by different values for the quantized Hall conductivity, $\sigma_{xy}$, by the presence or absence of spontaneous spin-polarization perpendicular to the direction of the Zeeman field, and by the presence or absence of a gap for spin-flip excitations. At small $\gamma$ (strong disorder), the electronic state is paramagnetic (denoted as PC in Fig. \[phasediag\]), there is no spin-polarization in the absence of Zeeman coupling, and the Hall conductivity is expected to jump from $0$ to $2 e^2/h$ as the filling factor $\nu$ crosses the $\nu=1$ line. For a small Zeeman coupling, there will be a small splitting between the majority-spin and minority-spin extended state energies and the zero-temperature Hall conductance should have a narrow intermediate $e^2/h$ plateau centered on $\nu=1$. However, we do not expect that this plateau will be observable at accessible temperatures, and have indicated this in Fig. \[phasediag\] by using a thick line to mark the $0$ to $2 e^2/h$ phase boundary. At somewhat larger $\gamma$ there is a phase transition at zero Zeeman energy between paramagnetic and ferromagnetic states (denoted as FC in Fig. \[phasediag\]). In our calculations this transition occurs at a larger value of $\gamma$ at $\nu=1$ than away from $\nu=1$. As $\gamma$ increases in 