${\rm PL}$ involutions on the nonorientable $2$-sphere bundle over $S\sp{1}$

PL INVOLUTIONS ON THE NONORIENTABLE 2-SPHERE BUNDLE OVER Sx

We show that there are exactly nine distinct PL involutions on the nonorientable 2-sphere bundle over Sl, up to PL equivalences. This, together with results of [1], [3] and [8], classifies all PL involutions on the 2sphere bundles over S1.

On Involutions of the 3-Sphere

On Ricci identities for submanifolds in the 2-osculator bundle

It is the purpose of the present paper to outline an introduction in theory of embeddings in the 2-osculator bundle. First, we recall the notion of 2-osculator bundle ([9], [2], [4]) and the notion of submani-folds in the 2-osculator bundle ([9]). A moving frame is constructed. The induced connections and the relative covariant derivation are discussed in the fourth and fifth section ([15], [16...

Circuits Over PP and PL

Wilson’s [19] model of oracle gates provides a framework for considering reductions whose strength is intermediate between truth-table and Turing. Improving on a stream of results by Beigel, Reingold, Spielman, Fortnow, and Ogihara [6, 9, 14], we prove that PL and PP are closed under NC1 reductions. This answers an open problem of Ogihara [14]. More generally, we show that NC k+1 = AC k and NC ...

The Real Cohomology Ring of a Sphere Bundle over a Differentiable Manifold