Splitting definably compact groups in o-minimal structures

An argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Further results for a one-dimensional linear thermoelastic equation with Dirichlet-Dirichlet boundary conditions

We show that a sequence of generalized eigenfunctions of a one-dimensional linear thermoelastic system with Dirichiet-Dirichlet boundary conditions forms a Riesz basis for the state Hilbert space. This develops a parallel result for the same system with Dirichlet-Neumann or Neumann-Dirichlet boundary conditions.

The relative abundance of Mortierella Coemans spp. in acid heath soils

The fungal population of two podzolized heath soils, one with and one without a hardpan, is characterized by a marked contrast in the species distribution of the genus Mortierella Linnemann.The dominant species in the hardpan soil is Mortierella ramanrniana (Möll) Linnemann, but when the pan is lacking it is replaced by Mortierella nana Linnemann. This is not a local effect, for an analysis of results from comparable English soils gives a parallel result. It can be stated that the dominance of one or other of these species is indicative of certain soil types.