# Research Reports

 1/2023 Métricas ad-invariantes em Álgebras de LieMarcos Ricardo Cavicchioli de Almeida  Este material ´e resultado de um trabalho de inicia¸c˜ao cient´ıfica, projeto de n´umero 2022/07595- 9 fomentado pela FAPESP. Para seu uso, ´e esperado que o estudante j´a tenha feito um curso b´asico de ´algebra linear e esteja familiarizado com espa¸cos vetoriais, transforma¸c˜oes lineares, produto interno, etc. Para uma revis˜ao desses assuntos, as referˆencias  e  s˜ao boas op¸c˜oes. Neste projeto, o objetivo principal foi o estudo de ´algebras de Lie munidas de m´etricas adinvariantes. Numa primeira instˆancia, se estudou a estrutura de uma ´algebra de Lie, que n˜ao ´e nada mais do que um espa¸co vetorial dotado de uma transforma¸c˜ao bilinear que satisfaz certas propriedades (chamada comumente de colchete de Lie), . Como primeiros exemplos foram trabalhados espa¸cos vetoriais not´aveis de matrizes, como sl(n,R), so(n,R), sp(n,R), que s˜ao ´algebras de Lie cl´assicas com m´etricas ad-invariantes , , . Com o estudo de formas bilineares, a ideia de m´etrica pode ser apresentada, bem como o estudo de ´algebras de Lie com uma abordagem mais abstrata, , , , . Atrav´es do c´alculo da forma de Cartan-Killing nos espa¸cos cl´assicos de matrizes, foi poss´ıvel verificar que o colchete de Lie ´e antissim´etrico. Uma m´etrica ´e uma forma bilinear n˜ao-degenerada que busca expandir a no¸c˜ao de produto interno em espa¸cos vetoriais, e o fato de uma m´etrica ser ad-invariante significa que o colchete de Lie define transforma¸c˜oes antissim´etricas com respeito `a m´etrica. Finalmente, se estudou no fim do projeto o processo de extens˜ao dupla introduzido por Favre e Santharoubane (), ainda que em alguns exemplos pontuais. Este processo permite construir ´algebras de Lie com m´etricas ad-invariantes a partir de ´algebras com a mesma estrutura, mas com dimens˜oes menores. Assim, foi poss´ıvel construir ´algebras de Lie de dimens˜oes baixas partindo de ´algebras abelianas de dimens˜ao 1 ou 2. rp-2023-01.pdf 2/2022 Tópicos de processos estocásticosVicenzo Bonasorte Reis Pereira , Élcio Lebensztayn rp-2022-022022.pdf 1/2022 Estimates for entropy numbers of multiplier operators of multiple seriesSérgio A. Córdoba, Jéssica Milaré, Sérgio A. TozoniThe asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system, on a probability space and uniformly bounded, is studied. For example, the orthonormal system can be obtained as the product of the functions of the Vilenkin system, Walsh system on a real sphere or of the trigonometric system on the unit circle. General upper and lower bounds for the entropy numbers are established by using Levy means of norms constructed using the orthonormal system. These results are applied to get upper and lower bounds for entropy numbers of specific multiplier operators, which generate, in particular cases, sets of finitely and infinitely differentiable functions, in the usual sense and in the dyadic sense. It is shown that these estimates have order sharp in various important cases. rp-2022-012022.pdf 1/2020 Extending Multivariate-t Semiparametric Mixed Models for Longitudinal data with Censored Responses and Heavy TailsThalita B. Mattos, Larissa A. Matos, Victor H. LachosIn this paper we extended the semiparametric mixed model for longitudinal censored data with normal errors to Student-t erros. This models allows exible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random e ects. Penalized likelihood equations are applied to derive the maximum likelihood estimates which appear to be robust against outlying observations in the sense of the Mahalanobis distance. We estimate nonparametric functions by using smoothing splines jointly estimate smoothing parameter by the EM algorithm. Finally, the performance of the proposed approach is evaluated through extensive simulation studies as well as application to dataset from AIDS study. rp-2020-01.pdf 6/2019 ON ARCS AND PLANE CURVESBeatriz Motta, Fernando TorresWe 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. rp-2019-06.pdf 5/2019 ON THE CURVE Y n = Xℓ(Xm + 1) OVER FINITE FIELDS IISaeed Tafazolian, Fernando TorresAbstract. Let F be the finite field of order q2. In this paper we continue the study in , ,  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. rp-2019-05.pdf 4/2019 Explosion in a Growth Model with Cooperative Interaction on an In nite GraphBruna de Oliveira Gonçalves, Marina VachkovskaiaIn 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 types. rp-2019-04.pdf 3/2019 A NEW SIMPLE PROOF FOR THE LUM-CHUA'S CONJECTUREVictoriano 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. rp-2019-03.pdf 2/2019 Estimates for n-widths of sets of smooth functions on complex spheresDeimer J. J. Aleans, Sergio A. TozoniIn 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. rp-2019-02.pdf 1/2019 Estudo de um Método Baseado em Autovalores Generalizados para o Subproblema de Região de ConfiançaJean Carlos A. Medeiros , Sandra Augusta SantosThe 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., based on Gander et al., 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. rp-2019-01.pdf