On the existence of positive solutions for a class of singular boundary value problems

In this paper, by the use of a fixed point theorem, many new necessary and sufficient conditions for the existence of positive solutions in C[0,1]∩C¹[0,1]∩C²(0,1) or C[0,1]∩C²(0,1) are presented for singular superlinear and sublinear second-order boundary value problems. Singularities at t=0, t=1 will be discussed.     

Numerical comparison of methods for solving second-order ordinary initial value problems

In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C²-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented.     

L-functions of second-order cusp forms

We discuss equivalent definitions of holomorphic second-order cusp forms and prove bounds on their Fourier coefficients. We also introduce their associated L-functions, prove functional equations for twisted versions of these L-functions and establish a criterion for a Dirichlet series to originate from a second order form. In the last section we investigate the effect of adding an assumption of periodicity to this criterion. 2000 Mathematics Subject Classification Primary—11F12, 11F66 G. Mason: Research supported in part by NSF Grant DMS 0245225. C. O’Sullivan: Research supported in part by PSC CUNY Research Award No. 65453-00 34.     

Wetting at a polymer gel–vapor interface; a system not influenced by a long-range van der Waals contribution

The exterior of a cross-linked polymer gel resembles a polymer brush. Such gel including its brush can be swollen by a good solvent. The detachment of the solvent–vapor (L–V) interface from the brush is controlled by the adsorption of polymer segments onto this interface and is to a very good approximation not influenced by long-range van der Waals contributions. A wetting transition in this system coincided with the adsorption–desorption transition for chains onto such L–V interface and has various unusual features. There are several indications that in practice this system should feature a second-order wetting transition.     

肉桂酸类分子非线性光学效应的理论研究

计算了一系列肉桂酸（苯基丙烯酸）类化合物的二阶非线性光学系数，系统地研究了取代基的电子性质和取代位置对其非线性光学性质的影响，并且探讨了该类化合物的饱和取代效应．研究表明，该类化合物的非线性光学性质与其分子结构有着密切的关系．     

On Two-point Boundary Value Problems for Second-order Difference Equation

In this paper, we aim to investigate the difference equation \begin{align*} \Delta^{2}y(t-1)+|y(t)|=0, \ \ \ \ \ t\in[1,T]_{\mathbb{Z}} \end{align*} with different boundary conditions $y(0)=0$ or $\Delta y(0)=0$ and $y(T+1)=B$ or $\Delta y(T)=B$,\ where\ $T\geq 1$ is an integer and $B\in\mathbb{R}$. We will show that how the values of $T$ and $B$ influence the existence and uniqueness of the solutions to the about problem. In details, for the different problems, the $TB$-plane explicitly divided into different parts according to the number of the solutions to the above problems. These parts of $TB$-plane for the value of $T$ and $B$ guarantee the uniqueness, the existence and the nonexistence of solutions respectively.     

Dynamic stability of carbonyl ylides

The non- and fluorine-substituted singlet carbonyl ylides are studied by using ab initio MCSCF calculations. The thermodynamic stability of the carbonyl ylides and the intramolecular stability to isomerization or fragmentation reaction coordinates is demonstrated in terms of the topological structure of the ab initio potential energy surfaces. The allylic resonance is found to be dynamically unstable, considering out-of-plane vibrational mode. The instability is studied by the symmetries of the low-lying excitations out of the MCSCF wave function.     

The application of the concept of reaction layer to the study of electrode processes coupled with the second-order chemical reactions at microelectrodes under steady-state conditions

Under steady-state conditions, the current equations of the second-order EC, ECE and DISPI reactions at microdisk, microspherical and microring electrodes are derived with the aid of the concept of the reaction layer. The conditions under which these equations would be valid are also discussed. Using these equations, methods to determine the kinetic parameters for the second-order EC, ECE and DISPI reactions are presented. The reduction of 2,6-diphenyl-pyrylium cation and oxidation of triphenylamine were investigated as examples of the second-order EC and ECE reactions.     

Effects of fluorine atoms on the optical nonlinear response of stilbene derivatives

The effect of fluorine atoms on the second-order optical nonlinear response of 4-nitro-4′-methoxy-trans-stilbene and polyene derivatives containing up to three double bonds (n=1, 2, 3) has been investigated within a semiempirical context (PM3 Hamiltonian). Experimental data reported in literature indicate a β_vec(SHG) value of 34, 47 and 76 (×10⁻³⁰ esu) for n=1, 2, 3, respectively, at 0.65 eV in CHCl₃ solutions. Our calculations show that fluorinating the nitrophenyl group the β_vec(SHG) is doubled. Further increase in the second-order nonlinear response can be obtained fluorinating the CC bond linkers between the aromatic moieties.Besides, the effect of different donor-acceptor pairs has been studied and the results interpreted in the two-state model context.Some results concerning the third-harmonic generation (THG), γ (THG), are presented, and a possible interpretation proposed.     

取代苯体系的二阶非线性光学性质:动力学李代数方法

提出了一种动力学李代数方法来研究取代苯体系的非线性光学性质. 对于给定的PPP模型(Pariser-Parr-Pople)哈密顿量, 生成了一个动力学李代数. 依据这些代数元构造出演化算子作为群参数的函数, 通过求解一组非线性微分方程能够得到这些群参数. 再按照统计力学中的密度算子公式给出取代苯分子体系偶极矩的统计平均值. 于是导出二阶极化率的表达式. 与其他量子力学计算结果比较, 表明这种动力学李代数方法在预言有机共轭分子的非线性光学性质上同样有用.