N electrons in a quantum dot: Two-point Pade approximants

We present analytic estimates for the energy levels of N electrons (N = 2 - 5) in a two-dimensional parabolic quantum dot. A magnetic field is applied perpendicularly to the confinement plane. The relevant scaled energy is shown to be a smooth function of the parameter \beta=(effective Rydberg/effective dot energy)^{1/6}. Two-point Pade approximants are obtained from the series expansions of the energy near the oscillator ($\beta\to 0$) and Wigner ($\beta\to\infty$) limits. The approximants are expected to work with an error not greater than 2.5% in the entire interval $0\le\beta < \infty$.Comment: 27 pages. LaTeX. 6 figures not include

Monomial transformations of the projective space

We prove that, over any field, the dimension of the indeterminacy locus of a rational transformation $f$ of $P^n$ which is defined by monomials of the same degree $d$ with no common factors is at least $(n-2)/2$, provided that the degree of $f$ as a map is not divisible by $d$. This implies upper bounds on the multidegree of $f$

Probing the geometry and motion of AGN coronae through accretion disc emissivity profiles

To gain a better understanding of the inner disc region that comprises active galactic nuclei it is necessary to understand the pattern in which the disc is illuminated (the emissivity profile) by X-rays emitted from the continuum source above the black hole (corona). The differences in the emissivity profiles produced by various corona geometries are explored via general relativistic ray tracing simulations. Through the analysis of various parameters of the geometries simulated it is found that emissivity profiles produced by point source and extended geometries such as cylindrical slabs and spheroidal coronae placed on the accretion disc are distinguishable. Profiles produced by point source and conical geometries are not significantly different, requiring an analysis of reflection fraction to differentiate the two geometries. Beamed point and beamed conical sources are also simulated in an effort to model jet-like coronae, though the differences here are most evident in the reflection fraction. For a point source we determine an approximation for the measured reflection fraction with the source height and velocity. Simulating spectra from the emissivity profiles produced by the various geometries produce distinguishable differences. Overall spectral differences between the geometries do not exceed 15 per cent in the most extreme cases. It is found that emissivity profiles can be useful in distinguishing point source and extended geometries given high quality spectral data of extreme, bright sources over long exposure times. In combination with reflection fraction, timing, and spectral analysis we may use emissivity profiles to discern the geometry of the X-ray source.Comment: 15 pages, 12 figures. Accepted for publication in MNRA