Representation of varieties in combinatory algebras

It is shown that the set of completions of algebras in a variety can be represented as the set of solutions of a single equation of the formA · X=B · X in the author's model of combinatory algebra.A andB are determined directly from the equations which present the variety. Conversely, the individual structures are realized as retracts and the algebraic operations as combinatory objects; these are reclaimable by fixed combinators from the individual solutionsX. These results can be extended to universal classes and to algorithmic classes.Presented by Walter Taylor.     

Mixed width-integrals of convex bodies

The mixed width-integrals are defined and shown to have properties similar to those of the mixed volumes of Minkowski. An inequality is established for the mixed width-integrals analogous to the Fenchel-Aleksandrov inequality for the mixed volumes. An isoperimetric inequality (involving the mixed width-integrals) is presented which generalizes an inequality recently obtained by Chakerian and Heil. Strengthened versions of this general inequality are obtained by introducing indexed mixed width-integrals. This leads to an isoperimetric inequality similar to Busemann’s inequality involving concurrent cross-sections of convex bodies.     

Enantioselective synthesis of 3-demethoxyaranciamycinone via asymmetric epoxidation

The tetracyclic olefin 1 was enantioselectively epoxidized using a molybdenum(VI)- oxodiperoxo complex containing a chiral lactamide. Conversion to the glycosides 8 and 9 showed the epoxide 2 to be the predominant (3 : 1) enantiomer.     

Calorimetric relaxation and glass transition in poly(propylene glycols) and its monomer

The glass transition temperature T_g of propylene glycol (PG) and poly(propylene glycols) (PPGs) of molecular weight up to 4000 has been measured by differential scanning calorimetry, and the activation energy and change in heat capacity ΔC_p have been determined in the glass transition range. The activation energy increases with an increase in the molecular weight of the polymer, and ΔC_p measured at a fixed heating rate decreases. The increase in T_g with molecular weight is remarkably more rapid for poly(propylene glycols) than for other polymers, and a limiting value of T_g is reached for a chain containing 20 monomer units. These results are discussed in terms of the Fox-Flory and the entropy theories. The calorimetric relaxation times are comparable with the extrapolated dielectric relaxation times. The initial increase of ΔC_p from PG to PPG 200 is attributed to the decrease of H-bonding sites from 12 in 3 monomers to 4 on polymerization to PPG 200 and further decrease with increase in molecular weight to an increasingly large amplitude of the β-process at T < T_g.     

Inequalities for Mean Circumscribing Simplices

The aim of this note is to strengthen and generalize an inequality of Sangwine–Yager regarding means of various quantities associated with the simplices circumscribing a convex body.     

On a generalization of Wick-powers for generalized free-fields in terms of Jacobi-fields

Within the frame of Jacobi-fields a generalization of Wick-powers of generalized free fields is proposed. The key notion is that of a (generalized) contraction map. Those contraction maps F which yield a relativistic quantum field A^F are characterized. Using some simplifying assumptions the general form of a contraction map F which yields a relativistic quantum field A^F is determined. Furthermore, those contraction maps F are characterized for which A^F is in the Borchers class of the generalized free field A we start with. The Wick-product of generalized free fields appears as a particular example of this construction.