Dissertationen zum Thema „Non-Euclidean spaces“

DE LA TESIS:

El contenido de esta tesis se enmarca dentro del Análisis Real. En particular, trata del estudio de ciertos problemas de la teoría de pesos, (una referencia clásica sobre esta teoría es el libro de J. García-Cuerva y J.L. Rubio de Francia [GR]). Nosotros consideramos, por este orden, tres problemas clásicos diferentes, que abarcan buena parte de la teoría de pesos:

(i) Estudio de las inclusiones para espacios con pesos y acotación de operadores integrales entre estos espacios.

(ii) Estudio de propiedades funcionales de espacios con pesos asociados a una reordenada decreciente de funciones.

(iii) Estudio de la acotación de operadores maximales asociados a regiones de aproximación entre espacios con pesos.

Todos estos problemas han sido tratados extensamente en la literatura. Nuestro enfoque ha sido el de extender estos resultados a espacios con la mínima estructura necesaria. Concretamente, hemos trabajado respectivamente en cada capítulo en los siguientes contextos:

(i) Espacios de medida arbitrarios.

(ii) Árboles.

(iii) Espacios de tipo homogéneo.

Puesto que un árbol puede ser a su vez un espacio de medida, o puesto que su frontera puede ser un espacio de tipo homogéneo, algunos resultados para espacios de medida y espacios de tipo homogéneo han sido aplicados a los árboles (véanse los capítulos primero y tercero). En cambio, en el capítulo segundo trabajamos exclusivamente en árboles.

Los espacios donde hemos desarrollado nuestra teoría no poseen, en general, ningún tipo de estructura algebraica. Por tanto, todos los resultados persiguen un objetivo común: la extensión de la teoría de pesos a espacios no euclidianos.

This research interprets and develops the 'conformal model of space' in a way appropriate for a graphics developer interested in the design of interactive software for exploring 2-dimensional non-Euclidean spaces. The conformal model of space extends the standard projective model – instead of adding just one extra dimension to standard Euclidean space, a second one is added that results in a Minkowski space similar to that of relativistic spacetime. Also, standard matrix algebra is replaced by geometric ( i.e. Clifford) algebra. The key advantage of the conformal model is that both Euclidean and non- Euclidean spaces are accommodated within it. Transformations in conformal space are generated by bivectors which are special elements of the geometric algebra. These induce geometric transformations in the embedded non Euclidean spaces. However, the relationship between the bivector generated transformations of the Minkowski modelling space and the geometric transformations they induce is extremely obscure. This thesis provides new analytical tools for determining the nature of this relationship. Their derivation was motivated by the need to successfully solve key implementation problems relating to navigation and in-scene mouse interaction. The analytic approaches developed not only successfully solved these problems but pointed the way to implementing other unplanned features. These include facilities for dynamically altering on-screen geometry as well as using multiple viewports to allow the user to interact with the same objects embedded in different geometries. These new analytical approaches could be powerful tools for solving future and as yet unforeseen implementation problems.

Thesis (M.S.) -- Central Connecticut State University, 1998.
Thesis advisor: Dr. Jeffrey McGowan. "...in partial fulfillment of the requirements for the degree of Master of Science." Includes bibliographical references (leaves [78-79]).

The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.

The main goal of this master's thesis is to research different approaches of rendering geometries and spaces in virtual reality. Learn more about the terms, non-Euclidean geometry and non-Euclidean spaces, their origin and different principles used in video game industry to simulate such geometries or spaces. Based on the research, a selection of an optimal API is needed for the implementation of such application. Application is designed to run on desktop computers with Microsoft Windows operating system. Application, in it's core, is a video game and the main goal of the player is to successfully complete each and every level of the game. These levels are designed in a specific way so that they each individually represent some form of non-Euclidean geometry or space.

Capes
Na busca por uma melhor representação da realidade tridimensional, as Geometrias não- Euclidianas oferecem uma alternativa ao euclidianismo clássico e um dos destaques e a Geometria Projetiva. Assim, o objetivo deste trabalho e, através de ilustrações, contribuir para a assimilação de definições como perspectiva, projeção e o principio da dualidade. E, a partir de resultados importantes como o Teorema de Desargues, o Teorema de Pappus e o Teorema de Pascal, queremos facilitar a compreensão e a visualização de algumas das técnicas de perspectiva que podem ser adaptadas para o uso na sala de aula pelos professores da Educação B ́ sica. A aplicação de uma oficina de Geometria Projetiva em uma turma do 6o ano do Ensino Fundamental e a avaliação dos resultados revelaram que o tema pode ser desenvolvido de maneira promissora com os estudantes na Educação B ́ sica, obtendo uma melhor compreensão do objeto real e associando-o ao conteúdo matemático envolvido.
In search for a better representation of three-dimensional reality, non-Euclidean Geometries offer an alternative to the classic euclidianism and the Projective Geometry is one of the highlights. The purpose of this word is contribute to the assimilation of definitions such as perspective, projection, and the principle of duality, through illustrations. And, from important results as Desargues’ Theorem, Pappus’ Theorem and Pascal’s Theorem, we want to facilitate understanding and viewing some of the perspective techniques that can be adapted for use in classroom by Basic Education teachers. The application of a workshop of Projective Geometry in a class of 6th grade of elementary school and the evaluation of the results revealed that the theme can be developed in a promising way with students in basic education, getting a better comprehension of the real object and associating it to the mathematical content involved.

Today, the main research field for the automotive industry is to find solutions for active safety. In order to perceive the surrounding environment, tracking nearby traffic objects plays an important role. Validation of the tracking performance is often done in staged traffic scenarios, where additional sensors, mounted on the vehicles, are used to obtain their true positions and velocities. The difficulty of evaluating the tracking performance complicates its development. An alternative approach studied in this thesis, is to record sequences and use non-causal algorithms, such as smoothing, instead of filtering to estimate the true target states. With this method, validation data for online, causal, target tracking algorithms can be obtained for all traffic scenarios without the need of extra sensors. We investigate how non-causal algorithms affects the target tracking performance using multiple sensors and dynamic models of different complexity. This is done to evaluate real-time methods against estimates obtained from non-causal filtering. Two different measurement units, a monocular camera and a LIDAR sensor, and two dynamic models are evaluated and compared using both causal and non-causal methods. The system is tested in two single object scenarios where ground truth is available and in three multi object scenarios without ground truth. Results from the two single object scenarios shows that tracking using only a monocular camera performs poorly since it is unable to measure the distance to objects. Here, a complementary LIDAR sensor improves the tracking performance significantly. The dynamic models are shown to have a small impact on the tracking performance, while the non-causal application gives a distinct improvement when tracking objects at large distances. Since the sequence can be reversed, the non-causal estimates are propagated from more certain states when the target is closer to the ego vehicle. For multiple object tracking, we find that correct associations between measurements and tracks are crucial for improving the tracking performance with non-causal algorithms.

This dissertation describes a method for modeling, simulating and real-time rendering piecewise linear approximations of generic non-Euclidean 3D spaces. The 3D rendering pipeline most commonly used, where one multiplies each vertex coordinate by a 4x4 matrix to project it on the screen does not work for all cases where the space does not obey Euclid’s postulates (non-Euclidean space). Furthermore, while other non-Euclidean rendering tools only work for a limited type of spaces, our approach allows us to model, simulate, and render any isometrically embeddable non-Euclidean space and eventual objects lying therein. We envision at least two main applications for our approach. The first for helping mathematicians get a better understanding of what arbitrary spaces look like (e.g., hyperconical space, hyper-spherical space, and so forth). The second for assisting physicists to visualize and simulate the effects of bent space (e.g., black holes, wormholes, Alcubierre drive, and so forth) on light, and on physical objects
Esta dissertação descreve um método para modelar, simular e renderizar aproximações lineares de espaços não Euclideanos de forma genérica e em tempo real. A técnica de renderização 3D mais comum, que multiplica a matriz de projeção 4 x 4 por cada vértice para determinar as coordenadas do respetivo pixel no ecrã, nem sempre funciona quando o espaço não obedece a um postulado de Euclides (espaço não-Euclideano). Além disso, enquanto outras ferramentas para renderizar espaços não-Euclideanos só funcionam para certos tipos de espaços, a nossa técnica permite modelar, simular e renderizar qualquer espaço não-Euclideano embebível isometricamente, bem como eventuais objetos nele existentes. Antevemos pelo menos dois usos para a nossa técnica. A primeira para ajudar matemáticos a compreender melhor o aspeto de espaços arbitrários (e.g., espaço hiper-cónico, espaço hiper-esférico, etc.). A segunda para ajudar físicos a visualizar e simular os efeitos de espaço curvo (e.g., buracos negros, buracos de minhoca, deformações Alcubierra drive, etc.) em luz e objetos físicos circundantes.

Since at least the Greek classic period that thought, within the western civilizations, has been sharing a strict relationship with Euclidean principles, which have influenced and characterized it, leading to a specific type of reasoning and identity. In turn, as expressions of the mind, the forms that we have been thinking about and have brought to material reality, have been following these same Euclidean principles. Thought has shared also a closed relationship with architecture and architecture space form. A relationship that became even more pronounced with the incoming phenomenon of the Industrial Revolution with its standardization and mass production techniques and technologies. Ever since, the majority of architecture spaces that we have been thinking about and eventually building, follow and share Euclidean-orthogonal principles and relationships. However, with the arrival of the 20th and 21st centuries’ Digital Revolutions and their novel representation, visualization and production techniques and technologies, the forms that we think about and manage to produce, have achieved an unprecedented range of freedom, in which both Euclidean and non-Euclidean free forms are considered. This happening opened a pertinent and relevant thinking and discussion on whether humans, in their nature and within a valid freedom of choice, tend to prefer the long settled Euclidean, orthogonal-based architecture spaces, with all the elements that such geometry implies, namely, the presence of angular, sharp edges and vertices, or, on the contrary, they tend to prefer non-Euclidean, curved, rounded architecture spaces. This thesis proposes to address the problem of the preference for form, namely architecture space form, divided in two sub-problems that the literature review helped to identify: aesthetic judgements and approach-avoidance decisions, two judgements that, in turn, may rely in two knowledge ‘databases’: a subjective-based one, build through our life time sensible and rational experiences, and, a more objective-based one, which hides behind our genetic legacy and lays on basic evolutionary defense functions or mechanisms. We will approach this thesis Problem and Research Question, through (i) a free discourse on historic key events that, through our evolutionary stages, may have contributed to the fact that we have been closer to some elements, namely forms and architecture form, with certain geometric characteristics, over others; (ii) the evolution of aesthetics and our basic evolutionary defense functions or mechanisms, through qualitative and quantitative research methodologies, (iii) state-of-the-art review on the topic of preference for elements with distinct geometric characteristics, and (iv) our own developed experimental user study on abstract architecture spaces with distinct geometric characteristics at the contour level, which, based on the thesis two sub-problems, tried to validate our raised hypotheses. The results of this thesis suggest that humans prefer abstract architecture spaces with curved, rounded elements, rather than those equipped with angular, sharp ones. On the other hand, they were inconclusive on whether we prefer Euclidean-orthogonal or full non-Euclidean abstract architecture spaces, possibly due to familiarity (“mere-exposure”) and ‘strangeness’ effects. These results validate and partial validate hypotheses ‘H1’ and ‘H2’, respectively, the two major hypotheses of this thesis.
Desde pelo menos o período grego clássico que o pensamento das civilizações ocidentais têm partilhado uma relação estreita com princípios Euclidianos, algo que tem influenciado e caracterizado este pensamento, orientando-o para uma forma específica de raciocínio e identidade. Por sua vez, enquanto expressões da mente, as formas que temos vindo a pensar e a trazer para a realidade material têm seguido estes mesmos princípios Euclidianos. Enquanto extensão do pensamento, o mesmo se tem aplicado à arquitetura, nomeadamente à sua forma. Esta relação tornou-se ainda mais pronunciada com a chegada do fenómeno da Revolução Industrial e as suas técnicas e tecnologias de estandardização e produção em série. Desde então, a maioria dos espaços de arquitetura que temos pensado e construído seguem e partilham princípios e relações ortogonais-Euclidianos. No entanto, com a entrada em cena das revoluções digitais dos séculos XX e XXI e as suas inovadoras técnicas e tecnologias de representação, visualização e produção, as formas que temos vindo a pensar e a conseguir produzir, atingiram um grau de liberdade sem precedentes, no qual entram em consideração tanto as formas Euclidianas como não-Euclidianas. Este acontecimento abriu uma discussão pertinente e relevante sobre se, na eventualidade de uma válida liberdade de escolha, os humanos efetivamente preferem os, tão presentes e enraizados, espaços de arquitetura ortogonais-Euclidianos, com todos os elementos que esta geometria implica, nomeadamente, a presença de arestas e vértices angulares e afiados, ou, pelo contrário, preferem espaços de arquitetura não-Euclidianos, com elementos curvos e arredondados. Esta tese de doutoramento propõe, neste sentido, abordar o problema da preferência pela forma, nomeadamente, pela forma dos espaços de arquitetura, dividido pelos dois sub-problemas que a revisão bibliográfica ajudou a identificar: Julgamentos estéticos e decisões de aproximação ou afastamento, dois julgamentos que, por sua vez, podem estar baseados em duas ‘bases de dados’ de conhecimento: uma subjetiva, construída ao longo das nossas experiências sensíveis e racionais, e, uma mais objetiva, que se esconde atrás do nosso legado genético e assenta em funções ou mecanismos básicos de defesa evolutiva. O Problema e a Pergunta de Investigação desta tese serão abordados através de (i) um discurso livre sobre eventos históricos chave que, ao longo da nossa evolução, possam ter contribuído para o facto de podermos ter mantido um maior grau de aproximação em relação a determinados elementos, nomeadamente, formas e a forma da arquitetura, com determinadas características geométricas, sobre outros; (ii) a evolução do estudo da estética e as nossas funções ou mecanismos básicos de defesa evolutiva, através metodologias de investigação qualitativa e quantitativa, (iii) a revisão do estado-da-arte sobre a preferência de elementos com características geométricas distintas e (iv) o desenvolvimento de um estudo experimental sobre espaços abstratos de arquitetura com características geométricas distintas ao nível do contorno, que, baseado nos dois sub-problemas desta tese, procurou validar as hipóteses de estudo levantadas. Os resultados desta tese sugerem que os humanos preferem espaços abstratos de arquitetura com elementos curvos e arredondados em relação àqueles dotados de elementos angulares e afiados. Por outro lado, foram inconclusivos quanto ao facto de podermos preferir espaços de arquitetura Euclidianos-ortogonais àqueles puramente não-Euclidianos. Estes resultados validam e validam parcialmente as hipóteses H1 e H2, respetivamente, as duas hipóteses principais desta tese de doutoramento.

The Vlasov-Poisson system is most commonly used to model the movement of charged particles in a plasma or of stars in a galaxy. It consists of a kinetic equation known as the Vlasov equation coupled with a force determined by the Poisson equation. The system in Euclidean space is well-known and has been extensively studied under various assumptions. In this paper, we derive the Vlasov-Poisson equations assuming the particles exist only on the 2-sphere, then take an in-depth look at particles which initially lie along a great circle of the sphere. We show that any great circle is an invariant set of the equations of motion and prove that the total energy, number of particles, and entropy of the system are conserved for circular initial distributions.
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