Area-Preserving Surface Diffeomorphisms

We prove some generic properties for $C^r$, $r=1, 2, ..., \infty$, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez \cite{FL03} on $S^2$ to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem

Testing surface area with arbitrary accuracy

Recently, Kothari et al.\ gave an algorithm for testing the surface area of an arbitrary set $A \subset [0, 1]^n$. Specifically, they gave a randomized algorithm such that if $A$'s surface area is less than $S$ then the algorithm will accept with high probability, and if the algorithm accepts with high probability then there is some perturbation of $A$ with surface area at most $\kappa_n S$. Here, $\kappa_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/\pi$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $\kappa_n$ with $1 + \eta$ for $\eta > 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.Comment: 5 page

A Simple Technique for Surface Area Determination Through Supercritical Co2 Adsorption

The measurement of specific surface area of porous materials has long been important in physical sciences and iscurrently growing in importance in applied and environmental science. Numerous systems have been developed forsurface area measurement by gas adsorption. Commercial systems are available which can measure a wide range ofabsolute surface area with relative ease. However, their cost is often prohibitive. In this study, an inexpensive apparatusfor surface area measurement has been set up to be used for measuring supercritical adsorption of CO2. The Ono-KondoLattice model was used to represent the adsorption isotherm and to determine the surface area. The results of surfacearea determination using CO2 adsorption combined with OK model have been compared to the numbers obtained fromnitrogen BET method. For surface area determination of zeolites and activated carbons, the new method give reasonableagreement results (within 10% deviation) compared to the results obtained from nitrogen BET method. In addition, thenew method also gives more reasonable results for surface area determination of coals. As known, the nitrogen BETmethod gives almost zero of coals’ surface area. This might due to the characteristic of the coals’ structure that might bechange (the pores are closed) during the cooling process in nitrogen BET method. Moreover, the new method can alsobe used to determine the surface area of porous materials using CO2 adsorption data at various temperatures withoutsacrificing their accuracy