Complete self-shrinkers with bounded the second fundamental form in Rn+1\mathbb{R}^{n+1}

Let $X:M^n\to \mathbb{R}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\leq S< C$ for some constant $C$, then $S=1$. Further we classify the $n$-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in $\mathbb{R}^{n+1}$, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.Comment: 17 pages, 0 figure, All comments are welcom

Development of a Transferable Reactive Force Field of P/H Systems: Application to the Chemical and Mechanical Properties of Phosphorene

ReaxFF provides a method to model reactive chemical systems in large-scale molecular dynamics simulations. Here, we developed ReaxFF parameters for phosphorus and hydrogen to give a good description of the chemical and mechanical properties of pristine and defected black phosphorene. ReaxFF for P/H is transferable to a wide range of phosphorus and hydrogen containing systems including bulk black phosphorus, blue phosphorene, edge-hydrogenated phosphorene, phosphorus clusters and phosphorus hydride molecules. The potential parameters were obtained by conducting unbiased global optimization with respect to a set of reference data generated by extensive ab initio calculations. We extend ReaxFF by adding a 60{\deg} correction term which significantly improves the description of phosphorus clusters. Emphasis has been put on obtaining a good description of mechanical response of black phosphorene with different types of defects. Compared to nonreactive SW potential [1], ReaxFF for P/H systems provides a huge improvement in describing the mechanical properties the pristine and defected black phosphorene and the thermal stability of phosphorene nanotubes. A counterintuitive phenomenon is observed that single vacancies weaken the black phosphorene more than double vacancies with higher formation energy. Our results also show that mechanical response of black phosphorene is more sensitive to defects for the zigzag direction than for the armchair direction. Since ReaxFF allows straightforward extensions to the heterogeneous systems, such as oxides, nitrides, ReaxFF parameters for P/H systems build a solid foundation for the reactive force field description of heterogeneous P systems, including P-containing 2D van der Waals heterostructures, oxides, etc

Why Some Verbs are Harder to Learn than Others – A Micro-Level Analysis of Everyday Learning Contexts for Early Verb Learning

Verb learning is important for young children. While most previous research has focused on linguistic and conceptual challenges in early verb learning (e.g. Gentner, 1982, 2006), the present paper examined early verb learning at the attentional level and quantified the input for early verb learning by measuring verb-action co-occurrence statistics in parent- child interaction from the learner’s perspective. To do so, we used head-mounted eye tracking to record fine-grained multimodal behaviors during parent-infant joint play, and analyzed parent speech, parent and infant action, and infant attention at the moments when parents produced verb labels. Our results show great variability across different action verbs, in terms of frequency of verb utterances, frequency of corresponding actions related to verb meanings, and infants’ attention to verbs and actions, which provide new insights on why some verbs are harder to learn than others

A Liouville theorem for the fractional Ginzburg-Landau equation

In this paper, we are concerned with a Liouville-type result of the nonlinear integral equation \begin{equation*} u(x)=\int_{\mathbb{R}^{n}}\frac{u(1-|u|^{2})}{|x-y|^{n-\alpha}}dy, \end{equation*} where $u: \mathbb{R}^{n} \to \mathbb{R}^{k}$ with $k \geq 1$ and $1<\alpha<n/2$. We prove that $u \in L^2(\mathbb{R}^n) \Rightarrow u \equiv 0$ on $\mathbb{R}^n$, as long as $u$ is a bounded and differentiable solution.Comment: 7 page

Anomalous phase transition of layered lepidocrocite titania nanosheets to anatase and rutile

In this study, phase transformations from lepidocrocite titania (L-TiO2) nanosheets to rutile (R-TiO2) and anatase (A-TiO2) have been systematically investigated as a function of the preparation conditions, such as pH and freeze-drying, and as a function of the temperature treatment. We have found that the transformation of (L-TiO2) into rutile takes place upon freeze-drying treatment. We report that temperature determined the final phase structure in the transition phase of the L-TiO2 nanosheets into TiO2 nanoparticles, while the pH determined the final morphology and particle size. On the basis of the experimental results, two different transition pathways of dissolution–recrystallization and topologically rolling transition have been proposed. Our results give a full map of phase transition and morphology evolution of L-TiO2 to R-TiO2/A-TiO2 that can provide guideline to new materials design, especially for photocatalysts