Research Reports

13/2018 Approximation of Differentiable and Analytic Functions by Splines on the Torus
J. G. Oliveira , S. A. Tozoni

We consider a continuous kernel K on the d-dimensional torus and we study the rate of convergence in Lq, of functions of the type f=K*ϕ where ϕ is a function in a Lp-space, by its interpolating sk-splines. The rate of convergence is obtained for functions in classes of Sobolev, of infinitely differentiable functions and of analytic functions, and it provides optimal error estimates of the same order as best trigonometric approximation, in several cases.


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12/2018 Extremal Norms for Fiber Bunched Cocycles
Jairo Bochi, Eduardo Garibaldi

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. ThemostusefultoolinthisareaisthecelebratedMañéLemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficientlystrongfiberbunching. Thereforeweextendtheclassicconcept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets. 


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11/2018 On Space Maximal Curves
Paulo César Oliveira, Fernando Torres

Any maximal curve X is equipped with an intrinsic embedding π : X → Pr which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve π(X) and tangent lines, in this paper we investigate the first positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and π(X) is contained in a cubic surface.
 


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10/2018 Locally Recoverable Codes From Algebraic Curves with Separated Variables
Carlos Munuera, Wanderson Tenório, Fernando Torres

 A Locally Recoverable code is an error-correcting code such that any erasure in a single coordinate of a codeword can be recovered from a small subset of other coordinates. We study Locally Recoverable Algebraic Geometry codes arising from certain curves defined by equations with separated variables. The recovery of erasures is obtained by means of Lagrangian interpolation in general, and simply by one addition in some particular cases.
 


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9/2018 The Multivariate Birnbaum-Saunders Distribution Based on a Asymmetric Distribution: EM-Estimation
Filidor Vilca, Camila Borelli Zeller, N. Balakrishnan

We derive here a multivariate generalization of the bivariate Birnbaum-Saunders (BS) distribution of Kundu et al. (2010) by basing it on the multivariate skew-normal (SN) distribution. The resulting multivariate Birnbaum-Saunders type distribution is an absolutely continuous distribution whose marginals are in the form of univariate Birnbaum-Saunders type distributions discussed by Vilca et al. (2011). We then study its characteristics and properties, such as the joint distribution function, marginal and conditional distributions. Next, we introduce a non-central multivariate BS distribution in order to present analytically a simple EM-algorithm for iteratively computing the maximum likelihood estimates of the model parameters, and compare the performance of this method with the estimation approach of Jamalizadeh and Kundu (2015). Moreover, the observed Fisher information matrix is analytically derived under the bivariate case, and some simulation studies and an application to a real data set are finally presented for the propose of illustrating the model and inferential results developed here.
 


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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.


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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.


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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.


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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.


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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.
 


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