Differential Invariants of Conformal and Projective Surfaces

We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Complexity and growth for polygonal billiards

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.Comment: 12 pages, 4 figure

Roughening and preroughening in the six vertex model with an extended range of interaction

We study the phase diagram of the BCSOS model with an extended interaction range using transfer matrix techniques, pertaining to the (100) surface of single component fcc and bcc crystals. The model shows a 2x2 reconstructed phase and a disordered flat phase. The deconstruction transition between these phases merges with a Kosterlitz-Thouless line, showing an interplay of Ising and Gaussian degrees of freedom. As in studies of the fully frustrated XY model, exponents deviating from Ising are found. We conjecture that tri-critical Ising behavior may be a possible explanation for the non-Ising exponents found in those models.Comment: 25 pages in RevTeX 3.0, seven uuencoded postscript figures, REPLACED because of submission error (figures were not included

Is surface melting a surface phase transition?

Monte Carlo or Molecular Dynamics calculations of surfaces of Lennard-Jones systems often indicate, apart from a gradual disordering of the surface called surface melting, the presence of a phase transition at the surface, but cannot determine the nature of the transition. In the present paper, we provide for a link between the continuous Lennard-Jones system and a lattice model. We apply the method for the (001) surface of a Lennard-Jones fcc structure pertaining to Argon. The corresponding lattice model is a Body Centered Solid on Solid model with an extended range of interaction, showing in principle rough, flat and disordered flat phases. We observe that entropy effects considerably lower the strength of the effective couplings between the atoms. The Argon (001) face is shown to exhibit a phase transition at T=70.5 +- 0.5 K, and we identify this transition as roughening. The roughening temperature is in good correspondence with experimental results for Argon.Comment: 17 pages REVTeX, 14 uuencoded postscript figures appende