PQ axiverse

String Axiverse

String theory suggests the simultaneous presence of many ultralight axions, possibly populating each decade of mass down to the Hubble scale 10−33eV. Conversely the presence of such a plenitude of axions (an “axiverse”) would be evidence for string theory, since it arises due to the topological complexity of the extra-dimensional manifold and is ad hoc in a theory with just the four familiar di...

Factoring N = pq

We discuss the problem of factoring N = pq and survey some approaches. We then present a specialized factoring algorithm that runs in time Õ(q0.31), which is comparable to the runtime Õ(p) of the factoring algorithm for integers of the form N = pq presented in [1]. We then survey the factoring algorithm of [1] and discuss the number of advice bits needed for it to run in polynomial time. Furthe...

The Dwarf Nova Pq Andromedae

Planarity Algorithms via PQ-Trees

We give a linear-time planarity test that unifies and simplifies the algorithms of Shih and Hsu and Boyer and Myrvold; in our view, these algorithms are really one algorithm with different implementations. This leads to a short and direct proof of correctness without the use of Kuratowski’s theorem. Our planarity test extends to give a uniform random embedding, to count embeddings, to represent...

Computing Surfaces via pq-Permutations

In algebraic topology, compact 2-dimensional manifolds are usually dealt through a well-defined class of words denoting polygonal presentations. In this paper, we show how to eliminate the useless bureaucracy intrinsic to word-based presentations by considering very simple combinatorial structures called pq-permutations. Thanks to their effectiveness, pq-permutations allow to define a rewriting...