Esta dissertação expõe sobre a resolubilidade do campo vetorial complexo L = ∂ /∂t +(a(x) + ib(x))∂/∂x, b ≢ 0 definido em Ωε = (-ε, ε) × S1, ε > 0, perto do conjunto característico Σ = {0} × S1, sendo a e b funções de classe C∞ em (- ε, ε) a valores reais. Os resultados apresentados mostram que a resolubilidade de L em uma vizinhança cheia de Σ depende da relação entre as ordens de anulamento de a e b em x = 0. / This dissertation deals with the solvability of complex vector fieldL = ∂ /∂t +(a(x) + ib(x))∂/∂x, b ≢ 0 defined on Ωε = (-ε ε) × S1, ε > 0, near the characteristic set Σ = {0} × S1, where a and b are C∞ real-valued functions in (- ε, ε). The presented results show hat solvability of L in a full neighborhood of Σ depends on the interplay between the order of vanishing of the functions and a and b at x = 0.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

Let G be a finite group and V be an orthogonal four-dimensional real representation space of G where the action of G is non-free. We give necessary and sufficient conditions for the existence of a G-equivariant vector field on the representation sphere of V in the cases G is the dihedral group, the generalized quaternion group and the semidihedral group in terms of decomposition of V into irreducible representations. In the case G is abelian, where the solution is already known, we give a more elementary solution.

This report discusses two methods of visualizing complex solutions of polynomials: modulus surfaces and vector fields. Both provide valuable information about the location of complex solutions and their multiplicity. A sketch of a proof of The Fundamental Theorem of Algebra utilizing modulus surfaces and complex analysis is also included. / text

The goal of this thesis is the development of a fast and robust algorithm that is able to detect patterns in flow fields independent from their orientation and adequately visualize the results for a human user. This thesis is an interdisciplinary work in the field of vector field visualization and the field of pattern recognition. A vector field can be best imagined as an area or a volume containing a lot of arrows. The direction of the arrow describes the direction of a flow or force at the point where it starts and the length its velocity or strength. This builds a bridge to vector field visualization, because drawing these arrows is one of the fundamental techniques to illustrate a vector field. The main challenge of vector field visualization is to decide which of them should be drawn. If you do not draw enough arrows, you may miss the feature you are interested in. If you draw too many arrows, your image will be black all over. We assume that the user is interested in a certain feature of the vector field: a certain pattern. To prevent clutter and occlusion of the interesting parts, we first look for this pattern and then apply a visualization that emphasizes its occurrences. In general, the user wants to find all instances of the interesting pattern, no matter if they are smaller or bigger, weaker or stronger or oriented in some other direction than his reference input pattern. But looking for all these transformed versions would take far too long. That is why, we look for an algorithm that detects the occurrences of the pattern independent from these transformations. In the second part of this thesis, we work with moment invariants. Moments are the projections of a function to a function space basis. In order to compare the functions, it is sufficient to compare their moments. Normalization is the act of transforming a function into a predefined standard position. Moment invariants are characteristic numbers like fingerprints that are constructed from moments and do not change under certain transformations. They can be produced by normalization, because if all the functions are in one standard position, their prior position has no influence on their normalized moments. With this technique, we were able to solve the pattern detection task for 2D and 3D flow fields by mathematically proving the invariance of the moments with respect to translation, rotation, and scaling. In practical applications, this invariance is disturbed by the discretization. We applied our method to several analytic and real world data sets and showed that it works on discrete fields in a robust way.

A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.

Made available in DSpace on 2014-06-11T19:27:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01Bitstream added on 2014-06-13T20:27:38Z : No. of bitstreams: 1 silva_mv_me_rcla.pdf: 927964 bytes, checksum: 1bf8757069fd7950b3ef35b7c13da6ba (MD5) / O objetivo deste trabalho é apresentar uma demonstração combinatória dos teore- mas de Índice de Poincaré, a saber: Sejam D um disco e γ seu bordo. Seja V um campo vetorial contínuo sobre D com pontos críticos isolados P1, P2, . . . , Pn pertencentes ao interior de D. Se V nunca se anula em γ, então W(γ) = I(P1) + I(P2) + . . . + I(Pn), onde I(Pi) é o índice do ponto crítico Pi e W(γ) o número de voltas de V sobre γ. Seja V um campo vetorial tangente contínuo sobre uma superfície compacta, co- nexa e orientável S. Então a soma dos índices dos pontos críticos de V é igual à característica de Euler de S. / bstract In this work we present a combinatorial proof for the Poincaré index theorems. Let V be a continuous vector field. Let D be a cell and γ its boundary. Supposing that V is not zero on γ, then W(γ) = I(P1) + I(P2) + . . . + I(Pn) where P1, P2, . . . , Pn are the critical points of V inside D, I(Pi) is the index of Pi, and W(γ) is the winding number of V on γ. Let V be a continuous tangent vector field on a compact, connected, orientable surface S. Then the sum of the indexes of the critical points of V equals the Euler characteristic of S.

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-02-22Bitstream added on 2014-06-13T19:54:22Z : No. of bitstreams: 1 moraes_jr_me_sjrp.pdf: 1163228 bytes, checksum: 853fa9bee4a6a3c25b24de14990f3221 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Consideramos dois casos principais de bifurcação de órbitas periódicas não hiperbólicas que dão origem a ciclos limite. Nosso estudo é feito para sistemas lineares por partes com três zonas em sua fórmula mais geral, que inclui situações sem simetria. Obtemos estimativas tanto para a amplitude como para o período do ciclo limite e apresentamos uma aplicação de interesse em engenharia: sistemas de controle. / We consider two main cases of bifurcation of non hyperbolic periodic orbits that give rise to limit cycles. Our study is done concerning piecewise linear systems with three zones in the more general formula that includes situations without symmetry. We obtain estimates for both the amplitude and the period of limit cycles and we present a applications of interest in engineering: control systems.

Made available in DSpace on 2014-06-11T19:26:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-28Bitstream added on 2014-06-13T19:06:23Z : No. of bitstreams: 1 morettijunior_a_me_sjrp.pdf: 762570 bytes, checksum: 59d4b94fad96e41726548c623175fe4e (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho temos como objetivo estudar o número e a distribuição de ciclos limites em sistemas diferenciais lineares por partes. Em particular estudamos o número de ciclos limites do sistema diferencial linear por partes planar ˙x = −y − ε φ ( x) , ˙y = x, onde ε 6= 0 é um parâmetro pequeno e φ é uma função periódica linear por partes ímpar de período 4 . Provamos que dado um inteiro arbitário positivo n, o sistema acima possui exatamente n ciclos limites na faixa |x| ≤ 2 (n + 1 ). Consequentemente, existem sistemas diferenciais lineares por partes contendo uma infinidade de ciclos limites no plano real. Inicialmente obtemos uma quota inferior par a o número destes ciclos limites na faixa | x| ≤ 2 (n + 1 ) via Teoria do Averaging . Em seguida , utilizando a Teoria de Campos de Vetores Rodados, verificamos que o sistema acima tem exatamente n ciclos limites na faixa | x| ≤ 2 (n + 1 ) / The main goal of this work aim to study the number and distribution of limit cycles in piecewise linear differential systems. In particular we consider the planar piecewise linear differential system ˙x = −y − ε φ ( x) , ˙y = x, where ε 6= 0 is a small parameter and φ is an odd piecewise linear periodic function of period 4 . We prove that given an arbitrary positive integer n, the system above has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 ) . Consequently, there are piecewise differential systems containing an infinite number of limit cycles in the real plane. First we get a lower bound on the number of limit cycles in the strip |x| ≤ 2 (n + 1 ) via Averaging Theory. In the following , using the Theory of Rotated Vector Fields, we see that above system has exactly n limit cycles in the strip | x| ≤ 2 (n + 1 )

Esta dissertação expõe sobre a resolubilidade do campo vetorial complexo L = ∂ /∂t +(a(x) + ib(x))∂/∂x, b ≢ 0 definido em Ωε = (-ε, ε) × S1, ε > 0, perto do conjunto característico Σ = {0} × S1, sendo a e b funções de classe C∞ em (- ε, ε) a valores reais. Os resultados apresentados mostram que a resolubilidade de L em uma vizinhança cheia de Σ depende da relação entre as ordens de anulamento de a e b em x = 0. / This dissertation deals with the solvability of complex vector fieldL = ∂ /∂t +(a(x) + ib(x))∂/∂x, b ≢ 0 defined on Ωε = (-ε ε) × S1, ε > 0, near the characteristic set Σ = {0} × S1, where a and b are C∞ real-valued functions in (- ε, ε). The presented results show hat solvability of L in a full neighborhood of Σ depends on the interplay between the order of vanishing of the functions and a and b at x = 0.