On the Rank of the Semigroup TE(X)

${\cal T}_X $ denotes the full transformation semigroup on a set $ X $. For a nontrivial equivalence $E$ on $X$, let \[ T_E (X) =\{ f\in {\cal T}_X : \forall \, (a,b)\in E,\, (af,bf)\in E \} . \] Then $T_E (X) $ is exactly the semigroup of continuous selfmaps of the topological space $X$ for which the collection of all $E$-classes is a basis. In this paper, we first discuss the rank of the homeomorphism group $G$, and then consider the rank of $T_E (X)$ for a special case that the set $X$ is finite and that each class of the equivalence $E$ has the same cardinality. Finally, the rank of the closed selfmap semigroup $\Gamma(X)$ of the space $X$ is observed. We conclude that the rank of $G$ is no more than 4, the rank of $T_E (X)$ is no more than 6 and the rank of $\Gamma(X)$ is no more than 5.     

Decentered twisted Gaussian Schell-model beams and their propagation through a misaligned first-order optical system

The propagation properties of decentered twisted Gaussian Schell-model (DTGSM) beams passing through a misaligned first-order optical system are studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived, which retain their form unchanged. It is shown that the DTGSM beams preserve their closed property. The second-order moments matrix and the Wigner distribution function evolve with the usual laws, whereas the first-order moments matrix varies, as if a ray passes through such system. The propagation of DTGSM beams through an aligned first-order optical system is treated as the limiting case that corresponds to the vanishing misalignment parameters.     

The Gain Properties of 1-D Active Photonic Crystal

The terminology ‘ID frequency’(w ID) is proposed after analyzing the 1D active photonic crystal based on the transfer matrix method. The relationship between wID and the structure parameters of the photonic crystal is investigated.     

On Set-Theoretical Solutions to Quantum Yang-Baxter Equation

The problem on the set-theoretical solutions to the quantum Yang-Baxter equation was presented byDrinfel'd as a main unsolved problem in quantum group theory. The set-theoretical solutions are a natural extensionof the usual (linear) solutions. In this paper, we not only give a further study on some known set-theoretical solutions(the Venkov's solutions), but also find a new kind of set-theoretical solutions which have a geometric interpretation.Moreover, the new solutions lead to the metahomomorphisms in group theory.     

Investigation of charge effects on poling and stability for corona-poled polymer films

The optimal corona-poling temperature of polymer films was accurately determined by measuring the temperature dependence of the in situ second-harmonic (SH) intensity profile under the applied poling electric field. The in situ SH intensity profile was first measured by probing both the surface voltages and the poling currents induced by surface/space charges for the corona-poled polymer films. Moreover, charge effects on the stability of the chromophore orientation were first studied by using the thermally stimulated discharge-current technique. PACS 72.20.Jv; 42.65.Ky; 73.61.Ph; 81.40.Tv     

MULTIPLICATION OF WEAK FUNCTIONS

This paper is a contionuation of [1] and is concerned with multiplication of weak functions. Here the weak functions are treated as generalized expansions in Hermite functions in [2].     

Imprinting by hot embossing in polymer substrates using a template of silica glass surface-structured by the ablation of LIBWE method

Laser-induced backside wet etching (LIBWE) of silica glass plates was performed to fabricate an imprinting template for hot embossing in polymer substrates such as polystyrene and silicone resin. Well-defined inverse surface-micropatterns of gratings and grid arrays on the substrates were produced by the hot embossing using a surface-structured silica glass as the template. These results indicate that the LIBWE method allows us to generate robust glass molding tools that exhibit the inverse shapes of the intended microstructures. PACS 52.38.Mf; 68.47.Mn; 81.05.Kf; 81.05.Lg; 83.50.Uv     

取代基效应(SCS)与乙烯-α-烯烃共聚物的序列结构Ⅲ.定量分析

本文在分析乙烯-α-烯烃共聚物序列结构时,把代表共单体反接单元的亚甲基(CH₂)βγ和γδ分别归属给了二元组VV和三元组EEV。与此同时,CH₂-δδ和δδ⁺归属给了二元组EE。表征乙烯长序列的CH₂-δ⁺δ⁺分属二元组EE和三元组EEE,对应峰强度在EE和EEE之间的分配是处理序列分布的关键。在本文中运用序列结构的Bovey关系和Randall统计进行演算,求得了修正值△的数学表达式,严格解和近似解。在规则链条件下得到的近似解与G.J.Ray的结果完全相同。当共单体含量较少时,谱峰强度Iδ⁺δ⁺的分配接近相等。在二元组和三元组的水平上,乙烯-α-稀烃共聚物的¹³C NMR谱中共有十三个峰,属于CH₂的有十个,属于CH的有三个。利用这些谱峰的强度数据可以建立一套计算公式,由此提供共聚物序列结构的全部信息。因此这是一个研究乙烯-α-烯烃共聚物序列结构的普适方法。     

X-ray absorption spectra of plasmas

In this paper we present a theoretical method to calculate the absorption spectra of hot dense plasmas. Based on our fully relativistic treatment incorporated with the quantum defect theory to handle the huge number of transition arrays from many configurations with high principal quantum number, we can calculate the absorption spectra for any element or multi-element plasmas with little computational efforts. We calculate the absorption spectra of C10H1605 plasmas, which are in good agreement with the experimental spectra. We can then provide diagnostic analysis for plasmas in relevant inertial confinement fusion (lCF) experiments; namely not only to determine plasmas' temperatures and densities, but also to provide the population densities of various ionic stages. Our theoretical method verified by "benchmark experiments" will be a basic tool to provide "precise" opacity data for the ICF research.``