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

An empirical model for protostellar collapse

We propose a new analytic model for the initial conditions of protostellar collapse in relatively isolated regions of star formation. The model is non-magnetic, and is based on a Plummer-like radial density profile as its initial condition. It fits: the observed density profiles of pre-stellar cores and Class 0 protostars; recent observations in pre-stellar cores of roughly constant contraction velocities over a wide range of radii; and the lifetimes and accretion rates derived for Class 0 and Class I protostars. However, the model is very simple, having in effect only 2 free parameters, and so should provide a useful framework for interpreting observations of pre-stellar cores and protostars, and for calculations of radiation transport and time-dependent chemistry. As an example, we model the pre-stellar core L1544.Comment: To appear in Astrophysical Journal, Jan 20th, 2001. 18 pages incl. 3 fig

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

Simulating star formation in molecular cloud cores I. The influence of low levels of turbulence on fragmentation and multiplicity

We present the results of an ensemble of simulations of the collapse and fragmentation of dense star-forming cores. We show that even with very low levels of turbulence the outcome is usually a binary, or higher-order multiple, system. We take as the initial conditions for these simulations a typical low-mass core, based on the average properties of a large sample of observed cores. All the simulated cores start with a mass of $M = 5.4 M_{\odot}$, a flattened central density profile, a ratio of thermal to gravitational energy $\alpha_{\rm therm} = 0.45$ and a ratio of turbulent to gravitational energy $\alpha_{\rm turb} = 0.05$. Even this low level of turbulence is sufficient to produce multiple star formation in 80% of the cores; the mean number of stars and brown dwarfs formed from a single core is 4.55, and the maximum is 10. At the outset, the cores have no large-scale rotation. The only difference between each individual simulation is the detailed structure of the turbulent velocity field. The multiple systems formed in the simulations have properties consistent with observed multiple systems. Dynamical evolution tends preferentially to eject lower mass stars and brown dwarves whilst hardening the remaining binaries so that the median semi-major axis of binaries formed is $\sim 30$ au. Ejected objects are usually single low-mass stars and brown dwarfs, yielding a strong correlation between mass and multiplicity. Our simulations suggest a natural mechanism for forming binary stars that does not require large-scale rotation, capture, or large amounts of turbulence.Comment: 20 pages, 12 figures submitted to A&

AN ANALYSIS OF THE YIELD-PRICE RISK ASSOCIATED WITH SPECIALTY CROPS

Specialty crops have been cited as means to diversify crop portfolios on the prairies. Lentils, a specialty crop, have high variability in yields and prices but are relatively uncorrelated with the yields and prices of other traditional Saskatchewan crops. In addition, yields and prices of lentils may be negatively correlated. These attributes have important but offsetting effects in crop portfolio selection. The objective of this article is to assess the relative profitability and riskiness of wheat and lentil rotations for a representative Saskatchewan farm and to select appropriate farmers who should consider production of lentils. The cumulative density function of net returns are simulated for both rotations assuming stochastic prices and yields. Stochastic dominance with respect to a function is used to identify the corresponding appropriate profile of agricultural producers for each crop rotation. The results indicate that lentils should be considered by a number of, but not all, Saskatchewan farmers.Crop Production/Industries, Risk and Uncertainty,

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