Fermi-Frenet coordinates for space-like curves

We generalize Fermi coordinates, which correspond to an adapted set of coordinates describing the vicinity of an observer's worldline, to the worldsheet of an arbitrary spatial curve in a static spacetime. The spatial coordinate axes are fixed using a covariant Frenet triad so that the metric can be expressed using the curvature and torsion of the spatial curve. As an application of Fermi-Frenet coordinates, we show that they allow covariant inertial forces to be expressed in a simple and physically intuitive way.Comment: 7 page

On the embedding of spacetime in five-dimensional Weyl spaces

We revisit Weyl geometry in the context of recent higher-dimensional theories of spacetime. After introducing the Weyl theory in a modern geometrical language we present some results that represent extensions of Riemannian theorems. We consider the theory of local embeddings and submanifolds in the context of Weyl geometries and show how a Riemannian spacetime may be locally and isometrically embedded in a Weyl bulk. We discuss the problem of classical confinement and the stability of motion of particles and photons in the neighbourhood of branes for the case when the Weyl bulk has the geometry of a warped product space. We show how the confinement and stability properties of geodesics near the brane may be affected by the Weyl field. We construct a classical analogue of quantum confinement inspired in theoretical-field models by considering a Weyl scalar field which depends only on the extra coordinate.Comment: 16 pages, new title and references adde

Riemannian Geometry of Noncommutative Surfaces

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further developments of the latter. The notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric-compatible, giving rise to the corresponding Riemann curvature. The latter also satisfies the noncommutative analogue of the first and second Bianchi identities. As examples, noncommutative analogues of the sphere, torus and hyperboloid are studied in detail. The problem of covariance under appropriately defined general coordinate transformations is also discussed and commented on as compared with other treatments.Comment: 28 pages, some clarifications, examples and references added, version to appear in J. Math. Phy

Soliton surfaces associated with symmetries of ODEs written in Lax representation

The main aim of this paper is to discuss recent results on the adaptation of the Fokas-Gel'fand procedure for constructing soliton surfaces in Lie algebras, which was originally derived for PDEs [Grundland, Post 2011], to the case of integrable ODEs admitting Lax representations. We give explicit forms of the \g-valued immersion functions based on conformal symmetries involving the spectral parameter, a gauge transformation of the wave function and generalized symmetries of the linear spectral problem. The procedure is applied to a symmetry reduction of the static $\phi^4$-field equations leading to the Jacobian elliptic equation. As examples, we obtain diverse types of surfaces for different choices of Jacobian elliptic functions for a range of values of parameters.Comment: 14 Pages, 2 figures Conference Proceedings for QST7 Pragu

Phase diagram of a model for a binary mixture of nematic molecules on a Bethe lattice

We investigate the phase diagram of a discrete version of the Maier-Saupe model with the inclusion of additional degrees of freedom to mimic a distribution of rodlike and disklike molecules. Solutions of this problem on a Bethe lattice come from the analysis of the fixed points of a set of nonlinear recursion relations. Besides the fixed points associated with isotropic and uniaxial nematic structures, there is also a fixed point associated with a biaxial nematic structure. Due to the existence of large overlaps of the stability regions, we resorted to a scheme to calculate the free energy of these structures deep in the interior of a large Cayley tree. Both thermodynamic and dynamic-stability analyses rule out the presence of a biaxial phase, in qualitative agreement with previous mean-field results

From Golden Spirals to Constant Slope Surfaces

In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces could be thought as the bi-dimensional analogue of the generalized helices. Some pictures are drawn by using the parametric equations we found.Comment: 11 pages, 8 figure

Improving the quality of Portuguese SMEs through competence evaluation

UIDB/04647/2020 UIDP/04647/2020Problem and goal. Since we are going through a key moment in terms of technological advances and their effect on the labour field, the investigation of the on-the-job training processes gains a predominant role. Moreover, the need to research an approach that promotes a balance between the four spheres of development of learning – economic, sustain-able, community and personal – is emerging. This study was conceived as a result of this ur-gency, aiming to respond to the following problem: to what extent does a training model, based on the VPL (Validation of Prior Learning) approach and supported by principles of the educational sciences, can contribute to individual empowerment and, consequently, to an improvement in the organizational quality of Portuguese SMEs (Small and Medium-sized Enterprises)? Methodology. The division of the research into two parts – Study 1 and Study 2 – al-lowed the collection and analysis of structural data (Study 1) that served as a basis for the defi- nition of a dynamic strategy (Study 2). Although undertaken at different times, they have thus proved to be complementary. In fact, the design of the training model was based, firstly, on the results obtained through exploratory interviews with key informants and, secondly, on the data obtained throughout the training process that involved ninety-two employees from ten Portuguese SMEs from diverse categories and sectors. Results. The training model, which emerged from the interaction between the theoreti-cal reflection and the practical context, comprises five components: distance learning, recog-nition of skills, language skills, intercultural skills and an individual action plan. Conclusion. This study, being an important contribution to the topic of on-the-job training in Portuguese SMEs, at the same time opens the way for the continuation of the research.publishersversionpublishe