, mostra se que ainda é possível limitar o grau desta curva em termos do grau da folheação e da regularidade do seu conjunto de singularidades. / An algebraic foliation of the projective plane over a field k can be given either by a vector field or a 1-form in P2, as dimension one and codimension one are the same notion since dim(P2) = 2. Then a natural question arises: How do vector fields and 1-forms in P2 relate? We will see that an 1-form ω is related with a vector field X belonging to the kernel of ω, that is, ω and X define the same foliation of the projective plane when ω(p)(X(p)) = 0 for all points p ∈P2. A second question concerns about the existence of algebraic curves that are invariant by a foliation of P2. Originally, Poincaré formulated the following problem: Is it possible to bound the degree of an invariant curve under a vector field in terms of the degree of the field? The problem has a negative answer, but by adding some hypothesis it can be reformulated in order to have a positive answer. If we assume that this invariant curve is smooth, we show that the degree of the curve is at most the degree of the vector field plus one. If an invariant curve is not smooth, we show that its degree can be limited in terms of regularity of its set of singularities.

Cette thèse concerne les homéomorphismes de surfaces.Soit f un difféomorphisme d'une surface M préservant l'aire et isotope à l'identité. Si f a un point fixe contractile isolé et dégénéré z0 avec un indice de Lefschetz égal à 1, et si l'aire de M est finie, nous prouverons au chapitre 3 que z0 est accumulé non seulement par des points périodiques mais aussi par des orbites périodiques au sens de la mesure. Plus précisément, la mesure de Dirac en z0 est la limite en topologie faible-étoile d'une suite de probabilités invariantes supportées par des orbites périodiques. Notre preuve est totalement topologique et s'applique au cas d'homéomorphismes en considérant l'ensemble de rotation local.Au chapitre 4, nous étudierons des homéomorphismes préservant l’aire et isotope à l’identité. Nous prouverons l’existence d'isotopies maximales particulières: les isotopies maximales à torsion faible. En particulier, lorsque f est un difféomorphisme ayant un nombre fini de points fixes tous non-dégénérés, une isotopie I joignant l'identité à f est à torsion faible si et seulement si pour tout point z fixé le long de I, le nombre de rotation (réel) ρ(I,z), qui est bien défini quand on éclate f en z, est contenu dans (-1,1). Nous démontrerons l'existence d'isotopies maximales à torsion faible, et nous étudierons la dynamique locale de feuilletages transverses à l'isotopie près des singularités isolées.Au chapitre 5, nous énoncerons une généralisation d'un théorème de Poincaré-Birkhoff local au cas où il existe des points fixes au bord. / This thesis concerns homeomorphisms of surfaces.Let f be an area preserving diffeomorphism of an oriented surface M isotopic to the identity. If f has an isolated degenerate contractible fixed point z0 with Lefschetz index one, and if the area of M is finite, we will prove in Chapter 3 that z0 is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at z0 is the limit in weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological and will works for homeomorphisms and is related to the notion of local rotation set.In chapter 4, we will define a kind of identity isotopies: torsion-low isotopies. In particular, when f is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy I of f is torsion-low if and only if for every point z fixed along the isotopy, the (real) rotation number ρ(I,z), which is well defined when one blows-up f at z, is contained in (-1,1). We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.In chapter 5, we will generalize a local Poincaré-Birkhoff theorem to the case where there exist fixed points on the boundary

Nous nous intéressons aux cristaux liquides nématiques, qui sont une phase de la matière intermédiaire entre les liquides et les solides cristallins. Ces états sont caractérisés par la présence de défauts ponctuels ou de ligne. Le but de cette thèse est d'apporter une contribution à l'étude mathématique des défauts, dans le cadre de la théorie variationnelle de Landau-de Gennes. Dans le premier chapitre, nous étudions les minimiseurs de l'énergie dans des domaines bornés de dimension deux. Lorsque la constante élastique tend vers zéro, les minimiseurs convergent vers une application localement harmonique, avec un nombre fini de singularités ponctuelles. Au voisinage de celles-ci, les minimiseurs sont biaxes (le molécules sont alignées localement dans plusieurs directions). Le deuxième chapitre est consacré à l'analyse asymptotique des minimiseurs en dimension trois, en supposant l'énergie majorée par le logarithme de la constante élastique. Comme dans le cas bidimensionnel, nous obtenons un résultat de compacité des minimiseurs, mais cette fois l'application limite peut présenter à la fois des singularités ponctuelles et de ligne. Nous donnons aussi des conditions suffisantes pour que l'hypothèse sur l'énergie évoquée précédemment soit satisfaite. Le troisième chapitre porte sur l'existence de minimiseurs à symétrie radiale dans une couronne en dimension trois. Enfin, dans le dernier chapitre nous présentons une obstruction topologique à l'existence de champs de vecteurs unitaires de faible régularité, sur des variétés à bord. Ce résultat constitue une étape préliminaire à l'étude de modèles variationnels pour les films nématiques sur une surface. / Nematic liquid crystals are an intermediate phase of matter, sharing properties with liquids and crystalline solids. They are composed of molecules which can flow freely, but tend to align locally along some preferred directions. Nematic phases exhibit defects, which can occur at isolated points or along lines, and are one of their mean features. This thesis mainly aims at discussing some mathematical results about defects and their generation, in the framework of the Landau-de Gennes theory. In the first chapter, we study minimizers of the energy functional in a bounded, smooth domain in dimension two. We show that, as the elastic constant tends to zero, minimizers converge to a locally harmonic map with a finite number of point singularities. Minimizers are biaxial in the core of defects (that is, more than one preferred direction of molecular alignment exists at a given point). Chapter two deals with the asymptotic analysis of minimizers in dimension three. We assume that the energy is comparable to the logarithm of the elastic constant and prove a compactness result. However, the limiting map is now allowed to have line singularities as well as point singularities. We also provide sufficient conditions for the logarithmic energy estimate to be satisfied. In chapter three, we study the existence of radially symmetric minimizers on spherical shells, in dimension three. Finally, in chapter four, we discuss a topological obstruction to the existence of unit vector fields of low regularity, on a compact manifold with boundary. This result can be understood as a first step in the analysis of some variational models for a surface coated with a thin nematic film.

La structure de l’oeil permet d’observer la microcirculation, grâce aux caméras de fond d’oeil. Ces appareils sont bon marché et couramment utilisés dans la pratique clinique, permettant le dépistage de maladies oculaires. La capacité des vaisseaux à adapter leur diamètre (autorégulation) afin de réguler le débit sanguin est importante dans la microcirculation. L’hémodynamique de l’oeil est impactée par la pression à l’intérieur du globe oculaire (IOP), qui est à son tour influencée par le flux sanguin oculaire. Les altérations de l’autorégulation et l’IOP jouent un rôle dans les maladies oculaires. La modélisation mathématique peut aider à interpréter l’interaction entre ces phénomènes et à mieux exploiter les données médicales disponibles. Dans la première partie, nous présentons un modèle simplifié d’interaction fluidestructure qui inclut l’autorégulation, appliqué à un reseau 3D obtenu par imagerie médicale. Les cellules musculaires lisses regulant le diamètre du vaisseau sont modélisés dans la structure. Ensuite, nous utilisons des équations de poroélasticité pour décrire le flux sanguin dans la choroïde, dans un modèle multi-compartiments de l’oeil. Cette approche permet de rendre compte de la transmission de la pulsatilité de la choroïde à la chambre antérieure, où l’IOP est mesurée. Nous présentons des résultats préliminaires sur la choroïde, l’humeur aqueuse et sur la choroïde couplée avec la vitrée. Enfin, nous présentons un modèle d’ordre réduit pour accélérer des simulations multi-physique. Des modèles de haute précision sont utilisés pour les compartiments d’intérêt et une représentation réduite de l’opérateur de Steklov-Poincaré est utilisée pour les autres compartiments. / The structure of the eye offers a unique opportunity to directly observe the microcirculation, by means, for instance, of fundus camera, which are cheap devices commonly used in the clinical practice. This can facilitate the screening of systemic deseases such as diabetes and hypertension, or eye diseases such as glaucoma. A key phenomenon in the microcirculation is the autoregulation, which is the ability of certain vessels to adapt their diameter to regulate the blood flow rate in response to changes in the systemic pressure or metabolic needs. Impairments in autoregulation are strongly correlated with pathological states. The hemodynamics in the eye is influenced by the intraocular pressure (IOP), the pressure inside the eye globe, which is in turn influenced by the ocular blood flow. The interest in the IOP stems from the fact that it plays a role in several eye-diseases, such as glaucoma. Mathematical modelling can help in interpreting the interplay between these phenomena and better exploit the available data. In the first part of the thesis we present a simplified fluid-structure interaction model that includes autoregulation. A layer of fibers in the vessel wall models the smooth muscle cells that regulate the diameter of the vessel. The model is applied to a 3D image-based network of retinal arterioles. In the second part, we propose a multi-compartments model of the eye. We use the equations of poroelasticity to model the blood flow in the choroid. The model includes other compartments that transmit the pulsatility from the choroid to the anterior chamber, where the measurements of the IOP are actually performed. We present some preliminary results on the choroid, the aqueous humor and on the choroid coupled with the vitreous. Finally, we present a reduced order modelling technique to speed up multiphysics simulations. We use high fidelity models for the compartments of particular interest from the modelling point of view. The other compartments are instead replaced by a reduced representation of the corresponding Steklov-Poincaré operator.

This diploma thesis deals with the issue of measuring the ambient temperature using a~single-mode optical fiber used as a temperature sensor. The thesis describes the basic knowledge about polarized light, its propagation in optical fiber and describes special fibers that preserve polarization. It also deals with the representation of polarization states on a Poincaré sphere and the use of Stokes and Jones vectors. In the practical part, several selected methods of optical signal supply to the temperature sensor are measured. Subsequently, the suitability of the given methods is evaluated from the measured results and the courses are compared with the records measured from real routes. This thesis aims to select a suitable connection of a temperature sensor for a long route of optical fiber and verify its functionality.

<div> <div> <div> <p>Resonant orbits in a multi-body environment have been investigated in the past to aid the understanding of perceived chaotic behavior in the solar system. The invariant manifolds associated with resonant orbits have also been recently incorporated into the design of trajectories requiring reduced maneuver costs. Poincaré sections are now also extensively utilized in the search for novel, maneuver-free trajectories in various systems. This investigation employs dynamical systems techniques in the computation and characterization of resonant orbits in the higher-fidelity Circular Restricted Three-Body model. Differential corrections and numerical methods are widely leveraged in this analysis in the determination of orbits corresponding to different resonance ratios. The versatility of resonant orbits in the design of low cost trajectories to support exploration for several planet-moon systems is demonstrated. The efficacy of the resonant orbits is illustrated via transfer trajectory design in the Earth-Moon, Saturn-Titan, and the Mars-Deimos systems. Lastly, Poincaré sections associated with different resonance ratios are incorporated into the search for natural, maneuver-free trajectories in the Saturn-Titan system. To that end, homoclinic and heteroclinic trajectories are constructed. Additionally, chains of periodic orbits that mimic the geometries for two different resonant ratios are examined, i.e., periodic orbits that cycle between different resonances are determined. The tools and techniques demonstrated in this investigation are useful for the design of trajectories in several different systems within the CR3BP. </p> </div> </div> </div>