58,033 research outputs found
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 () and Wigner
() limits. The approximants are expected to work with an error
not greater than 2.5% in the entire interval .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 of which is defined by monomials of the same
degree with no common factors is at least , provided that the
degree of as a map is not divisible by . This implies upper bounds on
the multidegree of
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
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