Reflections of universal algebras into semilattices, their Galois theories, and related factorization systems

Estabelecemos uma condição suficiente para a preservação dos produtos finitos, pelo reflector de uma variedade de álgebras universais numa subvariedade, que é, também, condição necessária se a subvariedade for idempotente. Esta condição é estabelecida, seguidamente, num contexto mais geral e caracteriza reflexões para as quais a propriedade de ser semi-exacta à esquerda e a propriedade, mais forte, de ter unidades estáveis, coincidem. Prova-se que reflexões simples e semi-exactas à esquerda coincidem, no contexto das variedades de álgebras universais e caracterizam-se as classes do sistema de factorização derivado da reflexão. Estabelecem-se resultados que ajudam a caracterizar morfismos de cobertura e verticais-estáveis em álgebras universais e no contexto mais geral já referido. Caracterizam-se as classes de morfismos separáveis, puramente inseparáveis e normais. O estudo dos morfismos de descida de Galois conduz a condições suficientes para que o seu par kernel seja preservado pelo reflector.

Spectral characterization of families of split graphs

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted.

The Jordan canonical form for a class of weighted directed graphs

Recently, Cardon and Tuckfield (2011) [1] have described the Jordan canonical form for a class of zero-one matrices, in terms of its associated directed graph. In this paper, we generalize this result to describe the Jordan canonical form of a weighted adjacency matrix A in terms of its weighted directed graph.

A creative approach to isometries integrating Geogebra and Italc with ‘paper and pencil’ environments

Creativity is recognized nowadays as a basic skill. However, the educational system fails in promoting their development. On the other hand, a growing acknowledgement of the importance of geometry emerges. Conceptual renewal, namely on isometries, requires new approaches based on mathematically significant tasks. The digital revolution has brought powerful tools but demands changes in the educational process. The use of Dynamic Geometry Environments (DGE), complementing ‘paper and pencil’, can contribute to provide rich learning environments, enhanced by Classroom Management Systems (CMS) such as iTALC. Indeed, the qualitative case study we carried out suggests that: the creation of an "atmosphere" of cooperation, collaboration and sharing seems to increase creativity dimensions; the use of DGE can facilitate the emergence of more creative productions; development of knowledge and geometrical capabilities seems to benefit from a complementary approach that combines DGE and ‘paper and pencil’ environments. Different approaches, with a more technological and exploratory nature seem to promote more favourable attitudes towards mathematics in general, and geometry, in particular.