The Fenchel-type inequality in the 3-dimensional Lorentz space and a Crofton formula

We generalize the Fenchel theorem to strong spacelike (which means that the tangent vector and the curvature vector span a spacelike 2-plane at each point) closed curves with index 1 in the 3-dimensional Lorentz space, showing that the total curvatures must be less than or equal to $2\pi$. A similar generalization of the Fary-Milnor theorem is also obtained. We establish the Crofton formula on the de Sitter 2-sphere which implies the above results.Comment: 9 pages, 4 figures. Comments are welcom

Observation of Floquet band topology change in driven ultracold Fermi gases

Periodic driving of a quantum system can significantly alter its energy bands and even change the band topology, opening a completely new avenue for engineering novel quantum matter. Although important progress has been made recently in measuring topological properties of Floquet bands in different systems, direct experimental measurement of Floquet band dispersions and their topology change is still demanding. Here we directly measure Floquet band dispersions in a periodically driven spin-orbit coupled ultracold Fermi gas. Using spin injection radio-frequency spectroscopy, we observe that the Dirac point originating from two dimensional spin-orbit coupling can be manipulated to emerge at the lowest or highest two dressed bands by fast modulating Raman laser frequencies, demonstrating topological change of Floquet bands. Our work will provide a powerful tool for understanding fundamental Floquet physics as well as engineering exotic topological quantum matter.Comment: 6 pages, 5 figure

Coating titania nanoparticles with epoxy-containing catechol polymers via Cu(0)-living radical polymerization as intelligent enzyme carriers

Immobilization of enzyme could offer the biocatalyst with increased stability and important recoverability, which plays a vital role in the enzyme’s industrial applications. In this study, we present a new strategy to build an intelligent enzyme carrier by coating titania nanoparticles with thermoresponsive epoxy-functionalized polymers. Zero-valent copper-mediated living radical polymerization (Cu(0)-LRP) was utilized herein to copolymerize N-isopropylacrylamide (NIPAM) and glycidyl acrylate (GA) directly from an unprotected dopamine-functionalized initiator to obtain an epoxy-containing polymer with terminal anchor for the “grafting to” or “one-pot” modification of titania nanoparticles. A rhodamine B-labeled laccase has been subsequently used as a model enzyme for successful immobilization to yield an intelligent titania/laccase hybrid bifunctional catalyst. The immobilized laccase has shown excellent thermal stability under ambient or even relatively high temperature above the lower critical solution temperature (LCST) at which temperature the hybrid particles could be facilely recovered for reuse. The enzyme activity could be maintained during the repeated use after recovery and enzymatic degradation of bisphenol A was proven to be efficient. The photocatalytic ability of titania was also investigated by fast degradation of rhodamine B under the excitation of simulated sunlight. Therefore, this study has provided a facile strategy for the immobilization of metal oxide catalysts with enzymes, which constructs a novel bifunctional catalyst that will be promising for the “one-pot” degradation of different organic pollutants

Monopoles and Landau-Ginzburg Models I

The end point of this series of papers is to construct the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is any compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. In the first paper, we exploit the framework of the gauged Landau-Ginzburg models to address two model problems for the (perturbed) Seiberg-Witten moduli spaces on either $\mathbb{C}\times\Sigma$ or $\mathbb{H}^2_+\times\Sigma$, where $\Sigma$ is any compact Riemann surface of genus $\geq 1$. Our first result states that any finite energy solution to the perturbed equations on $\mathbb{C}\times\Sigma$ is necessarily trivial. The second asserts that any small energy solutions on $\mathbb{H}^2_+\times\Sigma$ necessarily have energy decay exponentially in the spatial direction. These results will lead eventually to the compactness theorem in the second paper.Comment: 56 pages. v2. We add an appendix explaining the case for higher genus surfaces. v3. The statement of the main results are revised to incorporate higher genus surfaces as wel

Monopoles and Landau-Ginzburg Models II: Floer Homology

This is the second paper of this series. We define the monopole Floer homology for any pair $(Y,\omega)$, where $Y$ is a compact oriented 3-manifold with toroidal boundary and $\omega$ is a suitable closed 2-form. This generalizes the work of Kronheimer-Mrowka for closed oriented 3-manifolds. The Euler characteristic of this Floer homology recovers the Milnor torsion invariant of the 3-manifold by a theorem of Meng-Taubes.Comment: 147 pages. We add a finiteness resul