An m-convexity property for complementary pairs in the plane

Let A and B be closed nonempty sets in the plane with A B=R², A B=bdry A=bdry B=C. Assume that for every m 2 points in C, at least one of the corresponding segments is in A, at least one in B. Then each of A and B is locally expressible as a union of m–1 or fewer convex sets.     

Structural investigation of the promotional effect of hydrogen during the selective catalytic reduction of NOx with hydrocarbons over Ag/Al2O3 catalysts

In situ EXAFS has been used to examine the hydrogen effect on the selective catalytic reduction of NOx over silver/alumina catalysts. For all SCR conditions used, with or without co-reductant (H2 or CO), the catalyst structure remained the same. Significant changes in the catalyst were only found under reducing conditions. The enhanced activity found in the presence of hydrogen is thought to be due to a chemical effect and not the result of a change in the structure of the active site.     

A Krasnosel’skii-type theorem for paths of bounded length

Let S be a nonempty closed, simply connected set in the plane, and let α τ; 0. If every three points of 5 see a common point of S via paths of length at most α, then for some point s0 of S, s0 sees each point of S via such a path. That is, S is starshaped via paths of length at most α. Supported in part by NSF grant DMS-9207019     

Finitely starlike sets whose F-stars have positive measure

Let and assume that there is a countable collection of lines {L _i : 1 i} such that (int cl S) and ((int cl S) S) L _i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336.     

Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

Tilings whose members have finitely many neighbors

Let ? be a tiling of the plane such that each tile of ? meets at most finitely many other tiles. Then exactly one of the following must occur:

1. Uncountably many boundary points of ? belong to no nondegenerate edge of ?, hence ? has uncountably many singular points; or
2. Every boundary point of ? belongs to a nondegenerate edge of ?, moreover, ? has no singular points.

Furthermore, ifS is the set of singular points of ? andW={t:t∈bdry ? andt belongs to no nondegenerate edge of ?}, thenS=clW.     

Hybrid polyoxovanadates: anion-influenced formation of nanoscopic cages and supramolecular assemblies of asymmetric clusters

Two novel hybrid polyoxovanadates that are functionalized by (4-aminophenyl)arsonic acid ligands form upon the reduction of vanadates(V) in aqueous systems, whereby the underlying condensation reactions are influenced by the nature of the employed acid. In the presence of Cl(-) ions that derive from hydrochloric acid, a tetradecanuclear cage structure [V(IV)(14)O(16)(OH)(8)-(O(3)AsC(6)H(4)-4-NH(2))(10)](4-), whose cavity contains stabilizing halide ions and water molecules, is obtained. When nitric acid is used, a decanuclear [V(10)O(18)(O(3)AsC(6)H(4)-4-NH(2))(7)(DMF)(2)](5-) cluster can be isolated. The latter organizes into a hexagonal packing arrangement in the solid state.