Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Stability of Viscoelastic Wave Equation with Structural $\delta$-Evolution in $\mathbb{R}^{n}$

The aim of this paper is to study the Cauchy problem for the viscoelastic wave equation for structural $\delta$-evolution models. By using the energy method in the Fourier spaces, we obtain the decay estimates of the solution to considered problem.     

基于二维材料WX_2构建的范德华异质结的结构和性质及应变效应的理论研究

二维材料过渡金属硫属化物(TMDs),因其优越的物理化学特性及其在光电子器件、光催化等领域的潜在应用价值,得到了人们的广泛关注。基于TMDs材料可以构建具有不同性能的范德华(vdW)异质结,但构建的异质结由于其固有的能带带隙大小限制了其在全光谱上的响应,因而对其能带带隙调控变得十分重要。本文基于第一性原理方法系统地研究了WX_2 (X=S, Se, Te)从单层到体相的结构和性质,以及由此组装的vdW异质结构WS_2/WSe_2、WS_2/WTe_2和WSe_2/WTe_2的结构和性质以及应力应变对异质结构的能带带隙的影响。结果表明:结合HSE06泛函和自旋轨道耦合(SOC)效应的计算方案可以精确描述WX_2体系;异质结构WS_2/WSe_2,WS_2/WTe_2和WSe_2/WTe_2呈现type-II能带分类;在施加单轴或双轴的应力应变后,能带带隙大小发生相应改变,当晶格形变大于4%后,异质结构由半导体特性变成具有金属性。这些研究为光电子器件的设计提供了重要的指导意义。     

Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings

This paper is concerned with the mixed initial–boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects. We successfully establish uniform exponential and polynomial decay rates for the solutions to this initial–boundary value problem under much weak conditions concerning memory effects. More specifically, we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory-effect region includes a part of the system boundary, while the condition is a necessity in the previous literature; moreover, for the polynomial decay rates we only assume minimal conditions on the memory kernel function g, without the usual assumption of $g^{\prime }$ controlled by g.     

A theoretical study on tuning band gaps of monolayer and bilayer SnS2 and SnSe2 under external stimuli

Based on density functional theory, we systematically study the mechanical and electronic properties of monolayer and bilayer SnS₂ and SnSe₂. The electronic properties of these layers can be significantly tuned by applying in-plane strains and electric fields perpendicular to the sheets. The band gaps of monolayer SnS₂ and SnSe₂ slightly increase with the in-plane tensile strains, and they start to decrease after critical strains (5% for monolayer SnS₂ and 7% for monolayer SnSe₂). The band gaps of bilayer SnS₂ and SnSe₂ have a similar tendency to the monolayers with smaller critical strains (1% for bilayer SnS₂ and 2% for bilayer SnSe₂), which enables a semiconductor-to-metal transition at 10% strain for bilayer SnSe₂. We also find that an external electric field perpendicular to bilayer SnS₂ and SnSe₂ modulates their electronic band gaps. Semiconductor-to-metal transitions are achieved at the electric fields of 0.27 V/Å for bilayer SnS₂ and 0.13 V/Å for bilayer SnSe₂.     

Rocky coast processes: with special reference to the recession of soft rock cliffs

Substantial progress in research on the recession of coastal cliffs composed of soft materials has been made in recent years and data with higher accuracy have been accumulated. This paper provides the state of the art review in the recession studies and highlights two new findings obtained from the reanalysis of existing data. The review topics are: episodic and localized nature of cliff recession; the development of cliffline; the relationship between cliff height and recession rate; mechanisms of cliff toe erosion by waves; a fundamental equation for wave-induced toe erosion; factors controlling toe erosion; and slope instabilities and mass movements. The findings are presented on (1) the temporal change in cliffline recession mode and (2) the effect of beach sediment at the cliff base on the cliff erosion.