Let the field F contain all /?-power roots of unity for some prime p and suppose that the absolute Galois group G of F is a one-relator pro-p group. We use Merkurjev-Suslin's theorem on the power norm residue symbol to show that G is an extension of a Demushkin group by a free pro-/) group. Let F be a field, Fs its separable closure, and G = Ga\(Fs/F) its absolute Galois group. Let n be an inte...

-Some new oscillation criteria for the even order damped functional differential equation (a(t)z(n-1)(t)) ' ÷ p(t)lz(n-1)(Ol#x(n-a)(O ÷ q(t)f(xIaa(t)] . . . . . =Lq.~(t)]) = 0 are established, where ~ _> O. These criteria are an extension of some of the known results. 1. I N T R O D U C T I O N Recently, Grace and Lalli [1] discussed the oscillation of the nth-order functional differential equa...

The nth detour chromatic number, χn(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ (G) . We conjecture that χn(G) ≤ d τ(G) n e for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some suf...

Consider a sequence of N x N random nonnegative matrices in which each element depends on a vector u of parameters. The nth partial product is the random matrix formed by multiplying, from right to left, the first n of these random matrices in order. Under certain conditions, the elements of the nth partial product grow asymptotically exponentially as n increases, and the logarithms of the disc...

We study some properties of the Askey-Wilson polynomials (AWP) when q is a primitive Nth root of unity. For general four-parameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval i...