Rank and k-nullity of contact manifolds

We prove that the dimension of the 1-nullity distribution N(1) on a closed Sasakian manifold M of rankl is at least equal to 2l−1 provided that M has an isolated closed characteristic. The result is then used to provide some examples of k-contact manifolds which are not Sasakian. On a closed, 2n+1-dimensional Sasakian manifold of positive bisectional curvature, we show that either the dimension of N(1) is less than or equal to n+1 or N(1) is the entire tangent bundle TM. In the latter case, the Sasakian manifold Mis isometric to a quotient of the Euclidean sphere under a finite group of isometries. We also point out some interactions between k-nullity, Weinstein conjecture, and minimal unit vector fields.

Genetic Structure of the Florida Key Tree Cactus, Pilosocereus robinii, using Restriction Site associated DNA (RAD) markers

Rare plant conservation efforts must utilize current genetic methods to ensure the evolutionary potential of populations is preserved. One such effort involves the Key Tree Cactus, Pilosocereus robinii, which is an endangered columnar cactus native to the Florida Keys. The populations have precipitously declined over the past decade because of habitat loss and increasing soil salinity from rising sea levels and storm surge. Next-generation DNA sequencing was used to assess the genetic structure of the populations. Twenty individuals representative of both wild and extirpated cacti were chosen for Restriction Site Associated DNA (RAD) analysis. Samples processed using the HindIII and NotIII restriction enzymes produced 82,382,440 high quality reads used for genetic mapping, from which 5,265 Single Nucleotide Polymorphisms (SNPs) were discovered. The analysis revealed that the Keys’ populations are closely related with little population differentiation. In addition, the populations display evidence of inbreeding and low genetic diversity.