Universal Associated Legendre Polynomials and Some Useful Definite Integrals

We first introduce the universal associated Legendre polynomials, which are occurred in studying the non-central fields such as the single ring-shaped potential and then present definite integrals I_A^±(a, τ)=∫_-1⁺¹x^a[P_l'^m'(x)]²/(1±x)^τdx, a=0, 1, 2, 3, 4, 5, 6, τ=1, 2, 3, I_B(b, σ)=∫_-1⁺¹x^b[P_l'^m'(x)]²/(1- x²)^σdx, b=0, 2, 4, 6, 8, σ=1, 2, 3, and I_C^±(c, κ)=∫_-1⁺¹x^c[P_l'^m'(x)]²/[(1-x²)^κ(1±x)]dx, c=0, 1, 2, 3, 4, 5, 6, 7, 8, κ=1, 2. The superindices “±” in I_A^±(a, τ) and I_C^±(c, κ) correspond to those of the factor (1±x) involved in weight functions. The formulas obtained in this work and also those for integer quantum numbers l' and m' are very useful and unavailable in classic handbooks.     

On Integrals Involving Universal Associated Legendre Polynomials and Powers of the Factor (1-x2) and Their Byproducts

The associated Legendre polynomials play an important role in the central fields, but in the case of the non-central field we have to introduce the universal associated Legendre polynomials P_l'^m'(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential. We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor (1-x²)^-p-1 as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction. The calculations are obtained systematically using some properties of the generalized hypergeometric series.     

Color cyclic homology and Steinberg Lie color algebras

The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ?-cyclic homology group HC₁(R, ?) of the Γ-graded associative algebra R (which could be seemed as the generalization of cyclic homology group and the ?/2?-graded version of cyclic homology that was introduced by Kassel) to calculate the universal central extension of Steinberg Lie color algebras.     

酸碱体系pH统一公式及Mathematica精确求算

酸碱体系pH计算是大学分析化学理论课程学习的重点难点,教材涉及繁多的计算公式和近似条件,局限于常见的简单体系。采用酸离解常数（Ka）对酸、碱、两性物质的酸碱反应平衡进行统一描述,建立质子平衡方程,推导得到适用于所有酸碱体系的pH计算统一公式,并使用Mathematica数学软件,便捷地解决精确求解的问题。学习通法通式,并采用计算机精确计算pH,有助于学生整体掌握分析化学理论课程中pH计算的理论基础及解决复杂体系精确计算问题。     

Clifford分析中调和函数推广的Liouville定理

主要研究取值于泛Clifford代数C(Vn,n)上的调和函数的推广的Liouville定理,利用一些引理和正则函数的推广的Liouville定理,证明并得到了高维空间中调和函数的推广的Liouville定理.     

一体化网络中可证明安全的三方认证协议

为了有效地保证终端、接入交换路由器和认证中心的安全,本文提出了一体化网络中可证明安全的三方认证协议.该协议实现了终端和认证中心之间及接入交换路由器和认证中心之间的相互认证,不仅可以有效防止非授权终端接入网络,还可以防止伪造的认证中心和接入交换路由器对终端的欺骗.在BR扩展模型下,可证明该协议是安全的.通过性能分析得出,协议具有很高的效率.     

电力系统故障计算通用模型研究

提出了一种既适用于简单故障,又适用于复杂故障的计算机通用模型．该算法不需要考虑故障端口是横向或是纵向,也不需要考虑序网的串并联问题,而是将故障边界条件导出为相应的修正导纳矩阵,叠加在原三序网络导纳矩阵中,同时解决了变压器变比和相位位移问题．模型简单通用性强,适合于大规模电网故障计算．     

一种新的快速通用二维图像亚像素边缘检测算法

总结了目前二维灰度图像的亚像素边缘检测算法,针对它们存在的原理误差、计算复杂、用时长及不能通用等问题,提出了一种新的亚像素边缘检测算法.分析了3种基本边缘(阶跃型边缘、脉冲型边缘、屋脊型边缘)的特点,利用一维质心算法对这3种边缘特征计算上的通用性及简易性进行了二维推广,得到了一种具有快速通用性的亚像素边缘检测算法.并在此基础上针对提高离散化过程中的精度问题,引入了高斯卷积平滑的预处理方法;引入了Sobel算法对图像像素进行了筛选,进一步提高了计算速度.通过实验验证了此算法的有效性,并分析了误差产生的原因.     

A Universal Amplitude Ratio for the q≤4 Potts Model from a Solvable Lattice Model

The universal amplitude ratio R for the (q4)-state Potts model in two dimensions is determined by using results for the dilute A model in regime 1. The nature of the relationship between the Potts model and the dilute A model is discussed.     

混沌吸引子的普适转变

二维动力学映射混沌吸引子的片结构将随耗散性的减弱以普适的方式截断,最终趋于哈密顿极限。不稳定不变流形的异宿相交可以解释混沌吸引子片结构的突变行为,突变点具有普适的标度,哈密顿映射对于耗散方向的重正化理论可以给出这一普适标度。