On sumsets of convex sets

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i - a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5} |A|.Comment: 6 page

Absence of simulation evidence for critical depletion in slit-pores

Recent Monte Carlo simulation studies of a Lennard-Jones fluid confined to a mesoscopic slit-pore have reported evidence for ``critical depletion'' in the pore local number density near the liquid-vapour critical point. In this note we demonstrate that the observed depletion effect is in fact a simulation artifact arising from small systematic errors associated with the use of long range corrections for the potential truncation. Owing to the large near-critical compressibility, these errors lead to significant changes in the pore local number density. We suggest ways of avoiding similar problems in future studies of confined fluids.Comment: 4 pages Revtex. Submitted to J. Chem. Phy

Entropy-driven enhanced self-diffusion in confined reentrant supernematics

We present a molecular dynamics study of reentrant nematic phases using the Gay-Berne-Kihara model of a liquid crystal in nanoconfinement. At densities above those characteristic of smectic A phases, reentrant nematic phases form that are characterized by a large value of the nematic order parameter $S\simeq1$. Along the nematic director these "supernematic" phases exhibit a remarkably high self-diffusivity which exceeds that for ordinary, lower-density nematic phases by an order of magnitude. Enhancement of self-diffusivity is attributed to a decrease of rotational configurational entropy in confinement. Recent developments in the pulsed field gradient NMR technique are shown to provide favorable conditions for an experimental confirmation of our simulations.Comment: 10 pages, 5 figure

Existence, Regularity, and Properties of Generalized Apparent Horizons

We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.Comment: 16 pages, thoroughly revised, no major changes, to appear in Comm. Math. Phy

The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those of R. Schoen and S.-T. Yau. In recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a different proof of the full positive mass theorem in dimensions $3 \leq n < 8$. We pointed out that this theorem can alternatively be derived from our density argument and the positive energy theorem of the present paper.Comment: All comments welcome! Final version to appear in Comm. Math. Phy