The Fa,b,c conjecture, I

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and one part of the conjecture was proved. The conjecture completely determines all finite members of the family. Here we prove another part of the conjecture and foreshadow a paper which completes the proof of the other three parts.

The Fa,b,c conjecture is true, II

In 1977 a five-part conjecture was made about a family of groups related to trivalent graphs and subsequently two parts of the conjecture were proved. The conjecture completely determines all finite members of the family. Here we complete the proof of the conjecture by giving proofs for the remaining three parts. (c) 2006 Elsevier Inc. All rights reserved.

The critical dietary potassium concentration for induction of mineral disorders in non-lactating Welsh Mountain sheep

Diets with more than 30 g K/kg DM have previously been associated with hypomagnesaemia in grazing cattle, and to test whether such diets lead to mineral disorders in sheep, the absorption of Mg and other elements was investigated using experimental diets to which KC I had been added to provide 27, 29, 32 or 34 g K/kg DM. The apparent absorption, balance and apparent retention of Mg, and to a lesser extent Ca, were reduced for sheep offered the diets with 32 or 34 g K/kg DM. The absorption and retention of K, Na, P, Zn, Pb and Cd was not affected by treatment. The blood intracellular Ca concentration was reduced by the diets with 29, 32 or 34 g K/kg DM, compared to the diet with 27 g K/kg DM, but the concentration of other elements was unaffected. Blood plasma Ca concentration was increased at the highest level of K inclusion, providing evidence of mild hyperkalaemia and the involvement of Ca homeostatic mechanisms. It is concluded that Mg absorption by sheep will be impaired if the diet contains more than 30 g K/kg DM, equivalent to an intake of approximately 13 g K/d, but that a high K diet may be beneficial before parturition to accustom the sheep to Ca mobilization before lactation. (c) 2005 Elsevier B.V. All rights reserved.