  We confirm that a high rate of shear at the center is able to stabilize an
entire stellar disk against bar-forming (m=2) modes, irrespective of the dark
halo density. Our simulations of unstable power-law disks also yield the growth
rate and pattern speed of the dominant lop-sided (m=1) mode in close agreement
with that predicted by linear theory in our cleanest case. The one-armed modes,
which dominate models with extensive disks, can be controlled by tapering the
outer disk. Two-armed instabilities of hard-centered disks are more difficult
to identify because they are easily overwhelmed by particle noise.
Nevertheless, we have detected the predicted m=2 modes in simulations with very
large numbers of particles. Such bisymmetric instabilities in these disks are
provoked only by sharp edges and are therefore easily eliminated.
  We have constructed a cool disk model with an almost flat rotation curve and
a quasi-exponential density profile that is unambiguously stable. The halo in
this stable model has a large core radius, with the disk and bulge providing
almost all the rotational support in the inner parts.
