  We show that generalized eventually narrow sequences on a strongly
inaccessible cardinal $\kappa$ are preserved under the Cummings-Shaleh
non-linear iterations of the higher Hechler forcing on $\kappa$. Moreover
assuming GCH, $\kappa^{<\kappa}=\kappa$, we show that:
  (1) if $\kappa$ is strongly unfoldable,
$\kappa^+\leq\beta=\hbox{cf}(\beta)\leq \hbox{cf}(\delta)\leq\delta\leq\mu$ and
$\hbox{cf}(\mu)>\kappa$,then there is a cardinal preserving generic extension
in which
$$\mathfrak{s}(\kappa)=\kappa^+\leq\mathfrak{b}(\kappa)=\beta\leq\mathfrak{d}(\kappa)=\delta\leq
2^\kappa=\mu.$$ (2) if $\kappa$ is strongly inaccessible, $\lambda>\kappa^+$,
then in the generic extension obtained as the $<\kappa$-support iteration of
$\kappa$-Hechler forcing of length $\lambda$ there are no $\kappa$-towers of
length $\lambda$.

Logic