On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta function $$\zeta (s,a)=\sum_{r=0}^{\infty}(a+r)^{-s}\qquad\qquad a > 0$$ is investigated. It is shown that $\zeta (s,a)$ has no real zeroes $(s=\sigma,a)$ in the region $a >\frac{-\sigma}{2\pi e}+\frac{1}{4\pi e}\log (-\sigma) +1$ for large negative $\sigma$. In the region $0 < a < \frac{-\sigma}{2\pi e}$ the zeroes are asymptotically located at the lines $\sigma + 4a + 2m =0$ with integer $m$. If $N(p)$ is the number of real zeroes of $\zeta(-p,a)$ with given $p$ then $$\lim_{p\to\infty}\frac{N(p)}{p}=\frac{1}{\pi e}.$$ As a corollary we have a simple proof of Inkeri's result that the number of real roots of the classical Bernoulli polynomials $B_n(x)$ for large $n$ is asymptotically equal to $\frac{2n}{\pi e}$.Comment: 9 pages, 2 figure

In search for a perfect shape of polyhedra: Buffon transformation

For an arbitrary polygon consider a new one by joining the centres of consecutive edges. Iteration of this procedure leads to a shape which is affine equivalent to a regular polygon. This regularisation effect is usually ascribed to Count Buffon (1707-1788). We discuss a natural analogue of this procedure for 3-dimensional polyhedra, which leads to a new notion of affine $B$-regular polyhedra. The main result is the proof of existence of star-shaped affine $B$-regular polyhedra with prescribed combinatorial structure, under partial symmetry and simpliciality assumptions. The proof is based on deep results from spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro

On the Running of the Cosmological Constant in Quantum General Relativity

We present arguments that show what the running of the cosmological constant means when quantum general relativity is formulated following the prescription developed by Feynman.Comment: 5 page

Hydrodynamics of photoionized columns in the Eagle Nebula, M 16

We present hydrodynamical simulations of the formation, structure and evolution of photoionized columns, with parameters based on those observed in the Eagle Nebula. On the basis of these simulations we argue that there is no unequivocal evidence that the dense neutral clumps at heads of the columns were cores in the pre-existing molecular cloud. In our simulations, a variety of initial conditions leads to the formation and maintenance of near-equilibrium columns. Therefore, it is likely that narrow columns will often occur in regions with large-scale inhomogeneities, but that observations of such columns can tell us little about the processes by which they formed. The manner in which the columns in our simulations develop suggests that their evolution may result in extended sequences of radiation-induced star formation.Comment: 12 pages, 9 figures, Latex, MN macros, in press with MNRA

The summertime plankton community at South Georgia (Southern Ocean): comparing the historical (1926/27) and modern (post 1995) records.

The earliest comprehensive plankton sampling programme in the Southern Ocean was 32 undertaken during the early part of last century by Discovery Investigations to gain a 33 greater scientific understanding of whale stocks and their summer feeding grounds. An 34 initial survey was carried out around South Georgia during December 1926 and January 35 1927 to describe the distribution of plankton during the summer, and to serve as a 36 baseline against which to compare future surveys. We have reanalysed phytoplankton and 37 zooplankton data from this survey and elucidated patterns of community distribution and 38 compared them with our recent understanding of the ecosystem based on contemporary 39 data. Analysis of Discovery data identified five groups of stations with characteristic 40 phytoplankton communities which were almost entirely consistent with the original 41 analysis conducted by Hardy and Gunther (1935). Major groupings were located at the 42 western end of the island and over the northern shelf where Corethron spp. were 43 dominant, and to the south and east where a more diverse flora included high abundances 44 of Nitzschia seriata. Major zooplankton-station groupings were located over the inner 45 shelf which was characterised by a high abundance of Drepanopus forcipatus and in 46 oceanic water >500 m deep that were dominated by Foraminifera, Oithona spp., 47 Ctenocalanus vanus, and Calanoides acutus. Stations along the middle and outer shelf 48 regions to the north and west, were characterised by low overall abundance. There was 49 some evidence that groupings of stations to the north of the island originated in different 50 water masses on either side of the Southern Antarctic Circumpolar Current Front, the 51 major frontal system in the deep ocean close to South Georgia. However, transect lines 52 during 1926/27 did not extend far enough offshore to sample this frontal region 53 3 adequately. Interannual variability of zooplankton abundance was assessed from stations 54 which were sampled repeatedly during 7 recent British Antarctic Survey cruises (1995-55 2005) to the region and following taxonomic harmonization and numerical 56 standardization (ind. m-3), a subset of 45 taxonomic categories of zooplankton (species 57 and higher taxa) from 1926/27, were compared with similar data obtained during the 58 BAS cruises using a linear model. Initially comparisons were restricted to BAS stations 59 that lay within 40 km of Discovery stations although a comparison was also made using 60 all available data. Despite low abundance values in 1926/27, in neither comparison did 61 Discovery data differ significantly from BAS data. Calculation of the percentage 62 similarity index across cruises did not reveal any systematic differences in species 63 composition between 1926/27 and the present. In the light of ocean warming trends, the 64 existence of more subtle changes in species composition is not ruled out, but an absence 65 of finely resolved time-series data make this impossible to determine

First Observations of the Magnetic Field Geometry in Pre-stellar Cores

We present the first published maps of magnetic fields in pre-stellar cores, to test theoretical ideas about the way in which the magnetic field geometry affects the star formation process. The observations are JCMT-SCUBA maps of 850 micron thermal emission from dust. Linear polarizations at typically ten or more independent positions in each of three objects, L1544, L183 and L43 were measured, and the geometries of the magnetic fields in the plane of the sky were mapped from the polarization directions. The observed polarizations in all three objects appear smooth and fairly uniform. In L1544 and L183 the mean magnetic fields are at an angle of around 30 degrees to the minor axes of the cores. The L43 B-field appears to have been influenced in its southern half, such that it is parallel to the wall of a cavity produced by a CO outflow from a nearby T Tauri star, whilst in the northern half the field appears less disturbed and has an angle of 44 degrees to the core minor axis. We briefly compare our results with published models of magnetized cloud cores and conclude that no current model can explain these observations simultaneously with previous ISOCAM data.Comment: 13 pages, 3 figs, to appear in ApJ Letter