The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p r with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p and rank r. The q vectors of a basis of Hq co...

In the ORTHOGONAL VECTORS (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. TheOV Conjecture (OVC) posits that OV requires n time to solve, for all d = ω(logn). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and ...

A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which charact...