Lower Approximations by Fuzzy Consequence Operators

Peer ReviewedPostprint (author's final draft

Three aspects of bosonized supersymmetry and linear differential field equation with reflection

Recently it was observed by one of the authors that supersymmetric quantum mechanics (SUSYQM) admits a formulation in terms of only one bosonic degree of freedom. Such a construction, called the minimally bosonized SUSYQM, appeared in the context of integrable systems and dynamical symmetries. We show that the minimally bosonized SUSYQM can be obtained from Witten's SUSYQM by applying to it a nonlocal unitary transformation with a subsequent reduction to one of the eigenspaces of the total reflection operator. The transformation depends on the parity operator, and the deformed Heisenberg algebra with reflection, intimately related to parabosons and parafermions, emerges here in a natural way. It is shown that the minimally bosonized SUSYQM can also be understood as supersymmetric two-fermion system. With this interpretation, the bosonization construction is generalized to the case of N=1 supersymmetry in 2 dimensions. The same special unitary transformation diagonalises the Hamiltonian operator of the 2D massive free Dirac theory. The resulting Hamiltonian is not a square root like in the Foldy-Wouthuysen case, but is linear in spatial derivative. Subsequent reduction to `up' or `down' field component gives rise to a linear differential equation with reflection whose `square' is the massive Klein-Gordon equation. In the massless limit this becomes the self-dual Weyl equation. The linear differential equation with reflection admits generalizations to higher dimensions and can be consistently coupled to gauge fields. The bosonized SUSYQM can also be generated applying the nonlocal unitary transformation to the Dirac field in the background of a nonlinear scalar field in a kink configuration.Comment: 18 pages, LaTeX, minor typos corrected, ref updated, to appear in Nucl. Phys.

Collaboration and motivation in an online learning environment: students’ perceptions of collaborative activities and attitudes towards online learning

The report about distance learning in higher education in Portugal (2009) states that only a small percentage of HE enrolments are in DL courses, that the demand for the modality is growing, and that the Portuguese research in DL needs development, to support innovation in the modality. This study aimed to identify the collaborative activities that the 122 students enrolled in Masters’ courses between 2009-2012, in two Portuguese Universities, considered more motivating, their preferred type of assignment, the tools’ perceived ease of use, the social and cognitive aspects of teamwork, the tutor’s influence on teamwork and preferred team organization. The results indicate that the students feel comfortable participating, interacting and debating and that some collaborative activities such as designing projects, simulations, problem-based activities, discussions and written reports are more motivating than others. The results also show that the students have positive attitudes towards online learning, that online trust takes time to develop, that both face-to-face meetings and videoconference increase trust, though time flexibility is a practical advantage of online collaboration and that the way that collaborative assignments are designed can facilitate or hinder adequate collaboration