On the stability of circular orbits in galactic dynamics: Newtonian thin disks

The study of off-equatorial orbits in razor-thin disks is still in its beginnings. Contrary to what was presented in the literature in recent publications, the vertical stability criterion for equatorial circular orbits cannot be based on the vertical epicyclic frequency, because of the discontinuity in the gravitational field on the equatorial plane. We present a rigorous criterion for the vertical stability of circular orbits in systems composed by a razor-thin disk surrounded by a smooth axially symmetric distribution of matter, the latter representing additional structures such as thick disk, bulge and (dark matter) halo. This criterion is satisfied once the mass surface density of the thin disk is positive. Qualitative and quantitative analyses of nearly equatorial orbits are presented. In particular, the analysis of nearly equatorial orbits allows us to construct an approximate analytical third integral of motion in this region of phase-space, which describes the shape of these orbits in the meridional plane.Comment: 3 pages, 1 figure. In Proceedings of the MG13 Meeting on General Relativity, Stockholm University, Sweden, 1-7 July 2012. World Scientific, Singapore. Based on arXiv:1206.6501. in The Thirteenth Marcel Grossmann Meeting: On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (In 3 Volumes), chap. 438, pages 2346-2348 (2015

Vertical stability of circular orbits in relativistic razor-thin disks

During the last few decades, there has been a growing interest in exact solutions of Einstein equations describing razor-thin disks. Despite the progress in the area, the analytical study of geodesic motion crossing the disk plane in these systems is not yet so developed. In the present work, we propose a definite vertical stability criterion for circular equatorial timelike geodesics in static, axially symmetric thin disks, possibly surrounded by other structures preserving axial symmetry. It turns out that the strong energy condition for the disk stress-energy content is sufficient for vertical stability of these orbits. Moreover, adiabatic invariance of the vertical action variable gives us an approximate third integral of motion for oblique orbits which deviate slightly from the equatorial plane. Such new approximate third integral certainly points to a better understanding of the analytical properties of these orbits. The results presented here, derived for static spacetimes, may be a starting point to study the motion around rotating, stationary razor-thin disks. Our results also allow us to conjecture that the strong energy condition should be sufficient to assure transversal stability of periodic orbits for any singular timelike hypersurface, provided it is invariant under the geodesic flow.Comment: 13 pages, 4 figures; Accepted for publication in Physical Review

Enhancing web supported learning in higher education by adding a management layer to LMSs

There are many situations in the e-Learning experiences that can compromise the success of the courses. Many times simple reasons are great enough to motivate people to abandon them. For example, if someone does not execute a programmed activity inside the defined window of time, it can compromise the rest of the course to that person. In such situations it would be important that the teacher knew about the situation in useful time, to be able to take any corrective action. Another example could be presented, involving the professor and the learners. Let us assume that an activity A2 is programmed to be executed by the learners and that it depends on the previous knowledge of the result of the evaluation of a work submitted by the learners to the teacher (activity A1). If the teacher doesn’t inform the learners about their classification in useful time, that can compromise the execution of the activity A2. It seems to be necessary to use mechanisms of automatic management, in real time, of the envolvement of each participant in a distance learning course using LMS (Learning Management System). Such a functionality allows the detection of deviations to the scheduled activities planned for each actor. If it is the case, the referred mechanism can initiate the process of sending notifications to the relevant entities, enabling the correction of these deviations. Several organizations and consortiuns, involving the industry, governmental institutions and universities, are developing projects of standardization. It seemed important to us to see how the referred aspects were covered by those projects, and to perceive how it could be possible to articulate our work with the ones that are available from these organizations and consortiuns. This article describes the work that the authors are developing towards the specification of a layer for real-time management of user interactions with LMSs, during the operationalization of a course, and also includes a management meta-data model, related to that management layer

Modelling a layer for real-time management of interactions in web based distance learning

In the last few years, the University of Aveiro, Portugal, has been offering several distance learning courses over the Web, using e-learning platforms. Experience showed that different editions of a same course, using the same contents and structure, and having similar target learners, had different success rates. What would be the reason for that? A hypothesis was considered: The level of success could be directly related with the remote follow-up of the learners’ participation in the courses; the best results usually occur when the follow-up is closer. The existing e-learning platforms offer and the standardization works being developed by organizations and consortiums like IMS (IMS Global Learning Consortium, Inc), ADL SCORM (Advanced Distributed Learning Sherable Content Object Reference Model), IEEE LTSC LOM (Institute of Electrical and Electronic Engineers Learning Technologies Standard Committee Learning Object Metadata), ARIADNE (ARIADNE Foundation for the European Knowledge Pool), AICC CMI (Aviation Industry CBT Committee Computer Managed Instruction), etc, don’t cover the course monitorization concerns mentioned. Those projects were focused on aspects like contents and its delivery in the context of the execution of the courses’ activities. This is even true in the SCORM project that doesn’t include any reference to the management of the e-learning processes. Recently, in the context of the IMS Global Consortium, a new project designated IMS LD (Learning Design) is under development, providing a framework for the description of learning units under a three level model. In the most recently defined level, the C level, some functionalities related to notifications were proposed, expressing similar concerns to the ones that triggered our research. However, the extent at which IMS LD takes the functionalities is, from our point of view, not complete. This article describes a proposal of a reference model and functionalities towards a specification of a layer for real-time management of user interactions on LMSs, and its possible integration with the ADL SCORM standard proposal. The paper includes a discussion of the management metadata model for the LMS sub-system and how the integration of the management module under SCORM may be achieved