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.     

Thermal cyclization of (2-Ethynylphenyl)triazenes: facile synthesis of substituted cinnolines and isoindazoles

High-temperature intramolecular cyclization of N, N-dialkyl-N'-(4-substituted-2-ethynylphenyl)triazenes provides under neutral conditions both 6-substituted cinnolines and 5-substituted isoindazoles in moderate to excellent yields.     

The constant-volume-velocity nature of hearing aids: conclusions based on computer simulations

In the literature there are several references which imply that various parts of a hearing aid are sources of constant volume velocity. Reported herein are the findings of an investigation of the validity of such statements. A computer scheme, referenced elsewhere, for modeling in situ hearing aids was utilized to test the constant-volume-velocity hypothesis. In particular, capabilities of the receiver, ear hook, and earmold tip to deliver constant volume velocity were investigated via a computer. To facilitate such an investigation, a universal receiver/earmold model was created. This model was broken down into "source" and "load"at three locations: the receive output, output of the ear hook, and medial tip of the earmold. At each location comparisons were made between computed values of source and load impedance. The constant-volume-velocity hypothesis was assumed to be valid for those cases where source impedance was much, much greater than load impedance. Plots of such impedances show that, for the cases investigated, this rarely occurred, except over certain frequency bands. With the exception of in-the-ear hearing aids, these results appear to contradict inferences made in the literature about the constant-volume-velocity nature of hearing aids.     

Reactions of methylpyrylium,thiapyrylium, and flavylium salts with tetracyanoethylene

The reaction of 2- or 4-methyl-2,6-diphenylpyrylium perchlorate with tetracyanoethylene (TCNE) in pyridine gives 2- or 4-(2,3,3-tricyanopropylidene)-2,6-diphenylpyran in good yield. Similar results are obtained from 2- or 4-methyl-2,6-diphenylthiapyrylium and 4-methylflavylium perchlorates. In one case a stable charge-transfer salt is isolated from the reaction of a methylene base and TCNE. The reaction mechanism of dye formation is discussed.     

Dimethyldihydropyrene-dehydrobenzoannulene hybrids: studies in aromaticity and photoisomerization

The synthesis and study of dehydrobenzoannulene (DBA)-dimethyldihydropyrene (DDP) hybrids as models for the investigation of aromaticity in weakly diatropic systems is reported. Three new monofused DBA-DDP hybrids have been synthesized, and their NMR spectra are discussed with regard to quantifying the aromaticity remaining in multibenzene-fused DBAs. Nucleus-independent chemical shifts, determined at a series of locations for each compound, bond lengths, and (1)H and (13)C NMR chemical shifts were calculated and used to probe the aromaticity of these hybrids. Systems where more than one annulene/DBA is fused to the DDP core have also been obtained, and their potential use in photoinduced isomerization applications is discussed.