It is a challenging task to visualize the behavior of time-dependent 3D vector fields. Most of the time an overview of unsteady fields is provided via animations, but, unfortunately, animations provide only transient impressions of momentary flow. In this paper we present two approaches to visualize time varying fields with fixed geometry. Path lines and streak lines represent such a steady visualization of unsteady vector fields, but because of occlusion and visual clutter it is useless to draw them all over the spatial domain. A selection is needed. We show how bundles of streak lines and path lines, running at different times through one point in space, like through an eyelet, yield an insightful visualization of flow structure ('eyelet lines'). To provide a more intuitive and appealing visualization we also explain how to construct a surface from these lines. As second approach, we use a simple measurement of local changes of a field over time to determine regions with strong changes. We visualize these regions with isosurfaces to give an overview of the activity in the dataset. Finally we use the regions as a guide for placing eyelets.

No description available.

No description available.

The superline has one even and one odd coordinate. We consider the Lie superalgebra of contact vector fields on the superline. Its tensor density modules are a one-parameter family of deformations of the natural action on the ring of polynomials on the superline. They are parameterized by a complex number, and they are irreducible when this parameter is not zero. In this dissertation, we describe the annihilating ideals of these representations in the universal enveloping algebra of this Lie superalgebra by providing their generators. We also describe the intersection of all such ideals: the annihilator of the direct sum of the tensor density modules. The annihilating ideal of an irreducible non-zero left module is called a primitive ideal, and the space of all such ideals in the universal enveloping algebra is its primitive spectrum. The primitive spectrum is endowed with the Jacobson topology, which induces a topology on the annihilators of the tensor density modules. We conclude our discussion with a description of the annihilators as a topological space.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2018-03-28T11:58:56Z No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-03-29T11:29:24Z (GMT) No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-03-29T11:29:24Z (GMT). No. of bitstreams: 2 Dissertação - Jeferson Arley Poveda Contreras - 2018.pdf: 763599 bytes, checksum: 6800571168e0aa9de85d151e4c912725 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-03-07 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / The goal of this work is study limit cycles of piecewise smooth vector fields. First, we present the basic theory, passing through the areas of analysis, qualitative theory of differential equations and algebra. We also present basic concepts of Filippov fields, which are indispensable for the study of piecewise smooth fields. In chapter one, was the main topic, a general method for finding limit cycles will be described; in the second chapter limit cycles are found in a piecewise smooth vector field with non-degenerate center being perturbed by a piecewise polynomial vector field. In the fourth chapter, we study limit cycles in piecewise smooth Hamiltonian fields. / O objetivo deste trabalho é estudar ciclos limite de campos de vetores suaves por parte. Primeiro apresentaremos a teoria básica, passando pelas áreas de análise, teoria qualitativa das equações diferenciais e álgebra. Apresentamos também conceitos básicos de campos de Filippov, os quais são imprescindíveis para o estudo dos campos suaves por partes. No capítulo dos, como tópico principal, será descrito um método geral para encontrar ciclos limite; no segundo três são encontrados ciclos limites em um campo de vetores suave por partes com um centro não degenerado sendo perturbado por um polinômio. No quarto capitulo estudaremos os ciclos limites de campos de vetores Hamiltonianos por parte.

Submitted by Liliane Ferreira (ljuvencia30@gmail.com) on 2018-06-14T11:12:47Z No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-06-15T10:25:16Z (GMT) No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-06-15T10:25:16Z (GMT). No. of bitstreams: 2 Tese - João Lopes Cardoso Filho - 2018.pdf: 1729607 bytes, checksum: 8279e98ec23b68bab062f8c812957bf4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-05-18 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / In this work we exhibit canonical forms for 2D codimension one piecewise smooth vector Fields (PSVF). All possible orientations and codimension one scenarios were covered. Also the intrinsic objects that characterize each one of the canonical forms were presented. Also we present topological distinct canonical forms for a larger class for symmetric PSVF where the set of fixed points is contained in the variety os discontinuity. Finally we analyze the simultaneous occurrence of sliding and crossing limit cycle in the case where the piecewise linear vector fields presents a continuum of periodic orbits. / Neste trabalho exibiremos inicialmente as formas canônicas para campos vetoriais suaves por partes (PSVF) no plano. Todas os possíveis cenários de codimensão um são abordados. Também apresentamos formas canônicas topologicamente distintas para uma classe de PSVF com simetria onde o conjunto de pontos fixos está contido na variedade de descontinuidade. Finalmente, analisaremos a ocorrência simultânea de ciclos limite costurantes e deslizantes no caso linear por partes que apresentam um contínuo de órbitas periódicas.

Made available in DSpace on 2014-07-29T16:02:17Z (GMT). No. of bitstreams: 1 UBIRAJARA JOSe GAMA DE CASTRO.pdf: 516477 bytes, checksum: c0a8c62202b2da19be1a2dc69a29e416 (MD5) Previous issue date: 2011-02-24 / The main goal of this work is to study the behavior of Discontinuous Vector Fields in a neighborhood of a tipical singularity (tangency) using for this the regularization process developed by Teixeira and Sotomayor [9] and, using also, some technics of the Geometric Singular Perturbation Theory [2]. / O principal objetivo deste trabalho é estudar o comportamento numa vizinhança de uma singularidade típica (tangência) dos campos vetoriais descontínuos utilizando o processo de regularização desenvolvido por Teixeira e Sotomayor [9] e perturbações singulares [2].

[pt] Um campo de linhas sobre uma superfície é um mapa suave que atribui uma linha tangente a todos, exceto a um número finito de pontos. Esses campos modelam um número de propriedades geométricas e físicas, tais como as direções de curvatura principais nas superfícies ou o fluxo de tensão na elasticidade. Para entender um campo de linha, é usual estudar o comportamento de suas órbitas, que podem apresentar diferentes padrões. Para este fim, consideramos uma abordagem topológica que consiste em utilizar os pontos críticos e separatrices para decompor o campo em regiões de comportamento homogêneo. Focamos em campos que possuem uma estrutura de Morse–Smale. Isso permite operações como o cancelamento de pontos críticos controlados diretamente na decomposição de campo, o que é essencial para a remoção de ruído (simplificação da topologia) em campos provenientes de simulações ou amostragem de problemas do mundo real. Baseado na decomposição de um campo vetorial de Morse–Smale e no cancelamento de pontos críticos, Robin Forman introduziu uma definição discreta para esses campos. O presente trabalho fornece uma definição puramente combinatória para campos de linhas, os campos de linhas discretos, que implicam as construções discretas de Forman para campos de vetores por meio de uma nova representação destes. Campos de linhas discretos admitem uma decomposição que gera uma ponte entre os campos de linhas discretos e suaves, garantindo dessa forma a consistência topológica da definição. Também estabelecemos uma conexão entre um campo de linha discreto e um campo vetorial discreto, desse modo as ferramentas de campos de vetores podem ser usadas em campos de linhas. O trabalho fornece ainda um cancelamento topologicamente consistente de seus elementos críticos para um campo de linha discreto. / [en] A line field on a surface is a smooth map that assigns a tangent line to all but a finite number of points. Such fields model a number of geometric and physical properties, e.g. the principal curvature directions on surfaces or the stress flux in elasticity. They can be seen as a generalization of vector fields. To understand a line field, it is common to study the behavior of its orbits, which can have many different patterns. To this end, we consider a topological approach: we use the critical points and separatrices to decompose the field in regions of similar behavior. We focus on fields that have a Morse–Smale structure. This allows operations like the cancellation of critical points controlled directly in the field decomposition, which is essential for noise removal (topology simplification) on fields coming from simulations or sampling of real-world problems. Based on the decomposition of a Morse–Smale vector field and on cancellation of critical points, Robin Forman introduced a discrete definition for Morse-Smale vector fields. This thesis provides a purely combinatorial definition of line fields, the discrete line fields, entailing Forman s discrete constructions for vector fields through a new representation of these. Discrete line fields admit a (Morse–Smale type of) decomposition that generates a bridge between discrete and smooth line fields, thus guaranteeing the topological consistency of the definition. We also use double branched coverings to suspend discrete line fields to discrete vector fields, so that vector field tools can be used for discrete line fields. Finally we provide, for a discrete line field, a topologically consistent (Morse-like) cancellation of critical elements. This allows a simplification of the discrete line field topology retaining only the most significant features.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfillment of the requirements for the degree of Master of Science. Pretoria 1991. / Remote sensing can be used to produce maps of land-cover, but to be of use to the GIS community these maps must first be vectorized in an intelligent manner. Existing algorithms suffer from the defects of being slow, memory intensive and producing vast quantities of very short vectors. Furthermore if these vectors are thinned via standard algorithms, errors are introduced. The process of vectorizing raster maps is subject to major ambiguities. Thus an infinite family of vector maps ccrresponds to each raster map. This dissertation presents an algorithm for converting raster maps in a rapid manner to accurate vector maps with a minimum of vectors. The algorithm converts raster maps to vector maps using local information only, (a two by two neighbourhood). the method is "exact" in the sense that rasterizing the resulting polygons would produce exactly the same raster map, pixel for pixel. The method is "near optimal" in that it produces, in a local sense, that "exacb" vector map having the least number of vectors. The program is built around a home-grown object oriented Programming System (OOPS) for the C programming language. The main features of the OOPS system, (called OopCdaisy), are virtual and static methods, polymorphism, generalized containers, container indices and thorough error checking, The following general purpose objects are implemented with a large number of sophistiated methods :- Stacks, LIFO lists, scannable containers with indices, trees and 2D objects like points, lines etc. / AC2017

Estudamos uma classe de campos de vetores seccionalmente lineares no plano denotada por X. Tais campos aparecem frequentemente em modelos matemáticos aplicados à engenharia. Baseados no trabalho de J. Sotomayor e R. Garcia [SG03], impondo condições sobre as singularidades, órbitas periódicas e separatrizes, definimos um conjunto de campos de vetores que são estruturalmente estáveis em X. Provamos que esse conjunto é aberto, denso e tem medida de Lebesgue total em X, o qual é um espaço vetorial de dimensão finita. / We study a class of piecewise-linear vector fields in the plane denoted by X. These vector fields appear often in mathematical models applied to Engineering. Based on Jorge Sotomayor and Ronaldo Garcia paper [SG03], we impose conditions on singularities, periodic orbits and separatrices, to define a set of vector fields structurally stable in X. We give a proof that this set is open, dense and has full Lebesgue measure in X, that is a finite dimensional vector space.