From a Minkowski-type metric on $R_{+}^n$ satisfying the Einstein condition, we derive a nonlinear partial differential equation. In order to know the property of its solutions, we obtain an approximate solution with certain boundary conditions numerically by the finite element method, which will give us some clues to get theoretical solutions.

Regular general connections of conformal type provided for space-times are investigated. Using the Hessian-type tensors of time function, certain isometric decomposition theorems of space-times can be obtained. The null geodesics concerning the general connection of conformal type are defined and a criterion for its incompleteness is researched.

Let $k$ be a field of characteristic 0, $V$ an $n$-dimensional non-singular algebraic variety over $k,K$ the function field of $V$ and ${\text{Ω}}_{K/k}^1$ the module of differentials of $K$ over $k$. A closed differential $\omega \in {\text{Ω}}_{K/k}^1$ is called residue free if $re{s}_W(\omega )=0$ for any prime divisor $W$ of $V$ and a differential $\omega$ is called second kind if for any prime divisor $W$ , there exists an element ${\theta }_W\in K$ such that ${\nu }_W(\omega -d{\theta }_W)\ge 0$, where ${\nu }_W$ is the canonical valuation with respect to $W$. In this paper, we prove the following theorem: Let $\omega$ be a closed element of ${\text{Ω}}_{K/k}^1$. Then $\omega$ is residue free if and only if $\omega$ is of second kind.

We show that if $G$ is a 3-connected graph with radius $r$, then $r\le \frac{|V(G)|+15}{4}$.

It is shown how the atomic decomposition of tent spaces can be used to get a characterization of the weight functions $u,v$ for which the fractional maximal operators $M_s$ sends the weighted Lebesgue spaces $L_v^p$ into $L_u^q$ with .