Countable mixed Abelian groups with very nice full-rank subgroups

Measure concentration arguments are applied to get a power-type estimate for the dimension of almostl _p subspaces of isomorphs ofl _p ⁿ and for the length of almost-symmetric sequences under a nonlinear-type condition. Supported by the Fund for Basic Research Administered by the Israel Academy of Sciences and Humanities.     

Synthesis,spectroscopic properties and crystal structure of the eight-coordinated lead complex bis[thiocyanato]-1,4,7,10,13-pentaoxacyclopentadecano lead(II)

The complex [Pbc[15]crown-5](SCN)₂ was synthesized and characterized spectroscopically (IR and FIR range) and the crystal structure was determined by X-ray diffraction on a single crystal (orthorhombic space group Pmc2₁). The structure was solved by Patterson synthesis with least-square refinement. The final R-value is 0.0328. Two Pb-atoms and two unidentical halves of the ligand were found in the asymmetric unit. For each complexed lead cation and each half of the crown ether there exists a plane of reflexion, whereby C18 and C28 are disordered. There is, however, no centre of inversion between these unidentical halves. The elementary cell comprises four units and two different types of two centrally coordinated Pb cations. Both lead cations are eight-coordinated. In the case of Pb1 the eight donor centres are the five O-atoms of the ligand and three N-atoms of the thiocyanates. For Pb2 these are five oxygens of the ligand, one nitrogen and two sulphurs of the three thiocyanates. In this complex the thiocyanate ligands occur in both ionic and covalent states. The complexation of the crown ether and the two different species of thiocyanate anions are indicated too by typical shifted absorption bands in the IR spectra. The vibrations in the FIR region can be assigned to the interactions between lead and the donor groups.