We determine the isomorphic classes of 12-dimensionalsemisimple Hopf algebras over an algebraically closed field $k$ whose characteristic ch $k\neq 2,3$.

We are concerned with the question of whether a noncompact space with a nice local structure contains a ray, i.e., a closed homeomorph of $[0,1)$. We construct rays in incomplete locally path connected spaces, and also, in noncompact metrizable convex sets; as a consequence these spaces lack the fixed point property. On the other hand, we give an example of a noncompact (nonmetrizable) convex subset $C$ of a locally convex topological vector space $E$ which has the fixed point property.

We study a condition that allows us to characterize geodesic hyperspheres among all real hypersurfaces of quatemionic projective space.

We investigate the AR-property for convex sets in nonlocally convex linear metric spaces. We introduce the notion of LC-decomposability for convex sets and prove that any LC-decomposable convex set is an AR.

We investigate the group of f.p. homeomorphisms of an $n$-dimensional vector bundle $\xi$. In the case $n\geq 5$ and the base space of $\xi$ is countable dimensional, we show that every f.p. stable homeomorphism of $\xi$ can be approximated by f.p. $PL$ homeomorphisms with respect to the majorant topology. As an application we can show that if the base space is compact, then the group of f.p. $PL$ homeomorphisms of $\xi$ with the uniform topology has the mapping absorption property for maps from countable dimensional metric spaces into the group of f.p. homeomorphisms of $\xi$ which are PL on the unit open ball.

Almost-Einstein real hypersurfaces of quaternionic projective space, as defined in [3], can be characterized by a condition involving their curvature and Ricci tensors.

We study the self-equivalence groups associated with objects in (i) the track homotopy category over a fixed space $B$, (ii) the track homotopy category under a fixed space $A$ and (iii) the category of homotopy pairs. In each case a short exact sequence decomposition of the self-equivalence group is available. In the case of (i) the group is isomorphic to the group of fibre-homotopy self-equivalences of an associated fibration, the decomposition (in other form) is known and has been used as the basis of computations. We make sample computations in the simplest situations for (i), (ii), and (iii), in each case solving the extension problem that arises by considering secondary operations and determining the Toda-Hopf invariant of relevant tracks. We indicate that in certain cases such computations can be used to determine the self-equivalence group of a mapping cone.

We will show the existence of a duality on $CMC$-1 surfaces in hyperbolic 3-space, and we will show an analogue of the Osserman Inequality in terms of dual surfaces. Moreover, we will show that equality holds (in this analogue) if and only if all the ends of the surface are regular and embedded.

Let $f:X\rightarrow Y$ be a mapping between continua. Then $f$ induces two mappings $C(f):C(X)\rightarrow C(Y)$ and $2^{f}:2^{X}\rightarrow 2^{Y}$ in the natural way. In this paper, we shall study about the following question: Dose the correspondences $f\rightarrow C(f)$ and $f\rightarrow 2^{f}$ preserve or reverse what classes of mappings? When $Y$ is locally connected, many classes of mappings are preserved by these correspondences. We shall consider the classes of monotone, open, OM, confluent, quasi-monotone and weakly monotone mappings.

We prove that a compact connected nonmetrizable group contains a proper dense $\omega$-bounded subgroup. This is then used to show that every compact nonmetrizable group $G$ can be partitioned into $|G|$-many pairwise disjoint dense topologically homogeneous $\omega$-bounded subsets each of cardinal $|G|$ and each Haar nonmeasurable with full Haar outermeasure. This allows us to then generalize an observation of Kakutani and Oxtoby and to conclude that each infinite compact group $G$ may be partitioned into a collection of $|G|$-many pairwise disjoint dense subsets of full Haar outermeasure. A corollary of these results is that the Stone-Cech compactification $\beta X$ of an infinite discrete space $X$ may be partitioned into $|\beta X|$ pairwise disjoint $\omega$-bounded subsets each of size $|\beta X|$.