Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms

AbstractChebyshev polynomials of the third and fourth kinds, orthogonal with respect to (1 + x)12(1 − x)−12 and (1 − x)12(1 + x)−12, respectively, on [− 1, 1], are less well known than traditional first- and second-kind polynomials. We therefore summarise basic properties of all four polynomials, and then show how some well-known properties of first-kind polynomials extend to cover second-, third- and fourth-kind polynomials. Specifically, we summarise a recent set of first-, second-, third- and fourth-kind results for near-minimax constrained approximation by series and interpolation criteria, then we give new uniform convergence results for the indefinite integration of functions weighted by (1 + x)−12 or (1 − x)−12 using third- or fourth-kind polynomial expansions, and finally we establish a set of logarithmically singular integral transforms for which weighted first-, second-, third- and fourth-kind polynomials are eigenfunctions

Threshold behavior in metastable dissociation of multi-photon ionized thymine and uracil

Microsecond-timescale HNCO loss has been observed from single-color multi-photon ionized pyrimidine nucleobases in the gas phase. Photon energy thresholds for the metastable channels have been measured at 5.55 ± 0.02 eV for thymine and 5.57 ± 0.02 eV for uracil. We argue that these results can be attributed to accessing the molecules’ S1 states with additional vibrational energy matching the threshold energy for HNCO loss from the radical cation. Combined with previous photoionization energies, this enables the S1 adiabatic energies to be deduced: 3.67 ± 0.07 eV for thymine and 3.77 ± 0.07 eV for uracil. These values are consistent with recent calculations

Automatic near real-time selection of flood water levels from high resolution Synthetic Aperture Radar images for assimilation into hydraulic models: a case study

Flood extents caused by fluvial floods in urban and rural areas may be predicted by hydraulic models. Assimilation may be used to correct the model state and improve the estimates of the model parameters or external forcing. One common observation assimilated is the water level at various points along the modelled reach. Distributed water levels may be estimated indirectly along the flood extents in Synthetic Aperture Radar (SAR) images by intersecting the extents with the floodplain topography. It is necessary to select a subset of levels for assimilation because adjacent levels along the flood extent will be strongly correlated. A method for selecting such a subset automatically and in near real-time is described, which would allow the SAR water levels to be used in a forecasting model. The method first selects candidate waterline points in flooded rural areas having low slope. The waterline levels and positions are corrected for the effects of double reflections between the water surface and emergent vegetation at the flood edge. Waterline points are also selected in flooded urban areas away from radar shadow and layover caused by buildings, with levels similar to those in adjacent rural areas. The resulting points are thinned to reduce spatial autocorrelation using a top-down clustering approach. The method was developed using a TerraSAR-X image from a particular case study involving urban and rural flooding. The waterline points extracted proved to be spatially uncorrelated, with levels reasonably similar to those determined manually from aerial photographs, and in good agreement with those of nearby gauges

Modulated Amplitude Waves in Bose-Einstein Condensates

We analyze spatio-temporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits (``modulated amplitude waves'') on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.Comment: 5 pages, 4 figs, revtex, final form of paper, to appear in PRE (forgot to include \bibliography command in last update, so this is a correction of that; the bibliography is hence present again

History repeats? : the rise of the new middle classes in the developing world

1. Although the ability to detect chemical cues is widespread in many organisms, it is surprising how little is known about the role of chemical communication in avian life histories. Nowadays, growing evidence suggests that birds can use olfaction in several contexts. However, we still do not know the role of bird olfaction in one of the most important determinants of survival, predator detection. 2. Blue tits, Cyanistes caeruleus L., were exposed to chemical cues of: (i) mustelid (predator), (ii) quail (odorous control); or (iii) water (odourless control) inside the nest-box where they were provisioning 8-day-old nestlings. 3. We show that blue tits were able to detect the chemical cues and showed antipredatory behaviours to cope with the risk of predation. Birds delayed their entry to the nest-box, and they perched on the hole of the nest-box and refused to enter more times when they found predator scent than control scents inside the nest-box. In addition, birds decreased the time spent inside the predator-scented nest-box when feeding nestlings. 4. The discovery of the ability of birds to use chemical cues of predators to accurately assess predation may help to understand many aspects of bird life histories that have been neglected until now.Peer reviewe