Relatórios de Pesquisa

1/2018 The Generic Unfolding of a Codimension-Two Connection to a Two-Fold Singularity of Planar Filippov Systems
Douglas D. Novaes, Marco A. Teixeira, Iris O. Zeli

Generic bifurcation theory was classically well developed for smooth differential systems, establishing results for k-parameter families of planar vector
elds. In the present study we focus on a qualitative analysis of 2-parameter families, Za,ß of planar Filippov systems assuming that Zo;o presents a codimension- two minimal set. Such object, named elementary simple two-fold cycle, is characterized by a regular trajectory connecting a visible two-fold singularity to itself, for which the second derivative of the rst return map is nonvanishing. We analyzed the codimension-two scenario through the exhibition of its bifurcation diagram.

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16/2017 An Alphabetical Approach to Nivat´s Conjecture
Colle, C. F., Garibaldi, E.

Since techniques used to address the Nivat’s conjecture usually relies on Morse-Hedlund Theorem, an improved version of this classical result may mean a new step towards a proof for the conjecture. In this paper, we consider an alphabetical version of the Morse-Hedlund Theorem. Following methods highlighted by Cyr and Kra [1], we show that, for a configuration  2 AZ2 that contains all letters of a given finite alphabet A, if its complexity with respect to a quasi-regular set U  Z2 (a finite set whose convex hull on R2 is described by pairs of edges with identical size) is bounded from above by 1 2jUj + jAj ? 1, then  is periodic.

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15/2017 On Maximal Curves Related to Chebyshev Polynomials
Ahmad Kazemifard, Saeed Tafazolian, Fernando Torres

We study maximal curves arising from Chebyshev polynomials, where in particular some results from Garcia-Stichtenoth [4] are revisited and generalized.

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14/2017 Grüss-type Inequality by Means of a Fractional Integral
J. Vanterler da C. Sousa, D. S. Oliveira, E. Capelas de Oliveira

We use a fractional integral recently proposed to establish a generalization
of Gruss-type integral inequalities. We prove two theorems about these inequalities and
enunciate and prove other inequalities associated with this fractional operator.

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13/2017 On a Caputo-type Fractional Derivative
D.S. Oliveira, E. Capelas de Oliveira

In this work we present a new di erential operator of arbitrary order de ned
by means of a Caputo-type modi cation of the generalized fractional derivative recently
proposed by Katugampola. The generalized fractional derivative, when adequate limits
are considered, recovers the Riemann-Liouville and the Hadamard derivatives of arbitrary
order. Our di erential operator recovers as limiting cases the arbitrary order derivatives
proposed by Caputo and by Caputo-Hadamard. Some properties are presented, as well
the relation between this di erential operator of arbitrary order and the Katugampola
generalized fractional operator. As an application we prove the fundamental theorem of
fractional calculus associated with our operator.

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12/2017 Bayesian Inference for Zero-and/or-one Augmented Rectangular Beta Regression Models
Ana R.S. Santos, Caio L. N. Azevedo, Jorge L. Bazan, Juvêncio S. Nobre

In this paper, we developed a Bayesian inference for a zero-and/orone augmented rectangular beta regression model to analyze limitedaugmented data, under the presence of outliers. The proposed Bayesian tools were parameter estimation, model t assessment, model comparison, residual analysis and case in uence diagnostics, developed through
MCMC algorithms. In addition, we adapted available methods of posterior predictive checking using appropriate discrepancy measures.
Also, a comparison with the maximum likelihood estimation, previously proposed in the literature was performed, in terms of parameter recovery. We noticed that the results are quite similar, but the Bayesian approach is more easily implemented, including in uence diagnostics tools, besides also allowing incorporating prior information.
We conducted several simulation studies, considering some situations of practical interest, in order to evaluate the parameter recovery of the proposed model and estimation method, as well as the impact of transforming the observed zeros and ones along the use of non-augmented models. A psychometric real data set was analyzed to illustrate the
performance of the developed tools.

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11/2017 Fp2 - Maximal Curves with Many Automorphisms are Galois-Covered by the Hermitian Curve
Daniele Bartoli, Maria Montanucci , Fernando Torres

Let F be the finite field of order q2, q = ph with p prime. It is commonly
atribute to J.P. Serre the fact that any curve F-covered by the Hermitian curve Hq+1 :
yq+1 = xq + x is also F-maximal. Nevertheless, the converse is not true as the Giulietti-
Korchm´aros example shows provided that q > 8 and h ≡ 0 (mod 3). In this paper, we
show that if an F-maximal curve X of genus g ≥ 2 where q = p is such that |Aut(X)| >
84(g − 1) then X is Galois-covered by Hp+1. Also, we show that the hypothesis on the
order of Aut(X) is sharp, since there exists an F-maximal curve X for q = 71 of genus
g = 7 with |Aut(X)| = 84(7 − 1) which is not Galois-covered by the Hermitian curve

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10/2017 Generalized Weierstrass Semigroups and their Poincaré Series
J. J. Moyano-Fernández, W. Tenório , F. Torres

We investigate the structure of the generalized Weierstraß semigroups at several points on a curve defined over a finite field. We present a description of these
semigroups that enables us to deduce properties concerned with the arithmetical structure of divisors supported on the specified points and their corresponding Riemann-Roch
spaces. This characterization allows us to show that the Poincar´e series associated with generalized Weierstraß semigroups carry essential information to describe entirely their
respective semigroups.

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9/2017 Zero-one Augmented Beta and Zero Inflated Discrete Models with Heterogeneous Dispersion: An Application to Students’ Academic Performance
Hildete P. Pinheiro, Rafael P. Maia, Eufrásio A. Lima-Neto, Mariana Rodrigues-Motta

The purpose of this work is to present suitable statistical methods to study the performance of undergraduate students based on the incidence/proportion of failed courses. Some approaches are considered: in one of them the incidence of failed courses is modeled using zero in ated discrete distributions with heteroscedasticity, considering
the logarithm of the total number of courses as an o set; in another, the proportion of failed courses is modeled considering a zero-one augmented
beta distribution with heterogeneous dispersion parameter. A detailed residual analysis is performed to investigate the best model to  t the data. The zero-one augmented beta model with heteroscedasticity presents a better t based on residual analysis. Moreover, the model for the proportion brings us more information as it is straight
forward to interpret its results. The database consists of records of 3,699 students with Engineering major who entered the State University of Campinas, Brazil, from 2000 to 2005. The entrance exam scores, academic and demographic variables and their socioeconomic status are considered as covariates in the models.

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8/2017 Multidimensional Multiple Group IRT Models with Skew Normal Latent Trait Distributions
Juan L. Padilla, Caio L. N. Azevedo, Victor H. Lachos

Item response theory (IRT) models are one of the most important statistical tools for psychometric data analysis. Their applicability goes from educational
assessment to biological essays. The IRT models combine, at least, two sets of unknown quantities: the latent traits (person parameters) and item parameters (related to measurement instruments of interest). The multidimensional item response theory (MIRT) models are quite useful to analyze data sets involving multiple skills or latent traits, which occurs in many of the applications. However, most of the works in the literature consider the usual assumption of multivariate (symmetric) normal distribution to the latent traits and do not deal with the multiple group framework (few groups with many of subjects in each one). They, in general, consider a limited
number of model t assessment tools, and do not investigate the measurement instrument dimensionality in a detailed way, while also dealing with the model nonidenti ability in a di erent way than that we presented here and only for one group model. In this work, we propose a MIRT multiple group model with multivariate skew normal distributions for modeling the latent traits of each group under the centered parameterization, presenting simple and feasible conditions for model identi cation. A full Bayesian approach for parameter estimation, structural selection (model comparison and determination of the dimensionality of the measurement instrument) and model
t assessment are developed through Markov Chain Monte Carlo (MCMC) algorithms. The developed tools are illustrated through the analysis of a real data set related to the rst stage of the University of Campinas 2013 admission exam.

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