Research Reports

8/2018 Estimates for Entropy Numbers of Sets of Smooth Functions on the Torus Td
R. L. B. Stabile, S. A. Tozoni

In this paper, we investigate entropy numbers of multiplier operators of functions defined on the d-dimensional torus. In the first part, upper and lower bounds are established for entropy numbers of general multiplier operators bounded from Lp to Lq.  In the second part, we apply these results to study entropy numbers of sets of finitely differentiable functions, in particular  Sobolev classes, and sets of infinitely differentiable and analytic  functions, on the d-dimensional torus. We prove that, the estimates for the entropy numbers  are order sharp in various important situations.

PDF icon rp-2018-08.pdf
7/2018 A log Birnbaum-Saunders regression model based on the skew-normal distribution under the centred parameterization
Nathalia L. Chaves, Caio L. N. Azevedo, Filidor Vilca-Labra, Juvêncio S. Nobre

In this paper, we introduce a new regression model for positive and skewed data, a log Birnbaum-Saunders model based on the centred skew-normal distribution, and we present a several inference tools for this model. Initially, we developed a new version of skew-sinh-normal distribution and we describe some of its properties. For the proposed regression model, we carry out, through of the expectation conditional maximization (ECM) algorithm, the parameter estimation, model fit assessment, model comparison and residual analysis. Finally, our model accommodates more suitably the asymmetry of the data, compared with the usual log Birnbaum-Saunders model, which is illustrated through real data analysis.

PDF icon rp-2018-07.pdf
6/2018 A new Birnbaum-Saunders model based on the skew-normal distribution under the centred parameterization
Nathalia L. Chaves, Caio L. N. Azevedo, Filidor Vilca-Labra, Juvêncio S. Nobre

In this paper we introduce a new distribution for positive and skewed data by combining the Birnbaum-Saunders (BS) distribution and the centred skew-normal distribution. Several of its properties are developed. Our model accommodates both positively and negatively skewed positive data. Also, we show that our model circumvents some problems related to another BS distribution, based on the skew-normal distribution under the direct parameterization, previously presented in the literature. We developed both maximum likelihood (ML) and Bayesian estimation procedures, comparing them through a suitable simulation study. The convergence of the expectation conditional maximization (ECM) (for ML inference) and MCMC algorithms (for Bayesian inference) were veri ed and several factors of interest were compared in the parameter recovery study. In general, as the sample size increases, the results indicated that the Bayesian approach provided the most accurate estimates. Finally, our model accommodates the asymmetry of the data, compared with the usual BS model, which is illustrated through real data analysis.

PDF icon rp-2018-06.pdf
5/2018 Optimal Approximation by sk-Splines on the Torus
J. G. Oliveira, S. A. Tozoni

Fixed a continuous kernel K on the d-dimensional torus, we consider a generalization of the univariate sk-spline to the torus, associated with the kernel K. We prove an estimate which provides the rate of convergence of a given function by its interpolating sk-splines, in the norm of Lq for functions of convolution type f=K*ϕ where ϕ is a function in a Lp-space. The rate of convergence is obtained for functions f in Sobolev classes and this rate gives optimal error estimate of the same order as best trigonometric approximation, in a special case.

PDF icon rp-2018-5.pdf
4/2018 On the Ree Curve
Saeed Tafazolian, Fernando Torres

We point out a characterization of the Ree curve which involves the number of rational points, the genus, and the shape of two elements of the Weierstrass semigroup at a rational point.

PDF icon rp-2018-042018.pdf
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.

PDF icon rp-2018-03.pdf
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.

PDF icon rp-2018-02.pdf
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.

PDF icon rp-2018-01.pdf
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.

PDF icon rp-2017-16.pdf
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.

PDF icon rp-2017-15.pdf