The vanishing of L2 harmonic one-forms on based path spaces

We prove the triviality of the first L2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L2 two-forms, respectively; these are considered as extended Clark-Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L2 one-forms. This Laplacian is shown to have a spectral gap

The complexities and challenges of introducing electronic Ongoing Achievement Records in the pre-registration nursing course using PebblePad and hand-held tablets

This paper reports on a small pilot study aimed at eliciting the lecturer and student experience of using PebblePad to record the students' Ongoing Achievement Record (OAR) using hand-held tablets, at one university in England. Android tablets were purchased and attempts were made to transfer the OAR into the PebblePad system in an attempt to enhance the student experience of feedback from their via PebblePad, embed PebblePad learning technology in the practice component of the curriculum, enable the student to more readily engage in reflection and feedback with their personal tutor, practice education link and mentor, develop skills in the use of PebblePad and pilot the use of PebblePad in developing the Ongoing Achievement Record. Focus groups were carried out with students nurses (n=6) and lecturers (n=5) where participants were asked to discuss the successes and challenges of using PebblePad for the Ongoing Achievement Record, and suggest ways in which this strategy may be implemented more widely. Through a thematic analysis of the focus groups three broad themes of 'timing', 'technology literacy' and 'the technology' were identified. The findings from the study indicated that whilst this was not a positive experience on the whole for a number of reasons, there are lessons that can be learnt when attempting to introduce new ways of engaging with technology to enhance the student experience. Recommendations for implementing such an approach in the future are also presente

Anticommuting Variables, Fermionic Path Integrals and Supersymmetry

(Replacement because mailer changed `hat' for supercript into something weird. The macro `\sp' has been used in place of the `hat' character in this revised version.) Fermionic Brownian paths are defined as paths in a space para\-metr\-ised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This formula is used to give a proof of the Atiyah-Singer index theorem which is rigorous while being closely modelled on the supersymmetric proofs in the physics literature.Comment: 18 pages, KCL-TH-92-