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.     

Rates and mechanisms of indole and 3-Methylindole polymerization in cation exchange resin beads

Batch extraction and polymerization of indole and 3-methylindole (skatole) with acidic cation exchange resins functioned as pseudo first-order reactions. Activation energies of 3.4 and 5.7 keal/mole indicated greater steric hindrance to skatole polymerization, as would be expected from 2,2′ bonding of skatole compared to 3,2′ bonding for indole. Linear polymerization occurred, although a heterocyclic ring opened at the trimer stage of poly(indole). 2-Methylindole did not polymerize. General acid-base catalysis caused unreproducible partial depolytnerization when the polymers were extracted from resin beads. Thus, compounds chromatographically separated probably were of lower degree of polymerization than were the polymers formed in the resin matrix.     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.