In this paper, we study the complex contact structure of a twistor space over a self-dual, Einstein 4-manifold with nonzero scalar curvature. Although the existence of such a structure has been known and well utilized by researchers for several decades now, the Hermitian geometry resulting from the complex contact structure is still in the process of being fully developed. Here we give a characterization of such twistor spaces as those satisfying a curvature (and hence purely geometric) identity. Later, we describe how this result fits in with other areas of research in complex contact geometry.

For connected nilpotent groups, 7 is the lowest dimension where there are infinitely many non-isomorphic groups, and also where some groups have no discrete cocompact subgroups. Here one infinite family of 7-dimensional connected groups is studied, discrete cocompact subgroups H are found for some of them, and then the faithful simple quotients A of $C^{*}(\roman H)$ are identified. Such A are shown to be isomorphic to $C^{*}$-crossed products $C^{*}(\roman H_ 3,\Cal C(\Bbb T^3))$ generated by some intriguing effective minimal distal flows $(\roman H_3,\Bbb T^ 3)$, where $\roman H_3$ is the discrete 3-dimensional Heisenberg group.

In this paper, we deal with the problem of uniqueness of meromorphic functions sharing three values CM, and get rid of the restriction on the hyper-orders in a unicity theorem of K. Tohge. An example is provided to show that the result in this paper is best possible.

We consider a differential equation $f^{\left( n\right) }+A_{n-1}\left( z\right) f^{\left( n-1\right) }+...+A_{1}\left( z\right) f^{^{/}}+A_{0}\left( z\right) f=0,$ where $A_{0}\left( z\right) ,...,A_{n-1}\left( z\right) $ are entire functions with $A_{0}\left( z\right) \hbox{$/\hskip -11pt\equiv$}0$. Suppose that there exist a positive number $\mu ,$\ and a sequence $\left( z_{j}\right) _{j\in N}$ with $\stackunder{j\rightarrow +\infty }{\lim }z_{j}=\infty ,$ \ and also two real numbers $\alpha ,\beta $ $\left( \ 0\leq \beta \alpha \right) $\ such that \ $\left| A_{0}\left( z_{j}\right) \right| \geq e^{\alpha \left| z_{j}\right| ^{\mu }}\quad $and$% \quad \left| A_{k}\left( z_{j}\right) \right| \leq e^{\beta \left| z_{j}\right| ^{\mu }}$ as $\ j\rightarrow +\infty $ $\left( k=1,...,n-1\right) $. We prove that all solutions \ $f% \hbox{$/\hskip -11pt\equiv$}0$ of this equation are of infinite order. This result is a generalization of one theorem of Gundersen $\left( \left[ 3\right] ,\text{ }% p.\text{ }418\right) .$

We give here a number of examples of non-commutative harmonic oscillators ``in disguise'' that can be exactly solved by using the tensor product of the oscillator representation and the finite dimensional representation of $\SPE(\r)$, and its perturbations.

Let $\{\xi_{n}\}_{n\geq 0}$ denote an ergodic Markov chain with a finite state space $\mbox{\boldmath$\Xi$}=\{1,2,\cdots, s\}$. %with a transition matrix %$Q=(q_{jk})_{j,k\in\mbox{\boldmath$\Xi$}}$. For each $j,k\in\mbox{\boldmath$\Xi$}$, let $\{Y^{jk}_{n}\}_{n\geq 1}$ be a sequence of i.i.d. $\{-1,1\}$-valued random variables which are independent of $\{\xi_{n}\}$. %We assume that %$\{Y^{jk}_{n}\}_{n\geq 1}$ is the %sequence of identically distributed %random variables for each $j,k\in\mbox{\boldmath$\Xi$}$. We define the process $\{S_{n}\}_{n\geq 0}$ by $S_{0}=0$ and $S_{n}=S_{n-1}+Y^{\xi_{n-1}\xi_{n}}_{n}$ for $n\geq 1$. %The process $\{S_{n}\}_{n\geq 0}$ is known as %{\it a random walk defined on a finite %Markov chain}. Let $a$ be a positive integer. % and denote by %$[0,a]$ the interval consisting of the integers %$0,1,2,\cdots, a$. We denote by $T_{x}$ the first exit time of the process from the interval $[-x,a-x]$ for each $x=0,1,\cdots,a$. % $T_{x}=\inf\{n>0|x+S_{n}\notin [0,a]\}$ %for each $x\in [0,a]$. %Set We give an asymptotic behavior of the transition functions %of random walk defined on %a finite Markov chain with absorbing barriers $P_{jk}^{(n)}(x,y) =\mbox{\boldmath$P$}\{x+S_{n}=y; T_{x}>n; \xi_{n}=k |\xi_{0}=j\} % \label{asymptotic} $ as $n\rightarrow\infty$ for each $x,y\in [0,a]$ and all $j,k\in\mbox{\boldmath$\Xi$}$.

Let $f$ be a nonconstant entire function. %If $f(z)=z$ $\Longleftrightarrow $ $f'(z)=z$, and %$f'(z)=z$ $\Longrightarrow $ $f''(z)=z$, then $f\equiv f'$. In particular, If $f$, $f'$ and $f''$ have the same fixed points, then $f\equiv f'.$

By using the Darboux transformation in Soliton theory, we give the explicit construction for local isometric immersions of the Riemannian product $M^{n_1}_1(c_1)\times M^{n_2}_2(c_2)$ into space forms $M^m(c)$ with flat normal bundle via purely algebraic algorithm.

In this article E.\,Cartan's theorem about the existence of (local) totally geodesic sub\-mani\-folds with a prescribed tangent plane is generalized to manifolds which are equipped only with a linear connection; there is also given a global version of the theorem. The results are used for a short and geometric proof of the Theorem of Cartan-Ambrose-Hicks, and more generally of a generalization which is concerned with the existence of affine maps of arbitrary rank.