We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We compute the cohomology rings of these orbifolds when the dimension of the orbifold is even. These orbifolds are intimately related to the notion of small cover.

Assume $\mathsf{GCH}$ and that $\kappa$ is the first uncountable cardinal such that there is a non-free $\kappa$-free Abelian Whitehead group of cardinality $\kappa$. We prove that if all $\kappa$-free Abelian groups of cardinality $\kappa$ are Whitehead then $\kappa$ is necessarily an inaccessible cardinal.

We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.

Let $R$ be a ring which is $S$-compatible and $(S,\omega)$-Armendariz. In this paper, we investigate that the skew generalized power series ring $R[[S,\omega]]$ is a PF-ring if and only if for any two $S$-indexed subsets $P$ and $Q$ of $R$ such that $Q \subseteq ann_R (P)$ and there exists $a\in ann_R (P)$ such that $q a=q$ for all $q \in Q$. Further, we prove that if $R$ be a Noetherian ring then $R[[S,\omega]]$ is a PP-ring if and only if $R$ is a PP-ring.