Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A

Gluon sivers and experimental considerations for TMDs

The study and characterisation of transverse-momentum-dependent distribution functions (TMDs) is a major goal of the Electron-Ion Collider (EIC) physics programme. The study of gluon TMDs poses a greater challenge than for quark TMDs in DIS measurements, as gluons do not directly couple to photons. The study of D meson pairs has been proposed to provide access to gluon TMDs, but is demanding due to the rarity of D production. Here, we discuss the feasibility of such a measurement, and touch upon wider issues to be considered when measuring TMDs at the EIC.Comment: 4 pages, 2 figures, DIS 2012 conferenc

Molecular Hydrogen in the Lagoon: H2 line emission from Messier 8

The 2.12 micron v=1-0 S(1) line of molecular hydrogen has been imaged in the Hourglass region of M8. The line is emitted from a roughly bipolar region, centred around the O7 star Herschel 36. The peak H2 1-0 S(1) line intensity is 8.2 x 10E-15 erg s-1 cm-2 arcsec-2. The line centre emission velocity varies from -25 kms in the SE lobe to +45 kms in the NW lobe. The distribution is similar to that of the CO J=3-2 line. The H2 line appears to be shock-excited when a bipolar outflow from Herschel 36 interacts with the ambient molecular cloud. The total luminosity of all H2 lines is estimated to be ~ 16 Lsun and the mass of the hot molecular gas ~9 x 10E-4 Msun (without any correction for extinction).Comment: 11 pages, 4 figures (1 in colour). Submitted to Publications of the Astronomical Society of Australia, December 200

Naive entropy of dynamical systems

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal $\Gamma$-system has zero naive entropy in both senses, if $\Gamma$ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.Comment: 19 page