An irreducible open 3-manifold W is R 2-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane in W some component of W ? must have closure an embedded halfspace R 2 0; 1). In this paper it is shown that if M is a connected, P 2-irreducible, open 3-manifold such that 1 (M) is nitely generated and the universal covering space f M of M is R 2-irreducible, then eith...

let $g$ be a non-abelian group of order $p^n$‎, ‎where $nleq 5$ in which $g$ is not extra special of order $p^5$‎. ‎in this paper we determine the maximal size of subsets $x$ of $g$‎ ‎with the property that $xyneq yx$ for any $x,y$ in $x$ with‎ ‎$xneq y$‎.

In spectral clustering and spectral image segmentation, the data is partioned starting from a given matrix of pairwise similarities S. the matrix S is constructed by hand, or learned on a separate training set. In this paper we show how to achieve spectral clustering in unsupervised mode. Our algorithm starts with a set of observed pairwise features, which are possible components of an unknown,...

An irreducible open 3-manifold W is R-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane Π in W some component of W −Π must have closure an embedded halfspace R × [0,∞). In this paper it is shown that if M is a connected, P-irreducible, open 3-manifold such that π1(M) is finitely generated and the universal covering space M̃ of M is R-irreducible, then either ...

In this paper, we find lower bounds for the volumes of certain hyperbolic Haken 3manifolds. The theory of Jorgensen and Thurston shows that the volumes of hyperbolic 3-manifolds are well-ordered, but no one has been able to find the smallest one. The best known result for closed manifolds is that the smallest closed hyperbolic 3-manifold has volume > 0.16668, proven by Gabai, Meyerhoff, and Thu...