Slowly oscillating functions and a generalization of quasi-monotone coefficients

We first state a conjecture due to Yong concerning equality of two classes of functions, and then give examples to disprove the conjecture. Later we extend some theorems concerning integrability of Fourier series.     

Crystallization of polyethylene at elevated pressures

The crystallization from the melt of three sharp polyethylene fractions has been studied at 5 kbar. It has been shown that the thickness of so-called extended-chain lamellae is a function of time, temperature, and molecular weight. There is by no means just the fully extended molecular configuration present. Crystallization is qualitatively similar to that of chain-folded crystals at 1 bar, giving an optimum lamellar thickness which increases with time and decreasing supercooling. Fractional crystallization is widespread and is a major cause of disparate lamellar thickness. Isothermal thickening of lamellae during crystallization has been established directly. Morphological detail suggests further that layers can increase their thickness tenfold over their initial size.     

The numerical integration of a molecular integral using two different techniques

A two-electron integral which commonly occurs in molecular calculations is evaluated numerically using the different methods of Boys and Conroy and the results are discussed.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

Synthesis of azaspirocyclic compounds via organoiron complexes. Potential synthetic routes to histrionicotoxin and cephalotaxus alkaloids.

Reaction of cyclohexadienylium-Fe(CO)₃ complexes (7) and (8) with sodiomalono-nitrile gives the carbo-spirocyclic derivatives (9) and (10), whilst reaction with benzylamine and removal of iron leads to the azaspirocyclic enones (15) and (16), respectively.     

Multivalued groups, their representations and Hopf algebras

This paper introduces the concept ofn-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements. However, there are many examples that do not arise from this construction. We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra. In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group. Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group.