# Relatórios de Pesquisa

 4/2013 On Non-Smooth Perturbations of Degenerate or Non-Degenerate Planar CentersDouglas D. NovaesWe 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. 3/2013 Generalized Linear Mixed Models for Correlated Binary Data with T-linkDenise Reis Costa, Marcos O. PratesA 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. 2/2013 Limit Cycles for m-Piecewise Discontinuous Polynomial Liénard Differential EquationsJaume Llibre, Marco A. TeixeiraWe 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. 1/2013 Periodic Solutions of Discontinuous Second Order Differential SystemsJaume Llibre, Marco A. TeixeiraWe 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. 13/2012 Physical Assets Replacement: An Analytical ApproachIgor G. Cesca, Douglas D. NovaesThe economic life of an asset is the optimum length of its use-fulness, which is the moment that the asset’s expenses are minimum. In this paper, the economic life of physical assets, such as industry machine and equipment, can be interpreted as the moment that the minimum is reached by its equivalent property cost function, defined as the sum of all equivalent capital and maintenance costs during its life.Many authors in classical papers have used principles of engineering economic to solve the assets replacement problem. However, in the literature, the main attributes found were proved with intuitive ideas instead mathematical analysis. Therefore, in this paper the main goal is to study these principles of engineering economic with mathematical techniques.Here, is used non-smooth analysis to cla sify all the possibilities for the minimum of a class of equivalent property cost functions of assets. The minimum of these function gives the optimum moment for the asset to be replaced, i.e., its economic life. 12/2012 Bayesian Inference in Nonlinear Mixed-Effects Models Using Normal Independent DistributionsVíctor H. Lachos, Luis M. Castro, Dipak K. Dey 11/2012 Influence Diagnostics for Student-t Censored Linear Regression ModelsMonique B. Massuia, Celso R. B. Cabral, Larissa A. Matos, Víctor H. Lachos 10/2012 A multiple group Item Response Theory model with centred skew normal latent trait distributions under a Bayesian frameworkJosé R. S. Santos, Caio L. N. Azevedo, Heleno BolfarineThe multiple group IRT model (MGM) provides a useful framework for analyzing item response data from clustered respondents. In the MGM, the selected groups of respondents are of specific interest such thatgroup-specific population distributions need to be defined. An usual assumption for parameter estimation for this model, is to assume that the latent traits are random variables which follow possibly differentsymmetric normal distributions. However, many works suggest that this assumption does not apply in many cases. Furthermore, when this assumption does not hold, parameter estimates tend to be biased and misleadinginference can result. Therefore, it is important to model the distribution of the latent traits properly. In this paper we present an alternative latent traits modeling, for multiple group framework, based on theso-called skew-normal distribution. We name it SMGIRT model (skew multiple group IRT model). It extends the approach proposed by some authors in the literature. We use the centred parameterization. This approach ensures model identifiability. We propose and compare, concerning convergence issues, two MCMC algorithms for parameter estimation. A simulation study was performed in order to assess parameter recovery for the proposed modeland the selected algorithm concerning convergence issues. The results reveals that our proposed algorithm recovers properly all model parameters. Furthermore, we analyzed a real data set which presentsindication of asymmetry concerning the latent traits distribution. The results obtained by using our approach confirmed the presence of negative asymmetry of the latent traits distribution. Moreover, our modeloutperforms the usual symmetric normal MGM, leading to different conclusions concerning parameter estimation. 9/2012 $|S|$ Control Chart for Multivariate Process Variability Monitoring Based on Cornish-Fisher Correction and Meijer-G FunctionEmanuel P. Barbosa, Mario A. Gneri, Ariane MeneguettiThis paper presents an improved version of the generalized variance |S| control chart for multivariate process dispersion monitoring, based on the Cornish-Fisher formula for non-normality correctionof the normal based 3-sigma chart limits. Also, the exact sample distribution of |S| and its quantiles (chart exact limits) are obtained through the Meijer-G function, and an auxiliary control chart basedon the trace of V (standardized S matrix) is introduced. The performance of this corrected control chart is compared (in terms of false alarm risk) with the traditional normal based chart and the exactdistribution based chart (for dimensions d = 2 and d = 3). This study shows that the control limits corrections do remove the drawback of excess of false alarm associated with the traditional normalbased |S| control chart. The proposed new chart is illustrated with two numerical examples. 8/2012 Averaging Methods for Studying the Periodic Orbits of Discontinuous Differential SystemsJaume Llibre, Douglas D. Novaes, Marco A. TeixeiraThe main objective of this work is to extend the averaging method for studying the periodic orbits of a class of differential equations with discontinuous second member. Thus, overall results are presented to ensure the existence of limit cycles of such systems. Certainly these results represent new insights in averaging, in particular its relation with non smooth dynamical systems theory. An application is presented in careful detail.