| 23/2008 |
Martingales on Principal Fiber Bundles Pedro J. Catuogno, Simão Stelmastchuk Let $P(M,G)$ be a principal fiber bundle, $\omega$ be a connection form on $P(M,G)$ and $\nabla^{P}$ be a projectable connection on $P(M,G)$. The aim of this work is determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.  rp-2008-23.pdf |
| 22/2008 |
Função de Green, Equações Integrais e o Oscilador Harmônico Fracionário Ana Luisa Soubhia, Edmundo Capelas de Oliveira Apresenta-se um estudo da fun\c{c}\~ao de Green associada \`as equa\c{c}\~oes diferenciais do tipo el\'{\i}ptico. Constru\'{\i}mos a fun\c{c}\~ao de Green unidimensional, associada ao problema de Sturm-Liouville atrav\'es de duas metodologias, o chamado m\'etodo de Burkhardt, por quest\~oes hist\'oricas e o chamado m\'etodo de Sturm-Liouville. O caso de um oscilador harm\^onico unidimensional \'e apresentado como aplica\c{c}\~ao. Atrav\'es da metodologia das transformadas integrais, em particular a transformada de Fourier, discutimos o caso da equa\c{c}\~ao de Laplace num dom\'{\i}nio n\~ao limitado. Em resumo, apresentamos tr\^es maneiras distintas de se calcular a fun\c{c}\~ao de Green associada a um operador linear. Como uma aplica\c{c}\~ao da fun\c{c}\~ao de Green, introduzimos o conceito de equa\c{c}\~ao integral onde, em particular, no caso de uma equa\c{c}\~ao integral do tipo de Fredholm, a fun\c{c}\~ao de Green tem um papel importante, a saber, efetua a conex\~ao entre uma equa\c{c}\~ao diferencial ordin\'aria e a respectiva equa\c{c}\~ao integral. Atrav\'es da transformada de Laplace discutimos o chamado oscilador harm\^onico fracion\'ario de onde emerge naturalmente a fun\c{c}\~ao de Mittag-Leffler como uma extens\~ao natural da fun\c{c}\~ao exponencial. |
| 21/2008 |
One-Sided and Two-Sided Green´s Functions$^1$ R. Figueiredo Camargo, Ary O. Chiacchio, Edmundo Capelas de Oliveira We discuss the one-sided Green's function, associated with an initial value problem and the two-sided Green's function related to a boundary value problem. We present an especific calculation associated with a differential equation with constant coefficients. For both problems we present also the Laplace integral transform as another methodology to calculate these Green functions, and conclude which is the most convinient. An incursion in the so-called fractional Green's function is also presented. As an example we discuss the isotropic harmonic oscillator. |
| 20/2008 |
On Elliptic Problems Involving Critical Hardy-Sobolev Exponents and Sign-Changing Function Rodrigo da Silva Rodrigues In this paper, we deal with the existence and nonexistence of nonnegative nontrivial weak solutions for a class of degenerate quasilinear elliptic problems with weights and nonlinearity involving thecritical Hardy-Sobolev exponent and a sign-changing function. Some existence results are obtained by splitting the Nerahi manifold and by exploring some properties of the best Hardy-Sobolev constanttogether with an approach developed by Brezis and Nirenberg. rp-2008-20.pdf |
| 19/2008 |
Orthogonal Developments on Compact Symmetric Homogeneous Manifolds of Rank 1 Alexander Kushpel Sharp asymptotics of the norms of Fourier projections on compact homogeneous manifolds $M^d$ of rank 1, i.e., on $S^d$, $P^d(R)$, $P^d(C)$, $P^d(H)$, $P^{16}(Cay)$ are established. These results extend sharp asymptotic estimates found by Fejer [ 5] in the case of $S$ in 1910 and then by Gronwall [ 7] in 1914 in the case of $S^2$. rp-2008-19.pdf |
| 18/2008 |
Dynamical Spectral Sequences Over $Z_2$ Ketty A. de Rezende, M. R. da Silveira In this paper, we consider a sweeping method for a chain complex $C$ and its differential given by a connection matrix $\Delta$ over $\mathbb{Z}_2$ which determines an associated spectral sequence$(E^r,d^r)$. More specifically, a system spanning $E^r$ in terms of the original basis of $C$ is obtained as well as the identification of all differentials $d^r$. rp-2008-18.pdf |
| 17/2008 |
Bifurcation of Periodic Solutions for $C^5$ and $C^6$ Vector Fields in $R^4$ with Pure Imaginary Eigenvalues in Resonance 1:4 and 1:5 Jaume Llibre, Ana Cristina Mereu In this paper we study the bifurcation of families of periodic orbits at a singular point of a C^5 and C^6 differential system in R^4 with pure imaginary eigenvalues with resonance 1:4 and 1:5 respectively. From the singular point of the C^5 vector field with resonance 1:4 can bifurcate 0, 1, 2, 3, 4, 5 or 6 one-parameter families of periodic orbits. For the C^6 vector field with resonance 1:5, the maximal number of families of periodic orbits that bifurcate from this singular point is $40$. The tool for proving such a result is the averaging theory. rp-2008-17.pdf |
| 16/2008 |
Limit Cycles of the Generalized Polynomial Liénard Differential Equations Jaume Llibre, Ana Cristina Mereu, Marco A. Teixeira We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m greater than or equal to 1 there aredifferential equations of the form $\ddot{x}+f(x) \dot{x}+g(x)=0$, with f and g polynomials of degree n and m respectively, having at least [(n+m-1)/2] limit cycles, where [.] denotes the integer part function. rp-2008-16.pdf |
| 15/2008 |
Hopf bifurcation for vector fields in $R^4$ with pure imaginary eigenvalues in resonance 1:2 and 3:2 Jaume Llibre, Ana Cristina Mereu Assume that the linear part at a singular point of a C^k differential system with k=3,4,5 in R^4 has pure imaginary eigenvalues in resonance 1:2 when k=3,4 and in resonance 2:3 when k=5. If k=3 from this singular point it can bifurcate 0 or 1 one-parameter family of periodic orbits. If k=4 it can bifurcate 0, 1, 2, 3 or 4 one-parameter families of periodic orbits and if k=5 it can bifurcate 0, 1, 2, 3, 4, 5 or 6 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory. rp-2008-15.pdf |
| 14/2008 |
Spectral sequences in Conley's theory O. Cornea, Ketty A. de Rezende, M. R. da Silveira In this work we present an algorithm for a chain complex C and its differential given by a connection matrix which determines an associated spectral sequence (E,d). More specifically, a system spanning E in terms of the original basis of C is obtained as well as the identification of all differentials d. In exploring the dynamical implication of a nonzero differential, we prove the existence of a path joining two singularities in the case that a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow and which proves to be important in some applications. rp-2008-14.pdf |
