An indexed set of density bounds on lattice packings

It has long been known that the admissibility of a lattice with respect to a symmetric convex bodyB is equivalent to being a packing lattice for 1/2B. This fact is the basis of the interplay between the classical theory of the arithmetic minima of positive definite quadratic forms, on the one hand, and the dense lattice packing of spheres inR ⁿ , on the other.We give an indexed set of bounds _L (B) a _j , where 0 j n/2, on the lattice packing density ofB. The casej=0 reduces to the aforementioned long-known fact, andj=1 was proved by Elkies, Odlyzko, and Rush, and was used to obtain record high packing densities for various superballs. The new cases make possible the use of smaller primes in the construction of these dense packings.Supported by National Science Foundation grant DMS-9103233.     

The reactions of a dinuclear ferric complex (oxo) di-iron(III) triethylenetetraamminehexaacetate, Fe2O(ttha), with oxidizing and reducing free radicals. A pulse radiolysis study

The rate constants at which oxidizing and reducing radicals react with the dinuclear iron(III) complex Fe₂O(ttha)²⁻ were measured in neutral aqueous solution. The rate constants for reduction of the complex by ·CO₂^.− CH₃^.CHOH and O₂^.− were found to be comparable with rate constants previously measured in mononuclear iron(III) polyaminocarboxylate systems. Fe₂O(ttha)²⁻ reacts slowly with O₂^.− (k₈ = (1.2 ± 0.2) × 10⁴ dm³ mol⁻¹ s⁻¹) and, hence, is a relatively poor catalyst for the dismutation of superoxide radical. The hydrated electron reduces the complex at a diffusion-controlled rate in a process which consumes one proton: e_aq⁻ + Fe₂O(ttha)²⁻ → Fe₂^III,IIO(ttha)³⁻ The reduction by carbon-centered radicals produces a (III,II) mixed-valence complex with an absorption spectrum different from that of the Fe₂(II,III) species produced from reduction by the hydrated electron. The oxidizing radicals ^.OH and ·CO₃⁻ appear to act as reductants of the complex via ligand oxidation rather than by oxidation of the Fe₂^IIIO core to Fe₂^III,IVO. In the former case ligand attack appears to occur mainly at the methylene carbon of a glycinate group. The decarboxylation product, CO₂, was detected by its aquation reaction in the presence of a pH sensitive dye, bromthymol blue.