Polyharmonic and Related Kernels on Manifolds: Interpolation and Approximation

This article is devoted to developing a theory for effective kernel interpolation and approximation in a general setting. For a wide class of compact, connected $C^\infty$ Riemannian manifolds, including the important cases of spheres and SO(3), we establish, using techniques involving differential geometry and Lie groups, that the kernels obtained as fundamental solutions of certain partial differential operators generate Lagrange functions that are uniformly bounded and decay away from their center at an algebraic rate, and in certain cases, an exponential rate. An immediate corollary is that the corresponding Lebesgue constants for interpolation as well as for $L_2$ minimization are uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The kernels considered here include the restricted surface splines on spheres, as well as surface splines for SO(3), both of which have elementary closed-form representations that are computationally implementable. In addition to obtaining bounded Lebesgue constants in this setting, we also establish a "zeros lemma" for domains on compact Riemannian manifolds -- one that holds in as much generality as the corresponding Euclidean zeros lemma (on Lipschitz domains satisfying interior cone conditions) with constants that clearly demonstrate the influence of the geometry of the boundary (via cone parameters) as well as that of the Riemannian metric.Comment: Final manuscript. Contains a new zeros lemma for regions on compact Riemannian manifolds with Lipschitz boundaries -- constants are in terms of parameters from a cone condition. 2 figures. 41 page

SCUBA polarisation observations of the magnetic fields in the prestellar cores L1498 and L1517B

We have mapped linearly polarized dust emission from the prestellar cores L1498 and L1517B with the James Clerk Maxwell Telescope (JCMT) using the Submillimetre Common User Bolometer Array (SCUBA) and its polarimeter SCUBAPOL at a wavelength of 850um. We use these measurements to determine the plane-of-sky magnetic field orientation in the cores. In L1498 we see a magnetic field across the peak of the core that lies at an offset of 19 degrees to the short axis of the core. This is similar to the offsets seen in previous observations of prestellar cores. To the southeast of the peak, in the filamentary tail of the core, we see that the magnetic field has rotated to lie almost parallel to the long axis of the filament. We hypothesise that the field in the core may have decoupled from the field in the filament that connects the core to the rest of the cloud. We use the Chandrasekhar-Fermi (CF) method to measure the plane-of-sky field strength in the core of L1498 to be 10 +/- 7 uG. In L1517B we see a more gradual turn in the field direction from the northern part of the core to the south. This appears to follow a twist in the filament in which the core is buried, with the field staying at a roughly constant 25 degree offset to the short axis of the filament, also consistent with previous observations of prestellar cores. We again use the CF method and calculate the magnetic field strength in L1517B also to be 30 +/- 10 uG. Both cores appear to be roughly virialised. Comparison with our previous work on somewhat denser cores shows that, for the denser cores, thermal and non-thermal (including magnetic) support are approximately equal, while for the lower density cores studied here, thermal support dominates.Comment: 6 pages, 2 figures; accepted for publication by MNRA

Service user involvement in the evaluation of psycho-social intervention for self-harm: a systematic literature review

Background: The efficacy of interventions and treatments for self-harm is well researched. Previous reviews of the literature have highlighted the lack of definitively effective interventions for self-harm and have highlighted the need for future research. These recommendations are also reflected in clinical guidelines published by the National Institute for Health and Clinical Excellence (NICE, 2004) which also call for service user involvement in studies of treatment efficacy. Aims: A systematic review was undertaken to determine i) what contributions service users have made to the evaluation of psychosocial interventions ii) by what methods have service users been involved iii) in what ways could service user involvement supplement empirical evidence for interventions

Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on \Omega \subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi are quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J. on Math. Ana

Estimates of Radiation by Superluminal Neutrinos

We show that the more energetic superluminal neutrinos with quadratically dispersed superluminalities \delta=\beta^2-1, for \beta=v/c where v is the neutrino velocity, also lose significant energy to radiation to the \nu+e^-+e^+ final state in travelling from CERN to Gran Sasso as has been shown to occur for those with constant superluminality by Cohen and Glashow if indeed \delta \simeq 5\times 10^{-5}. In addition, we clarify the dependence of such radiative processes on the size of the superluminality.Comment: 6 pages, no figures; text re-arranged for journal purposes; improved references; published version(title changed by Editors