Spin Gaps in a Frustrated Heisenberg model for CaV4_4O9_9

I report results of a density matrix renormalization group (DMRG) study of a model for the two dimensional spin-gapped system CaV$_4$O$_9$. This study represents the first time that DMRG has been used to study a two dimensional system on large lattices, in this case as large as $24\times 11$, allowing extrapolation to the thermodynamic limit. I present a substantial improvement to the DMRG algorithms which makes these calculations feasible.Comment: 10 pages, with 4 Postscript figure

Quadrature domains and kernel function zipping

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be zipped down to a strikingly small data set. It is also proved that the kernel functions associated to a quadrature domain must be algebraic.Comment: 13 pages, to appear in Arkiv for matemati

A new look at microlensing limits on dark matter in the Galactic halo

The motivation for this paper is to review the limits set on the MACHO content of the Galactic halo by microlensing experiments in the direction of the Large Magellanic Cloud. This has been prompted by recent measurements of the Galactic rotation curve, which suggest that the limits have been biassed by the assumption of an over-massive halo. The paper first discusses the security of the detection efficiency calculations which are central to deriving the MACHO content of the Galactic halo. It then sets out to compare the rotation curves from various halo models with recent observations, with a view to establishing what limits can be put on an all-MACHO halo. The main thrust of the paper is to investigate whether lighter halo models which are consistent with microlensing by an all-MACHO halo are also consistent with recent measures of the Galactic rotation curve. In this case the population of bodies discovered by the MACHO collaboration would make up the entire dark matter content of the Galactic halo. The main result of this paper is that it is easy to find low mass halo models consistent with the observed Galactic rotation curve, which also imply an optical depth to microlensing similar to that found by the MACHO collaboration. This means that all-MACHO halos cannot be ruled out on the basis of their observations. In conclusion, limits placed on the MACHO content of the Galactic halo from microlensing surveys in the Magellanic Clouds are inconsistent and model dependent, and do not provide a secure basis for rejecting an all-MACHO halo.Comment: 8 pages, 4 figures, accepted for publication in A&