Unions of three starshaped sets in R2

LetS be a compact, simply connected set inR ². If every boundary point ofS is clearly visible viaS from at least one of the three pointsa, b, c, thenS is a union of three starshaped sets whose kernels containa, b, c, respectively. The result fails when the number three is replaced by four.As a partial converse, ifS is a union of three starshaped sets whose kernels containa, b, c, respectively, then the set of points in the boundary ofS clearly visible from at least one ofa, b, orc is dense in the boundary ofS.Supported in part by NSF grant DMS-8705336.     

Preparation and characterization of dealuminated metakaolin and its use in the transformation of waste plastics to aromatic hydrocarbons

Acid activated metakaolins (AAMKs) have been prepared by calcination of the natural clay at 600 degrees C to provide a metakolin which was then leached at 80 degrees C for 3 h using 1M, 2M, 3M, and 6M HCl. These materials were characterized and their ability to transform the off gases from HDPE decomposition into useful aromatic species was evaluated. The amount of adsorbed water and the number of acid sites increased with the severity of acid treatment. Variable temperature DRIFTS spectroscopy of pyridine treated samples revealed that both Br?nsted and Lewis acid centers were present until 425 degrees C. Pyridine bonded to the Lewis acid centers was more thermally stable. The AAMKs were all selective to the production of toluene with respectable, but lesser, amounts of xylenes and trimethylbenzenes. This selectivity contrasts with that of acid leached and pillared smectites which are selective toward trimethylbenzene.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.