Populations often have very complex hierarchical structure. Therefore, it is crucial in genetic monitoring and conservation biology to have a reliable estimate of the pattern of population subdivision. F(ST)'s for pairs of sampled localities or subpopulations are crucial statistics for the exploratory analysis of population structures, such as cluster analysis and multidimensional scaling. Howe...

Let F denote the finite field with two elements. We describe a construction of partitions of F , for n = 2 − 1, m ≥ 4, into cosets of pairwise distinct Hamming codes (we call such codes nonparallel) of length n. We give a lower bound for the number of different such partitions.

Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f (n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f (n) = (n3/2).

We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M , F is isotopic to a least-area surface F (g), then F is incompressible.

We show that for any closed surface F with χ(F ) −4 (or χ(F ) −2), there exist graphs that triangulate the torus or the Klein bottle (or the projective plane) and that quadrangulate F . We also give a sufficient condition for a graph triangulating a closed surface to quadrangulate some other surface. © 2006 Elsevier Inc. All rights reserved.

A regular linear line complex is a three-parameter set of lines in space, whose Plücker vectors lie hyperplane, which not tangent to the Klein quadric. Our main result bound O(n1/2m3/4+m+n) for number incidences between n and m points F3, where F field, n≤char(F)4/3 positive characteristic. Zahl has recently observed that bichromatic pairwise coming from two distinct complexes account nonzero s...

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

We present a deterministic linear time algorithm to find a set of five points that stab a set of n pairwise intersecting disks in the plane. We also give a simple construction with 13 pairwise intersecting disks that cannot be stabbed by three points.

We show that there is some absolute constant c > 0, such that for any union-closed family F ⊆ 2, if |F| ≥ ( 1 2 − c)2n, then there is some element i ∈ [n] that appears in at least half of the sets of F . We also show that for any union-closed family F ⊆ 2, the number of sets which are not in F that cover a set in F is at most 2, and provide examples where the inequality is tight.