On the Distribution of the Sum of n Non-Identically Distributed Uniform Random Variables

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the characteristic function, we derive explicit formulae for the distribution of the sum of n non-identically distributed uniform random variables in both the continuous and the discrete case. The results, though involved, have a certain elegance. As examples, we derive from our general formulae some special cases which have appeared in the literature.Comment: 20 page

MANAGING NUTRIENT LOSSES: SOME EMPIRICAL RESULTS ON THE POTENTIAL WATER QUALITY EFFECTS

Over-application of manure on cropland can cause water quality degradation. This paper reports a modeling approach for assessing tradeoffs among manure storage and handling systems as they relate to the nutrient loadings in cropland runoff, including nitrate losses to groundwater. The CREAMS simulation model provided estimates of nutrient losses. A linear optimization model was used to determine the income-nutrient loss tradeoffs. Six-month storage was profitable for farmers with average-size dairy herds, but compared to daily spreading caused increased nitrate leaching through the soil to groundwater resources. Twelve-month storage systems decreased farm profitability while decreasing the total nitrogen losses from farm fields.Environmental Economics and Policy,

A Gamma Ray Burst with a 220 Microsecond Rise Time and a Sharp Spectral Cutoff

The Gamma Ray Burst GRB920229 has four extreme and unprecedented properties; a rise in brightness with an e-folding time scale of $220 \pm 30 \mu s$, a fall in brightness with an e-folding time scale of $400 \pm 100 \mu s$, a large change in spectral shape over a time of $768 \mu s$, and a sharp spectral cutoff to high energies with $\Delta E/E = 18$. The rapid changes occur during a spike in the light curve which was seen 0.164 s after the start of the burst. The spectrum has a peak $\nu F_{\nu}$ at 200 keV with no significant flux above 239 keV, although the cutoff energy shifts to less than 100 keV during the decay of the spike. These numbers can be used to place severe limits on fireball models of bursts. The thickness of the energy production region must be smaller than $\sim 66 km$, ejected shells must have a dispersion of the Lorentz factor of less than roughly 1% along a particular radius, and the angular size of the radiation emission region is of order 1 arc-minute as viewed from the burst center. The physical mechanism that caused the sharp spectral cutoff has not been determined.Comment: 20 pages, 3 figures, Submitted to ApJ

Spectral Densities of Response Functions for the O(3) Symmetric Anderson and Two Channel Kondo Models

The O(3) symmetric Anderson model is an example of a system which has a stable low energy marginal Fermi liquid fixed point for a certain choice of parameters. It is also exactly equivalent, in the large U limit, to a localized model which describes the spin degrees of freedom of the linear dispersion two channel Kondo model. We first use an argument based on conformal field theory to establish this precise equivalence with the two channel model. We then use the numerical renormalization group (NRG) approach to calculate both one-electron and two-electron response functions for a range of values of the interaction strength U. We compare the behaviours about the marginal Fermi liquid and Fermi liquid fixed points and interpret the results in terms of a renormalized Majorana fermion picture of the elementary excitations. In the marginal Fermi liquid case the spectral densities of all the Majorana fermion modes display a |omega| dependence on the lowest energy scale, and in addition the zero Majorana mode has a delta function contribution. The weight of this delta function is studied as a function of the interaction U and is found to decrease exponentially with U for large U. Using the equivalence with the two channel Kondo model in the large U limit, we deduce the dynamical spin susceptibility of the two channel Kondo model over the full frequency range. We use renormalized perturbation theory to interpret the results and to calculate the coefficient of the ln omega divergence found in the low frequency behaviour of the T=0 dynamic susceptibility.Comment: 26 pages, 18 figures, to be published in Eur. Phys. J.

Neutrino Capture and r-Process Nucleosynthesis

We explore neutrino capture during r-process nucleosynthesis in neutrino-driven ejecta from nascent neutron stars. We focus on the interplay between charged-current weak interactions and element synthesis, and we delineate the important role of equilibrium nuclear dynamics. During the period of coexistence of free nucleons and light and/or heavy nuclei, electron neutrino capture inhibits the r-process. At all stages, capture on free neutrons has a larger impact than capture on nuclei. However, neutrino capture on heavey nuclei by itself, if it is very strong, is also detrimental to the r-process until large nuclear equilibrium clusters break down and the classical neutron-capture phase of the r-process begins. The sensitivity of the r-process to neutrino irradiation means that neutrino-capture effects can strongly constrain the r-process site, neutrino physics, or both. These results apply also to r-process scenarios other than neutrino-heated winds.Comment: 20 pages, 17 figures, Submitted to Physical Review

Axistationary perfect fluids -- a tetrad approach

Stationary axisymmetric perfect fluid space-times are investigated using the curvature description of geometries. Attention is focused on space-times with a vanishing electric part of the Weyl tensor. It is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution. The existence of a rigidly rotating perfect fluid, generalizing the interior Schwarzschild metric is proven. Theorems are stated on Petrov types and electric/magnetic Weyl tensors.Comment: 12 page