一道数学奥林匹克问题的简证及进一步研究

中等数学2008年第11期数学奥林匹克问题高235： 已知实数a，b，c，满足a十b＋c=1，a^2＋b^2＋c^2=1。求证：a^5＋b^5＋c^5≤1 原解答太繁，本文先给出①的一个简证．     

与角内一定点有关的一组几何题

引例设过P（2，1）的直线m与x轴，y轴的正方向分别交于A．B，求△AOB的面积S的最小值．     

反射式飞行时间质谱仪结合纳升电喷雾电离源研究金属簇合物

An experiment facility has been set up for the study of metal cluster compounds in our laboratory, which consists of a nano-electrospray ionization source, an ion transmission and focus system, and a reflectron time-of-flight mass spectrometer. Taking advantage of the nano-electrospray ionization source, polyvalent ions are usually produced in the “ionization” process and the obtained mass resolution of the equipment is over 8000. The molecular ion peaks of metal cluster compounds [Au₂₀(PPhpy₂)₁₀Cl₂](SbF₆)₄, where PPhpy₂=bis(2-pyridyl)phenylphosphine, and [Au₆Ag₂(C)L₆](BF₄)₄, where L=2-(diphenylphosphino)-5-methylpyridine, are distinguished in the respective mass spectrum, accompanied by some fragment ion peaks. In addition, the mass-to-charge ratios of the parent ions are determi-nated. Preliminary results suggest that the device is a powerful tool for the study of metal cluster compounds. It turns out that the information obtained by the instrumentation serves as an essential supplement to single crystal X-ray diffraction for structure characterization of metal cluster compounds.     

一道二阶常系数非齐次微分方程的八种解法

针对一道研究生入学考试试题,即求微分方程y″-3y′+2y=2xex的通解,利用待定系数法、常数变易法、算子法、降阶法、积分因子法、拉普拉斯变换法以及MATLAB软件分别给出八种解法.     

用于团簇负离子的共线光电子速度成像仪

描述了包含共线的光电子速度成像仪，以及与双阀激光溅射团簇源联用的时间飞行质谱装置．为了避免预混气体，两个脉冲阀和一个反应通道被用来产生反应负离子．共线的光电子成像仪采用了改进的速度成像透镜系统，它的能量分辨率优于3%．此外还报道了Si₄^-团簇在532和355 nm，以及Si₃C^- 团簇在532 nm的光电子速度成像．从实验图像中可以获得光电子能谱和各向异性参数．从Si₄^-团簇能谱上获得了中性的Si₄团簇的基态和第一激发态的振动频率分别为330和312 cm^-1．初步的实验结果证明这套光电子成像装置对研究团簇负离子的电子结构以及光脱附动力学非常有效.     

基于VO2和石墨烯实现hBN声子极化激元和自发发射率的主动调谐

鉴于极化激元对介电环境异常敏感的特性,本文中提出了双曲材料六方氮化硼(hBN)和石墨烯与相变材料二氧化钒(VO₂)组成的异质结构,用来研究hBN声子极化激元(PhPs)的主动可调谐性.研究结果表明,通过控制hBN/VO₂异质结构中VO₂相变可实现对hBN PhPs的主动调谐,获得主动可调谐的自发发射(SE)率.当在hBN/VO₂异质结构中添加石墨烯时,会在hBN双曲线带内耦合产生双曲等离子体-声子极化激元(HPPPs),而在双曲线带外产生表面等离子体-声子极化激元(SPPPs),通过控制VO₂相态和调节石墨烯化学势亦可实现石墨烯/hBN/VO₂异质结构的耦合色散及SE率的主动调谐.该研究为使用诸如相变材料和石墨烯等功能材料调谐各向异性光学材料与光的相互作用机制提供了理论指导.     

与角内一定点有关的一组几何题

引例设过P(2,1)的直线m与x轴,y轴的正方向分别交于A,B,求△AOB的面积S的最小值,此题一般的解法是用直线的斜截式,将面积表示成斜率k的函数,用判别式法或重要不等式。下面给出一种较简单的解法。解显然,直线m的截距a,b都为正,设直线的     

Strong chirality in twisted bilayer α-MoO3

Chiral structures are promising in many applications, such as biological sensing and analytical chemistry, and have been extensively explored. In this paper, we theoretically investigate the chiral response of twisted bilayer α-MoO₃. Firstly, the analytical formula for the transmissivity is derived when the structure is illuminated with circularly polarized plane waves. Furthermore, the results demonstrate that the twisted bilayer α-MoO₃ can excite the strong chirality with the maximum circular dichroism (CD) of 0.89. In this case, the chirality is due to the simultaneous breaking the rotational symmetry and mirror symmetry, which originates from the relative rotation of two α-MoO₃ layers. To better understand the physical mechanism, the polarization conversion between the left-hand circular polarization (LCP) and right-hand circular polarization (RCP) waves is discussed as well. Moreover, it is found that the structure can maintain the strong chirality (CD> 0.8) when the twisted angle varies from 69° to 80°, which effectively reduces the strictness in the requirement for rotation angle. In addition, the CD can be larger than 0.85 when the incidence angle of circularly polarized plane wave is less than 40°, implying that the chirality is robust against the angle of incidence. Our work not only provides an insight into chirality induced by the twisted bilayer α-MoO₃, but also looks forward to applications in biological sensing.