A remark on homogeneous affine varieties and related matters

In this note we give an example of affine quotient $G/H$ where $G$ is an affine algebraic group over an algebraically closed field of characteristic 0 and $H$ is a unipotent subgroup not contained in the unipotent radical of $G$. Some remarks about symmetric algebras of centralizers of nilpotent elements in simple Lie algebras, in particular cases, are added.Comment: 23 pages - from page 15 to 22 are presented programs for calculatio

Nucleosynthesis of light element isotopes in evolved stars experiencing extended mixing

We present computations of nucleosynthesis in red giants and asymptotic giant branch stars of Population I experiencing extended mixing. The assumed physical cause for mass transport is the buoyancy of magnetized structures, according to recent suggestions. The peculiar property of such a mechanism is to allow for both fast and slow mixing phenomena, as required for reproducing the spread in Li abundances displayed by red giants and as discussed in an accompanying paper. We explore here the effects of this kind of mass transport on CNO and intermediatemass nuclei and compare the results with the available evidence from evolved red giants and from the isotopic composition of presolar grains of AGB origin. It is found that a good general accord exists between predictions and measurements; in this framework we also show which type of observational data best constrains the various parameters. We conclude that magnetic buoyancy, allowing for mixing at rather different speeds, can be an interesting scenario to explore for explaining together the abundances of CNO nuclei and of Li.Comment: 8 pages, 7 figures, proceeding of 'The Origin of the Elements Heavier than Fe' September 24-28, 2008, Torino, Italy. PASA (accepted for publication

On a variety related to the commuting variety of a reductive Lie algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities

On the Need for Deep Mixing in AGB Stars of Low Mass

The photospheres of low-mass red giants show CNO isotopic abundances that are not satisfactorily accounted for by canonical stellar models. The same is true for the measurements of these isotopes and of the $^{26}$Al/$^{27}$Al ratio in presolar grains of circumstellar origin. Non-convective mixing, occurring during both Red Giant Branch (RGB) and Asymptotic Giant Branch (AGB) stages is the explanation commonly invoked to account for the above evidence. Recently, the need for such mixing phenomena on the AGB was questioned, and chemical anomalies usually attributed to them were suggested to be formed in earlier phases. We have therefore re-calculated extra-mixing effects in low mass stars for both the RGB and AGB stages, in order to verify the above claims. Our results contradict them; we actually confirm that slow transport below the convective envelope occurs also on the AGB. This is required primarily by the oxygen isotopic mix and the $^{26}$Al content of presolar oxide grains. Other pieces of evidence exist, in particular from the isotopic ratios of carbon stars of type N, or C(N), in the Galaxy and in the LMC, as well as of SiC grains of AGB origin. We further show that, when extra-mixing occurs in the RGB phases of population I stars above about 1.2 $M_{\odot}$, this consumes $^3$He in the envelope, probably preventing the occurrence of thermohaline diffusion on the AGB. Therefore, we argue that other extra-mixing mechanisms should be active in those final evolutionary phases.Comment: Accepted for publication on "The Astrophysical Journal Letters

Deep inside low-mass stars

Low-mass stars exhibit, at all stages of their evolution, the signatures of complex physical processes that require challenging modeling beyond standard stellar theory. In this review, we recall the most striking observational evidences that probe the interaction and interdependence of various transport processes of chemicals and angular momentum in these objects. We then focus on the impact of atomic diffusion, large scale mixing due to rotation, and internal gravity waves on stellar properties on the main sequence and slightly beyond.Comment: Invited Review to be published in the proceedings of the IAU Symposium 252 "The Art of Modelling stars in the 21st Century" - Sanya - China - April 200

Hydrodynamical stellar models including rotation, internal gravity waves and atomic diffusion. I. Formalism and tests on Pop I dwarfs

In this paper, we develop a formalism in order to incorporate the contribution of internal gravity waves to the transport of angular momentum and chemicals over long time-scales in stars. We show that the development of a double peaked shear layer acts as a filter for waves, and how the asymmetry of this filter produces momentum extraction from the core when it is rotating faster than the surface. Using only this filtered flux, it is possible to follow the contribution of internal waves over long (evolutionary) time-scales. We then present the evolution of the internal rotation profile using this formalism for stars which are spun down via magnetic torquing. We show that waves tend to slow down the core, creating a "slow" front that may then propagate from the core to the surface. Further spin down of the surface leads to the formation of a new front. Finally we show how this momentum transport reduces rotational mixing in a 1.2Msun, Z=0.02 model, leading to a surface lithium abundance in agreement with observations in the Hyades.Comment: 14 pages, accepted for publication in A&