A simplified proof of a theorem about group representations

It is a standard theorem of group representation theory that the dimension of an irreducible representation is a divisor of the order of the group. This paper gives a new, relatively simple proof, intended to make the theorem understandable to readers unfamiliar with algebraic integers.     

Synthesis and characterization of poly-β-hydroxybutyrate. II. Synthesis of D-poly-β-hydroxybutyrate and the mechanism of ring-opening polymerization of β-butyrolactone

Synthesis of the naturally occurring polyester, D -poly-β-hydroxybutyrate (PHB) was accomplished by using an optically active monomer. Polymerization of D -(+)-β-butyrolactone (β-BL) of 73% optical purity with a catalyst system of Et₃Al–H₂O produced a polymer with a similar optical activity and essentially identical to the natural polymer as isolated from bacterial cells. This paper describes the synthesis and characterization of this optically active polyester along with a suggested mechanism to account for the observed stereospecific polymerization of β-BL with this catalyst system.     

Effect of shear on human insulin in zinc suspension

Human insulin in zinc suspension was used as a model protein to test the effect of shear on the settling rate of proteins, a possible inference for protein denaturation. The rate of settling was determined directly in a spectropho-tometer. Shear effects are important in retaining the activity of proteins and are present in bubble, foam, and droplet protein fractionation processes. A sim pletest, such as that conducted here, mayeven be useful for monitoring changes in protein structure caused by commercial shipping of the protein. The settling ratefor insulin was continously monitored in theoriginal bottle by spectrophotometric absorbance changes as a function of time. A settling curve was determined following each shear experiment, which included shaking the “worked” insulin solution in a vortex mixer for different lengths of time. It was determined, when comparing long shaking times with short ones, that the initial settling rate was less for the long-term shaking of the insulin samples and greater for the short-term shaking. The secondary effects of light and heat, along with shaking, a pparently did not produce differences from shaking alone.     

Reactions of fused dihydro-1,2,4-thiadiazoles with isoselenocyanates giving 6aλ4-thia-1,3,4,6-tetraazapentalene derivatives and 5,10-dihydro-1,2,4-thiaselenazolo[4,5-b][2,4]benzodiazepines

3-Methyl-6,7-dihydro-5H-1,2,4-thiadiazolo[4,5-a]pyrimidine (1) reacted with isoselenocyanates with elimination of acetonitrile and concomitant addition of two molecules of the isoselenocyanate to give 2,3-di-substituted-6,7-dihydro-5H-2aλ⁴-thia-2,3,4a,7a-tetraazacyclopent[cd]indene-1(2H),4(3H)-diselones (6a)–(6j). 3-Methyl-5,10-dihydrobenzo[e]-1,2,4-thiadiazolo[4,5-a][1,3]diazepine (3) likewise reacted with alkyl isoselenocyanates to give the 2,3-dialkyl-5-10-dihydro-2aλ⁴-thia-2,3,4a,10a-tetraazapentaleno[3,3a,4-gh]benzocycloheptene-1,4-diselones (9a)–(9h), but reaction of (3) with aryl isoselenocyanates took place with elimination of acetonitrile and incorporation of one molecule of the aryl isoselenocyanate in the product to give 3-arylimino-5,10-dihydro-1,2,4-thiaselenazolo[4,5-b][2,4]benzodiazepines (10a)–(10h). Structure (10) is a new heterocyclic system. The pyrimidine (1) and the diazepine (3) reacted with aryl isoselenocyanates at room temperature in solvents of low polarity to give zwitterion 1:1 addition compounds (7) and (12), respectively. NMR studies reveal that the thiaselenazoles (10) react in solution with aryl isoselenocyanates to give diaryl diselones (11) in a reversible process involving a Dimroth rearrangement. © 1996 John Wiley & Sons, Inc.     

An electron microscope study of tin dioxide and antimony-doped tin dioxide

Crystals of tin dioxide, SnO₂, have been grown pure or doped with a few percent of antimony using vapor growth methods in order to investigate the microstructures of reduced and oxidized SnO₂. They were examined by X-ray diffraction and by optical and electron microscopy. SnO₂ crystals were found to contain few faults, but the antimony-doped crystals were extensively twinned in some regions. Reduction of SnO₂ crystals to yield CS phases was unsuccessful. These results are discussed in terms of the known crystal chemistry of the oxides involved.     

High-pressure silicate pyrochlores,Sc2Si2O7 and In2Si2O7

The silicate compounds Sc₂Si₂O₇ and In₂Si₂O₇ have been converted from thortveitite type to pyrochlore type at 1000°C, 120 kbar, with resulting cell constants of 9.287(3) and 9.413(3) Å, respectively. Invariant reflection intensities in the X-ray powder diffraction patterns allowed precise absorption corrections to be made, and refinement of thermal parameters and of the single structural parameter x gave values of 0.4313(21) and 0.4272(15), respectively. The corresponding six-coordinate SiO distances were 1.761(7) and 1.800(5) Å, and the average eight-coordinate distances for ScO₈ and InO₈ were 2.267 and 2.275 Å. Values of structure-refined bond lengths for compounds containing six-coordinate silicon are surveyed, and overall weighted average octahedral distances of 1.782(14) Å for SiO and 2.520(18) Å for OO are derived. Pyrochlore phases were not produced from rare-earth disilicate or monosilicate phases subjected to the same reaction conditions as the Sc and In compounds.