We have reviewed the chemistry and cooling behaviour of low-density (n < 10 4 cm −3 ) primordial gas and devised a cooling model wich involves 19 collisional and 9 radiative processes and is applicable for temperatures in the range (1K< T < 10 8 K). We derived new fits of rate coefficients for the photo-attachment of neutral hydrogen, the formation of molecular hydrogen via H − , charge exchange between H 2 and H + , electron detachment of H − by neutral hydrogen, dissociative recombination of H + 2 with slow electrons, photodissociation of H + 2 , and photodissociation of H 2 . Further it was found that the molecular hydrogen produced through the gas-phase processes, H + 2 + H → H 2 + H + , and H − + H → H 2 + e − , is likely to be converted into its para configuration on a faster time scale than the formation time scale. We have tested the model extensively and shown it to agree well with former studies. We further studied the chemical kinetics in great detail and devised a minimal model which is substantially simpler than the full reaction network but predicts correct abundances. This minimal model shows convincingly that 12 collisional processes are sufficient to model the H, He, H + , H − , He + , He ++ , and H 2 abundances in low density primordial gas for applications with no radiation fields.

This paper presents a continuation of our efforts to numerically study accretion disks that are misaligned (tilted) with respect to the rotation axis of a Kerr black hole. Here we present results of a global numerical simulation which fully incorporates the effects of the black hole spacetime as well as magnetorotational turbulence that is the primary source of angular momentum transport in the flow. This simulation shows dramatic differences from comparable simulations of untilted disks. Accretion onto the hole occurs predominantly through two opposing plunging streams that start from high latitudes with respect to both the black-hole and disk midplanes. This is due to the aspherical nature of the gravitational spacetime around the rotating black hole. These plunging streams start from a larger radius than would be expected for an untilted disk. In this regard the tilted black hole effectively acts like an untilted black hole of lesser spin. Throughout the duration of the simulation, the main body of the disk remains tilted with respect to the symmetry plane of the black hole; thus there is no indication of a Bardeen-Petterson effect in the disk at large. The torque of the black hole instead principally causes a global precession of the main disk body. In this simulation the precession has a frequency of 3(M ⊙ /M ) Hz, a value consistent with many observed low-frequency quasi-periodic oscillations. However, this value is strongly dependent on the size of the disk, so this frequency may be expected to vary over a large range.

Head-on collisions of kink and antikink solitons are investigated numerically in the classical onedimensional h(4'-1 )2 model. It is shown that whether a kink-antikink interaction settles to a bound state or a two-soliton solution depends "fractally" on the impact velocity. We discuss the results using the framework of perturbation theory which helps to clarify the nature of the fractal structure in terms of resonances with the internal shape mode oscillations. We also review the technique of collective coordinates used to reduce the infinite-dimensional system to one with just two degrees of freedom. Although we do not expect exact agreement by using such a simplification, we show that the reduced system bears a striking qualitative resemblance to the full infinite-dimensional system, reproducing the fractal structure. The maximum Lyapunov exponents are computed for the bound-state oscillations and found to be -0.3 for both the full and reduced systems, demonstrating the chaotic nature of the bound state.

We have developed a method of solving for multi-species chemical reaction flows in non-equilibrium and self-consistently with the hydrodynamic equations in an expanding FLRW universe. The method is based on a backward differencing scheme for the required stability when solving stiff sets of equations and is designed to be efficient for three-dimensional calculations without sacrificing accuracy. In all, 28 kinetic reactions are solved including both collisional and radiative processes for the following nine separate species: H, H + , He, He + , He ++ , H − , H + 2 , H 2 , and e − . The method identifies those reactions (involving H − and H + 2 ) ocurring on the shortest time scales, decoupling them from the rest of the network and imposing equilibrium concentrations to good accuracy over typical cosmological dynamical times. Several tests of our code are presented, including radiative shock waves, cosmological sheets, conservation constraints, and fully three-dimensional simulations of CDM cosmological evolutions in which we compare our method to results obtained when the packaged routine LSODAR is substituted for our algorithms.