Neste trabalho utilizamos a tecnica de construcao de campos de vetores controlados para obter estimativas do valor da filtracao de uma aplicacao polinomial $Theta:R^n,0 ightarrowR^p,0$ para que a familia $f_t=f+tTheta$ seja $C^ell$-$mathcal{G}$-trivial, bi-lipschitz trivial ou topologicamente trivial, onde $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ ou $mathcal{K}$ e $f:R^n,0 ightarrow R^p,0$ e um germe de aplicacao polinomial satisfazendo uma condicao de nao-degeneracao com relacao a algum poliedro de Newton. Obtemos tambem resultados sobre a trivializacao $C^ell$-modificada para familias de aplicacoes semi-quase-homogeneas de classe $C^{ell + 1}$, e familias de funcoes Newton nao-degeneradas de classe $C^{ell + 1}$. / In this work we use controlled vector fields to obtain estimates for the filtration of a polynomial map-germ $Theta:R^n,0 ightarrowR^p,0$ such that the family $f_t=f+tTheta$ is $C^ell$-$mathcal{G}$-trivial, bi-Lipschitz trivial, or topologicaly trivial, where $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ or $mathcal{K}$ and $f:R^n,0 ightarrowR^p,0$ is a polynomial map-germ satisfying a non-degeneracy condition. Results are also obtained on the modified $C^ell$-trivialization for families of semi-wheighted homogeneous maps of class $C^{ell+1}$ with an isolated sigularity at the origin, and families of Newton non-degenerate functions of class $C^{ell+1}$.

Consider a polyhedral surface in three-space that has the property that it can change its shape while keeping all its polygonal faces congruent. Adjacent faces are allowed to rotate along common edges. Mathematically exact flexible surfaces were found by Connelly in 1978. But the question remained as to whether the volume bounded by such surfaces was necessarily constant during the flex. In other words, is there a mathematically perfect bellows that actually will exhale and inhale as it flexes? For the known examples, the volume did remain constant. Following an idea of Sabitov, but using the theory of places in algebraic geometry (suggested by Steve Chase), Connelly et al. showed that there is no perfect mathematical bellows. All flexible surfaces must flex with constant volume. We built several models to illustrate the above theory, in particular, we built a model of the cubeoctahedron after a suggestion by Walser. This model is cut at a line of symmetry, pops up to minimize its energy stored by 4 rubber bands in its interior, and in doing so also maximizes its volume. Three MATLAB programs were written to illustrate how the cuboctahedron is obtained by truncation, how the physical cuboctahedron moves and how the motion of the cubeoctahedron can be described if self-intersection is possible.

Neste trabalho utilizamos a tecnica de construcao de campos de vetores controlados para obter estimativas do valor da filtracao de uma aplicacao polinomial $Theta:R^n,0 ightarrowR^p,0$ para que a familia $f_t=f+tTheta$ seja $C^ell$-$mathcal{G}$-trivial, bi-lipschitz trivial ou topologicamente trivial, onde $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ ou $mathcal{K}$ e $f:R^n,0 ightarrow R^p,0$ e um germe de aplicacao polinomial satisfazendo uma condicao de nao-degeneracao com relacao a algum poliedro de Newton. Obtemos tambem resultados sobre a trivializacao $C^ell$-modificada para familias de aplicacoes semi-quase-homogeneas de classe $C^{ell + 1}$, e familias de funcoes Newton nao-degeneradas de classe $C^{ell + 1}$. / In this work we use controlled vector fields to obtain estimates for the filtration of a polynomial map-germ $Theta:R^n,0 ightarrowR^p,0$ such that the family $f_t=f+tTheta$ is $C^ell$-$mathcal{G}$-trivial, bi-Lipschitz trivial, or topologicaly trivial, where $ellgeq 1$, $mathcal{G}=mathcal{R}$, $mathcal{C}$ or $mathcal{K}$ and $f:R^n,0 ightarrowR^p,0$ is a polynomial map-germ satisfying a non-degeneracy condition. Results are also obtained on the modified $C^ell$-trivialization for families of semi-wheighted homogeneous maps of class $C^{ell+1}$ with an isolated sigularity at the origin, and families of Newton non-degenerate functions of class $C^{ell+1}$.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well.

Etant donné trois entiers k, s et a, nous prouvons dans le premier chapitre qu'il existe un graphe k-régulier fini (resp. un graphe k-régulier connexe fini) dont le groupe d'automorphismes a exactement s orbites sur l'ensemble des sommets et a orbites sur l'ensemble des arêtes si et seulement si (s,a)=(1,0) si k=0, (s,a)=(1,1) si k=1, s=a>0 si k=2, 0< s <= 2a <= 2ks si k>2. (resp. (s,a)=(1,0) si k=0, (s,a)=(1,1) si k=1 ou 2, s-1<=a<=(k-1)s+1 et s,a>0 si k>2.) Nous étudions les polyèdres convexes de R³ dans le second chapitre. Pour tout polyèdre convexe P, nous notons Isom(P) l'ensemble des isométries de R³ laissant P invariant. Si G est un sous-groupe de Isom(P), le f_G-vecteur de P est le triple d'entiers (s,a,f) tel que G ait exactement s orbites sur l'ensemble sommets de P, a orbites sur l'ensemble des arêtes de P et f orbites sur l'ensemble des faces de P. Remarquons que (s,a,f) est le f_{id}-vecteur (appelé f-vecteur dans la littérature) d'un polyèdre si ce dernier possède exactement s sommets, a arêtes et f faces. Nous généralisons un théorème de Steinitz décrivant tous les f-vecteurs possibles. Pour tout groupe fini G d'isométries de R³, nous déterminons l'ensemble des triples (s,a,f) pour lesquels il existe un polyèdre convexe ayant (s,a,f) comme f_G-vecteur. Ces résultats nous permettent de caractériser les triples (s,a,f) pour lesquels il existe un polyèdre convexe tel que Isom(P) a s orbites sur l'ensemble des sommets, a orbites sur l'ensemble des arêtes et f orbites sur l'ensemble des faces. La structure d'incidence I(P) associée à un polyèdre P consiste en la donnée de l'ensemble des sommets de P, l'ensemble des arêtes de P, l'ensemble des faces de P et de l'inclusion entre ces différents éléments (la notion de distance ne se trouve pas dans I(P)). Nous déterminons également l'ensemble des triples d'entiers (s,a,f) pour lesquels il existe une structure d'incidence I(P) associée à un polyèdre P dont le groupe d'automorphismes a exactement s orbites de sommets, a orbites d'arêtes et f orbites de sommets.

A polyhedral graph G is called PCC if every vertex of G has strictly positive combinatorial curvature and the graph is not a prism or antiprism. In this thesis it is shown that the maximum order of a 3-regular PCC graph is 132 and the 3-regular PCC graphs which match that bound are enumerated. A new PCC graph with two 39-faces and 208 vertices is constructed, matching the number of vertices of the largest PCC graphs discovered by Nicholson and Sneddon. A conjecture that there are no PCC graphs with faces of size larger than 39 is made, along with a proof that if there are no faces of size larger than 122, then there is an upper bound of 244 on the order of PCC graphs. / Graduate

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

No description available.

Neste trabalho iremos apresentar uma releitura de um resultado clássico da topologia, na visão da topologia algébrica e em sua notação atual. A demonstração deste, apresentada por Cauchy (1813), é comentada de maneira crítica em Lima (1985a) e para esta apresentação destacaremos as definições, teoremas e entes básicos para o seu entendimento. / In this work we will present a rereading of a classic topology result, in the view of the algebraic topology in its current notation. The proof of this, presented by Cauchy (1813), is critically commented on Lima (1985a) for which we will present the definitions, theorems, basic entities for their understanding.