Relatórios de Pesquisa

2/2019 Estimates for n-widths of sets of smooth functions on complex spheres
Deimer J. J. Aleans, Sergio A. Tozoni

In this work we investigate n-widths of multiplier operators defined for functions on a complex sphere and bounded from L^p into L^q. We study lower and upper estimates for the n-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of such operators. As application we obtain, in particular, estimates for the Kolmogorov n-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on a complex sphere, in L^q, which are order sharp in various important situations.

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1/2019 Estudo de um Método Baseado em Autovalores Generalizados para o Subproblema de Região de Confiança
Jean Carlos A. Medeiros , Sandra Augusta Santos

The trust-region methods are iterative methods for numerically solving minimization problems, not only unconstrained but also constrained ones. They consist of defining a quadratic model for the objective function f from a current point x^k and establishing a closed ball centered on x^k and with radius Δ; this neighborhood around x^k is called trust region, because in this region we will trust that the model generates a good approximation for the objective function; then each iteration will have a subproblem of minimizing the model subject to the trust region, thereby generating a sequence of approximations to the solution of the problem, ie the objective function minimizer. Recently, Adachi et al.[1], based on Gander et al.[4], developed a method adressing the subproblem in a non-iterative way, solving only one generalized eigenvalue problem. This work investigates the usage of this strategy for solving low dimensional unconstrained minimization problems. The visual appeal provides an additional tool for exploring the geometric features of this approach.

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15/2018 Estimates for n-widths of Multiplier Operators of Multiple Walsh Series
Sergio A. Córdoba, Sérgio A. Tozoni

Estimates for Kolmogorov and Gelfand n-widths of multiplier operators of multiple Walsh series are obtained. Upper and lower bounds are established for n-widths of general multiplier operators. These results are applied to get upper and lower bounds for n-widths of specific multiplier operators, which generate sets of finitely and infinitely differentiable functions in the dyadic sense. It is shown that these estimates have order sharp in various important cases.

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14/2018 Dynamical Obstruction to the Existence of Continuous Sub-Actions for Interval Maps with Regularly Varying Property
Eduardo Garibaldi, Irene Inoquio-Renteria

In ergodic optimization theory, the existence of sub-actions is an important tool in the study of the so-called optimizing measures. For transformations with regularly varying property, we highlight a class of moduli of continuity which is not compatible with the existence of continuous sub-actions. Our result relies fundamentally on the local behavior of the dynamics near a fixed point and applies to interval maps that are expanding outside an neutral fixed point, including Manneville-Pomeau and Farey maps.

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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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