We introduce the notion of special spherical symmetry and classify the complete regular minimal surfaces in the three sphere having this symmetry. We also show that the Clifford torus is the unique embedded minimal torus in three sphere possessing special spherical symmetry.

La estabilidad de las pendientes rocosas se considera crucial para la seguridad pública en las carreteras, así como también para la seguridad del personal y los equipos que trabajan en los cortes de roca. La inestabilidad y fallas en taludes rocosos se producen debido a muchos factores, como la geometría, las discontinuidades geológicas, el material del talud, y las condiciones climáticas severas. Así también como las cargas externas, las fuertes precipitaciones y la sismicidad que podrían jugar un papel importante en la falla. En esta tesis se desarrolla el método cinemático y el método empírico de “Índice de Riesgo” (tomado del inglés “Hazard Índex”) para la evaluación de la estabilidad del talud en la carretera Tarapoto- Yurimaguas, donde el talud corresponde a una roca residual tipo arenisca de baja resistencia, con alta precipitación por ser zona tropical y ubicada dentro de una zona sísmica. Se identifican los posibles tipos de falla mediante el análisis cinemático basado en la dirección del rumbo y buzamiento de las discontinuidades, se analiza las fallas encontradas con respecto al factor sísmico, y la condición de la estabilidad con el método empírico “Hazard Index” con respecto al factor de precipitación para así identificar la condición de estabilidad y mostrar el resultado. La tesis destaca algunas limitaciones de los métodos utilizados. / The stability of rocky slopes is considered crucial for public safety on the roads, as well as for the safety of personnel and teams working on rock cuts. Rock slope instability and failure occur due to many factors such as geometry, geological discontinuities, slope material, and severe weather conditions. As well as external loads, heavy precipitation, and seismicity that could play an important role in the failure. In this thesis we develop the kinematic method and the empirical method of "hazard Index" for evaluating the stability of the slope on the Tarapoto-Yurimaguas road, where the slope corresponds to a residual rock Low resistance sandstone type, with high precipitation as it is a tropical zone and located within a seismic zone. Possible types of failure are identified through kinematic analysis based on the direction of heading and dip of the discontinuities, the failures found are analyzed with respect to the seismic factor, and the condition of stability with the empirical method "Hazard Index" with respect to to the precipitation factor to identify the stability condition and show the result. The thesis highlights some limitations of the methods used. / Tesis

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-06-18T20:37:32Z No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T17:47:02Z (GMT) No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-07-06T17:54:01Z (GMT) No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) / Made available in DSpace on 2015-07-06T17:54:32Z (GMT). No. of bitstreams: 1 Dissertação-Euderley de C Nunes.pdf: 25821021 bytes, checksum: 4f3d1be54be4ba8e8d8c720131d61fb6 (MD5) Previous issue date: 2015-03-04 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Aiming to contribute to the teaching of basic education, this work will present a simple approach through the use of a well-known technique in antiquity, which is the stereographic projection. This paper will deal with the studies developed by Georg Friedrich Bernhard Riemann (1826-1866), which demonstrates how to design stereographically a sphere on a plane, called the complex plane. For this, we will show that the use of complex numbers has great relevance for understanding of the techniques commonly used in the development of cartography and other areas. We will present the set of complex numbers and then de ne the stereographic projection and some of its main properties, where we use the Geogebra software version 5.0, seeing that the software produces 3D animations, which will support in understanding the stereographic projection and of their properties by the high school students and teachers. Thus, this research will serve as a motivating element for students and teachers that seek to improve their knowledge because the study by Riemann is based on complex numbers which are studied in the course of primary education. / Com o objetivo de contribuir com o ensino da educação básica, este trabalho apresentará através de uma abordagem simples o uso de uma técnica muito conhecida na antiguidade, que é a projeção estereográ ca. Este trabalho abordará os estudos desenvolvidos por Georg Friedrich Bernhard Riemann (1826-1866), que demonstra como projetar estereogra camente uma esfera sobre um plano, denominado de plano complexo. Para isso, mostraremos que o uso dos números complexos terá grande relevância para compreendermos uma das técnicas mais usadas no desenvolvimento da cartogra a e outras áreas. Apresentaremos o conjunto dos números complexos e em seguida de niremos a projeção estereográ ca e algumas de suas principais propriedades, onde faremos o uso do software Geogebra versão 5.0, visto que este software produz animações em 3D, que servirão de suporte para a compreensão da projeção estereográ ca e de suas respectivas propriedades por parte dos alunos e professores do ensino médio. Com isso, esta pesquisa servirá de elemento motivador para alunos e professores que busquem aprimorar seus conhecimentos, pois o estudo desenvolvido por Riemann tem como base os números complexos que são estudados no decorrer do ensino básico.

Submitted by FERNANDO MARQUES CALISTER null (fcalister@gmail.com) on 2016-10-02T01:33:35Z No. of bitstreams: 1 Representações dos Números Complexos e Transformações de Mobius.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-10-05T16:28:55Z (GMT) No. of bitstreams: 1 calister_fm_me_sjrp.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) / Made available in DSpace on 2016-10-05T16:28:55Z (GMT). No. of bitstreams: 1 calister_fm_me_sjrp.pdf: 617587 bytes, checksum: e9bbc4361adf7d335874ab0c7f3fdc3f (MD5) Previous issue date: 2016-08-19 / 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. / 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 Möbius Transformations and the Stereographic Projection.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2017-08-08T13:29:35Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Anne Caroline da Silva - Dissertação - CTG.pdf: 5307858 bytes, checksum: e576e8a612fb53ea26dea1d4597d3ee9 (MD5) / Made available in DSpace on 2017-08-08T13:29:35Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Anne Caroline da Silva - Dissertação - CTG.pdf: 5307858 bytes, checksum: e576e8a612fb53ea26dea1d4597d3ee9 (MD5) Previous issue date: 2016-07-27 / CAPES / Na pedreira Combritas, localizada no distrito de Pão de Açúcar, município de Taquaritinga do Norte, Pernambuco, procedeu-se a análise estrutural da rocha, susceptível de aplicação ao bom planejamento das aberturas das faces livres, consistindo no mapeamento geológico de detalhe das principais descontinuidades presentes no maciço a partir da coleta de dados. Possibilitou-se, assim, a projeção das feições geométrico-estruturais e as prováveis orientações de instabilidade, que definem a ruptura natural da rocha, isto é, o bloco de partição. O objetivo foi a análise das características geomecânicas evidentes na face livre e a forma como está orientada. Os prováveis tipos de rupturas foram verificados nas cinco faces livres estudadas, através das técnicas de projeções estereográficas e cônicas das descontinuidades – famílias de fraturas, foliações, tipo bandamento e xistosidade – expressivos do bloco de partição natural da rocha. Assim, buscou-se a definição das superfícies levantante, alongante e trincante, e suas interseções nas faces livres já existentes. As interpretações das projeções identificaram a tendência do maciço a rupturas em cunha. Os marcadores de deformação apontaram para o estágio frágil-rúptil dos litotipos do maciço, enquanto o bloco de partição permitiu apontar a melhor abertura da face livre, que leva à otimização de carga, essencialmente evitando-se taludes negativos e repés, reduzindo os custos financeiros. As cunhas de partição caracterizaram as dificuldades registradas nas faces livres já abertas na pedreira de agregados para a construção civil. Em essência, há obliquidade entre a foliação da rocha e as famílias de fraturas mais frequentes, traduzindo-se, em geral, com alongamentos segundo a direção NNW-SSE até NE-SW, com mergulhos para o sentido leste do que se conclui que a abertura da face livre será otimizada se respeitar a direção daquele alongamento, mergulhando do sentido leste para oeste. Alternativamente, deve ser respeitada a direção da superfície alongante da rocha, de tal modo que a face livre verticalizada, nessa situação, deverá gerar menos taludes negativos e passivos ambientais. / The quarry Combritas, located in Pão de Açucar district, Taquaritinga of Norte city, Pernambuco, proceeded to the structural rock analysis that can be applied to good planning of the openings of the free faces, consisting of geological detail mapping of major discontinuities present in the mass from the data collection. It is possible, therefore, the projection of the geometric and structural features and likely directions of instability, which define the natural break the rock, that is, the partition block. The objective was to analyze the geomechanical characteristics evident on the free face and how it is oriented. The probable types of breaks were observed in five free faces studied, through the techniques of stereographic projections and conic of discontinuities - families fractures, foliation, type banding and foliation - expressive natural partition block of rock. Thus, we sought to define the boosting surfaces alongante and trincante, and their intersections on existing free faces. Interpretations of the projections identified the trend of massive breaks the wedge. Deformation marker pointed to the fragile, brittle stage of the mass lithologies while the partition block allowed to point the best opening of the free face, which leads to the load optimization essentially avoiding negative slopes and baize, reducing financial costs . The partition wedges characterized the difficulties recorded in the free faces now open in the quarry aggregates for the construction industry. In essence, there is obliquity between the foliation of the rock and the families of more frequent fractures, resulting in general with stretching according to NNW-SSE to NESW, with dips to the east direction than it is concluded that opening of the free face will be optimized to respect the direction of that stretch, plunging the eastbound west. Alternatively, it should be respected the direction of alongante rock surface, so that the free face vertically in this situation should generate less negative slope and environmental liabilities.

Dans cette thèse, nous explorons premièrement la notion de directionnalité lors de la conception de repères d'ondelettes du plan. Cette propriété, qui semble essentielle pour la vision biologique, donne lieu à une meilleure représentation des contours d'objets dans les décompositions d'images utilisant ces repères. Elle génère en outre une redondance supplémentaire qui exploitée à bon escient, permet par exemple de réduire les effets d'un bruit additif (gaussien). Nous montrons également comment cette directionnalité, généralement perçue comme un paramètre figé, peut être adaptée localement aux éléments d'une image. Nous définissons ainsi le concept d'analyse d'images multisélective. Dans ce cadre, des règles de récurrence héritées d'une analyse multirésolution circulaire associent des ondelettes d'une certaine sélectivité angulaire pour générer des ondelettes de plus faible directionnalité jusqu'à l'obtention d'une ondelette totalement isotrope. Dans le cas d'un repère d'ondelettes linéaire, ces différents niveaux de sélectivité ont la possibilité de s'ajuster localement au contenu d'une image. Nous constatons par ailleurs que cette adaptabilité fournit de meilleures reconstructions que les méthodes à sélectivité fixe lors d'approximations non linéaires d'images. Cette thèse traite ensuite du problème de l'analyse de données représentées sur la sphère. Il a été établi précédemment [AV99] que la transformée continue en ondelettes (CWT) s'étend à cet espace par l'emploi d'une dilatation stéréographique respectant la compacité de S2. Dans certain cas, il est utile de réduire la redondance de cette transformée, ne fut ce que pour faciliter le traitement des données dans l'espace multi-échelle généré. Nous étudions par conséquent comment créer des repères sphériques semi-continus, où seule l'échelle est échantillonnée, et totalement discrétisés. Nous tirons parti dans ce dernier cas de grilles sphériques équi-angulaires et des règles de quadrature associées pour obtenir des conditions suffisantes à la reconstruction des fonctions analysées. Les capacités d'analyse et de synthèse de repères d'ondelettes DOG sont également testées sur des exemples de données sphériques. Une dernière partie de ce document est dédiée à l'étude d'un objet physique étonnant : la couronne solaire. Cette couche extérieure du soleil est observée depuis 1996 par l'expérience EIT à bord du satellite SoHO dans différentes longueurs d'onde de l'ultraviolet lointain. La compréhension physique des multiples phénomènes apparaissant dans la couronne solaire passe par le traitement automatique des images EIT. Dans cette tâche, nous nous limitons à deux problèmes particuliers. Nous utilisons premièrement la CWT et sa capacité à analyser la régularité locale d'une image pour gommer les traces laissées par les rayons cosmiques, majoritairement non solaires, sur les enregistrements EIT. Deuxièmement, la couronne solaire contient des éléments de faible taille (<60arcsec) nommés points brillants (ou BPs pur Bright points). Ceux-ci trouvent leur origine dans l'échauffement local du plasma coronal sous l'action du champs magnétique solaire. En abordant une approche similaire à celle développée en [Bij99], nous étudions comment sélectionner et caractériser ces BPS en décomposant une image en ses objets constitutifs. Ces derniers sont issus de tubes de maxima dans la description multi-échelle de l'image, c'est-à-dire d'une généralisation discrète des lignes de maxima de la CWT.

Title: Descriptive geometry applications in cartography Author: Jana Vlčková Department: Department of Mathematics Education Supervisor: RNDr. Vlasta Moravcová, Ph.D., Department of Mathematics Education Abstract: This thesis is focused on constructions of geographic nets in maps and serves as a base for studies and teaching for teachers and students of secondary schools or universities, which are focused on the cartography field. The thesis contains seven chapters, the first of which describes basic terms used in the thesis. Other chapters deal with the analysis of chosen cartography projections, whereas first three chapters concerns with the projection of the globe into a plane, another two chapters concerns with a cylindrical surface and the last one concerns with a conical surface. Constructive tasks, which serve for practising the subject matter, are enclosed for each chapter. Key words: cartographic projection, azimuthal projection, ortographic projection, gno- monic projection, stereographic projection, cylindrical projection, conical projection 1

We focus on the two overlapping areas of preference queries and dataset summarization. A (linear) preference query specifies the relative importance of the attributes in a dataset and asks for the tuples that best match those preferences. Dataset summarization is the task of representing an entire dataset by a small, representative subset. Within these areas, we focus on three important sub-problems, significantly advancing the state-of-the-art in each. We begin with an investigation into a new formulation of preference queries, identifying a neglected and important subclass that we call threshold projection queries. While literature typically constrains the attribute preferences (which are real-valued weights) such that their sum is one, we show that this introduces bias when querying by threshold rather than cardinality. Using projection, rather than inner product as in that literature, removes the bias. We then give algorithms for building and querying indices for this class of query, based, in the general case, on geometric duality and halfspace range searching, and, in an important special case, on stereographic projection. In the second part of the dissertation, we investigate the monochromatic reverse top-k (mRTOP) query in two dimensions. A mRTOP query asks for, given a tuple and a dataset, the linear preference queries on the dataset that will include the given tuple. Towards this goal, we consider the novel scenario of building an index to support mRTOP queries, using geometric duality and plane sweep. We show theoretically and empirically that the index is quick to build, small on disk, and very efficient at answering mRTOP queries. As a corollary to these efforts, we defined the top-k rank contour, which encodes the k-ranked tuple for every possible linear preference query. This is tremendously useful in answering mRTOP queries, but also, we posit, of significant independent interest for its relation to myriad related linear preference query problems. Intuitively, the top-k rank contour is the minimum possible representation of knowledge needed to identify the k-ranked tuple for any query, without apriori knowledge of that query. We also introduce k-regret minimizing sets, a very succinct approximation of a numeric dataset. The purpose of the approximation is to represent the entire dataset by just a small subset that nonetheless will contain a tuple within or near to the top-k for any linear preference query. We show that the problem of finding k-regret minimizing sets—and, indeed, the problem in literature that it generalizes—is NP-Hard. Still, for the special case of two dimensions, we provide a fast, exact algorithm based on the top-k rank contour. For arbitrary dimension, we introduce a novel greedy algorithm based on linear programming and randomization that does excellently in our empirical investigation. / Graduate / 0984

[pt] Esta dissertação visa mostrar que a construção de alguns mapas, chamados mapas conformes, pode ser expressa por funções complexas e essa relação será mostrada ao longo do texto. Inicialmente são apresentadas as coordenadas esféricas utilizadas por geógrafos e matemáticos e a construção de um mapeamento da esfera terrestre no plano, projeção estereográfica. Nas seções seguintes, são apresentadas: definições e propriedades das funções complexas com ênfase em suas interpretações geométricas; alguns mapas gerados pelas funções exponencial, logarítmica e trigonométricas complexas; a relação entre função exponencial e o Mapa de Mercator; algumas características de uma função elíptica; a relação entre uma função elíptica e o Mapa Pierce Quincuncial. / [en] This master thesis aims to show that the construction of some maps, called conformal maps, can be expressed by complex functions and this relation will be shown through the text. First it will be presented the spherical coordinates used for geographers and mathematicians, and the construction of a mapping of the terrestrial sphere in the plane, stereographic projection. In the following sections, they are presented: Definitions and properties of complex functions with emphasis on their geometric interpretations; Some maps generated by the exponential, logarithmic and complex trigonometric functions; The relationship between exponential function and the Mercator Map; Some characteristics of an elliptical function; The relationship between an elliptical function and the Quincuncial Pierce Map.

In this thesis, we present some new results in distribution theory for both discrete and continuous random variables, together with their motivating applications. We start with some results about the Multivariate Gaussian Distribution and its characterization as a maximizer of the Strichartz Estimates. Then, we present some characterizations of discrete and continuous distributions through ideas coming from optimal transportation. After this, we pass to the Simpson's Paradox and see that it is ubiquitous and it appears in Quantum Mechanics as well. We conclude with a group of results about discrete and continuous distributions invariant under symmetries, in particular invariant under the groups $A_1$, an elliptical version of $O(n)$ and $\mathbb{T}^n$. As mentioned, all the results proved in this thesis are motivated by their applications in different research areas. The applications will be thoroughly discussed. We have tried to keep each chapter self-contained and recalled results from other chapters when needed. The following is a more precise summary of the results discussed in each chapter. In chapter \ref{chapter 2}, we discuss a variational characterization of the Multivariate Normal distribution (MVN) as a maximizer of the Strichartz Estimates. Strichartz Estimates appear as a fundamental tool in the proof of wellposedness results for dispersive PDEs. With respect to the characterization of the MVN distribution as a maximizer of the entropy functional, the characterization as a maximizer of the Strichartz Estimate does not require the constraint of fixed variance. In this chapter, we compute the precise optimal constant for the whole range of Strichartz admissible exponents, discuss the connection of this problem to Restriction Theorems in Fourier analysis and give some statistical properties of the family of Gaussian Distributions which maximize the Strichartz estimates, such as Fisher Information, Index of Dispersion and Stochastic Ordering. We conclude this chapter presenting an optimization algorithm to compute numerically the maximizers. Chapter \ref{chapter 3} is devoted to the characterization of distributions by means of techniques from Optimal Transportation and the Monge-Amp\`{e}re equation. We give emphasis to methods to do statistical inference for distributions that do not possess good regularity, decay or integrability properties. For example, distributions which do not admit a finite expected value, such as the Cauchy distribution. The main tool used here is a modified version of the characteristic function (a particular case of the Fourier Transform). An important motivation to develop these tools come from Big Data analysis and in particular the Consensus Monte Carlo Algorithm. In chapter \ref{chapter 4}, we study the \emph{Simpson's Paradox}. The \emph{Simpson's Paradox} is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, the statistical implications are deep. Basic examples appear in arithmetic, geometry, linear algebra, statistics, game theory, sociology (e.g. gender bias in the graduate school admission process) and so on and so forth. In our new results, we prove the occurrence of the \emph{Simpson's Paradox} in Quantum Mechanics. In particular, we prove that the \emph{Simpson's Paradox} occurs for solutions of the \emph{Quantum Harmonic Oscillator} both in the stationary case and in the non-stationary case. We prove that the phenomenon is not isolated and that it appears (asymptotically) in the context of the \emph{Nonlinear Schr\"{o}dinger Equation} as well. The likelihood of the \emph{Simpson's Paradox} in Quantum Mechanics and the physical implications are also discussed. Chapter \ref{chapter 5} contains some new results about distributions with symmetries. We first discuss a result on symmetric order statistics. We prove that the symmetry of any of the order statistics is equivalent to the symmetry of the underlying distribution. Then, we characterize elliptical distributions through group invariance and give some properties. Finally, we study geometric probability distributions on the torus with applications to molecular biology. In particular, we introduce a new family of distributions generated through stereographic projection, give several properties of them and compare them with the Von-Mises distribution and its multivariate extensions. / Thesis / Doctor of Philosophy (PhD)