Cátia Regina de Oliveira Quilles Queiroz

The paper’s aims are threefold. First, to identify the Fuchsian groups associated with the $\{4\lambda, 4\}$ and $\{6\lambda, 3\}$ hyperbolic tessellations in quaternion orders, where $\lambda = 3^{n}$ . Second, to provide the bases of the maximal quaternion orders associated with such dense tessellation families. Third, to provide the algebraic structure for constructing hyperbolic lattices or geometrically uniform signal constellations leading to a complete algebraic labeling. These tessellations are relevant for being denser and having inherent mathematical properties, making them a favorite in several classical and quantum applications. However, to the best of our knowledge, a systematic approach and results are needed to put the applications of such relevant mathematical structures in perspective. Hence, the contributions of this paper are related to the presentation of the corresponding Fuchsian groups $\Gamma _{4\lambda }$ and $\Gamma _{6\lambda }$ derived from quaternion algebras $\mathcal {A}$ over a number field $\mathbb {K}$ whose primitive elements are established in Theorems 10 and 20, respectively. Consequently, Theorems 13 and 14, and Theorems 23 and 24 establish the bases of the maximal orders related to the $\{4\lambda, 4\}$ and $\{6\lambda,3\}$ tessellations, respectively, leading to the algebraic labeling to be complete. Therefore, these results provide the algebraic structures for constructing the corresponding hyperbolic lattices.

This article aims to understand the influence and consequences of academic monitoring in the relationship between undergraduates and mathematics in the context of a federal university. The methodology used was the case study, with a mixed research approach. For this, a semi-structured questionnaire was applied to forty-one students enrolled in two classes of Calculus I and Mathematics, offered for the Biological Sciences, Mathematics and Biotechnology courses in the second half of 2019. When seeking an understanding of the student-discipline relationship, it was found that most students do not frequently participate in monitoring sessions or seek help from the professor outside the classroom. Furthermore, even though establishing good relations with the monitors and professors, there is no appropriation of the mathematical content, which becomes an obstacle in the transition phase between High School and Higher Education. Therefore, there is a need to raise students' awareness to encourage more active participation in monitoring, communicating their demands to the university. Simultaneously, there is a need for awareness among professors and university administration regarding the significance of monitoring, aiming to not only increase availability but also provide more flexible schedules. In another aspect, it is observed that the contact with the monitoring allows the monitor and the teacher to carry out a diagnosis of the difficulties, seeking new teaching strategies and methodologies. We hope that the present study favors the understanding of the theme of academic tutoring focused on Mathematics disciplines.

Esta pesquisa teve por objetivo apresentar uma sequência didática para o ensino de números inteiros baseada na aplicação de um jogo de tabuleiro educativo que foi proposto pelos pesquisadores como um produto educacional. Tal proposta surgiu da necessidade de se buscar alternativas para transpor a Matemática formal para uma linguagem lúdica e atrativa a estudantes da Educação Básica. A pesquisa buscou fazer aplicações práticas do referido jogo em salas de aula de 7º ano do Ensino Fundamental Anos Finais, a fim de verificar a eficácia do jogo a partir de uma perspectiva socioconstrutivista. Como principais suportes teóricos, foi adotada a abordagem proposta por Lev Semyonovich Vygotsky, que leva em conta a interação entre indivíduos em diferentes níveis de desenvolvimento cognitivo no processo de aprendizagem, bem como os estudos do educador matemático Cristiano Alberto Muniz que aponta os jogos como um instrumento de mediação entre as dimensões intrapessoais e socioculturais durante os processos de ensino-aprendizagem da Matemática. Os resultados preliminares desta pesquisa indicam que o jogo proposto cumpre seu objetivo de ser divertido e interativo, embora haja a possibilidade de que estudos futuros possam ser aprofundados a partir dos resultados experienciados durante pesquisa.

This paper aims to construct signal sets from quotient rings of the quaternion over a real number field associated with the arithmetic Fuchsian group $\Gamma _{4g}$ , where $g$ is the genus of the associated surface. These Fuchsian groups consist of the edge-pairing isometries of the regular hyperbolic polygons (fundamental region) $P_{4g}$ , which tessellate the hyperbolic plane $\mathbb {D}^{2}$ . The corresponding tessellations are the self-dual tessellations $\{4g, 4g\}$ . Knowing the generators of the quaternion orders which realize the edge-pairings of the polygons, the signal points of the signal sets derived from the quotient rings of the quaternion orders are determined. It is shown by examples the relevance of adequately selecting the ideal in the maximal order to construct the signal sets satisfying the property of geometrical uniformity. The labeling of such signals is realized by using the mapping by set partitioning concept to solve the corresponding Diophantine equations (extreme quadratic forms). Trellis coded modulation and multilevel codes whose signal sets are derived from quotient rings of quaternion orders are considered possible applications.