The sign-rank of a matrix A with entries in {−1,+1} is the least rank of a real matrix B with Aij · Bij > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F (x, y)]x,y for a specific function F : {−1, 1} × {−1, 1} → {−...

The purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a Hilbert space is expressible as a harmonic matrix-valued function. We show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix-valued function. We also prove that for any natural number, n, a harmonic function with values in the ra...

This paper investigates the arithmetic of fractional ideals and the infrastructure of the principal ideal class of a purely cubic function eld of unit rank one. We rst describe how irreducible polynomials split into prime ideals in purely cubic function elds of nonzero unit rank. This decomposition behavior is used to compute so-called canonical bases of fractional ideals; such bases are very s...