Thesis (M.C.S.)--Carleton University, 2004. / Includes bibliographical references (p. 70-72). Also available in electronic format on the Internet.

Thesis (M.Sc.)--Memorial University of Newfoundland. / Bibliography : leaves 62-63. Also available online.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-04-13Bitstream added on 2014-06-13T20:55:43Z : No. of bitstreams: 1 mialich_fr_me_sjrp.pdf: 707207 bytes, checksum: 0335d58e9114dcc029b4d78ec2a82f51 (MD5) / Este trabalho tem como tema central Poliedros e o Teorema de Euler. Foi feita uma breve análise da definição de Poliedros e apresentadas algumas considerações históricas a respeito dos Poliedros e o Teorema de Euler. Foram abordadas duas versões/demonstrações do Teorema de Euler, a primeira para poliedros convexos, e a segunda, conhecida como de Teorema de Euler segundo Cauchy (que engloba certos poliedros não convexos, que são homeomorfos à esfera). Ainda, como consequência do Teorema de Euler, foi demonstrado o teorema da existência de apenas cinco poliedros regulares, conhecidos como Poliedros de Platão. Analisou-se também o conteúdo/ensino de Poliedros em certos documentos oficiais (PCN, Currículo do Estado de SP, Matrizes do SARESP e ENEM). Por último foi elaborada uma proposta de atividades explorando poliedros, o Teorema de Euler e os conteúdos de área, volume e planificação, bem como análise e resolução de algumas questões do SARESP e ENEM (relativas a tais conteúdos), utilizando, para melhor compreensão e visualização, o software matemático Poly, a fim de construir uma aprendizagem mais significativa para os alunos. Com o desenvolvimento do trabalho foi possível compreender melhor a definição de poliedros, o Teorema de Euler e refletir um pouco sobre o desenvolvimento das pesquisas Matemáticas a partir de alguns aspectos históricos. Através da análise do conteúdo, em certos documentos oficiais, pode-se verificar que o assunto/tema tratado no trabalho faz parte desses documentos e têm sido cobrados em avaliações, mas, em geral, de forma bastante simples. Observamos também, que em algumas questões analisadas os enunciados não estavam muito claros, o que pode confundir os alunos / This work has as its central theme Polyhedra and Euler's Theorem. We made a brief analysis of the definition of Polyhedra and some historical considerations about Polyhedra and Euler's Theorem. We considered two versions/proofs of Euler's Theorem, the first for convex polyhedra, and the second, known as Euler's Theorem according to Cauchy (which includes certain nonconvex polyhedra that are homeomorphic to the sphere). Also, as a consequence of Euler's theorem, it was demonstrated the theorem of existence of only five regular polyhedra, known as Plato’s Polyhedra. We also analyzed the content/teaching of Polyhedra in certain official documents (PCN, SP State Curriculum, SARESP matrices and ENEM). Finally we presented a proposal of activities exploring polyhedra, Euler's Theorem, the contents area, volume and planning, as well as an analysis and resolution of some questions from SARESP and ENEM (for such contents), by using, for better understanding and visualization, the Poly mathematical software in order to build a more meaningful learning for students. With the development of this work we got a better understand of the definition of polyhedra, the Euler's Theorem and we reflected a little on research and development of mathematics from some historical aspects. By analysing the content in certain official documents, it can be seen that the subject/topics covered in this work are parts of these documents and have been rated in tests, but generally in a quite simple form. We also observed that in some questions discussed, the statements were not very clear, which can confuse the students

Orientador: Ermínia de Lourdes Campello Fanti / Banca: João Carlos Viera Sampaio / Banca: Flávia Souza Machado da Silva / O PROFMAT - Programa de Mestrado Profissional em Matemática em Rede Nacional é coordenado pela Sociedade Brasileira de Matemática e realizado por uma rede de Instituições de Ensino Superior. / Resumo: Este trabalho tem como tema central Poliedros e o Teorema de Euler. Foi feita uma breve análise da definição de Poliedros e apresentadas algumas considerações históricas a respeito dos Poliedros e o Teorema de Euler. Foram abordadas duas versões/demonstrações do Teorema de Euler, a primeira para poliedros convexos, e a segunda, conhecida como de Teorema de Euler segundo Cauchy (que engloba certos poliedros não convexos, que são homeomorfos à esfera). Ainda, como consequência do Teorema de Euler, foi demonstrado o teorema da existência de apenas cinco poliedros regulares, conhecidos como Poliedros de Platão. Analisou-se também o conteúdo/ensino de Poliedros em certos documentos oficiais (PCN, Currículo do Estado de SP, Matrizes do SARESP e ENEM). Por último foi elaborada uma proposta de atividades explorando poliedros, o Teorema de Euler e os conteúdos de área, volume e planificação, bem como análise e resolução de algumas questões do SARESP e ENEM (relativas a tais conteúdos), utilizando, para melhor compreensão e visualização, o software matemático Poly, a fim de construir uma aprendizagem mais significativa para os alunos. Com o desenvolvimento do trabalho foi possível compreender melhor a definição de poliedros, o Teorema de Euler e refletir um pouco sobre o desenvolvimento das pesquisas Matemáticas a partir de alguns aspectos históricos. Através da análise do conteúdo, em certos documentos oficiais, pode-se verificar que o assunto/tema tratado no trabalho faz parte desses documentos e têm sido cobrados em avaliações, mas, em geral, de forma bastante simples. Observamos também, que em algumas questões analisadas os enunciados não estavam muito claros, o que pode confundir os alunos / Abstract: This work has as its central theme Polyhedra and Euler's Theorem. We made a brief analysis of the definition of Polyhedra and some historical considerations about Polyhedra and Euler's Theorem. We considered two versions/proofs of Euler's Theorem, the first for convex polyhedra, and the second, known as Euler's Theorem according to Cauchy (which includes certain nonconvex polyhedra that are homeomorphic to the sphere). Also, as a consequence of Euler's theorem, it was demonstrated the theorem of existence of only five regular polyhedra, known as Plato's Polyhedra. We also analyzed the content/teaching of Polyhedra in certain official documents (PCN, SP State Curriculum, SARESP matrices and ENEM). Finally we presented a proposal of activities exploring polyhedra, Euler's Theorem, the contents area, volume and planning, as well as an analysis and resolution of some questions from SARESP and ENEM (for such contents), by using, for better understanding and visualization, the Poly mathematical software in order to build a more meaningful learning for students. With the development of this work we got a better understand of the definition of polyhedra, the Euler's Theorem and we reflected a little on research and development of mathematics from some historical aspects. By analysing the content in certain official documents, it can be seen that the subject/topics covered in this work are parts of these documents and have been rated in tests, but generally in a quite simple form. We also observed that in some questions discussed, the statements were not very clear, which can confuse the students / Mestre

Understanding the dynamic behavior of particulate materials is extremely important to many industrial processes with a wide range of applications ranging from hopper flows in agriculture to tumbling mills in the mining industry. Thus simulating the dynamics of particulate materials is critical in the design and optimization of such processes. The mechanical behavior of particulate materials is complex and cannot be described by a closed form solution for more than a few particles. A popular and successful numerical approach in simulating the underlying dynamics of particulate materials is the discrete element method (DEM). However, the DEM is computationally expensive and computationally viable simulations are typically restricted to a few particles with realistic particle shape or a larger number of particles with an often oversimplified particle shape. It has been demonstrated for numerous applications that an accurate representation of the particle shape is essential to accurately capture the macroscopic transport of particulates. The most common approach to represent particle shape is by using a cluster of spheres to approximate the shape of a particle. This approach is computationally intensive as multiple spherical particles are required to represent a single non-spherical particle. In addition spherical particles are for certain applications a poor approximation when sharp interfaces are essential to capture the bulk transport behavior. An advantage of this approach is that non-convex particles are handled with ease. Polyhedra represent the geometry of most convex particulate materials well and when combined with appropriate contact models exhibit realistic transport behavior to that of the actual system. However detecting collisions between the polyhedra is computationally expensive, often limiting simulations to only a few thousand of particles. Driven by the demand for real-time graphics, the Graphical Processor Unit (GPU) offers cluster type performance at a fraction of the computational cost. The parallel nature of the GPU allows for a large number of simple independent processes to be executed in parallel. This results in a significant speed up over conventional implementations utilizing the Central Processing Unit (CPU) architecture, when algorithms are well aligned and optimized for the threading model of the GPU. This thesis investigates the suitability of the GPU architecture to simulate the transport of particulate materials using the DEM. The focus of this thesis is to develop a computational framework for the GPU architecture that can model (i) tens of millions of spherical particles and (ii) millions of polyhedral particles in a realistic time frame on a desktop computer using a single GPU. The contribution of this thesis is the development of a novel GPU computational frame- work Blaze-DEM, that encompasses collision detection algorithms and various heuristics that are optimized for the parallel GPU architecture. This research has resulted in a new computational performance level being reached in DEM simulations for both spherical / Thesis (PhD)--University of Pretoria, 2015. / Mechanical and Aeronautical Engineering / PhD / Unrestricted

No description available.

Computer Aided Design systems intended for three dimensional solid modelling have traditionally used geometric representations incompatible with established representations in computer vision. The utilization of object models built using these systems require a representation conversion before they can be used in automatic sensing systems. Considerable advantages follow from building a combined CAD and sensing system based on a common geometric model. For example, a library of objects can be built up and its models used in vision and touch sensing system integrated into an automated assembly line to 'discriminate between objects and determine- orientation and distance. This thesis studies a representation scheme, the dual spherical representation, useful in geometric modelling and machine recognition. We prove that the representation uniquely represents genus 0 polyhedra. We show by,example that our representation is not a strict dual of the vertex connectivity graph, and hence is not necessarily ambiguous. However, we have not been able to prove that the representation is unambiguous. An augmented dual spherical representation which is unique for general polyhedra is presented. This graph theoretic approach to polyhedra also results in an elegant method for decomposition of polyhedra into combinatorially convex parts. An algorithm implementation details and experimental results for recognition of polyhedra using a large field tactile sensor are given. A theorem relating the edges in the dual spherical representation and the edge under perspective projection is proved. Sensor fusion using visual and tactile sensory inputs is proposed to improve recognition rates. / Master of Science

In this thesis, we study foldings of orthogonal polygons into orthogonal polyhedra. The particular problem examined here is whether a paper cutout of an orthogonal polygon with fold lines indicated folds up into a simple orthogonal polyhedron. The folds are orthogonal and the direction of the fold (upward or downward) is also given. We present a polynomial time algorithm to solve this problem. Next we consider the same problem with the exception that the direction of the folds are not given. We prove that this problem is NP-complete. Once it has been determined that a polygon does fold into a polyhedron, we consider some restrictions on the actual folding process, modelling the case when the polyhedron is constructed from a stiff material such as sheet metal. We show an example of a polygon that cannot be folded into a polyhedron if folds can only be executed one at a time. Removing this restriction, we show another polygon that cannot be folded into a polyhedron using rigid material.

Two types of problems are studied in this thesis: reconstruction and visualization of polygons and polyhedra. <br /><br /> Three problems are considered in reconstruction of polygons and polyhedra, given a set of projection characteristics. The first problem is to reconstruct a closed convex polygon (polyhedron) given the number of visible edges (faces) from each of a set of directions <em>S</em>. The main results for this problem include the necessary and sufficient conditions for the existence of a polygon that realizes the projections. This characterization gives an algorithm to construct a feasible polygon when it exists. The other main result is an algorithm to find the maximum and minimum size of a feasible polygon for the given set <em>S</em>. Some special cases for non-convex polygons and for perspective projections are also studied. <br /><br /> For reconstruction of polyhedra, it is shown that when the projection directions are co-planar, a feasible polyhedron (i. e. a polyhedron satisfying the projection properties) can be constructed from a feasible polygon and vice versa. When the directions are covered by two planes, if the number of visible faces from each of the directions is at least four, then an algorithm is presented to decide the existence of a feasible polyhedron and to construct one, when it exists. When the directions see arbitrary number of faces, the same algorithm works, except for a particular sub-case. <br /><br /> A polyhedron is, in general, called equiprojective, if from any direction the size of the projection or the projection boundary is fixed, where the "size" means the number of vertices, edge, or faces. A special problem on reconstruction of polyhedra is to find all equiprojective polyhedra. For the case when the size is the number of vertices in the projection boundary, main results include the characterization of all equiprojective polyhedra and an algorithm to recognize them, and finding the minimum equiprojective polyhedra. Other measures of equiprojectivity are also studied. <br /><br /> Finally, the problem of efficient visualization of polyhedra under given constraints is considered. A user might wish to find a projection that highlights certain properties of a polyhedron. In particular, the problem considered is given a set of vertices, edges, and/or faces of a convex polyhedron, how to determine all projections of the polyhedron such that the elements of the given set are on the projection boundary. The results include efficient algorithms for both perspective and orthogonal projections, and improved adaptive algorithm when only edges are given and they form disjoint paths. A related problem of finding all projections where the given edges, faces, and/or vertices are not on the projection boundary is also studied.

Inverse systems, inverse limit spaces, and bonding maps are defined. An investigation of the properties that an inverse limit space inherits, depending on the conditions placed on the factor spaces and bonding maps is made. Conditions necessary to ensure that the inverse limit space is compact, connected, locally connected, and semi-locally connected are examined. A mapping from one inverse system to another is defined and the nature of the function between the respective inverse limits, induced by this mapping, is investigated. Certain restrictions guarantee that the induced function is continuous, onto, monotone, periodic, or open. It is also shown that any compact metric space is the continuous image of the cantor set. Finally, any compact Hausdorff space is characterized as the inverse limit of an inverse system of polyhedra.