An exact method for determining the critical percolation probability, pc, for a number of two-dimensional site and bond problems is described. For the site problem on the plane triangular lattice pc = ½. For the bond problem on the triangular, simple quadratic, and honeycomb lattices, pc=2 sin (118π),12,1−2 sin (118π), respectively. A matching theorem for the mean number of finite clusters on certain two-dimensional lattices, somewhat analogous to the duality transformation for the partition function of the Ising model, is described.

Si may play an important role in bone formation and connective tissue metabolism. Although biological interest in this element has recently increased, limited literature exists on the Si content of foods. To further our knowledge and understanding of the relationship between dietary Si and human health, a reliable food composition database, relevant for the UK population, is required. A total of 207 foods and beverages, commonly consumed in the UK, were analysed for Si content. Composite samples were analysed using inductively coupled plasma-optical emission spectrometry following microwave-assisted digestion with nitric acid and H 2 O 2 . The highest concentrations of Si were found in cereals and cereal products, especially less refined cereals and oat-based products. Fruit and vegetables were highly variable sources of Si with substantial amounts present in Kenyan beans, French beans, runner beans, spinach, dried fruit, bananas and red lentils, but undetectable amounts in tomatoes, oranges and onions. Of the beverages, beer, a macerated whole-grain cereal product, contained the greatest level of Si, whilst drinking water was a variable source with some mineral waters relatively high in Si. The present study provides a provisional database for the Si content of UK foods, which will allow the estimation of dietary intakes of Si in the UK population and investigation into the role of dietary Si in human health.

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The relationship between the excluded-volume problem for a discrete random walk on a lattice and the corresponding Ising model of ferromagnetism is investigated. Systematic methods are presented for the construction of rigorous lower bounds to the limit ;u=lim ri^0 o(c n+ i/c n ), where c n is the number of w-step self-avoiding walks on a given lattice. In this way Temperley's conjecture that n=coth(J/kTc),where Tc is the Curie temperature of the corresponding Ising-model ferromagnet, is disproved. The series c n for various two-and three-dimensional lattices have been enumerated exactly for values of n from ten to twenty. Extrapolation of these series, by procedures known to be valid from exact Ising-model results, yields more accurate values of ju than Wall's statistical calculations and also shows that c n r^j n 0l yiP' where a-1/3 for plane lattices and a-1/7 for three-dimensional lattices. This means that the entropy of the wth "link" of a polymer molecule in solution should vary as SSn -k lny,-}-ket/n. The relevance of these results to the interpretation of the boundary tension of the Ising model, to the critical behavior of gases, and to the mean square size of a polymer molecule is discussed briefly. 7