Research Reports

3/2018 Limit Cycles Bifurcating From Discontinuous Polynomial Pertubations of Higher Dimensional Linear Differential Systems
Jaume Llibre, Douglas D. Novaes, Iris O. Zeli

We study the periodic solutions bifurcating from periodic orbits of linear differential systems x0 = Mx, when they are perturbed inside a class of discontinuous piecewise polynomial differential systems with two zones. More precisely, we study the periodic solutions of the differential system x0 = Mx + "F n 1 (x) + "2Fn2 (x);
in Rd+2 where " is a small parameter, M is a (d+2) x (d+2) matrix having one pair of pure imaginary conjugate eigenvalues, m zeros eigenvalues, and d - m
non{zero real eigenvalues. For solving this problem we need to extend the averaging theory for studying periodic solutions to a new class of non{autonomous d + 1-dimensional discontinuous piecewise smooth differential system.

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2/2018 Normal Forms of Bireversible Vector Fields
P. H. Baptistelli, M. Manoel, I.O. Zeli

In this paper we adapt the method of [P. H. Baptistelli, M. Manoel and I. O. Zeli. Normal form theory for reversible equivariant vector elds. Bull. Braz. Math. Soc., New Series 47 (2016), no. 3, 935-954] to obtain normal forms of a class of smooth bireversible vector elds. These are vector elds reversible under the action of two linear involution and whose linearization has a nilpotent part and a semisimple part with purely imaginary eigenvalues. We show that these can be put formally in normal form preserving the reversing symmetries and their linearization. The approach we use is based on an algebraic structure of the set of this type of vector elds. Although this can lead to extensive
calculations in some cases, it is in general a simple and algorithmic way to compute the normal forms. We present some examples, which are Hamiltonian systems without resonance for one case and other cases with certain resonances.

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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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