Long range substituent effects relfected in the 13C NMR spectra of allyl alcohols and their derivatives

Acylation of allyl alcohols induces strong carbon shifts, shielding γ and deshielding δ effects. These shifts are a consequence of through-bond polarization of the olefinic carbons. Allyl ethers show similar, but milder perturbations. The olefinic carbon shifts reveal a strong concentration dependence. This effect, in greatly reduced form, is also observed in non-oxygenated alkenes. Hydrogen bonding by alcohols enhances the chemical shift difference in olefinic carbon pairs by increased polarization.     

Stereochemical studies—XXV: Conformational equilibria of 2-substituted 1,1-dialkylcyclohexanes

The positions of the conformational equilibria in a series of 1,1,2-trisubstituted cyclohexanes have been determined by ¹H NMR. The gauche-interaction of substituents in the 1,1-dimethyl series are, in general, very close to those of monosubstituted cyclohexanes. However, in the spiro-compounds 5 and 6 the equatorial conformers are somewhat destabilized by gauche-interactions though still enthalpy preferred. In contrast, the enthalpy preferred conformer in the 1,1-diethyl series is the axial one.     

Construction of hjelmslev planes from (t,k)-nets

A (t, k)-net is an abstract generalization of the incidence structures which occur as the point and line neighborhoods of a finite Hjelmslev plane. A (t, k)-net contains ‘substructures’ which are nets of ordert and degreek. Every (t, r) Hjelmslev plane (brieflyH-plane) can be constructed from a suitable collection of (t, r+1)- and (t, r)-nets. A (t, r)H-plane or (t, k)-net is called extremal provided: each two points which are joined by more than two lines are joined by preciselyt lines and dually. IfB is a ‘properly’ extremal (t, r)H-plane (means both 2 andt≠2 occur among the joining numbers), thent is even; andr=2 orr=1+(t/2). All the 3-uniform [J. Combinat. Theory 9, 267–288 (1970, this Zbl.204, 210)] (4, 2)H-planes are examples. Two further examples are constructed in the paper: an (8, 2) translationH-planeC and an (8, 2) projectiveH-planeD, none of whose affineH-planes are translationH-planes. All point neighborhoods fromC andD and all line neighborhoods fromD are isomorphic to a give (8, 3)-netE;E is constructed by considering the subspaces of a 64-point symplectic geometry overZ ₂.E is also used to answer (affirmatively) the question of the existence of proper fairly near affineH-planes [J. Combinat. Theory 16A, 34–50 (1974)].     

The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.