厚薄板通用三角形位移元

本文构造出了一种具有九个自由度的厚薄板通用三角形位移单元，并给出了单元刚度矩阵显式。这种单元以其简洁的常规位移元列式可在相当宽的板厚变化范围内（包括板厚为零）获得很高的计算精度，其结果可与相应的矩形单元相比。而且不会出现剪切自锁。     

特征值问题的预变换方法(I): 杨辉三角阵变换与二阶PDE 特征多项式

本文提出一类求解特征值问题的下三角预变换方法, 目标是通过相似变换后矩阵下三角元素平方和明显减少、且变换后的特征值及其特征向量较易求解, 使变换后的对角线可作为全体特征值很好的一组初值, 其作用如同对于解方程组找到好的预条件子, 加速迭代收敛. 以二阶PDE 数值计算为例,对于以Laplace 方程为代表的特征波向量组及正交多项式组有广泛的应用前景.
杨辉三角是我国古代数学家的一项重要成就. 本文引入杨辉三角矩阵作为预变换子, 给出一般矩阵用杨辉三角矩阵作为左、右预变换子时变为上三角矩阵的充要条件, 给出了元素为行指标二次多项式的两个矩阵类(三对角线阵与五对角线阵) 中特征值何时保持二次多项式的充要条件, 并应用于构造新的二元PDE 正交多项式.     

与$\zeta(2k 1)$函数有关的两类快速收敛级数

Values of new series sum（（（2n-1）!ζ（2n））/（2n ＋ 2k）!）α2n from n=1 to ∞,sum（（（2n-1）!ζ（2n））/（2n＋2k ＋1）!）β2n from n=1 to ∞ are given concerning ζ（2k ＋ 1）,where k is a positive integer,α can be taken as 1,1/2,1/3,2/3,1/4,3/4,1/6,5/6 and β can be taken as 1,1/2.Some previous results are included as special cases in the present paper and new series converges more rapidly than those exsiting results for α = 1/3,or α = 1/4,or α = 1/6.     

Physalis method for heterogeneous mixtures of dielectrics and conductors: Accurately simulating one million particles using a PC

Prosperetti’s seminal Physalis method, an Immersed Boundary/spectral method, had been used extensively to investigate fluid flows with suspended solid particles. Its underlying idea of creating a cage and using a spectral general analytical solution around a discontinuity in a surrounding field as a computational mechanism to enable the accommodation of physical and geometric discontinuities is a general concept, and can be applied to other problems of importance to physics, mechanics, and chemistry. In this paper we provide a foundation for the application of this approach to the determination of the distribution of electric charge in heterogeneous mixtures of dielectrics and conductors. The proposed Physalis method is remarkably accurate and efficient. In the method, a spectral analytical solution is used to tackle the discontinuity and thus the discontinuous boundary conditions at the interface of two media are satisfied exactly. Owing to the hybrid finite difference and spectral schemes, the method is spectrally accurate if the modes are not sufficiently resolved, while higher than second-order accurate if the modes are sufficiently resolved, for the solved potential field. Because of the features of the analytical solutions, the derivative quantities of importance, such as electric field, charge distribution, and force, have the same order of accuracy as the solved potential field during postprocessing. This is an important advantage of the Physalis method over other numerical methods involving interpolation, differentiation, and integration during postprocessing, which may significantly degrade the accuracy of the derivative quantities of importance. The analytical solutions enable the user to use relatively few mesh points to accurately represent the regions of discontinuity. In addition, the spectral convergence and a linear relationship between the cost of computer memory/computation and particle numbers results in a very efficient method. In the present paper, the accuracy of the method is numerically investigated by example computations using one dielectric particle, one isolated conductor particle, one conductor particle connected to an external source with imposed voltage, and two conductor/dielectric particles with strong interactions. The efficiency of the method is demonstrated with one million particles, which suggests that the method can be used for many important engineering applications of broad interest.     

Integrating across Pascal's triangle

Sums across the rows of Pascal's triangle yield n² while certain diagonal sums yield the Fibonacci numbers which are asymptotic to ?n where ? is the golden ratio. Sums across other diagonals yield quantities asymptotic to cn where c depends on the directions of the diagonals. We generalize this to the continuous case. Using the gamma function, we generalize the binomial coefficients to real variables and thus form a generalization of Pascal's triangle. Integration over various families of lines and curves yields quantities asymptotic to cx where c is determined by the family and x is a parameter. Finally, we revisit the discrete case to get results on sums along curves.     

On the Helly Number for Line Transversals to Parallel Triangles

We prove two results about the problem of finding the Helly number for line transversals to a family of parallel triangles in the plane: (1) If each three triangles of a family of parallel right triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all     

一个求解一类投影问题的总体线性收敛的迭代法

所求的解就是c在p上的投影。 对于问题(1.1),He基于求解线性互补问题的投影收缩(PC)法,把投影问题转化为等价的广义线性互补问题,提出了一个求解这类问题的迭代方法。 原始的PC方法只能证明迭代是全局收敛的,而无法估计其收敛速度。为此,[4]和[5]对原始的PC方法作了改进,提出了固定步长的PC法并证明了其收敛速度是线性的。但在实际应用中,固定步长的PC法比原始的PC法慢的多,而且在求步长时,还要估计约束矩阵范数的大小。 本文基于[5]的思想,对于(1.1)提出了一个新的PC方法,该方法是全局线性收敛的。 本文中用到的符号说明如下:x_i表示x的第i个分量。如果u∈(?)且Ω(?)(?)为凸闭集,则P_Ω[u]定义为u到Ω上的投影。特别地,u_+定义为u到非负卦限(?)上的投影,对于一个正定矩阵G,范数||u++G表示(u~TGu)(?)。     

杨辉三角形的若干性质

研究了杨辉三角中的D av id星恒等式,给出了n阶星恒等式的定义,证明了n(n 3)阶星恒等式的存在性,并且给出了构造n阶星恒等式的方法.