Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For {\it arbitrary} self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich's extension, the answer is easily extracted from the general result. In combination with \cite{BKD}, a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.Comment: 27 pages, 2 figures; to appear in Manuscripta Mathematic

Exercise, Service and Support: Client Experiences of Physical Activity Referral Schemes(PARS)

Physical activity referral schemes (PARS) represent one of the most prevalent interventions in the fight against chronic illness such as coronary heart disease and obesity. Despite this, issues surrounding low retention and adherence continue to hinder the potential effectiveness of such schemes on public health. This article reports on the second stage of a larger investigation into client experiences of PARS focusing specifically on findings from five client-based focus groups and interviews with five Scheme Organisers. The resulting analysis reveals three main factors impacting participant perceptions of the quality of service and support received: the organisation of PARS provision, client engagement with the PARS community and the nature and extent of client support networks. The article demonstrates that staff have a considerable role to play in engaging clients in the PARS system and that Scheme Organisers should give serious thought to ensuring that clients have valuable and sustainable networks of support. Furthermore, it is suggested that Scheme Organisers need to facilitate a system in which staff are genuinely engaged with the needs of clients and are able to provide individualised programmes of physical activity

Falling of a quantum particle in an inverse square attractive potential

Evolution of a quantum particle in an inverse square potential is studied by analysis of the equation of motion for 〈r2〉. In such a way we identify the conditions of falling of a particle into the center. We demonstrate the existence of a purely quantum limit of falling, namely, a particle does not fall, when the coupling constant is smaller than a certain critical value. Also the time of falling of a particle into the center is estimated. Although there are no stationary energy levels for this potential, we show that there are quasi-stationary states which evolve with 〈r2〉 being constant in time. Our results are compared with measurements of neutral atoms falling in the electric field of a charged wire. Modifications of the experiment, which may help in observing quantum limit of falling, are proposed