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- The explicit formula for Gauss-Jordan elimination applied to flexible systemsPublication . Tran, Nam van; Justino, Júlia; Berg, Imme van denFlexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers. The neutrices model imprecisions, giving rise to calculation rules extending informal error calculus. Stability conditions for flexible systems are given in terms of relative imprecision and size of determinants. We then apply the explicit formula for the elements of the successive intermediate matrices of the Gauss-Jordan elimination procedure to find the solution of flexible systems, keeping track of the error terms at every stage. The solution respects the original imprecisions in the right-hand side and is the same as the one given by Cramer’s rule.
- On The Explicit Formula For Gauss-Jordan EliminationPublication . Van Tran, Nam; Justino, Júlia; Van den Berg, Imme
- The Expansion of Pedagogical Alignment: A Step for the Learning SuccessPublication . Justino, Júlia; Rafael, SilvianoThe adaptation of higher education institutions to the changing needs of employers and society demands critical and creative ability in the elaboration of new curricula. The implementation of these new curricula requires a change in methodology to the learner-centered approach that is not easy at first and requires a deeper reflective and analytical attitude from the teacher. How it starts and how should be applied is a recurring issue at conferences and is the subject of this paper. The first step consists of building the training structure to be implemented, been usual to build a structure which supports contents, learning outcomes and assessment methodology. But this is not sufficiently comprehensive to provide guidance and reflection to the teacher about the learning activities to be implemented nor the learning itself, even with the support of an IT tool to maintain the focus and establish the development phases of the pedagogical structure. The new feature presented in this paper is the expansion of the training structure to all educational elements needed for the implementation of the student-centered approach. An example is given within the context of a course unit of a short-cycle tertiary educational programme (2 years – 180 ECTS), providing the syllabus planning, the learning outcomes setting, linking them with their ground level of cognition, the teaching-learning activities setting for each specific content, the resources to students allocation and/or creation, structuring them on an IT platform, the pedagogical strategies setting and the evaluation method definition. This planning allows to structure the whole teaching-learning process of the course unit, considering the available timeframe, the application environment or context, the students’ competences to exercise and the teacher’s pedagogical skills.
- Caso de estudo: aplicação de técnicas de aprendizagem ativa no ensino da Matemática de um CTESP em PBLPublication . Justino, Júlia; Rafael, SilvianoO presente artigo decorre de um estudo realizado à aplicação da metodologia cen-trada no aluno de uma nova unidade curricular de matemática desagregada do ambiente de aprendizagem por projetos de um novo curso técnico superior profis-sional no ensino superior. Salientam-se os principais objetivos gerais pretendidos e indicam-se as técnicas pedagógicas que se consideraram adequadas para serem aplicadas neste contexto. O estudo segue uma abordagem quantitativa, baseada em inquéritos e complementada por ações típicas de abordagem qualitativa. São refe-ridos alguns aspetos relativos à atitude dos alunos, individualmente e em grupo, antes da aplicação da metodologia e depois da aplicação das técnicas pedagógicas referidas. Apresenta-se o método de avaliação e a forma como foi utilizado. São apresentados e discutidos os resultados do desempenho escolar onde se destaca uma taxa de sucesso de 81%.
- A DSS for Bicriteria Location ProblemsPublication . Fernandes, Sérgio; Captivo, Maria Eugénia; Clímaco, JoãoIn this paper, we present a new interactive decision support system (DSS) – named SABILOC – in action. The DSS is aimed at supporting decision-making concerning bicriteria location models and it was specially designed to deal with problems in which the facilities to be located have environmental impacts. We point out three advantages of SABILOC when compared with other DSS for location problems available in the literature. Firstly, an interactive procedure is used allowing the decision-maker to obtain non-dominated solutions for bicriteria location models. Secondly, a multiattribute analysis module allows the decision-maker to proceed with a more detailed analysis of a subset of solutions selected from the first interactive phase. Finally, the embedment of a Geographic Information System into SABILOC makes the system much more useful in real world location problems. In order to better describe and validate the potentialities and functionalities of SABILOC, we present a case study of a real world problem applied to waste transfer station siting.
- Invertible flexible matricesPublication . Justino, JúliaMatrices with coefficients having uncertainties of type o (.) or O(.), called flexible matrices, are studied from the point of view of nonstandard analysis. The uncertainties of the afore-mentioned kind will be given in the form of the so-called neutrices, for instance the set of all infinitesimals. Since flexible matrices have uncertainties in their coefficients, it is not possible to define the identity matrix in an unique way and so the notion of spectral identity matrix arises. Not all nonsingular flexible matrices can be turned into a spectral identity matrix using Gauss-Jordan elimination method, implying that that not all nonsingular flexible matrices have the inverse matrix. Under certain conditions upon the size of the uncertainties appearing in a nonsingular flexible matrix, a general theorem concerning the boundaries of its minors is presented which guarantees the existence of the inverse matrix of a nonsingular flexible matrix.
- Cramer’s Rule applied to flexible systems of linear equationsPublication . Justino, Júlia; van den Berg, ImmeSystems of linear equations, called flexible systems, with coefficients having uncertainties of type o (.) or O (.) are studied. In some cases an exact solution may not exist but a general theorem that guarantees the existence of an admissible solution, in terms of inclusion, is resented. This admissible solution is produced by Cramer’s Rule; depending on the size of the uncertainties appearing in the matrix of coefficients and in the constant term vector some adaptations may be needed.