We extend the two-dimensional Cartesian shapelet formalism to d-dimensions. Concentrating on the three-dimensional case, we derive shapelet-based equations for the mass, centroid, root-mean-square radius, and components of the quadrupole moment and m
oment of inertia tensors. Using cosmological N-body simulations as an application domain, we show that three-dimensional shapelets can be used to replicate the complex sub-structure of dark matter halos and demonstrate the basis of an automated classification scheme for halo shapes. We investigate the shapelet decomposition process from an algorithmic viewpoint, and consider opportunities for accelerating the computation of shapelet-based representations using graphics processing units (GPUs).
We report two detections of deuterated molecular hydrogen (HD) in QSO absorption-line systems at $z > 2$. Toward J2123-0500, we find $N$(HD) $= 13.84 pm 0.2$ for a sub-DLA with metallicity $simeq 0.5Z_{odot}$ and $N$(H$_2$) = $17.64 pm 0.15$ at $z =
2.0594$. Toward FJ0812+32, we find $N$(HD) $= 15.38 pm 0.3$ for a solar-metallicity DLA with $N$(H$_2$) = $19.88 pm 0.2$ at $z = 2.6265$. These systems have ratios of HD to H$_2$ above that observed in dense clouds within the Milky Way disk and apparently consistent with a simple conversion from the cosmological ratio of D/H. These ratios are not readily explained by any available model of HD chemistry and there are no obvious trends with metallicity or molecular content. Taken together, these two systems and the two published $z > 2$ HD-bearing DLAs indicate that HD is either less effectively dissociated or more efficiently produced in high-redshift interstellar gas, even at low molecular fraction and/or solar metallicity. It is puzzling that such diverse systems should show such consistent HD/H$_2$ ratios. Without clear knowledge of all the aspects of HD chemistry that may help determine the ratio HD/H$_2$, we conclude that these systems are potentially more revealing of gas chemistry than of D/H itself and that it is premature to use such systems to constrain D/H at high-redshift.
