A molecular mechanics force field for the cobalt corrinoids

A force field for the cobalt (III) corrinoids (derivatives of vitamin B₁₂) for use with a modified version of the molecular mechanics program 2(87) has been developed empirically around 19 cobalt corrinoid crystal structures. Bond lengths, bond angles and torsional angles are reproduced with r.m.s. differences of 0.01 Å, 2.4 °, and 4.2 °, respectively, within the standard deviation of the mean of these parameters found in the solid state. The axial ligand occupying the lower coordination site in the cobalamins, 5,6-dimethylbenzimidazole, is shown to have very limited rotational freedom and is constrained by the downward-pointing b and d propionamide side chains of the corrin ring. Strain-energy profiles for rotation of the side chains of the corrin ring show the existence of several local energy minima and this explains the observed variability in the orientations of these side chains in the solid state. The known change in conformation which occurs in the C ring when the e side chain is epimerized from the lower to the upper face of the corrin ring in cyano-13-epicobalamin is correctly predicted, provided the starting conformation of the C ring is unbiased. A study of cyano-8-epicobalamin indicates that an analogous conformational change does not occur in the B ring and the epimerized d side chain assumes an equatorial orientation relative to the corrin ring. Parameters for the Co---C bond in alkylcobalamins were developed and the structure of methyl- and adenosylcobalamin are accurately reproduced. An examination of the strain energy consequences of rotation of the adenosyl ligand about the Co---C bond identifies a number of low-energy conformations at least two of which, in which adenosyl lies over the “southern” and “eastern” portions of the corrin ring, respectively, have been previously deduced from NMR observations. Coordinated neopentyl in neopentylcobalamin is much more hindered to rotation about the Co---C bond and the lowest conformation finds two γ(C) atoms straddling the upwardly projecting C46 methyl group of the corrin.     

Lattice paths in regions with the catalan property

If α = {α₀, α₁,…, α_n} and β = {β₀, β₁,…, β_n} are two non-decreasing sets of integers such that α₀ = 0 < β₀, α_n < β_n = n, and α_i < i < β_i for 1 ? i ? n ? 1, let L denote the set of lattice points (p, q) such that 0 ? p ? n and α_p ? q ? β_p. We determine all such regions L with the property that the number of lattice paths from (0, 0) to (p, p) in L is the Catalan number$\text{(p + 2)}{}^{\mathrm{?1}}\text{(}\text{2p+2}\text{p+1}\text{)}$ for 0 ? p ? n.     

Dispersion in annular climbing film flow

The dispersion of a tracer injected as a pulse into a climbing liquid film is investigated for a series of water and air flow rates, and for a number of different electrolyte tracers. It is found that at all flow rates the observed concentration distribution depends on the nature of the tracer. This observation is explained in terms of two effects: molecular diffusion in a viscous sub-layer and ion fractionation associated with droplet formation at the gas-liquid interface. The overall dispersive characteristics of the system are described in terms of a mathematical model assuming dispersed plug flow in both the film and entrained droplets with interchange between these phases. This model is fitted to experimental tracer concentration distributions using a non-linear least-squares regression procedure. The parameter values obtained from the fitting procedure are studied to determine trends with flow rates and tracer properties. Values for a film dispersion parameter, $\text{P}{}_{\mathrm{f}}$, are found to correlate significantly with the molecular diffusion coefficients of the tracers. Consistent values for an ion fractionation coefficient, $\text{k}{}_{\mathrm{if}}$, are also obtained.     

A mathematical model of an anthrax epizoötic in the Kruger National Park

An outbreak of anthrax in a single homogeneous geographical region is modelled. The spread of the disease to neigbouring regions is also simulated and the model is compared with data from the 1970 epizoötic in the Kruger National Park.Two possible driving mechanisms for the epizoötic are proposed and examined, viz. environmental contamination or direct contact between vulnerable animals and fresh carcases. In the latter case, it is shown that the model exhibits a threshold effect, in that an epizoötic will only arise if the vulnerable population exceeds a threshold value which is determined in terms of one of the parameters of the model.     

On nonlinear nuclear operators

A class of operators is introduced and referred to as nonlinear nuclear operators. As in the case of linear nuclear operators this class of nonlinear operators is defined by means of a tensor product of two topological vector spaces. It is then shown that, as in the linear case, a series representation of the operator is valid. The deviation of the nonlinear theory from the linear case is discussed and an example in the framework of generalized stochastic processes is given.     

Approximation and the spectral multiplicity of special automorphisms

Summary Conditions have been given [7] under which a special automorphism over an automorphism admitting a simple approximation again admits a simple approximation, and so has simple spectrum.In this paper using different techniques to those employed in [7], we obtain improved results in the same direction. Specifically, conditions are given for a special automorphism over an automorphism, which admits either a simple approximation, or an approximation with suitable speed, to have bounded spectral multiplicity. Furthermore, we obtain as a corollary a result on primitive automorphisms, which partially generalises a result appearing in [4].     

Polystyrene impregnated with beta-diphenylglyoxime, a selective reagent for palladium

alpha- and beta-diphenylglyoximes impregnated on macroreticular polystyrene (XAD-2) were tested as adsorptive reagents and one of them (the beta-isomer) was found to adsorb palladium in preference to platinum (selectivity factor 20) and nickel (factor > 100).     

Affine invariant convergence results for Newton's method

We present a unified derivation of affine invariant convergence results for Newton's method. Initially we derive affine invariant forms of the perturbation lemma and a mean value theorem. With their aid we obtain an optimal radius of convergence for Newton's method, from which further radius of convergence estimates follow. From the Newton-Kantorovitch theorem we derive other estimates of the radius of convergence. We discuss estimation of the parameters in the expressions we have derived.     

The use of threshold energy differences in fast neutron activation analysis for oxygen and fluorine

A method is described for the simultaneous determination of oxygen and fluorine; the latter has a lower neutron-activation threshold energy so that the fluorine-to-oxygen¹⁶N formation ratio changes after partial moderation of the incident 14.7 MeV neutron beam. In mixtures containing approximately equal amounts of fluorine and oxygen, these elements can each be determined with a precision of ca. ±1%; as expected, errors in the determination of a minor component (ca. 17%) are larger, being ca. ±5%.