| 2/2010 |
Conley´s Spectral Sequence via the Sweeping Algorithm Margarida P. Mello, Ketty A. de Rezende, M. R. da Silveira In this article we consider a spectral sequence $(E^r; d^r)$ associated to a filtered Morse-Conley chain complex $(C; \Delta)$ where $\Delta$ is a connection matrix. The underlying motivation is to understand connection matrices under continuation. We show how the spectral sequence is completely determined by a family of connection matrices. This family is obtained by a sweeping algorithm for $\Delta$ over fields $F$ as well as over $Z$. This algorithm constructs a sequence of similar matrices $\Delta^0=\Delta$, $\Delta^1$, ..., where each matrix is related to the others via a change-of-basis matrix. Each matrix $\Delta^r$ over $F$ (resp., over $Z$) determines the vector space (resp., $Z$-module) $E^r$ and the differential $d^r$. We also prove the integrality of the final matrix $\Delta^R$ produced by the sweeping algorithm over $Z$ which is quite surprising, mainly because the intermediate matrices in the process may not have this property. Several other properties of the change-of-basis matrices as well as the intermediate matrices $\Delta^r$ are obtained. The sweeping algorithm and the computation of the spectral sequence $(E^r; d^r)$ are implemented in the software Mathematica.  rp-2010-2.pdf |
| 1/2010 |
On Equivalent Expressions for the Faraday´s Law of Induction Fábio G. Rodrigues In this paper we give a rigorous proof of the equivalence of some different forms of Faraday’s law of induction clarifying some misconceptions on the subject and emphasizing that many derivations of this law appearing in textbooks and papers are only valid under very special circunstances and not satisfactory under a mathematical point of view. rp-2010-1.pdf |
| 32/2009 |
A Pluzhnikov´s Theorem, Brownian Motions and Martingales in Lie Group with Skew-Symmetric Connections Simão Stelmastchuk Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem. rp-2009-32.pdf |
| 31/2009 |
A Characterization of Einstein Manifolds Simão Stelmastchuk Let $(M,g)$ be any Riemannian manifold. Our goal is to show that if $g$ and Ricci tensor $r_{g}$ are no locally constant, if, locally, their product is non-negative (respectively, non-positive), and if its scalar curvature $s_{g}$ is non-negative (respectively, non-positive), then $(M,g)$ is an Einstein manifolds. This result is a generalization of the characterization for compacts Einstein manifolds given by Hilbert. rp-2009-31.pdf |
| 30/2009 |
An Equivalence Between Harmonic Sections and Sections that are Harmonic Maps Simão Stelmastchuk Let $\pi:(E,\nabla^{E}) \rightarrow (M,g)$ be an affine submersion with horizontal distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a Riemannian manifold. Let $\sigma$ be a section of $\pi$, namely, $\pi \circ \sigma = Id_{M}$. It is possible to study the harmonic property of section $\sigma$ in two ways. First, we see $\sigma$ as a harmonic map. Second, we see $\sigma$ as harmonic section. In the Riemannian context, it means that $\sigma$ is a critical point of the vertical functional energy. Our main goal is to find conditions to the assertion: $\sigma$ is a harmonic map if and only if $\sigma$ is a harmonic section. rp-2009-30.pdf |
| 29/2009 |
Stochastic Characterization of Harmonic Sections and a Liouville Theorem Simão Stelmastchuk Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G, P)$ be an associate fiber bundle. Our interest is to study harmonic sections of the projection $\pi_E$ of $E$ into $M$. Our first purpose is to give a stochastic characterization of harmonic section from $M$ into $E$ and a geometric characterization of harmonic sections with respect to its equivariant lift. The second purpose is to show a version of Liouville theorem for harmonic sections and to prove that section $M$ into $E$ is a harmonic section if and only if it is parallel. rp-2009-29.pdf |
| 28/2009 |
Minimizing Orbits in the Discrete Aubry-Mather Model Eduardo Garibaldi, Philippe Thieullen We consider a generalization of the Frenkel-Kontorova model in higher dimension. We give a wider applicability to Aubry’s theory by studying models with vector-valued states over a one dimensional chain. This theory has a lot of similarities with Mather’s twist approach over a multidimensional torus. Weakening the standard hypotheses used in one dimensional, we investigate properties (like boundness of jumps and definability of a rotation vector) of a special class of strong ground states: the calibrated configurations.The main mathematical tool is to cast the study the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax-Oleinik problems and interpret their solutions as discrete viscosity solutions in the same spirit of Hamilton-Jacobi methods. With reduced hypotheses, we reproduce in this discrete setting some classical results of the Lagrangian Aubry-Mather theory. In particular, we obtain a graph property for the Aubry set, representation formulas for calibrated sub-actions and the existence of separating sub-actions. rp-2009-28.pdf |
| 27/2009 |
Duality for Poset Codes Allan de Oliveira Moura, Marcelo Firer In this work we extend Wei's Duality Theorem, relating the generalized Hamming weight hierarchy of a code to the hierarchy of the dual code, to the scope of codes with poset metrics. As a consequence of this duality theorem we prove some results concerning discrepancy of codes and chain condition for generalized weights. rp-2009-27.pdf |
| 26/2009 |
On the Variations of the Betti Numbers of Regular Levels of Morse Flows Maria Alice Bertolim, Ketty A. de Rezende, O. Manzoli Neto, G. M. Vago We generalize results in [C-dR] by completely describing how the Betti numbers of the boundary of an orientable manifold vary after attaching a handle, when the homology coefficients are in Z, Q, R or Z/pZ with p prime. Next we consider the Ogasa invariant associated with handle decompositions of manifolds. We make use of the above results in order to obtain upper bounds for the Ogasa invariant of product manifolds. rp-2009-26.pdf |
| 25/2009 |
On the Linear Transport Subject to Random Velocities and Initial Conditions Fábio A. Dorini, Maria Cristina C. Cunha This paper deals with the random linear transport equation for which the velocity and the initial condition are random functions. Expressions for the density and joint density functions of the transport equation solution are given. We also verify that in the Gaussian time-dependent velocity case the probability density function of the solution satisfies a convection-diffusion equation with a time-dependent diffusioncoefficient. Examples are included. rp-2009-25.pdf |
