An interesting example for spectral invariants

In "Illinois J. of Math. {\bf 38} (1994) 653--678", the heat operator of a Bismut superconnection for a family of generalized Dirac operators is defined along the leaves of a foliation with Hausdorff groupoid. The Novikov-Shubin invariants of the Dirac operators were assumed greater than three times the codimension of the foliation. It was then showed that the associated heat operator converges to the Chern character of the index bundle of the operator. In "J. K-Theory {\bf 1} (2008) 305--356", we improved this result by reducing the requirement on the Novikov-Shubin invariants to one half of the codimension. In this paper, we construct examples which show that this is the best possible result.Comment: Third author added. Some typos corrected and some material added. Appeared in Journal of K Theory, Volume 13, in 2014, pages 305 to 31

Effects of Magnetic Field Strength and Orientation on Molecular Cloud Formation

We present a set of numerical simulations addressing the effects of magnetic field strength and orientation on the flow-driven formation of molecular clouds. Fields perpendicular to the flows sweeping up the cloud can efficiently prevent the formation of massive clouds but permit the build-up of cold, diffuse filaments. Fields aligned with the flows lead to substantial clouds, whose degree of fragmentation and turbulence strongly depends on the background field strength. Adding a random field component leads to a "selection effect" for molecular cloud formation: high column densities are only reached at locations where the field component perpendicular to the flows is vanishing. Searching for signatures of colliding flows should focus on the diffuse, warm gas, since the cold gas phase making up the cloud will have lost the information about the original flow direction because the magnetic fields redistribute the kinetic energy of the inflows.Comment: 11 pages, 9 figures, accepted by Ap

Gravitational Collapse in Turbulent Molecular Clouds. II. Magnetohydrodynamical Turbulence

Hydrodynamic supersonic turbulence can only prevent local gravitational collapse if the turbulence is driven on scales smaller than the local Jeans lengths in the densest regions, a very severe requirement (Paper I). Magnetic fields have been suggested to support molecular clouds either magnetostatically or via magnetohydrodynamic (MHD) waves. Whereas the first mechanism would form sheet-like clouds, the second mechanism not only could exert a pressure onto the gas counteracting the gravitational forces, but could lead to a transfer of turbulent kinetic energy down to smaller spatial scales via MHD wave interactions. This turbulent magnetic cascade might provide sufficient energy at small scales to halt local collapse. We test this hypothesis with MHD simulations at resolutions up to 256^3 zones, done with ZEUS-3D. We first derive a resolution criterion for self-gravitating, magnetized gas: in order to prevent collapse of magnetostatically supported regions due to numerical diffusion, the minimum Jeans length must be resolved by four zones. Resolution of MHD waves increases this requirement to roughly six zones. We then find that magnetic fields cannot prevent local collapse unless they provide magnetostatic support. Weaker magnetic fields do somewhat delay collapse and cause it to occur more uniformly across the supported region in comparison to the hydrodynamical case. However, they still cannot prevent local collapse for much longer than a global free-fall time.Comment: 32 pages, 14 figures, accepted by Ap

Onset of Fast Magnetic Reconnection in Partially Ionized Gases

We consider quasi-stationary two-dimensional magnetic reconnection in a partially ionized incompressible plasma. We find that when the plasma is weakly ionized and the collisions between the ions and the neutral particles are significant, the transition to fast collisionless reconnection due to the Hall effect in the generalized Ohm's law is expected to occur at much lower values of the Lundquist number, as compared to a fully ionized plasma case. We estimate that these conditions for fast reconnection are satisfied in molecular clouds and in protostellar disks.Comment: 19 pages, 1 figure, 1 tabl

Ambipolar Diffusion-Mediated Thermal Fronts in the Neutral ISM

In a thermally bistable medium, cold, dense gas is separated from warm, rareified gas by thin phase transition layers, or fronts, in which heating, radiative cooling, thermal conduction, and convection of material are balanced. We calculate the steady-state structure of such fronts in the presence of magnetic fields, including the processes of ion-neutral drift and ion-neutral frictional heating. We find that ambipolar diffusion efficiently transports the magnetic field across the fronts, leading to a flat magnetic field strength profile. The thermal profiles of such fronts are not significantly different from those of unmagnetized fronts. The near uniformity of the magnetic field strength across a front is consistent with the flat field strength-gas density relation that is observed in diffuse interstellar gas.Comment: 17 pages, 12 figures, 1 table, accepted for publication in Ap

Evolution of Unmagnetized and Magnetized Shear Layers

We present numerical simulations of the growth and saturation of the Kelvin-Helmholtz instability in a compressible fluid layer with and without a weak magnetic field. In the absence of a magnetic field, the instability generates a single eddy which flattens the velocity profile, stabilizing it against further perturbations. Adding a weak magnetic field - weak in the sense that it has almost no effect on the linear instability - leads to a complex flow morphology driven by MHD forces and to enhanced broadening of the layer, due to Maxwell stresses. We corroborate earlier studies which showed that magnetic fields destroy the large scale eddy structure through periodic cycles of windup and resistive decay, but we show that the rate of decay decreases with decreasing plasma resistivity, at least within the range of resistivity accessible to our simulations. Magnetization increases the efficiency of momentum transport, and the transport increases with decreasing resistivity.Comment: 17 pages, 14 figures, to be published in ApJ April 20, 2008, v677n

Age Constraints for an M31 Globular Cluster from Main Sequence Photometry

We present a color-magnitude diagram (CMD) of the globular cluster SKHB-312 in the Andromeda galaxy (M31), obtained with the Advanced Camera for Surveys on the Hubble Space Telescope. The cluster was included in deep observations taken to measure the star formation history of the M31 halo. Overcoming a very crowded field, our photometry of SKHB-312 reaches V ~ 30.5 mag, more than 1 mag below the main sequence turnoff. These are the first observations to allow a direct age estimate from the turnoff in an old M31 cluster. We analyze its CMD and luminosity function using a finely-spaced grid of isochrones that have been calibrated using observations of Galactic clusters taken with the same camera and filters. The luminosity difference between the subgiant and horizontal branches is ~0.2 mag smaller in SKHB-312 than in the Galactic clusters 47 Tuc and NGC 5927, implying SKHB-312 is 2-3 Gyr younger. A quantitative comparison to isochrones yields an age of 10 +2.5/-1 Gyr