A modularity criterion for Klein forms, with an application to modular forms of level 13

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ₁(N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ₁(13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).     

Chiral selective tunneling induced graphene nanoribbon switch

An armchair graphene nanoribbon switch has been designed based on the principle of the Klein paradox. The resulting switch displays an excellent on-off ratio performance. An anomalous tunneling phenomenon, in which electrons do not pass through the graphene nanoribbon junction even when the conventional resonance condition is satisfied, is observed in our numerical simulations. A selective tunneling rule is proposed to explain this interesting transport behavior based on our analytical results. Based on this selective rule, our switch design can also achieve the confinement of an electron to form a quantum qubit.      

巧用滴瓶设计系列化学实验

利用常见化学仪器滴瓶设计系列化学实验——钠与乙醇反应、铝与NaOH溶液反应、铜与浓、稀硝酸反应以及铁的析氢腐蚀和吸氧腐蚀等, 改进后的实验装置简约, 便于携带, 易于操作, 现象明显, 无污染, 也为教学赢得时间。     

A novel foam based separation strategy for extracting minute target impurities

The purpose of this study was to extract impurities from compounds using a simple separatory bottle to purify target compounds with a foam column and allow for the further characterization of impurities. Charged dyes were used as target compounds due to the ease of detection of dyes and isolated impurities. Foaming agents were used in a glass bottle with a modified cap to separate a target impurity using an appropriately charged ligand. By passing N₂ gas through the solution, the surfactants sodium dodecyl sulfate and cetylpyridinium chloride generated foams that separated the dyes, Methylene blue and Orange G, respectively, from a solution containing both dyes. Sodium dodecyl sulfate condensed Methylene blue from the solution with high purity while cetylpyridinium chloride condensed Orange G with less purity. A range of concentrations (0.01–0.5 mmol/L) of dyes were used for separation. The condensability (volume and/or concentration) of the target compound increased as its concentration decreased. This novel separation method is a simple, rapid, inexpensive, and effective way to prepare samples and allows for the characterization of these impurities using sensitive analytical detection techniques.     

双室聚丙烯低温反应瓶的设计制造及在硅芴合成中的应用

设计制造了瓶底有一大一小2个室的聚丙烯（PP）塑料反应瓶,将其用于活泼的金属有机反应中硅芴的合成。 首先将制备的2,2′-二锂基联苯（2）转移到小室中,然后将SiCl4溶液注入大室内,再将反应瓶倾斜,使得化合物2的溶液分次进入大室进行反应,保证反应在SiCl4充分过量条件下进行,可大大降低螺硅芴（4）副反应产物的生成,使9,9-苯基硅芴的产率由20%提高至35%~40%,SiCl4的用量降至4倍摩尔量,从而也降低了产物后处理中柱分离的处理量。     

Bound states of the Klein－Gordon and Dirac equations for scalar and vector harmonic oscillator potentials

We give the exact bound states of the Klein－Gordon and Dirac equations with equal scalar and vector harmonic oscillator potentials.     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

On certain spanning subgraphs of embeddings with applications to domination

We prove the existence of certain spanning subgraphs of graphs embedded in the torus and the Klein bottle. Matheson and Tarjan proved that a triangulated disc with n vertices can be dominated by a set of no more than n/3 of its vertices and thus, so can any finite graph which triangulates the plane. We use our existence theorems to prove results closely allied to those of Matheson and Tarjan, but for the torus and the Klein bottle.     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.