On Bismut Flat Manifolds

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-K\"ahler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.Comment: In this 3rd version, we add a lemma on Hermitian surfaces with flat Riemannian connection. References are updated and typos correcte

On Frankl and Furedi's conjecture for 3-uniform hypergraphs

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi in \cite{FF} conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In this paper, we give some partial results for this conjecture.Comment: 19 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1211.650

Correlating Interlayer Spacing and Separation Capability of Graphene Oxide Membranes in Organic Solvents.

Membranes synthesized by stacking two-dimensional graphene oxide (GO) hold great promise for applications in organic solvent nanofiltration. However, the performance of a layer-stacked GO membrane in organic solvent nanofiltration can be significantly affected by its swelling and interlayer spacing, which have not been systematically characterized. In this study, the interlayer spacing of the layer-stacked GO membrane in different organic solvents was experimentally characterized by liquid-phase ellipsometry. To understand the swelling mechanism, the solubility parameters of GO were experimentally determined and used to mathematically predict the Hansen solubility distance between GO and solvents, which is found to be a good predictor for GO swelling and interlayer spacing. Solvents with a small solubility distance (e.g., dimethylformamide, N-methyl-2-pyrrolidone) tend to cause significant GO swelling, resulting in an interlayer spacing of up to 2.7 nm. Solvents with a solubility distance larger than 9.5 (e.g., ethanol, acetone, hexane, and toluene) only cause minor swelling and are thus able to maintain an interlayer spacing of around 1 nm. Correspondingly, GO membranes in solvents with a large solubility distance exhibit good separation performance, for example, rejection of more than 90% of the small organic dye molecules (e.g., rhodamine B and methylene blue) in ethanol and acetone. Additionally, solvents with a large solubility distance result in a high slip velocity in GO channels and thus high solvent flux through the GO membrane. In summary, the GO membrane performs better in solvents that are unlike GO, i.e., solvents with large solubility distance

Manifolds with positive orthogonal Ricci curvature

In this paper we study the class of compact K\"ahler manifolds with positive orthogonal Ricci curvature: $Ric^\perp>0$. First we illustrate examples of K\"ahler manifolds with $Ric^\perp>0$ on K\"ahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of K\"ahler manifolds which admit metrics of $Ric^\perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^\perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four