The critical group from a cryptographic perspective

The critical group of a graph is an abelian group that arisesin several contexts, and there are some similarities with thegroups that are used in cryptography. We construct a familyof graphs with critical groups that are cyclic, and discussthe associated computational problems using algorithms basedon the theory of ‘chip-firing’.     

Specht modules and chromatic polynomials

An explicit formula for the chromatic polynomials of certain families of graphs, called bracelets', is obtained. The terms correspond to irreducible representations of symmetric groups. The theory is developed using the standard bases for the Specht modules of representation theory, and leads to an effective means of calculation.     

A procedure for measuring the flexibility of single wood-pulp fibres

A way of determining the flexibility of wood-pulp fibres is developed, which involves i) a precise measurement of the topology of single-fibres by using a confocal laser scanning microscope and ii) the measurement of the elastic modulus of the fibres by using a single-fibre fatigue cell. Reported in this paper are the initial results of tests carried out on black spruce fibres, which have been subjected to three different levels of mechanical refining energy, namely ∼1100, 2300, and 3500 kWh/t. It is found that the fibre flexibility rises significantly between the first and second energy levels, but it does not change to the same degree between the second and third ones. The described procedure of measuring the flexibility of fibres may be used to establish the appropriate refiner energy necessary for the production of a specific grade of paper. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 1, pp. 115–128, January–February, 2006.     

Viscosity effect on the structural compactness of latex flocs formed under weak depletion attractions

Dilute aqueous dispersions of colloidal polystyrene latex spheres were flocculated by adding a nonadsorbing polymer sample, poly(acrylic acid). The structural compactness of the flocs thus formed was characterized in terms of their mass fractal dimension using the small-angle static light scattering technique. It was found that with low poly(acrylic acid) concentrations and thus weak depletion attraction forces, the dispersion medium viscosity had a marked effect on the floc structure. An increase in the viscosity led to formation of denser flocs. This was revealed in three sets of depletion flocculation experiments: (a) adjusting the background electrolyte concentration at a fixed level of poly(acrylic acid), (b) using water and 30% (w/w) glycerol as the respective solvents, and (c) inducing latex flocculation with two poly(acrylic acids) of different molecular weights at the respective critical polyacid concentrations. Direct force measurements were made with atomic force microscopy to isolate the influence of viscosity on floc structure from that of interparticle interaction energies. We conclude that the formation of denser flocs with increasing medium viscosity can be attributed to the reduced diffusivity of particles in the solution. The latter resulted in an enhanced rate of floc restructuring (through relaxation of attached particles) relative to floc growth.     

Effect of temperature on the mechanism of retention of fullerenes in liquid chromatography using various alkyl bonded stationary phases

Summary The temperature-dependency of the separation of fullerenes in liquid chromatography (LC) has been examined using various alkyl bonded stationary phases. It has been found that a maximum retention temperature exists with long alkyl bonded stationary phases, whereas there is no similar effect with the newly synthesized alkyl bonded phases which have two phenyl groups at the base of the bonded phase. The interpretation of the retention behavior of fullerenes in the low temperature region on alkyl bonded stationary phases is discussed using information obtained by CP-MAS solid-state NMR spectroscopy and LC.     

Poly(styrene-b-2-vinylpyridine-1-oxide) diblock copolymers: 3. Microemulsions in water/propan-2-ol/toluene mixtures

Pseudo-ternary phase diagrams have been constructed for the three-component solvent system (toluene+water+propan-2-ol) containing diblock copolymers of poly(styrene-b-2-vinylpyridine-1-oxide). Microemulsions have been shown to form on the water-rich side of the phase diagram, in the region of the phase boundary without polymer. Dynamic light-scattering experiments have led to droplet size values in the region of 100 nm, with the size depending strongly on the propan-2-ol/water concentration, as well as the amount of solubilised toluene in the core. Viscometry experiments have been carried out to measure polymer aggregation numbers in the microemulsion droplets, and interfacial tension measurements have shown that in the absence of propan-2-ol (effectively a cosurfactant) the limiting value of the oil/water interfacial tension, even an saturation adsorption of the copolymer is 20 mNm^–1. However, addition of propan-2-ol reduces the interfacial tension to the very low values generall commensurate with microemulsion formation.     

Molecular shape recognition capability of liquid-crystal bonded phases in reversed-phase high performance liquid chromatography

Summary The chromatographic retention behaviour of two liquidcrystal bonded phases have been evaluated using polycyclic aromatic hydrocarbons (PAHs) as the probe samples in reversed-phase high performance liquid chromatography (RP-HPLC). The results clearly indicate that these phases have better planarity and shape recognition capabilities than commercially-avaialble polymeric octadecylsilica (ODS) phases whose strong planarity and shape selectivities were found earlier. It can also be concluded from the chromatographic observations that the shape recognition capability of these phases is dependent on both mobile phase composition and column temperature, but that the effect of mobile phase and temperature on the shape selectivity work independently. The retention behaviour can be explained by changes in the phase structure with changes of eluent composition and temperature.     

Learning Mathematics: a Critical Appraisal of the Developments of the Past Twelve Years—with Special Reference to the Primary Stage

In this country major changes in the infant stage of education began a long time ago and have gradually spread right through the primary school. In this article a discussion of some specific investigations that children can profitably follow, and their mathematical implications, are presented.