let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

A classic result of Johnson and Lindenstrauss asserts that any set of n points in d-dimensional Euclidean space can be embedded into k-dimensional Euclidean space—where k is logarithmic in n and independent of d—so that all pairwise distances are maintained within an arbitrarily small factor. All known constructions of such embeddings involve projecting the n points onto a spherically random k-...

Following guidance from the Organizing Committee, the authors give a brief introduction to the theory of spaces which are resolvable in the sense introduced by Hewitt (1943). The new results presented here are these. (A) A countably compact regular Hausdorff space without isolated points is w-resolvable--that is, it admits an infinite family of pairwise disjoint dense subsets. (B) Among Tychono...

We present Joshua 4.0, the newest version of our open-source decoder for parsing-based statistical machine translation. The main contributions in this release are the introduction of a compact grammar representation based on packed tries, and the integration of our implementation of pairwise ranking optimization, J-PRO. We further present the extension of the Thrax SCFG grammar extractor to piv...

We prove several dichotomy theorems which extend some known results on σ -bounded and σ -compactpointsets. In particular we show that, given a finite number of Δ1 equivalence relations F1 , . . . , Fn , any Σ1 set A of the Baire space either is covered by compact Δ1 sets and lightface Δ1 equivalence classes of the relations Fi , or A contains a superperfect subset which is pairwise Fi -inequiva...

Given a compact 2–dimensional manifold M we classify all continuous flows φ without wandering points on M . This classification is performed by finding finitely many pairwise disjoint open φ−invariant subsets {U1, U2, . . . , Un} of M such that ⋃n i=1 Ui = M and each Ui is either a suspension of an interval exchange transformation, or a maximal open cylinder made up of closed trajectories of φ.

Consider the space Hom(Z n , G) of pairwise commuting n-tuples of elements in a compact Lie group G. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingulariza-tion of the generic component of Hom(Z n , G), which allows us to derive formulas for its singular and equivariant cohomology in terms of the Lie algebra of a maximal torus in G and the ...