多项式组特征列的矩阵求法

基于矩阵多元多项式的带余除法,给出了代数情形多项式组特征列的一种新求法,并举例验证了这种方法的有效性.     

平面扇形域问题的新正交关系和变分原理

通过构造新的对偶向量, 用空间的环向坐标数学上比拟Hamilton体系的时间变量, 在平面弹性扇形域问题中导出了一个斜对角Hamilton算子. 该算子具有主对角元为零, 斜对角元是非零对称算子的结构特性. 得到两个独立的、对称的子正交关系. 恰当选择对偶向量后, 直角坐标系下各向同性平面弹性问题的新正交关系被推广到 极坐标情形. 根据控制微分方程的弱(积分)形式及相应的边界条件, 建立了对应边值问题的变分原理, 并提出了相应的泛函表达式.     

Hilbert空间线性二次最优控制问题中的一个算子的可逆性

对于如下出现在Hilbert空间线性二次最优控制问题中的线性算子■其中F_3,F_5是自伴算子．本文得到了它具有有界逆的充分必要条件,并举例验证了结果的有效性．     

2×2阶上三角算子矩阵的谱扰动

研究了Hilbert空间H⊕K上的2×2阶上三角算子矩阵MC=(A O C B)当A,B给定,C为任意有界线性算子时,对MC的点谱、剩余谱、连续谱的扰动分别给出了描述.     

矩形中厚板Hamilton正则方程的解析解

运用Fourier分析方法，建立了对边简支的矩形中厚板弯曲问题的完备的辛本征展开. 借助于Mathematica软件的帮助，得到了来源于矩形中厚板问题的Hamilton算子的本征函数. 接着证明了本征函数系的完备性，这为使用分离变量法求解相应问题提供了理论保证；进而运用完备性定理，得到了问题的解析解；一个数值算例验证了结果的正确性.     

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.     

On the feasibility of variable separation method based on Hamiltonian system for plane magnetoelectroelastic solids

The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.     

一类无穷维Hamilton算子本征函数系的完备性

研究了Sturm-Liouvile偏微分方程导出的无穷维Hamilton算子的本征值问题.证明了导出的无穷维Hamilton算子族本征函数系的完备性,为对此类方程应用基于Hamilton体系的分离变量法提供了理论基础.最后举例说明了结果的有效性.     

$2\times 2$阶上三角算子矩阵的谱扰动

研究了Hilbert空间H(?)K上的2×2阶上三角算子矩阵Mc=(AO CB)当A,B 给定,C为任意有界线性算子时,对Mc的点谱、剩余谱、连续谱的扰动分别给出了描述．     

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2×2 symmetric operator matrices is proved. The result is applied to 4×4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.