Consensus with Max Registers

We consider the problem of implementing randomized wait-free consensus from max registers under the assumption of an oblivious adversary. We show that max registers solve m-valued consensus for arbitrary m in expected O(log^* n) steps per process, beating the Omega(log m/log log m) lower bound for ordinary registers when m is large and the best previously known O(log log n) upper bound when m is small. A simple max-register implementation based on double-collect snapshots translates this result into an O(n log n) expected step implementation of m-valued consensus from n single-writer registers, improving on the best previously-known bound of O(n log^2 n) for single-writer registers

Algorithmic Algebraic Geometry and Flux Vacua

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure

Exploring the Vacuum Geometry of N=1 Gauge Theories

Using techniques of algorithmic algebraic geometry, we present a new and efficient method for explicitly computing the vacuum space of N=1 gauge theories. We emphasize the importance of finding special geometric properties of these spaces in connecting phenomenology to guiding principles descending from high-energy physics. We exemplify the method by addressing various subsectors of the MSSM. In particular the geometry of the vacuum space of electroweak theory is described in detail, with and without right-handed neutrinos. We discuss the impact of our method on the search for evidence of underlying physics at a higher energy. Finally we describe how our results can be used to rule out certain top-down constructions of electroweak physics.Comment: 35 pages, 2 figures, LaTe