We extend the log 2 (N; m; P ) network proposed by Shyy and Lea to base k. We give a unifying proof (instead of three separate cases as done by Shyy and Lea) for the condition of being strictly nonblocking, and a simpler expression of the result. We compare the number of crosspoints for log k (N; m; p) over various k.

In this paper, we prove Dirichlet’s theorem that, given any pair h, k with (h, k) = 1, there are infinitely many prime numbers congruent to h (mod k). Although this theorem lies strictly within the realm of number theory, its proof employs a range of tools from other branches of mathematics, most notably characters from group theory and holomorphic functions from complex analysis.

A minor-closed class of matroids is (strongly) fractal if the number n-element in dominated by excluded minors. We conjecture that when K an infinite field, K-representable strongly fractal. prove sparse paving with at most k circuit-hyperplanes a least three. The minor-closure spikes (with k≥5) satisfies strictly weaker condition: 2t-element However, there are only finitely many minors ground ...

We study the interplay between canonical heights and endomorphisms of an abelian variety A over a number field k. In particular we show that whenever the ring of endomorphisms defined over k is strictly larger than Z there will be Q-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of A(k).1

A family F of k-graphs is called non-principal if its Turán density is strictly smaller than that of each individual member. For each k ≥ 3 we find two (explicit) k-graphs F and G such that {F,G} is ∗This author is partially supported by NSF grant DMS-0400812.