任意二阶张量的特征张量及其特性分析

黄永念曾对一个二阶张量引进了一个特征张量的概念,且利用它给出了一个常系数常微分方程组得显式解表达式.最近发现这种特征张量是并矢形式.利用这种并矢表示可以大大简化张量的运算工作.     

圆孔径低交叉极化改进型偏置双镜天线

本文导出改进型偏置双镜天线孔镜场交叉极化的代数表示式,及确定映象函数的一阶偏微分方程。利用叠代法可把映象函数的解算简化为一阶常微分方程的求解问题。利用这些公式可以快速选定满足交叉极化给定要求的几何参数,并确定映象函数,从而避免了反复多次解算二阶偏微分方程或一阶偏微分方程组和重新构造主镜的大量复杂的计算工作。     

Schr(o)dinger方程的时空有限元方法与守恒性

对非线性Schroedinger常微分方程，利用常微分方程连续有限元法证明了能量守恒；对非线性Schroedinger偏微分方程利用时空都连续的全离散有限元方法证明了能量积分守恒和利用空间连续、时间问断的有限元法得到电荷近似守恒，误差为高阶量．并在数值计算上探讨了守恒性和近似程度。结果与理论相吻合．     

一类Poincaré方程的中心焦点判定

本文根据计算焦点量的一类递推公式和Poincare对称原理,以计算机代数为工具,得 到了一类Poincare方程的焦点量公式,并给出了原点为中心的条件.     

常系数线性常微分方程组的显式解

本文利用张量分析给出了常系数线性常微分方程组和n阶常系数线性常微分方程初值问题一般解的显式表示,包括特征根有重根时的情况.实际上本文给出了计算矩阵exp[At]的元素的一般公式.这种方法不仅在公式表示上简洁方便,而且更适用于计算机的程序设计,大大加快了运算速度.     

复微分方程组的m分量-可允许解

本文给出了复域内一类代数微分方程组的ｍ分量－可允许解之定义，并利用Ｎｅｖａｎｌｉｎｎａ值分布理论讨论了这类微分方程组的这种解的存在性，得到了更精确、更一般的结果．     

关于Getz-Jacobson方法的若干讨论

考虑一类右端函数为多项式情形的常微分方程组初值问题的奇异性.先指出Getz-Jacobson所提出的代数方法的某些不足,然后提出一个新的方法,使得更加方便地讨论此类问题解的爆破性质.最后应用这种方法成功地解决了来自化学反应的一个实例.     

基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

常系数线性微分方程组的基解矩阵的一种新求法

在求解常系数线性微分方程组时,关键是基解矩阵的计算.给出了利用哈密顿—凯莱定理计算基解矩阵的一种方法,并通过实例说明了这种方法的特点和在简化计算方面的有效性.     

用齐次化原理解线性非齐次微分方程

齐次化原理是求解线性非齐次偏微分方程的一种方法。本文利用这种方法求解线性非齐次常微分方程,并推导出解的一般公式。     

CALCULATION OF FOCAL VALUES FOR E_3~1 SYSTEMS

1IlltroductionInthispaper)weconsiderthenumberoflimitcyclesforthefollowingcubicpolynomialdifferentialsystemForthepurposeofanalysingthenumberoflimitcycles,inawaylitisnecessarytocalculatethefocalvalues.Alotofworkonquadraticsystemshavebeendone.Now,manymathematicianshaveturntheirinteresttocubicsystems.However,itismoredifficultforacubicsystems,becausethealgorithmcomplicacyforcalculatingthefocalvalueisexponentiallyincreasing.Attemptoperformthecalculationbyhandisnotpossible.Someonehasstudiedsomespec…     

Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.     

Multiple limit cycles bifurcation from the degenerate singularity for a class of three-dimensional systems

In this paper, bifurcation of small amplitude limit cycles from the degenerate equilibrium of a three-dimensional system is investigated. Firstly, the method to calculate the focal values at nilpotent critical point on center manifold is discussed. Then an example is studied, by computing the quasi-Lyapunov constants, the existence of at least 4 limit cycles on the center manifold is proved. In terms of degenerate singularity in high-dimensional systems, our work is new.     

非负矩阵谱半径的新界值

讨论n×n阶非负矩阵的谱半径估计,给出估计其界值的新方法,该方法易于计算且能得到比"非负不可约矩阵谱半径的估计"一文中精确度更高的界,并用数值例子验证了这种方法的有效性.     

Solution of a stochastic Darcy equation by polynomial chaos expansion

This paper deals with solving a boundary value problem for the Darcy equation with a random hydraulic conductivity field.We use an approach based on polynomial chaos expansion in a probability space of input data.We use a probabilistic collocation method to calculate the coefficients of the polynomial chaos expansion. The computational complexity of this algorithm is determined by the order of the polynomial chaos expansion and the number of terms in the Karhunen–Loève expansion. We calculate various Eulerian and Lagrangian statistical characteristics of the flow by the conventional Monte Carlo and probabilistic collocation methods. Our calculations show a significant advantage of the probabilistic collocation method over the directMonte Carlo algorithm.     

关于多值解析函数的教学研究

复变函数论中的多值函数教学是一个难点.钟玉泉先生的教材在这一难点的处理方面是较成功的.他通过例题,介绍多值函数分成单值分支的方法,介绍求函数值的方法,并对这些方法进行了总结,得出的结论是:当给定初值后,只有通过连续变化才能得到其它点的函数值.这一点和传统的代入法求函数值完全不同.然而该教材就在这一总结之后,又用代入法求Arcsin2.我们认为这自我否定了刚刚建立起来的求值方法,扰乱了读者的思想.本文通过对Arcsinz分成单值解析分支的讨论,对求Arcsin2提出了新的教材处理方案,以期和读者商榷.     

基于神经网络的SND近似计算

利用查表确定标准正态分布的函数值非常有限,这给工程应用带来很多不便。文章讨论了基于神经网络计算标准正态分布函数值的方法、数学原理、网络构造和学习过程。示例表明,计算简洁、方便,准确率能达到10^-6。     

A Chebyshev polynomial method for line integrals with singularities

In this paper we consider a Chebyshev polynomial method for the calculation of line integrals along curves with Cauchy principal value or Hadamard finite part singularities. The major point we address is how to reconstruct the value of the integral when the parametrization of the curve is unknown and only empirical data are available at some discrete set of nodes. We replace the curve by a near‐minimax parametric polynomial approximation, and express the integrand by means of a sum of Chebyshev polynomials. We make use of a mapping property of the Hadamard finite part operator to calculate the value of the integral. This revised version was published online in June 2006 with corrections to the Cover Date.     

基于模糊环境下的最大偏差法和TODIM法的多属性应急选址决策

犹豫模糊集(HFS)是一个元素的隶属度可以用多个实数值表示的集合.该方法通常用于处理决策者在具有多个评价指标、方案、变量等多个值时犹豫不决的情况.本文基于犹豫不决的模糊环境,提出了一种新的混合方法.首先,基于最大偏差法确定属性权重;其次,利用TODIM方法计算各方案的优势度.通过对这些优势度的聚合,可以进一步得到每个方...     

Predictor homotopy analysis method and its application to some nonlinear problems

The purpose of this paper is to present a kind of analytical method so-called Predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential equations with boundary conditions. This method is very useful especially for those boundary value problems which admit multiple solutions and furthermore is capable to calculate all branches of the solutions simultaneously. As illustrative examples, the method is checked by the model of mixed convection flows in a vertical channel and a nonlinear model arising in heat transfer which both admit multiple (dual) solutions.