On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

The solution of a convection–diffusion–reaction equation arising from cathodic reduction in a pulsating crack

This paper derives the convection–diffusion-reaction equation governing the reaction between the dissolved oxygen in sea-water and the steel walls of a pulsating crack. By the neglect of the diffusion term it is shown that an exact solution of the convection-reaction equation can be obtained. A numerical method for the solution of the complete convection–diffusion-reaction equation is derived by the use of finite differences. The numerical computation of the initial transient and the final periodic steady-state values is also discussed.     

Liquid crystalline hexa-amides and side chain polysiloxanes of azacrown [18]-N6

The hexa-4-dodecyloxybenzoyl derivative 1 of azacrown [18]-N₆ was originally reported to have a 'tubular' mesophase on the basis of its large central ring and 6-fold symmetry. Starting in the mesophase, annealing of 1 under a cover slip results in formation of a new crystalline phase that melts directly to an isotropic liquid at the temperature previously observed for the mesophase to isotropic transition. Thus the phase behaviour of 1 is kinetically controlled. The analogous hexa-3,4-bisdodecyloxybenzoyl derivative 2 of azacrown [18]-N₆ has no kinetic limitations to its phase changes and has an enantiotropic columnar liquid crystalline phase. We have synthesized side chain copolysiloxanes with a (CH₂)₁₁ spacer and 75-84 per cent by weight of the same 4-dodecyloxybenzoyl-[18]-N₆ mesogen. The polysiloxanes also display a liquid crystalline phase.     

Dye laser excitation of the fluorosulfate radical. Totally symmetric upper state fundamentals

Upper state vibronic levels associated with totally symmetric fundamentals of the fluorosulfate radical were populated by means of dye laser excitation. The analysis of the resulting fluorescence confirmed the assignment of these levels. Theoretical intensities using the Green's function approach to vibronic coupling were calculated and compared to the experimental values.     

The magnetic susceptibilities of cobaltocene and chromocene

The magnetic susceptibilities of cobaltocene and chromocene have been measured between 83 and 293 K; for cobaltocene the results suggest an appreciable orthorhombic splitting of the ²∏(σ²πδ⁴ ground state, whilst for chromocene the data provide further support for the ³Δ(σδ³) ground state previously deduced.     

Investigation of novel mediators for a glucose biosensor based on metal picolinate complexes

The metal complexes [Os(byp)(2)(pic)](+) and [Ru(byp)(2)(pic)](+) where byp is 2,2'-bipyridine and HPic is o-picolinic acid were synthesised and characterised using spectroscopic and electrochemical techniques. These complexes were then evaluated as mediators for a glucose oxidase (GOx)-based biosensor. Results demonstrate the electrocatalytic behaviour of both metal couples towards regeneration of the flavoprotein GOx (FADH(2)) group, when co-immobilised with glucose oxidase. Surface immobilisation was achieved by potential cycling in aqueous solutions of the metal complexes at a glucose oxidase (GOx)/Nafion modified electrode. This proved successful in terms of catalytic efficiency and stability of redox sites. Kinetic parameters associated with both enzymatic and mediator reactions were estimated and the stability/performance properties of the sensor were tested.     

SCF-π-electron calculations using orthogonalised atomic orbitals

Modifications of the SCF-LCAO-π-MO method analysed in the previous paper are described in which provision is made for the incorporation of Variable Bond Order and Variable Electronegativity procedures. A comparison is made with the results of other π-electron calculations and values are reported for twenty hydrocarbon systems.     

Synthesis with migrating phenylthio groups: connective routes to 2-phenylthioallyl alcohols and 2-phenylthio enones

The title compounds may be made from bis(phenylthio) acetals and aldehydes by routes involving PhS migration.     

The role of angular momentum in collision-induced vibration-rotation relaxation in polyatomics

Vibrational relaxation of the 6(1) level of S(1)((1)B(2u)) benzene is analyzed using the angular momentum model of inelastic processes. Momentum-(rotational) angular momentum diagrams illustrate energetic and angular momentum constraints on the disposal of released energy and the effect of collision partner on resultant benzene rotational excitation. A kinematic "equivalent rotor" model is introduced that allows quantitative prediction of rotational distributions from inelastic collisions in polyatomic molecules. The method was tested by predicting K-state distributions in glyoxal-Ne as well as J-state distributions in rotationally inelastic acetylene-He collisions before being used to predict J and K distributions from vibrational relaxation of 6(1) benzene by H(2), D(2), and CH(4). Diagrammatic methods and calculations illustrate changes resulting from simultaneous collision partner excitation, a particularly effective mechanism in p-H(2) where some 70% of the available 6(1)-->0(0) energy may be disposed into 0-->2 rotation. These results support the explanation for branching ratios in 6(1)-->0(0) relaxation given by Waclawik and Lawrance and the absence of this pathway for monatomic partners. Collision-induced vibrational relaxation in molecules represents competition between the magnitude of the energy gap of a potential transition and the ability of the colliding species to generate the angular momentum (rotational and orbital) needed for the transition to proceed. Transition probability falls rapidly as DeltaJ increases and for a given molecule-collision partner pair will provide a limit to the gap that may be bridged. Energy constraints increase as collision partner mass increases, an effect that is amplified when J(i)>0. Large energy gaps are most effectively bridged using light collision partners. For efficient vibrational relaxation in polyatomics an additional requirement is that the molecular motion of the mode must be capable of generating molecular rotation on contact with the collision partner in order to meet the angular momentum requirements. We postulate that this may account for some of the striking propensities that characterize polyatomic energy transfer.