On fiber diameters of continuous maps

We present a surprisingly short proof that for any continuous map $f : \mathbb{R}^n \rightarrow \mathbb{R}^m$, if $n>m$, then there exists no bound on the diameter of fibers of $f$. Moreover, we show that when $m=1$, the union of small fibers of $f$ is bounded; when $m>1$, the union of small fibers need not be bounded. Applications to data analysis are considered.Comment: 6 pages, 2 figure

Three theorems on phrase structure grammars of type 1

It is shown that the class of languages generated by type 1 phrase structure grammars is not enlarged by allowing end markers, that this class is closed under the operation of intersection, and that those languages representable by linear bounded automata belong to this class