  We exhibit a global bound for the Lyubeznik numbers of a ring of prime
characteristic. In addition, we show that for a monomial ideal, the Lyubeznik
numbers of the quotient rings of its radical and its polarization are the same.
Furthermore, we present examples that show striking behavior of the Lyubeznik
numbers under localization. We also show related results for generalizations of
the Lyubeznik numbers.
