Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector

Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where V =\gfq^n and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in \GL_n(\gfq). In fact, we determine \gfq[V \oplus V^*]^G for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of "doubly parametrized" series of group actions whose invariant rings are known to have a uniform description.Comment: 16 page

An Algorithm to Calculate Optimal Homogeneous Systems of Parameters

AbstractWhen a homogeneous system of parametersf1,...,fnis chosen for a graded algebraA, it is important for subsequent computations that the degrees, deg(fi), are as small as possible. More precisely, one would like the product or the sum of the degrees to be minimal, depending on the application.This article investigates which degree vectors can occur as the degrees of a homogeneous system of parameters. From this, an algorithm is derived which constructs an optimal homogeneous system of parameters. Here the notion of what is considered asoptimalis part of the input. An important application is the case whereAis the invariant ring of a finite linear group. There is an implementation of the algorithm in Magma which applies to this case

Crystallinity versus mass-loss rate in Asymptotic Giant Branch stars

Infrared Space Observatory (ISO) observations have shown that O-rich Asymptotic Giant Branch (AGB) stars exhibit crystalline silicate features in their spectra only if their mass-loss rate is higher than a certain threshold value. Usually, this is interpreted as evidence that crystalline silicates are not present in the dust shells of low mass-loss rate objects. In this study, radiative transfer calculations have been performed to search for an alternative explanation to the lack of crystalline silicate features in the spectrum of low mass-loss rate AGB stars. It is shown that due to a temperature difference between amorphous and crystalline silicates it is possible to include up to 40% of crystalline silicate material in the circumstellar dust shell, without the spectra showing the characteristic spectral features. Since this implies that low mass-loss rate AGB stars might also form crystalline silicates and deposit them into the Interstellar Medium (ISM), the described observational selection effect may put the process of dust formation around AGB stars and the composition of the predominantly amorphous dust in the Interstellar Medium in a different light. Our model calculations result in a diagnostic tool to determine the crystallinity of an AGB star with a known mass-loss rate.Comment: accepted by A&A, 10 pages, 11 figure

Modelling the alumina abundance of oxygen-rich evolved stars in the Large Magellanic Cloud

In order to determine the composition of the dust in the circumstellar envelopes of oxygen-rich asymptotic giant branch (AGB) stars we have computed a grid of modust radiative-transfer models for a range of dust compositions, mass-loss rates, dust shell inner radii and stellar parameters. We compare the resulting colours with the observed oxygen-rich AGB stars from the SAGE-Spec Large Magellanic Cloud (LMC) sample, finding good overall agreement for stars with a mid-infrared excess. We use these models to fit a sample of 37 O-rich AGB stars in the LMC with optically thin circumstellar envelopes, for which 5$-$35-$\mu$m Spitzer infrared spectrograph (IRS) spectra and broadband photometry from the optical to the mid-infrared are available. From the modelling, we find mass-loss rates in the range $\sim 8\times10^{-8}$ to $5\times10^{-6}$ M$_{\odot}\ \mathrm{yr}^{-1}$, and we show that a grain mixture consisting primarily of amorphous silicates, with contributions from amorphous alumina and metallic iron provides a good fit to the observed spectra. Furthermore, we show from dust models that the AKARI [11]$-$[15] versus [3.2]$-$[7] colour-colour diagram, is able to determine the fractional abundance of alumina in O-rich AGB stars.Comment: 22 pages, 17 figures, accepted MNRA