The affine ring of any cyclic quotient singularity has countably many natural graded ring structures. In two dimensional case, we construct every such graded ring in terms of Q-coefficient divisor on $P^{1}$ which is naturally associated with a resolution of the singularity.

In this paper we show that if $ H\star$ $A$ is a bicrossproduct Hopf algebra then $(H\star A, \sigma)$ is braided if and only if $\sigma$ has a unique form: $\sigma(h\otimes a, g\otimes b)=\sum\beta(h_{1}, g_{1})\omega(h_{2}, g_{2(-1)})\omega(h_{3}, b_{1})\alpha(a_{1}, b_{2})\tau(a_{2}, g20)$ such that $\beta$, $\omega$, $\tau$ and $\alpha$ satisfy certain compatible conditions. The result is applied to a certain bicrossproduct of $H$ and $H^{cop}$, where $H$ is a Hopf algebra with bijective antipode.

In this paper, we study Higson compactifications of the half-open interval obtained by expanding and contracting the base space. We show that the Higson coronas of the half-open interval obtained by these operations are indecomposable continua. Moreover, we show that the Stone-Cech compactification can be approximated by such Higson compactifications.

Let $k$ be an algebraically closed field of characteristic $0$. We show that all Hopf algebras of dimension 15, 21 or 35 over $k$ are necessarily semisimple. We also prove that Hopf algebras of dimension 25 or 49 are either semisimple or pointed. This concludes the full classification of Hopf algebras of the above mentioned dimensions. We also classify pointed Hopf algebras of dimension $pq^{2}$, where $p\neq q$ are prime numbers, and semisimple Hopf algebras of dimension 45.

Let $\phi: M\rightarrow N$ be a pseudohermitian immersion ([6]) of a compact strictly pseudoconvex $CR$ manifold $M$ into a totally umbilical real hypersurface $N$, of nonzero mean curvature of $(\Vert H\Vert\neq 0)$, of a complex Hopf manifold $\bm{C}H^{n}$, tangent to the Lee field $B_{0}$ of $\bm{C}H^{n}$. If $B_{0}$ is orthogonal to the $CR$ structure of $N$ and $E(\phi) \gt \mathop{V\!ol}(M)/[(1+\Vert H\Vert^{2})\Vert H\Vert^{2}]$ then $\phi$ is an unstable harmonic map.