We present a generalized degree for maps between manifolds via framed bordism theory. We study when the degree of a map can be considered as an element of a homotopy group of spheres. Finally we apply this tools to prove a result concerning the generalized Hopf's invariant.

A space $X$ is said to have property $B$ if every infinite open cover $\mathcal{U}$ of $X$ has an open refinement $mathcal{V}$ such that every point $x\in X$ has a neighborhood $W$ with $|\{V\in \mathcal{V}:W\cap V\neq\emptyset\}| \lt |mathcal{V}|$. It is proved that a locally Lindelof space is paracompact iff it has property $B$.

This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Dirichlet boundary condition in the $characteristic$ case. Intuitively, our result may be stated as follows: One can construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space until it "$dies$" at which time it reaches the set where the absorption phenomenon occurs.

In this note, we shall classify all the dual Lie algebra structures induced by multiplicative Poisson tensors on an arbitrary dimensional Heisenberg Lie group.

Let $B_{n}(\alpha)$ be the class of functions of the form $f(z)=\frac{a_{-p}}{z^{p}}+\sum_{k=0}^{\infty}$ $(a_{-p}\neq 0, p\in N=\{1,2, \cdots\})$ which are regular in the punctured disc$U^{*}=\{z:0 \lt |z| \lt 1\}$ and satisfying ${\rm Re}\{\frac{D^{n+1}f(z)}{D^{n}f(z)}-(p+1)\} \lt -\alpha$ $(n\in N_{0}=\{0,1, \cdots\}, |z| \lt 1,0\leqq\alpha \lt p)$, where $D^{n}f(z)=\frac{a_{-p}}{z^{p}}+\sum_{m=1}^{\infty}(p+m)^{n}a_{m-1}z^{m-1}$ It is proved that $B_{n+1}(\alpha)\subset B_{n}(\alpha)$. Since $B_{0}(\alpha)$ is the class of meromorphically $p$-valent starlike functions of order $\alpha$, all functions in $B_{n}(\alpha)$ are $p$-valent starlike. Further a property preserving integrals is considered.

We accomplish the Kaehler version of Choi's characterizations of rotationally symmetric manifolds.

We prove the following theorem: Let $f:X\rightarrow R^{n}$ and $g:Y\rightarrow R^{n}$ be any maps of compacta $X$ and $Y$ into the Euclidean $n$-space $R^{n}$, $n\geqq 5$. Suppose that $\dim(X\times Y) \lt n$ and that 2 $\dim X+$$\dim Y \lt 2n-1$. Then for every $\epsilon \gt 0$ there exist maps $f^{\prime}$:$X\rightarrow R^{n}$ and $g^{\prime}$:$Y\rightarrow R^{n}$ such that $d(f, f^{\prime}) \lt \epsilon,$d(g, g^{\prime}) \lt \epsilon$ and $f^{\prime}(X)()g^{\prime}(Y)$$=\emptyset$.