Symmetry Properties of Autonomous Integrating Factors

We study the symmetry properties of autonomous integrating factors from an algebraic point of view. The symmetries are delineated for the resulting integrals treated as equations and symmetries of the integrals treated as functions or configurational invariants. The succession of terms (pattern) is noted. The general pattern for the solution symmetries for equations in the simplest form of maximal order is given and the properties of the associated integrals resulting from this analysis are given.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Gauge Variant Symmetries for the Schr\"odinger Equation

The last multiplier of Jacobi provides a route for the determination of families of Lagrangians for a given system. We show that the members of a family are equivalent in that they differ by a total time derivative. We derive the Schr\"odinger equation for a one-degree-of-freedom system with a constant multiplier. In the sequel we consider the particular example of the simple harmonic oscillator. In the case of the general equation for the simple harmonic oscillator which contains an arbitrary function we show that all Schr\"odinger equations possess the same number of Lie point symmetries with the same algebra. From the symmetries we construct the solutions of the Schr\"odinger equation and find that they differ only by a phase determined by the gauge.Comment: 12 page

Linking academic emotions and student engagement: mature-aged distance students’ transition to university

This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of Further and Higher Education on 2013, available online: https://www.tandfonline.com/doi/full/10.1080/0309877X.2014.895305Research into both student engagement and student emotions is increasing, with widespread agreement that both are critical determinants of student success in higher education. Less researched are the complex, reciprocal relationships between these important influences. Two theoretical frameworks inform this paper: Pekrun’s taxonomy of academic emotions and Kahu’s conceptual framework of student engagement. The prospective qualitative design aims to allow a rich understanding of the fluctuating and diverse emotions that students experience during the transition to university and to explore the relationships between academic emotions and student engagement. The study follows 19 mature-aged (aged 24 and over) distance students throughout their first semester at university, using video diaries to collect data on their emotional experiences and their engagement with their study. Pre and post-semester interviews were also conducted. Findings highlight that different emotions have different links to engagement: as important elements in emotional engagement, as inhibitors of engagement and as outcomes that reciprocally influence engagement. There are two key conclusions. First, student emotions are the point of intersection between the university factors such as course design and student variables such as motivation and background. Second, the flow of influence between emotions, engagement, and learning is reciprocal and complex and can spiral upwards towards ideal engagement or downwards towards disengagement and withdrawal.Publishe