Volume growth, eigenvalue and compactness for self-shrinkers

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in \ir{3} obtained by Colding-Minicozzi under weaker conditions.Comment: 17 page

Filtration and transport of heavy metals in graphene oxide enabled sand columns

A fixed-bed sand column with graphene oxide (GO) layer was used to remove heavy metals (Cu(II) and Pb(II)) from an aqueous solution injected under steady flow. Due to the time constrained kinetic process of heavy metal sorption to GO, removal efficiency was affected by the injection flow rate. When injection flow rate changed from 1 to 5 mL min−1, the removal efficiency of the two metals decreased from 15.3% to 10.3% and from 26.7% to 19.0% for Cu(II) and Pb(II), respectively. Provided a fixed concentration of heavy metals in the injected flow, an increase in GO in column from 10 to 30 mg resulted in an sharp increase in the removal efficiency of Pb(II) from 26.7% to 40.5%. When Cu(II) and Pb(II) were applied simultaneously, the removal efficiency of the two metals was lower than when applied by individually. GO-sand column performance was much better for the removal of Pb(II) than for Cu(II) in each corresponding treatment. When breakthrough curve (BTC) data were simulated by the convection-dispersion-reaction (CDER) model, the fittings for Cu in every treatment were better than that of Pb in corresponding treatment. Considering the small amount of GO used to enable the sand columns that resulted in a great increase in k value, compared to the GO-free sand columns, the authors propose GO as an effective adsorption media in filters and reactive barriers to remove Pb(II) from flowing water

Two Photon Transition Form Factor of cˉc\bar{c}c Quarkonia

The two photon transition of $\bar{c}c$ quarkonia are studied within a covariant approach based on the consistent truncation scheme of the quantum chromodynamics Dyson-Schwinger equation for the quark propagator and the Bethe--Salpeter equation for the mesons. We find the decay widths of $\eta_{c}^{} \to \gamma\gamma$ and $\chi_{c0,2}^{} \to \gamma\gamma$ in good agreement with experimental data. The obtained transition form factor of $\eta_{c}^{} \to \gamma\gamma^{\ast}$ for a wide range of space-like photon momentum transfer squared is also in agreement with the experimental findings of the BABAR experiment. As a by-product, the decay widths of $\eta_{b}^{},\chi_{b0,2}^{} \to \gamma\gamma$ and the transition form factor of $\eta_{b}^{}, \chi_{c0,b0}^{} \to\gamma\gamma^{\ast}$ are predicted, which await for experimental test

Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.Comment: 31 pages. Comments are welcome