Relatórios de Pesquisa

10/2013 Ruelle Operator Duality for Coupled Smooth Markov Maps of the Circle
Vincent Pit

Let TL and TR be two smooth surjective Markov maps of the circle, with TR expansive, coupled in such a way that there exists an extension (C, TC ) whose first factor is TL−1 and the second factor of TC is TR . Let AL piecewise continuous and AR piecewise absolutelycontinuous be two respective potentials. We show that, when those potentials are in involutionby a smooth kernel W on C, there is an explicit isomorphism between eigenfunctions of the Ruelle operator of (TL , AL ) and eigendistributions of the Ruelle operator of (TR , AR ) for the same eigenvalue. This gives a regularity result for eigendistributions of transfer operators associated with non-maximal eigenvalues.

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9/2013 Academic performance of students from entrance to graduation via quasi U-statistics: a study at a Brazilian research university
Rafael Pimentel Maia, Hildete P. Pinheiro, Aluísio Pinheiro

We present novel methodology to assess undergraduate students’ performance. The proposed methods are based on measures of diversity and on the decomposability of quasi U-statistics to define average distances between and within groups. They have been employed as an alternative to the classic analysis of variance especially when the assumption of normality is not met. The quasi U-statistics nonparametric method can handle tests for interaction and uses jackknife to get p-values for the tests. The nonparametric method also results in smaller error variances, illustrating its robustness against model misspecication.

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8/2013 A Nonsmooth Two-Sex Population Model
Eduardo Garibaldi, Marcelo Sobottka

This paper considers a two-dimensional logistic model to study populations with two genders.The growth behavior of a population is guided by two coupled ordinary differential equations given by a non-differentiable vector field whose parameters are the secondary sex ratio (the ratio of males to females at time of birth), inter- and intra-gender competitions, fertility rates and a mating function. Using geometrical techniques, we analyze the singularities and the basin of attraction of the system, determining the relationships between the parameters for which the system presents an equilibrium point. In particular, we describe conditions on the secondary sex ratio and discuss the role of the median number of female sexual partners of each male for the conservation of a two-sex species.

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7/2013 Application of prediction models using fuzzy sets: an Bayesian inspired approach
Felipo Bacani, Laécio C. Barros

A fuzzy inference framework based on fuzzy relations is explored and applied to a real set of simulated forecasts and experimental data referring to temperature and humidity in specific coffee crop sites in Brazil. In short, the used model consists of fuzzy relations over possibility distributions, resulting in a fuzzy model analog to a Bayesian inference process. The application of the fuzzy model to temperature and humidity data resulted in a set of revised forecasts, which were later compared to the correspondent set of experimental data using two different statistical measures of accuracy, MAPE (mean absolute percentage error) and Willmott D. Statistical results were confronted to the original simulated forecast fit to experimental data, showing that the methodology was, in most cases, able to improve the specialist’s forecasts in both statistical measures.

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6/2013 Augmented mixed beta regression models for periodontal proportion data
Diana M. Galvis, Dipankar Badyophadyay, Víctor H. Lachos

Continuous (clustered) proportion data often arise in various domains of medicine andpublic health where the response variable of interest is a proportion (or percentage) quantify-ing disease status for the cluster units, ranging between zero and one. However, due to thepresence of relatively disease-free as well as highly diseased subjects in any study, the proportion values can lie in the interval [0, 1]. While Beta regression can be adapted for assessing covariate effects here, it’s versatility is often challenged due to the presence/excess of zeros and ones because the Beta support lies in the interval (0, 1). To circumvent this, we augment the probabilities of zero and one with the Beta density, controlling for the clustering effect.Our approach is Bayesian with the ability to borrow information across various stages of thecomplex model hierarchy, and produces a computationally convenient framework amenable toavailable freeware. The marginal likelihood is tractable, and can be used to develop Bayesiancase-deletion influence diagnostics based on q-divergence measures. Both simulation studiesand application to a real dataset from a clinical periodontology study quantify the gain in modelfit and parameter estimation over other adhoc alternatives, and provide quantitative insight intoassessing the true covariate effects on the proportion responses.

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5/2013 La Construcción de la Definición Axiomática de Probabilidad
Mario A. Gneri
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4/2013 On Non-Smooth Perturbations of Degenerate or Non-Degenerate Planar Centers
Douglas D. Novaes

We provide sufficient conditions for the existence of limit cycles of non–smooth perturbed planar centers, when the set of discontinuity is an algebraic variety. It is introduced a mechanism which allows us to deal with such system, even in higher dimension. The main tool used in this paper is the averaging method. Two applications are given in careful detail.

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3/2013 Generalized Linear Mixed Models for Correlated Binary Data with T-link
Denise Reis Costa, Marcos O. Prates

A critical issue in modelling binary response data is the choice of the links. We introducea new link based on the Student t-distribution (t-link) for correlated binary data. The t-link relates to the common probit-normal link adding one additional parameter which purely controlsthe heaviness of the tails of the link. We propose an interesting EM algorithm for computingthe maximum likelihood for generalized linear mixed t-link models for correlated binary data.In contrast with recent developments (Tan et al., 2007; Meza et al., 2009), this algorithm usesclosed-form expressions at the E-step, as opposed to Monte Carlo simulation. Our proposedalgorithm rely on available formulas for the mean and variance of a truncated multivariatet-distribution. To illustrate the new methodology, a real data set on respiratory infection inchildren and a simulation study are presented.

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2/2013 Limit Cycles for m-Piecewise Discontinuous Polynomial Liénard Differential Equations
Jaume Llibre, Marco A. Teixeira

We provide lower bounds for the maximum number of limit cycles for the m–piecewise discontinuous polynomial differential equations x = y + sgn(gm (x, y))F (x), y = −x, where the zero set of the function sgn(gm (x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane in sectors of angle 2π/m, and sgn(z) denotes the sign function.

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1/2013 Periodic Solutions of Discontinuous Second Order Differential Systems
Jaume Llibre, Marco A. Teixeira

We provide sufficient conditions por the existence of periodic solutions of some classes of autonomous and non–autonomous second order differential equations with discontinuous right–hand sides. In the plane the discontinuities considered are given by the straight lines either x = 0, or xy = 0. Two applications of these results are made, one to control systems with variable structure, and the other to small external periodic excitation of a discontinuous nonlinear oscillator.

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