Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥ d2 − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d+ 1)2/2 for degrees d = 2q − 1 where q is an odd prime p...

If G is a finitely generated profinite group then the verbal subgroup Gq is open. In a d-generator finite group every product of qth powers is a product of f(d, q) qth powers. 20E20, 20F20.

Legendre's chi-function, Xn(z) = 2"D¡tlo zlk+l/(2k + 1)" , is reexpanded in a power series in powers of log z . The expansion obtained is well suited for the computation of Xn(z) in the two cases of real z close to 1, and z = ela , a e R . For n = 2 and n = 3 , the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses...

The K-length of a form f in K[x1, . . . , xn], K ⊂ C, is the smallest number of d-th powers of linear forms of which f is a K-linear combination. We present many results, old and new, about K-length, mainly in n = 2, and often about the length of the same form over different fields. For example, the K-length of 3x − 20xy + 10xy is three for K = Q( √ −1), four for K = Q( √ −2) and five for K = R.

A detailed investigation of the scaling properties of the fully finite O(n) systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like r(-d-sigma), below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an ...