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Item type: Item , Formulação e resolução de problemas: uma alternativa metodológica para o desenvolvimento do pensamento crítico(Universidade Federal de Goiás, 2025-12-08) Lourenço, Arthur Alexandre Ramos; Lopes, Janice Pereira; Lopes, Janice Pereira; Santos, Maria Bethânia Sardeiro dos; Costa, Maraiza OliveiraThis paper presents the trajectory and results of a bibliographic study whose objective was to identify and analyze academic productions in the context of Mathematics Education that address problem posing and problem solving, and their possible relationships with the development of students’ critical thinking. To this end, open-access institutional repositories were consulted, considering master’s theses published between 2020 and 2025. The study focused on the analysis of three works, selected based on inclusion and exclusion criteria established to delimit the research corpus. Among the analyzed works were: a master’s thesis that investigated problem posing and problem solving related to conscious water consumption at school; another that analyzed the creation of mathematical problems from flyers and advertisements in the teaching of Financial Mathematics; and a third that discussed contextualization in the teaching of Mathematics in a rural school. As a result of the investigation and systematic analysis of these studies, it was possible to map the thematic scenarios that supported the productions, highlighting aspects that indicate certain convergences among the analyzed works, including the presence of elements such as contextualization, the development of problem situations, an emphasis on activities aligned with students’ everyday contexts, as well as the analysis of social practices related to the teaching of Mathematics. Based on the data obtained, both from the individual analysis of each study and from the comparative analysis among them, a systematic overview of characteristics that seem to structure and guide reflections and practices of problem posing and problem solving in the mathematics classroom is presented. Moreover, the results show that the act of posing and solving problems tends to foster meaningful student engagement and to bring the teaching of Mathematics closer to the social aspects of students’ lived experiences, which, consequently, may contribute to the development of their critical thinking when facing situations, both inside and outside school, whose understanding depends on mathematical knowledge.Item type: Item , Approximate proximal methods for variational inequalities on Hadamard manifolds(2025) Bento, Glaydston de Carvalho; Ferreira, Orizon Pereira; Papa Quiroz, Erik AlexIn this paper, we present an approximate proximal point method for addressing the variational inequality problem on Hadamard manifolds, and we analyse its convergence properties. The proposed algorithm exhibits inexactness in two aspects. Firstly, each proximal subproblem is approximated by utilizing the enlargement of the vector field under consideration, and subsequently, the next iteration is obtained by solving this subproblem while allowing for a suitable error tolerance. As an illustrative application, we develop an approximate proximal point method for nonlinear optimization problems on Hadamard manifolds.Item type: Item , Subsampled cubic regularization method for finite-sum minimization(2025) Gonçalves, Max Leandro NobreThis paper proposes and analyses a subsampled Cubic Regularization Method (CRM) for solving finite-sum optimization problems. The new method uses random subsampling techniques to approximate the functions, gradients and Hessians in order to reduce the overall computational cost of the CRM. Under suitable hypotheses, first- and second-order iteration-complexity bounds and global convergence analyses are presented. We also discuss the local convergence properties of the method. Numerical experiments are presented to illustrate the performance of the proposed scheme.Item type: Item , An inexact proximal point method with quasi-distance for quasiconvex multiobjective optimization problems on Riemannian manifolds(2025) Upadhyay, Balendu Bhooshan; Poddar, Subham; Ferreira, Orizon Pereira; Jen-Chih, YaoIn this paper, we investigate a class of unconstrained multiobjective optimization problems in the framework of Riemannian manifolds (abbreviated as, MOP-RM), where the components of the objective function are assumed to be locally Lipschitz and quasiconvex. By employing the powerful tool of Mordukhovich limiting subdifferential, we introduce an inexact proximal point algorithm with quasi-distance (abbreviated as, IPPMQ-RM), to solve MOP-RM. Moreover, we establish the well-definedness of the sequence generated by the IPPMQ-RM algorithm. Based on two different versions of error criteria, we introduce two variants of IPPMQ-RM, namely, IPPMQ-RM1 and IPPMQ-RM2. We deduce that the cluster points of the sequences generated by the IPPMQ-RM1 and IPPMQ-RM2 algorithms are Pareto-Mordukhovich critical points of MOP-RM. Further, we derive that if the components of the objective function of MOP-RM are geodesic convex, then these cluster points become Pareto efficient solutions of MOP-RM. We establish the finite termination of the IPPMQ-RM1 and IPPMQ-RM2 algorithms. By employing MATLAB R2023b, a non-trivial numerical example has been furnished to illustrate the effectiveness of the proposed algorithms, namely, IPPMQ-RM1 and IPPMQ-RM2. Moreover, to demonstrate that the sequence generated by the IPPMQ-RM algorithm converges faster than the algorithms existing in the literature, we furnish several non-trivial numerical examples in the framework of well-known Riemannian manifolds.Item type: Item , Quasilinear elliptic problems via nonlinear rayleigh quotient(2025) Carvalho, Marcos Leandro Mendes; Gasiński, Leszek; Santos Júnior, João Rodrigues dos; Silva, Edcarlos Domingos daIt is established existence and multiplicity of solution for the following class of quasilinear elliptic problems:{−ΔΦu = 𝜆a(x)|u|q−2u + |u|p−2u, x ∈ Ω,u = 0, x ∈ 𝜕Ω,where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, 1 < q < 𝓁 ≤ m < p < 𝓁∗ and Φ : ℝ → ℝ is suitable N-function.The main feature here is to show whether the Nehari method can be applied to find the largest positive number 𝜆∗ > 0in such a way that our main problem admits at least two distinct solutions for each 𝜆 ∈ (0, 𝜆∗ ). Furthermore, using somefine estimates and some extra assumptions on Φ, we prove the existence of at least two positive solutions for 𝜆 = 𝜆∗and 𝜆 ∈ (𝜆∗, 𝜆) where 𝜆 > 𝜆∗