Relatórios de Pesquisa

13/2012 Physical Assets Replacement: An Analytical Approach
Igor G. Cesca, Douglas D. Novaes

The 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.

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12/2012 Bayesian Inference in Nonlinear Mixed-Effects Models Using Normal Independent Distributions
Víctor H. Lachos, Luis M. Castro, Dipak K. Dey
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11/2012 Influence Diagnostics for Student-t Censored Linear Regression Models
Monique B. Massuia, Celso R. B. Cabral, Larissa A. Matos, Víctor H. Lachos
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10/2012 A multiple group Item Response Theory model with centred skew normal latent trait distributions under a Bayesian framework
José R. S. Santos, Caio L. N. Azevedo, Heleno Bolfarine

The 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.

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9/2012 $|S|$ Control Chart for Multivariate Process Variability Monitoring Based on Cornish-Fisher Correction and Meijer-G Function
Emanuel P. Barbosa, Mario A. Gneri, Ariane Meneguetti

This 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.

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8/2012 Averaging Methods for Studying the Periodic Orbits of Discontinuous Differential Systems
Jaume Llibre, Douglas D. Novaes, Marco A. Teixeira

The 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.

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7/2012 Perturbed Damped Pendulum: Finding Periodic Solutions
Douglas D. Novaes

Using the equation of motion of the damped pendulum, we introduce the averaging method on the study of periodic solutions of dynamical systems with small perturbation. We provide sufficient conditions for the existence of periodic solutions of the perturbed damped pendulum with small oscillations having equations of motion\[\ddot{\T}=-a\T-b\dot{\T}+\e f(t,\T,\dot{\T}),\]where $a>0,\,b>0$ and $\e$ are real parameters, with $a=g/l$, $g$ the acceleration of the gravity, $l$ the length of the rod and $b$ the damping coefficient. Here the parameters $b$ and $\e$ are small and the smooth function $f$ is $T$--periodic in $t$. The averaging theory provides a useful means to study dynamical systems, accessible to Master and PhD students.

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6/2012 On The Periodic Solutions of a Generalized Smooth and Non-Smooth Perturbed Planar Double Pendulum with Small Oscillations
Jaume Llibre, Douglas D. Novaes, Marco A. Teixeira

We provide sufficient conditions for the existence of periodic solutions of the smooth and non-smooth perturbed planar double pendulum with small oscillations having equations of motion\[\begin{array}{l}\ddot{\T}_{1}=-a\T_{1}+\T_{2}+\e \left(F_1(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_2(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{1}})\right),\\\ddot{\T}_{2}=b\T_{1}-b\T_{2}+\e \left(F_3(t,\T_1,\dot {\T}_1,\T_2, \dot{\T}_2)+F_4(t,\T_1,\dot {\T}_1,\T_2, \dot {\T}_2){\rmsgn}(\dot{\T_{2}})\right),\end{array}\]where $a>1,b>0$ and $\e$ are real parameters. Here the parameter $\e$ is small and the smooth functions $F_i$ for $i=1,2,3,4$ define the perturbation which are periodic functions in $t$ and in resonance $p_{i}$:$q_{i}$ with some of the periodic solutions of the unperturbed double pendulum, being $p_{i}$ and $q_{i}$ relatively prime positive integers.

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5/2012 On The Periodic Solutions of a Perturbed Double Pendulum
Jaume Llibre, Douglas D. Novaes, Marco A. Teixeira

We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion\[\begin{array}{l}\ddot{\T}_{1}=-2a\T_{1}+a\T_{2}+\e F_1(t,\T_1,\dot \T_1,\T_2, \dot \T_2),\\\ddot{\T}_{2}=2a\T_{1}-2a\T_{2}+\e F_2(t,\T_1,\dot \T_1,\T_2, \dot\T_2),\end{array}\]where $a$ and $\e$ are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length $l$. In fact $a=g/l$ where $g$ is the acceleration ofthe gravity. Here the parameter $\e$ is small and the smooth functions $F_1$ and $F_2$ define the perturbation which are periodic functions in $t$ and in resonance $p$:$q$ with some of the periodicsolutions of the unperturbed double pendulum, being $p$ and $q$ positive integers relatively prime.

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4/2012 Minimização irrestrita usando gradientes conjugados e regiões de confiança
John Lenon C. Gardenghi, Sandra A. Santos

This work focus on the conjugate gradient method to solve the trust region sub-problem for unconstrained minimization. We aim to describe an intuitive and detailed study about this method, starting from an introduction to methods of conjugate directions, some necessary requisites and tools for understanding the conjugate gradient method and its integration with the trust region strategy for unconstrained minimization. The computational implementation of the method using the CAS Maxima enabled the numerical experiments, which validated the study and the implementation done and allowed a comparison between conjugate gradient and Leverberg-Marquardt for least squares problems.

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