# Relatórios de Pesquisa

 13/2014 Aubry set for Asymptotically Sub-Additive PotentialsEduardo Garibaldi, João Tiago Assunção GomesGiven a topological dynamical systems $(X, T)$, consider a sequence of continuous potentials $F := \{f_n: X → \mathbb{R}\}_{n\geq 1}$ that is asymptotically approached by sub-additive families. In a generalized version of ergodic optimization theory, one is interested in describing the set $M_{\rm max}(F)$ of $T$-invariant probabilities that attain the following maximum value ${\rm max} \{\lim_{ n\to\infty} \frac{1}{n} \int f_n d\mu : \mu\ {\rm is}\ T{\rm -invariant\ probability}\}$. For this purpose, we extend the notion of Aubry set, denoted by $\Omega(F)$. Our main result provide a sufficient condition for the Aubry set to be a maximizing set, i. e., $\mu$ belongs to $M_{\rm max}(F)$ if, and only if, its support lies on $\Omega(F)$. Furthermore, we apply this result to the study of the generalized spectral radius in order to show the existence of periodic matrix configurations approaching this value. 12/2014 Censored Mixed-Effects Models for Irregularly Observed Repeated Measures with Applications to HIV Viral LoadsLarissa A. Matos, Luis M. Castro, Víctor H. LachosIn some AIDS clinical trials, the HIV-1 RNA measurements are collected irregularly over time and are often subject to some upper and lower detection limits, depending on the quantification assays. Linear and nonlinear mixed-effects models, withmodifications to accommodate censored observations, are routinely used to analyze this type of data Vaida & Liu (2009); Matos et al. (2013a). This paper presents a framework for fitting LMEC/NLMEC with response variables recorded at irregular intervals. To address the serial correlation among the within-subject errors, a damped exponential correlation structure is considered in the random error and an EM-type algorithm is developed for computing the maximum likelihood estimates,obtaining as a byproduct the standard errors of the fixed effects and the likelihood value. The proposed methods are illustrated with simulations and the analysis of two real AIDS case studies. 11/2014 Robust Mixture Regression Modeling Based on Scale Mixtures of Skew-Normal DistributionsCamila Borelli Zeller, Celso R. B. Cabral, Víctor H. LachosThe traditional estimation of mixture regression models is based on the assumption of normality (symmetry) of component errors and thus is sensitive to outliers, heavy-tailed errors and/or asymmetric errors. In this work we present a proposal to deal with these issues simultaneously in the context of the mixture regression by extending the classic normal model byassuming that the random errors follow a scale mixtures of skew-normal distributions. This approach allows us to model data with great flexibility, accommodating skewness and heavy tails. The main virtue of considering the mixture regression models under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation whichallows easy implementation of inference. We develop a simple EM-type algorithm to perform maximum likelihood inference of the parameters of the proposed model. In order to examine the robust aspect of this flexible model against outlying observations, some simulation studies are also been presented. Finally, a real data set is analyzed, illustrating the usefulness of the proposed method. 10/2014 Bifurcations of mutually coupled equations in random graphsEduardo Garibaldi, Tiago PereiraWe study the behavior of solutions of mutually coupled equations in heterogeneous random graphs. Heterogeneity means that some equations receive many inputs whereas most of the equations are given only with a few connections. Starting from a situation where the isolated equations are unstable, we prove that the a heterogeneous interaction structure leads to the appearance of stable subspaces of solutions. Moreover, we show that, for certain classes of heterogeneous networks, increasing the strength of interaction leads to a cascade of bifurcations in which the dimension of the stable subspace of solu-tions increases. We explicitly determine the bifurcation scenario in terms of the graph structure. 9/2014 A Note on Certain Maximal CurvesSaeed Tafazolian, Fernando TorresWe characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely x m y a + y n + x b = 0. We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type y q + y = x m via class field theory. 8/2014 Lyapunov Graphs for Circle Valued Morse FunctionsKetty A. de Rezende, Guido G. E. Ledesma, Oziride Manzoli Neto, Gioia M. VagoWithin the context of Novikov theory, the Conley index is to used to obtain results for circular Morse flows on compact n-manifold. Examples are provided for orientable and non-orientable surfaces via a complete characterization of circular Morse digraphs. A generalization of these results is presented for smooth flows associated to circle valued Lyapunov functions. 7/2014 Augmented mixed models for clustered proportion dataDipankar Bandyopadhyay, Diana M. Galvis, Víctor H. LachosOften in biomedical research, we deal with continuous (clustered) proportion responses ranging between zero and one quantifying the disease status of the cluster units. Interestingly, the study population might also consist of relatively disease-free as well as highly diseased subjects, contributing to proportion values in the interval [0, 1]. Regression on a variety of parametric densities with support lying in (0, 1), such as beta regression, can assess important covariate effects. However, they are deemed inappropriate due the presence of zeros and/or ones. To evade this, we introduce a class of general proportion density (GPD), and further augment the probabilities of zero and one to this GPD, controlling for the clustering. Our approach is Bayesian, and presents a computationally convenient framework amenable to available freeware.Bayesian case-deletion influence diagnostics based on q-divergence measures are automatic from the MCMC output. The methodology is illustrated using both simulation studies and application to a real dataset from a clinical periodontology study. 6/2014 Censored Linear Regression Models for Irregularly Observed Longitudinal Data using the Multivariate-t DistributionAldo M. Garay, Luis M. Castro, Jacek Leskow, Víctor H. LachosIn AIDS studies it is quite common to observe viral load measurements collected irregularlyover time. Moreover, these measurements can be subjected to some upper and/or lower detection limitsdepending on the quantification assays. A complication arises when these continuous repeated measureshave a heavy-tailed behavior. For such data structures, we propose a robust structure for a censoredlinear model based on the multivariate Student-t distribution. To compensate for the autocorrelationexisting among irregularly observed measures, a damped exponential correlation structure is employed.An efficient EM-type algorithm is developed for computing the maximum likelihood estimates, obtaining as a by-product the standard errors of the fixed effects and the log-likelihood function. The proposed algorithm uses closed-form expressions at the E-step, that rely on formulas for the mean and variance of a truncated multivariate Student-t distribution. The methodology is illustrated through an application to an HIV-AIDS study and several simulation studies. 5/2014 Further examples of maximal curves which cannot be covered by the Hermitian curveSaeed Tafazolian, Arnoldo Teheran-Herrera, Fernando TorresWe construct examples of curves defined over the finite field Fq6 which are covered by the GK-curve. Thus such curves are maximal over Fq6 although they cannot be covered by the Hermitian curve for q > 2. We also give examples of maximal curvesthat cannot be Galois covered by the Hermitian curve over the finite field Fq2n with n > 3 odd and q > 2. We point out some applications to codes related to an array coming from telescopic semigroups. 4/2014 Improving Shewhart-type Generalized Variance Control Charts for Multivariate Process Variability Monitoring using Cornish-Fisher Quantile Correction, Meijer-G Function and Other ToolsEmanuel P. Barbosa, Mario A. Gneri, Ariane MeneguettiThis paper presents an improved version of the Shewhart-type generalized variance |S| control chart for multivariate Gaussian process dispersion monitoring, based on the Cornish-Fisher quantile formula for non-normality correction of the traditional nor-mal 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 (inverse Mellin-Barnes integral transform), and an auxiliary control chart based on the trace of the standardized S matrix is introduced in order to avoid non detection of certain changes in the process variance-covariance Σ matrix.The performance of the proposed CF-corrected control chart is compared, considering false alarm risk (using analytical and simulation tools), with the traditional normal based chart and with the exact distributed based chart (for dimensions d = 2 and d = 3). This study shows that the proposed control limit corrections do remove the drawback of excess of false alarm associated with the traditional normal based |S| control chart.The proposed new chart (with its corresponding auxiliary chart) is illustrated with two numerical examples.