https://repositorio.ufu.br/handle/123456789/5473 Mon, 28 Sep 2026 20:34:13 GMT 2026-09-28T20:34:13Z https://repositorio.ufu.br/handle/123456789/50413 Title: O grafo funcional de uma família de polinômios sobre uma extensão quadrática de um corpo finito Abstract: Let $X$ be a finite set and $f: X \rightarrow X$ an arbitrary function. The iteration of $f$ over $X$ yields a dynamical system that can be represented as a graph. In fact, the functional graph of $f$ is the directed graph $\mathscr{G}(f)=(\mathcal{V}, \mathcal{E})$, where $\mathcal{V}=X$ and $\mathcal{E}=\{\langle x, f(x)\rangle \mid x \in X\}$. The obtained dynamical system has applications in many areas such as cryptography, communications and physics. Let $\mathbb{F}_q$ be a finite field with $q$ elements of odd characteristic. In this work, we will fully describe the dynamics of the function $f(X)=c\left(X^{q+1} \pm a X^2\right)$ over the finite field $\mathbb{F}_{q^2}$, for $a \in\{ \pm 1\}$ and $c \in \mathbb{F}_q^*$. We will also provide partial results for $a \in \mathbb{F}_q^* \backslash\{ \pm 1\}$. Mon, 30 Jun 2025 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/50413 2025-06-30T00:00:00Z https://repositorio.ufu.br/handle/123456789/50366 Title: Análise combinatória: fundamentos teóricos e propostas didáticas para o ensino das técnicas de contagem na Educação Básica Abstract: The transition from direct counting to structured combinatorial reasoning represents a milestone in the development of mathematical thinking. However, the traditional didactic approach in Basic Education often reduces Combinatorics to the mechanical application of formulas, leading to learning difficulties and recurring student errors. Given this scenario, the main objective of this dissertation is to present the theoretical foundations of Combinatorics and to structure didactic proposals applicable to Basic Education, aiming to improve teaching practices and to strengthen student learning. The theoretical framework is developed across two chapters: the first addresses the Fundamental Methods of Counting (Additive and Multiplicative Principles, simple groupings, circular permutations, permutations with repetition, and combinations with repetition); the second explores Advanced Counting Approaches (Principle of Inclusion-Exclusion, Derangements, Kaplansky’s Lemmas, Reflection Principle, Pascal’s Triangle, Binomial Theorem, and Leibniz’s Multinomial Theorem). Finally, the third chapter presents the Educational Product: an educational booklet featuring inquiry-based learning sequences aligned with the BNCC, progressively covering the early and final years of Elementary School through High School, employing resources ranging from concrete manipulatives to path counting on rectangular grids (lattice paths). Thu, 10 Sep 2026 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/50366 2026-09-10T00:00:00Z https://repositorio.ufu.br/handle/123456789/50342 Title: Corpos de funções algébricas e códigos AG em dois pontos Abstract: In this dissertation, we study algebraic function fields and their application to the construction of algebraic–geometric codes, with emphasis on two-point codes on the norm–trace curve. We present the necessary theoretical foundations, including places, valuations, divisors, and the Riemann–Roch Theorem, and discuss relevant examples such as Hermitian codes. Finally, we investigate the parameters of two-point codes on the norm–trace curve, highlighting their advantages over classical one-point codes. Tue, 24 Feb 2026 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/50342 2026-02-24T00:00:00Z https://repositorio.ufu.br/handle/123456789/50285 Title: Estudo numérico do modelo de Giesekus em escoamento bidimensional com obstáculo Abstract: This study develops and analyzes a numerical method for simulating two-dimensional viscoelastic flows described by the Giesekus constitutive model, with emphasis on the flow around a circular cylinder confined between parallel plates. The mathematical formulation is based on the Navier-Stokes equations for incompressible fluids, written in the vorticity-velocity form, and for the implementation of the immersed obstacle, the Immersed Boundary Method is adopted, allowing the use of fixed Cartesian meshes without the need to generate meshes adjusted to the boundary. The spatial discretization is performed using high-order compact schemes, and the temporal integration is done using the fourth-order Runge-Kutta method. For the numerical stabilization of viscoelastic regimes, the log-conformation technique is used for the evolution of the polymer stress tensor, which is especially necessary in regimes associated with high Weissenberg numbers. The solution of Poisson's equations is obtained using a multigrid method, ensuring computational efficiency. Initially, the numerical methodology is verified by simulating classical Newtonian flows, comparing the drag coefficients and the wake structure to results available in the literature. Next, the viscoelastic flow between parallel plates is analyzed, investigating the influence of the constitutive parameters of the Giesekus model. Lastly, we move the study to the viscoelastic flow around a confined circular cylinder, evaluating the impact of fluid elasticity on the problem. The results indicate that the drag coefficient in viscoelastic flow past a confined cylinder strongly depends on the model parameters. For a fixed Reynolds number, the drag coefficient increases with the viscosity ratio beta, while it decreases as Re increases, preserving a trend similar to the Newtonian regime. In addition, increasing the mobility parameter alpha_G is associated with a reduction in the drag coefficient, with a more pronounced effect for lower values of this parameter. Thu, 26 Feb 2026 00:00:00 GMT https://repositorio.ufu.br/handle/123456789/50285 2026-02-26T00:00:00Z