Research Reports

Beatriz Motta, Fernando Torres

We investigate complete plane arcs which arise from the set of rational points of certain non-Frobenius classical plane curves over finite fields. We also point out direct consequences on the Griesmer bound for some linear codes.

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Saeed Tafazolian, Fernando Torres

Abstract. Let F be the finite field of order q2. In this paper we continue the study
in [20], [19], [18] of F-maximal curves defined by equations of type yn = xℓ(xm + 1).
For example new results are obtained via certain subcovers of the nonsingular model of
vN = ut2
− u where q = tα,  ≥ 3 odd and N = (tα + 1)/(t + 1). We do observe that
the case  = 3 is closely related to the Giulietti-Korchm´aros curve.

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4/2019 Explosion in a Growth Model with Cooperative Interaction on an In nite Graph
Bruna de Oliveira Gonçalves, Marina Vachkovskaia

In this paper we study explosion/non-explosion of a continuous time growth process with cooperative interaction on Z+. We consider symmetric neighborhood and di erent types of rate functions and prove that explosion occurs for exponential rates, but not for polynomial. We also present some simulations to illustrate the explosion

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Victoriano Carmona, Fernando Fernández-Sánchez, Douglas D. Novaes.

In this paper, using the theory of inverse integrating factor, we provide a new simple proof for the Lum-Chua's conjecture, which says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In addition, we prove that if this limit cycle exists, then it is hyperbolic and its stability is characterized in terms of the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle has not been pointed out before.

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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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