Finite and uniform stability of sphere coverings

By attaching cables to the centers of the balls and certain intersections of the boundaries of the balls of a ball covering ofE ^d with unit balls, we can associate to any ball covering a collection of cabled frameworks. It turns out that a finite subset of balls can be moved, maintaining the covering property, if and only if the corresponding finite subframework in one of the cabled frameworks is not rigid. As an application of this cabling technique we show that the thinnest cubic lattice sphere covering ofE ^d is not finitely stable. The first two authors were partially supported by the Hungarian National Science Foundation under Grant No. 326-0413.     

On hyperplanes and polytopes

We call a convex subsetN of a convexd-polytopePE ^d ak-nucleus ofP ifN meets everyk-face ofP, where 0<k<d. We note thatP has disjointk-nuclei if and only if there exists a hyperplane inE ^d which bisects the (relative) interior of everyk-face ofP, and that this is possible only if [d+2/2]kd–1. Our main results are that any convexd-polytope with at most 2d–1 vertices (d3) possesses disjoint (d–1)-nuclei and that 2d–1 is the largest possible number with this property. Furthermore, every convexd-polytope with at most 2d facets (d3) possesses disjoint (d–1)-nuclei, 2d cannot be replaced by 2d+2, and ford=3, six cannot be replaced by seven.Partially supported by Hung. Nat. Found. for Sci. Research number 1238.Partially supported by the Natural Sciences and Engineering Council of Canada.Partially supported by N.S.F. grant number MCS-790251.     

M...HNR interactions in imino-bound diaryltriazene complexes: structure and fluxionality

The nominally square-planar coordination of the d(8) complexes [MClL(1)L(2)(p-XC(6)H(4)NNNHC(6)H(4)X-p)](M = Rh, L(1)= L(2)= CO, X = H, Me, Et or F; M = Ir, L(1)= L(2)= CO, X = Me; M = Pd or Pt, L(1)= Cl, L(2)= PPh(3), X = Me; M = Pd, L(1)L(2)=eta(3)-C(3)H(5), X = Me), with the triazene N-bonded via the imine group, is supplemented by an axial M...H-N interaction involving the terminal amino group.     

The d2/d3 alkyne redox pair [WF2(PhC identical to CPh)Tp']z (z = +1 or 0): missing links in a 'redox family tree'

The d2/d3 redox pair [WF2(PhC identical to CPh)Tp']z [z = +1 or 0, Tp' = hydrotris(3,5-dimethylpyrazolyl)borate] is the missing link in a 'redox family tree' relating the d6 tricarbonyls [M(CO)3L]- to the d2 trihalides [MX3L] (M = Mo or W, L = Cp or Tp') by a series of stepwise reactions involving sequential one-electron oxidation followed by ligand substitution.