Families of spherical caps: spectra and ray limit

We consider a family of surfaces of revolution ranging between a disc and a hemisphere, that is spherical caps. For this family, we study the spectral density in the ray limit and arrive at a trace formula with geodesic polygons describing the spectral fluctuations. When the caps approach the hemisphere the spectrum becomes equally spaced and highly degenerate whereas the derived trace formula breaks down. We discuss its divergence and also derive a different trace formula for this hemispherical case. We next turn to perturbative corrections in the wave number where the work in the literature is done for either flat domains or curved without boundaries. In the present case, we calculate the leading correction explicitly and incorporate it into the semiclassical expression for the fluctuating part of the spectral density. To the best of our knowledge, this is the first calculation of such perturbative corrections in the case of curvature and boundary.Comment: 28 pages, 7 figure

Variability of behaviour in electricity load profile clustering: who does things at the same time each day?

UK electricity market changes provide opportunities to alter households' electricity usage patterns for the benet of the overall electricity network. Work on clustering similar households has concentrated on daily load proles and the variability in regular household behaviours has not been considered. Those households with most variability in reg- ular activities may be the most receptive to incentives to change timing. Whether using the variability of regular behaviour allows the creation of more consistent groupings of households is investigated and compared with daily load prole clustering. 204 UK households are analysed to nd repeating patterns (motifs). Variability in the time of the motif is used as the basis for clustering households. Dierent clustering algorithms are assessed by the consistency of the results. Findings show that variability of behaviour, using motifs, provides more consistent groupings of households across dierent clustering algorithms and allows for more ecient targeting of behaviour change interventions