In computer graphics technique, the 3D graphic pipeline flow has two processing modules: Geometry module and Rendering module. The geometry module supports vertex coordinate transformation, vertex lighting computation, backface-culling, pre-clipping, and clipping functions. Clipping module clips the outside part of objects by view volume boundaries. Adding clipping module into geometry module will make 3D graphics pipeline flow more efficiency. Due to the sequential parsing nature of clipping, it causes the challenges to implement clipping function in hardware design. This paper implements a dual-path clipping engine placed after the Vertex Shader in geometry module and supports OpenGL-ES 2.0 specification. With the clipping engine, it reduces the unnecessary operations in 3D graphics pipeline flow and makes the performance efficient. The pipelined and shared hardware design is proposed to improve the area cost and throughput of the interpolation operation in clipping engine. The two vertices in/out clipping method is proposed in this paper. Users have more different choices of clipping algorithms for hardware implementation with respect to the performance and hardware limitation.

The focus of this study is scattering matrices in the framework of quantum graphs,more precisely the matrices which describe equi-transmission. They are unitary andHermitian and are independent of the energies of the associated system. In the firstarticle it is shown that in the case where reflection does not occur, such matrices existonly in even dimensions. A complete description of the matrices in dimensions 2, 4,and 6 is given. In dimension 6, 60 five-parameter families are obtained. The 60 matricesyield a combinatorial bipartite graph K62. In the second article it is shown that whenreflection is allowed, the standard matching conditions matrix is equi-transmitting forany dimension n. All equi-transmitting matrices up to order 6 are described. For oddn (3 and 5), the standard matching conditions matrix is the only equi-transmitting matrix.For even n (2, 4 and 6) there exists other equi-transmitting matrices apart fromthose equivalent to the standard matching conditions. All such additional matriceshave zero trace.

Nagrinėjami atsitiktiniai sankirtų grafai G(n,m,p)jų viršūnių laipsnių skirstinius. Įrodyta, kad grafo viršūnės laipsnis turi Binominį pasiskirstymą. Rasta išraiška tikimybės p, kad dvi grafo viršūnės renkasi bendrą objektą. / Random intersection graphs audits vertex degree distributions are viewed. Its proved, vertex degree has Binomial distribution. Probability p that two vertex of graph chooses common object is find.

Thesis (Ph. D.)--University of Oregon, 2008. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 40-41). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

vii, 41 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We generalize a theorem of Zhu relating the trace of certain vertex algebra representations and modular invariants to the arena of vertex super algebras. The theorem explains why the space of simple characters for the Neveu-Schwarz minimal models NS( p, q ) is modular invariant. It also expresses negative products in terms of positive products, which are easier to compute. As a consequence of the main theorem, the subleading coefficient of the singular vectors of NS( p, q ) is determined for p and q odd. An interesting family of q -series identities is established. These consequences established here generalize results of Milas in this field. / Adviser: Arkady Vaintrob

Orientador: Cristiano Torezzan / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T15:05:42Z (GMT). No. of bitstreams: 1 Monticeli_AndreRodrigues_M.pdf: 7043231 bytes, checksum: 683696a5c1b284a08a9d19c54647edaa (MD5) Previous issue date: 2010 / Resumo: Neste trabalho abordamos o problema de descrever o conjunto solução de um sistema de inequações lineares. Este problema está fortemente relacionado com o problema clássico da enumeração de vértices de um poliedro. Descrevemos o método de Fourier-Motzkin que pode ser utilizado para eliminar variáveis de um sistema de inequações lineares e projetar a região de solução num espaço de dimensão menor. Mostramos como o problema da enumeração de vértices pode ser convertido em um problema de encontrar o fecho convexo do conjunto de pontos dual ao sistema de inequações lineares, uma vez encontrado um ponto interior factível. Alguns algoritmos para o fecho convexo de um conjunto finito de pontos e também para encontrar um ponto interior factível são estudados. Nosso interesse, além de listar os vértices e as faces é também visualizar a região de solução utilizando um programa computacional. Para tanto propomos um método que constrói a lista dos vértices e faces do poliedro definido por um dado sistema de inequações lineares e grava o resultado num arquivo de texto puro com extensão obj, que é compatível com os principais softwares de visualização gráfica 3D. O método foi implementado no Octave e diversos testes foram feitos, analisando o custo computacional e possíveis dificuldades que podem surgir devido a erros numéricos ou falta de memória / Abstract: In this work we approach the problem of describing the solution of a system of linear inequalities. This problem is closely related to the classical problem known as vertex enumeration. We describe the method of Fourier-Motzkin, that can be used to eliminate variables in a system of linear inequalities, projecting its solution in a lower dimensional space. We show how the vertex enumeration problem can be converted into an equivalent problem of finding the convex hull of a set of dual points, once found a feasible interior point. Some algorithms for convex hull and also for finding a feasible interior point are studied. Our interest is not only to store the vertices and faces but also visualize the correspondent polyhedron using a computer graphics software. In this way we propose a method that stores the polyhedron's vertices and faces and output the results into a plain text _le with extension obj, which is a geometric definition file format that can be opened with all major 3D graphics software. The method was implemented in Octave and several tests were made, analyzing the computational cost and possible difficulties that may arise due to numerical errors or memory requirements / Mestrado / Matematica / Mestre em Matemática

Most of the research on domination focuses on vertices dominating other vertices. In this paper we consider vertexedge domination where a vertex dominates the edges incident to it as well as the edges adjacent to these incident edges. The minimum cardinality of a vertex-edge dominating set of a graph G is the vertex-edge domination number γve(G). We present bounds on γve(G) and relationships between γve(G) and other domination related parameters. Since any ordinary dominating set is also a vertex-edge dominating set, it follows that γve(G) is bounded above by the domination number of G. Our main result characterizes the trees having equal domination and vertex-edge domination numbers.

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one that are γ t-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γt-critical graph for k≤8 and provide an example which shows that the maximum diameter is in general at least 5k/3 - O(1).

This essay will present the vertex pursuit game of cops and robbers and the problem that made it famous: Meinyel's conjecture. The conjecture stood unproved from 1987 until 2010 when Łuczak and Prałat proved the conjecture with their "Zig-Zag Theorem", which is also covered in the essay.

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