Additive formulae of theta functions and modular equations of degree five

Ramanujan recorded additive formulae of theta functions that are related to modular equations of degree five. We present elementary and simple proofs.     

李三系与Laurent多项式代数F[t,t~(-1)]

本文通过讨论Laurent多项式代数及其导子代数的对合自同构确定了一类具体的无限维单李三系, 并且提供了一种利用Novikov代数上自然的李代数结构来构造李三系的方法.     

Hydrogen bubbles and formation of nanoporous silicon during electrochemical etching

Many nanoporous Si structures, including those formed by common electrochemical etching procedures, produce a uniformly etched nanoporous surface. If the electrochemical etch rate is slowed down, details of the etch process can be explored and process parameters may be varied to test hypotheses and obtain controlled nanoporous and defect structures. For example, after electrochemical etching of heavily n‐doped (R = 0.05–0.5 Ω·cm) silicon 〈100〉 single crystals at a current density of 10 mA cm^?2 in buffer oxide etch (BOE) electrolyte solution, defect craters containing textured nanopores were observed to occur in ring‐shaped patterns. The defect craters apparently originate at the hydrogen/BOE bubble interface, which forms during hydrogen evolution in the reaction. The slower hydrogen evolution due to low current density and high BOE viscosity allows sufficient bubble residence time so that a high defect density appears at the bubble edges where local reaction rates are highest. Current‐carrying Si? OH species are most likely responsible for the widening of the craters. Reducing the defect/doping density in silicon lowers the defect concentration and thereby the density of nanopores. Measurements of photoluminescence lifetime and intensity show a distinct feature when the few nanopores formed at the ring edges are isolated from each other. Overall features observed in the photoluminescence intensity by XPS strongly emphasize the role of surface oxide that influences these properties. Copyright © 2005 John Wiley & Sons, Ltd.     

完全单半群上同余的两种刻画

给出了完全单半群的相容组和同余结的定义,并利用它们刻画了完全单半群上的同余.     

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.     

On Determining the Percolation Reynolds Number and the Characteristic Linear Dimension for Ideal and Fictitious Porous Media

Various versions of representations of the percolation Reynolds number for porous media with isotropic and anisotropic flow properties are considered. The formulas are derived and the variants are analyzed with reference to model porous media with a periodic microstructure formed by systems of capillaries and packings consisting of spheres of constant diameter (ideal and fictitious porous media, respectively). A generalization of the Kozeny formula is given for determining the capillary diameter in an ideal porous medium equivalent to a fictitious medium with respect to permeability and porosity and it is shown that the capillary diameter is nonuniquely determined. Relations for recalculating values of the Reynolds number determined by means of formulas proposed earlier are given and it is shown that taking the microstructure of porous media into account, as proposed in [1, 2], makes it possible to explain the large scatter of the numerical values of the Reynolds number in processing the experimental data.     

一种岩石损伤本构模型在地下强爆炸中的应用

为了描述地下爆炸波的传播,建立了一种考虑塑性硬化、剪胀和损伤软化效应的岩石本构模型,并应用于地下强爆炸自由场的数值计算,获得的速度和位移波形以及峰值速度衰减曲线等与国外地下试验测量数据和同类计算相比十分接近,从而验证了本构模型的有效性。     

后处理工艺制备多级孔Ti-BEA沸石及其环氧化催化性能

基于脱铝多级孔BEA沸石与二氯二茂钛的固相反应,开展了钛掺杂量可调的多级孔Ti-beta后处理工艺制备研究.对制备的多级孔Ti-beta样品的理化性质进行了表征,包括X射线衍射、氮气吸附脱附测试、扫描电镜、透射电镜、紫外可见吸收光谱和紫外拉曼光谱等.结果表明,多级孔BEA沸石具有较好的化学稳定性,脱铝-钛化的后处理过程未对样品多级孔结构产生明显影响. 以环己烯和十二烯的烯烃环氧化为探针反应表征了合成多级孔Ti-beta与纯相微孔Ti-beta沸石的催化性能.结果表明,在小分子环己烯的环氧化反应中,多级孔Ti-beta沸石的催化活性(转化率59.4%)与微孔Ti-beta相当(转化率57.9%);但是在较大分子十二烯的催化反应中,多级孔结构Ti-beta材料的催化性能(转化率11.1%)明显优于纯相微孔材料(转化率6.8%),且产物中环氧化物选择性更高(分别为60.3%和37.8%).     

The Critical Effect of Niobium Doping on the Formation of Mesostructured TiO2: Single‐Crystalline Ordered Mesoporous Nb‐TiO2 and Plate‐like Nb‐TiO2 with Ordered Mesoscale Dimples

Highly ordered mesoporous niobium‐doped TiO₂ with a single‐crystalline framework was prepared by using silica colloidal crystals with ca. 30 nm in diameter as templates. The preparation of colloidal crystals composed of uniform silica nanoparticles is a key to obtain highly ordered mesoporous Nb‐doped TiO₂. The XPS measurements of Nb‐doped TiO₂ showed the presence of Nb⁵⁺ and correspondingly Ti³⁺. With the increase in the amount of doped Nb, the crystalline phase of the product was converted from rutile into anatase, and the lattice spacings of both rutile and anatase phases increased. Surprisingly, the increase in the amount of Nb led to the formation of plate‐like TiO₂ with dimpled surfaces on one side, which was directly replicated from the surfaces of the colloidal silica crystals.     

李color三系的Frattini子系的性质

给出了李color三系的Frattini子系的定义,得到了李color三系的Frattini子系的一些性质·特别的,证明了李color三系T有分解T=T_1⊕T_2⊕…⊕T_m,则φ(T)有分解φ(T)=φ(T_1)⊕φ(T_2)⊕…⊕φ(T_m).