On the covering multiplicity of lattices

Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI _n, A_n, D_n, E_n and their duals for small values ofn. Although it appears thatCM(I _n)=2 ⁿ⁻¹ ifn≤33, asn → ∞ we haveCM(I _n)∼2.089... ⁿ . The results have application to numerical integration.     

The antipode construction for sphere packings

Summary A construction for sphere packings is introduced that is parallel to the anticode construction for codes. This provides a simple way to view Vardy's recent 20-dimensional sphere packing, and also produces packings in dimensions 22, 44–47 that are denser than any previously known.Oblatum 18-VII-1994 & 7-XII-1994     

The essential selfcommutator of a subnormal operator

In this paper an easier proof is obtained of Alexandru Aleman's extension of an inequality of Axler and Shapiro for subnormal operators to the essential norm. The method is applied to show that a hyponormal operator whose essential spectrum has area zero must be essentially normal.

     

The evaluation and use of properties of individual ions in slution

A critical examination is given of extra-thermodynamic methods for evaluation of properties of individual ions in solution. The ways in which such data may be applied, or are required in various problems concerning specific properties of electrolyte solutions and in the fields of kinetics, electrochemistry, colloid and interfacial chemistry, are outlined. Special attention is given to the problem of evaluation of individual ionic thermodynamic functions for the hydration of ions but, in some cases, corresponding data for nonaqueous solutions are available for discussion and are of interest in a comparative way. The reliability of the various methods is assessed and estimates of numerical values for various individual ionic functions are tabulated. The question of scales of individual ionic radii, on which some of the derived data depend, is also examined. The dependence of various properties for anions and for cations on ionic radius is discussed and data are plotted comparatively. It is shown that for most, but not all, properties, the values for anions are more sensitive to increasing ionic radius than are the values for cations. This behavior, as well as the exceptions, is of theoretical interest. The significance of ion-specific behavior at interfaces is considered and the importance of interactions between ions of like sign of charge is indicated.Presented at the Symposium on Electrolytes, University of Reading, September 13, 1977.     

Radical cyclization of sugar-derived beta-(alkynyloxy)acrylates: synthesis of novel fused ethers

[formula: see text] Tributyltin radical mediated cyclization of carbohydrate-derived beta-(alkynyloxy)acrylates leading to highly functionalized cis- and trans-fused bicyclic ethers of various ring sizes is described. The efficacy of the radical cyclization is nicely illustrated in the iterative construction of a trans-fused tricyclic tetrahydropyran.