Superlinear advantage for exact quantum algorithms

A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is: how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms. For total Boolean functions in the query model, the biggest known gap was just a factor of 2: PARITY of N inputs bits requires $N$ queries classically but can be computed with N/2 queries by an exact quantum algorithm. We present the first example of a Boolean function f(x_1, ..., x_N) for which exact quantum algorithms have superlinear advantage over the deterministic algorithms. Any deterministic algorithm that computes our function must use N queries but an exact quantum algorithm can compute it with O(N^{0.8675...}) queries.Comment: 20 pages, v6: small number of small correction

The synoptic problem: on Matthew's and Luke's use of Mark

In New Testament studies, the synoptic problem is concerned with the relationships between the gospels of Matthew, Mark and Luke. Assuming Markan priority, we investigate the relationship between the words in Mark that are retained unchanged by Matthew and those that are retained unchanged by Luke. This is done by mapping the sequence of words in Mark into binary time series that represent the retention or non-retention of the individual words, and then carrying out a variety of logistic regression analyses