In this paper, we shall consider normal forms for derivations in ${\rm AI}_{\xi}^{-}$, where ${\rm AI}_{\xi}^{-}$ is a system introduced by Arai (cf. [4]) and its consistency implies the consistency of Feferman's ${\rm ID}_{\xi}$ (cf. [6]). We shall give two normal form theorems for derivations in ${\rm AI}_{\xi}^{-}$. One (Theorem 1) implies the consistency of ${\rm AI}_{\xi}^{-}$. The other (Theorem 2) implies the $\omega$-consistency of ${\rm AI}_{\xi}^{-}$.

We study closed connected topological or smooth 4-manifolds fibering over a surface in terms of classifying spaces, characteristic classes, and intersection forms.

Kenneth Kunen showed that there is no non-trivial elementary embedding of the universe $V$ into itself. We generalize this theorem by proving that there is no non-trivial generic elementary embedding of $V$ into itself.

The notions of $n$-shape for compact pairs and proper $n$-shape for locally compact spaces were introduced in [1] and [2], respectively. In this paper, strengthening $n$-shape of pairs, we define the notion of relative $n$-shape of compact pairs. By constructing a functor from the proper $n$-shape category to the relative $n$-shape category, we prove that for locally compact spaces $X$ and $Y$ with $\dim\leq n+1$, if $n$-${\rm Sh}_{p}(X)=n$-${\rm Sh}_{p}(Y)$ then $n$-${\rm Sh}(FX, EX)=n$-${\rm Sh}(FY,EY)$ rel. $(EX,EY)$ and $n$-${\rm Sh}(CX, \{\infty\})=n$-${\rm Sh}(CY, \{\infty\})$ rel. $(\{\infty\}, \{\infty\})$, where $FX$ is the Freudenthal compactification of $X$ with $EX$ the ends, and $CX=X\cup\{\infty\}$ is the one-point compactification of X. As corollaries, (1) if $X$ is connected $SUV^{n}$ and $\dim X\leq n+1$, then $FX\in UV^{n}$, (2) if $X$, $Y\subset\mu_{\infty}^{n+1}=\mu^{n+1}\backslash \{*\}$ are $Z$-sets and $n$-${\rm Sh}_{p}(X)=n$-${\rm Sh}_{p}(Y)$, then $\mu_{\infty}^{n+1}\backslash X$ is homeomorphic $(\approx)$ to $\mu_{\infty}^{n+1}\backslash Y$.

Let $f:M\rightarrow\tilde{M}$ be an isometric immersion of a Riemannian manifold $M$ into a Riemannian manifold $\tilde{M}$. We study the geometry of submanifolds under various assumptions with respect to the first curvature $\tilde{\lambda}_{1}$ and the second curvature $\tilde{\lambda}_{2}$ of $\tilde{\sigma}=f\circ\sigma$ in $\tilde{M}$ for a helix $\sigma$ in $M$.

We study for various kinds of geometric stmctures on Riemannian manifolds whether these stmctures induoe on totally geodesic submanifolds stmctures of the same kind.

Assume that both $X$ and $Y$ are non-compact locally compact spaces. Let $\delta(X\times Y)$ be a compactification of $X\times Y$ such that $\delta(X\times Y)\geq\omega X\times\omega Y$, where $\omega X$ and $\omega Y$ are the one-point compactifications of $X$ and $Y$, respectively. Then J. L. Blasco [2] proved the theorem that $\delta(X\times Y)$ is not a weakly singular compactification of $X\times Y$ if $X$ is pseudcompact. In this paper we give an alternative, simpler proof for the above theorem. Furthermore, in the case $X$ is either a non-separable metrizable space or a separable metrizable space with a non-compact quasi-component space $Q(X)$ and $d(Y)\leq d(X)$, where $d(X)$ is the density of $X$, for any compact space $S$ we establish a theorem that $X\times Y$ has a singular compactification with $S$ as a remainder if and only if $X$ has a singular compactification with $S$ as a remainder.