Photolytic closure of 4,5-diphenyl-oxazol-2-ones to phenanthro[9,10-d]oxazol-2-ones and an improved synthesis of benzoins

The synthesis of phenathrenes substituted by nitrogen and oxygen atoms in position 9 and 10 by photolysis of 4,5-diphenyl-oxazol-2-ones (cf. Scheme 4) is described. Moreover an improved general procedure for the synthesis of benzoins in aprotic solvent (tetra-t-butylammonium cyanide in DMF/DMSO) is developed.     

Convex optimization and lagrange multipliers

The duality theorem of linear programming is used to prove several results on convex optimization. This is done without using separating hyerplane theorems.This work was supported in part by a grant from Investors in Business Education.     

Isotactic sequence lengths by ozonation of poly(propylene oxide)

Ozonation followed by lithium aluminum hydride reduction cleaved high molecular weight isotactic poly(propylene oxide) to crystalline polyglycols. From the melting point and molecular weight of the latter, the molar freezing point depression produced by end groups is found to be ca. 18°C./mole, as compared to that estimated for poly(ethylene glycols), K_f = 12°C./mole, from earlier data. By assuming syndiotactic placements (or other irregularities) would produce the same molar depression, the melting point of isotactic poly(propylene oxides) produced by various catalysts has been used to estimate the isotactic sequence lengths.     

Electron impact induced fragmentations and rearrangements of some phenyl substituted phosphine oxides

Electron impact induced fragmentations and rearrangements of 1-phenyl-2,2,3-trimethylphosphetane 1-oxide, 1-phenylphospholane 1-oxide and 1-phenylphosphorinane 1-oxide were investigated. A process common to all of these phosphine oxides was electron impact induced formation of new carbon-carbon bonds involving the phenyl rings. Evidence for such rearrangements included observation of hydrocarbon fragment ions containing seven or more carbons.     

Contractive projections with contractive complement in Lp space

Using the concepts of conditional expectation and independence of subalgebras, we characterize those contractive projections, P, on L_p, over a probability measure space, having the property that I − P is contractive. By contractive projection we mean a linear operator, P, on the Lebesgue space, L_p, 1 < p < ∞, ≠2, with P² = P, ∥ = 1.     

Extended-chain crystals. VI. Annealing of polyethylene under elevated pressure

Polyethylene crystals of different degrees of perfection were annealed at 5.1 kb pressure for 20 hr at various temperatures and analyzed by electron microscopy, thermal analysis, and density determination. No annealing took place until the temperature was close to the melting point of the starting material. Up to 235°C increasing solidstate annealing was observed. Mixed crystals of up to 0.989 g/cm³ density and 1500 Å thickness in the chain direction could be produced. At slightly higher temperature recrystallization to extended-chain crystals rather than annealing occurred. The annealing process at atmospheric pressure seems to be similar in nature, but takes much longer for comparable perfection. From a comparison of annealing and crystallization it is concluded that polymer crystallization goes through a stage of internally imperfect order during which most of the observed chain extension occurs. Estimates of this outer imperfect layer of a growing crystal place its depth at 30,000 Å.     

Continual measurements in the spacetime formation of nonrelativistic quantum mechanics

We study the nature and effects of some continual measurements in nonrelativistic quantum mechanics, a concept introduced by R. P. Feynman in his path integral formulation of quantum mechanics. We prove the existence in various senses of mathematically rigorous objects formally equivalent to the path integrals associated with such measurements, by means of both a limiting definition and more general techniques; and we derive some quantitative results concerning the effects on the wave-function and the numerical value of certain associated probabilities.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.