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Dynamical chiral-symmetry breaking in asymptotically free theories with a slowly running coupling is analyzed. When the confinement scale A is much less than the cutoff M beyond which the theory cannot be used in isolation, the dynamical mass X(p) starts from a value = A for momenta p < A and falls slowly for a significant range p > A. It then takes on the asymptotic form (\np) a /p 2 where a > 1. In hypercolor theories this behavior generates sufficiently large fermion masses for a higher value of Mthan naively expected. This in turn could allow for an adequate suppression of flavor-changing neutral currents.

Following recent work on the quantum Hall effect on S 4 , we solve the Landau problem on the complex projective spaces CP k and discuss quantum Hall states for such spaces. Unlike the case of S 4 , a finite spatial density can be obtained with a finite number of internal states for each particle. We treat the case of CP 2 in some detail considering both Abelian and nonabelian background fields. The wavefunctions are obtained and incompressibility of the Hall states is shown. The case of CP 3 is related to the case of S 4 .

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