惯性定律的新证法

本文从实对称矩阵的角度,给出了惯性定律的一种新证法。这种证法可使我们从已给的一个实二次型f(x1,x2,……,xn)=∑ni,j=1aijxixj的矩阵A=（aij)n&;#215;n的诸元素,来直接研究这个二次型的性质的这一体系更加完整。同时,本文还大大地改进了雅可比方法,使雅可比的方法更加完美,应用更加广泛。     

完全次对称雅可比矩阵的某些性质

是在对完全对称雅可比矩阵及相应次对称矩阵对比研究的基础上,导出了完全次对称雅可比矩阵的特征值和相应特征向量之间的某些十分有趣的性质.     

可逆保面积映象对称周期轨道线性雅可比矩阵的一般结构

对于二维可逆保面积DeVogelaere映象给出n次迭代线性雅可比矩阵和对称偶周期轨道线性雅可比矩阵的一般表示式,从而导出对称周期轨道线性雅可比矩阵的一般结构。证明了一般的可逆保面积映象具有相同的结构,并从所得的一般结构讨论了可逆保面积映象对称周期轨道分岐的一般行为。     

一类矩阵多项式的秩特征

给出了一类矩阵多项式的秩特征定理及它的多种证法.     

关于正定矩阵一不等式的简单证明

设A=(aij)是一n阶正定实对称矩阵,本文用代数方法证明了|A|≤a11a22…ann,当且仅当A是对角矩阵时等号成立.且证法简单.     

具有二项式型多项式下三角矩阵的性质

n 1阶下三角方阵Ln[x]定义为:(Ln[x])ij=(?)i-j(x)l(i,j)(如果i≥j),否则为0,且满足条件l(i,k)l(k,j)=l(i,j)(k-j i-j)和 ,即二项式型多项式函数矩阵.n 1阶方阵Ln定义为:当i≥j时,(Ln)ij=l(i,j),否则为0.本文研究了比Pascal函数矩阵及Lah矩阵更广泛的一类矩阵Ln[x]与Ln,得到了更一般的结果和一些组合恒等式.     

幂零-中心化方法的一个应用

试图丰富谱任意符号模式矩阵类.给出了一个新的含有2n个非零元的符号模式矩阵,并运用幂零-中心化方法与幂零-雅可比方法分别研究了该模式的所有母模式是谱任意的.进一步证明了该模式是极小谱任意的.最后比较了两种证明方法的联系与区别.     

阻尼牛顿法迭代中病态问题趋于稳定的渐变(瞬变)过程分析

<正>1引言设映像F:DR~n→R~n,考虑非线性方程组F(x)=0,x∈DR~n,其中F(x)=(f_1(x),f_2(x),…,f_n(x))T,分量f_i(x):R~n→R(i=1,2,…,n)是连续可微实值函数.目前,非线性方程组求解的数值方法有牛顿法、同伦型法、单纯形法与胞腔排除法等[1]~[3]牛顿法是一种非常实用的计算方法,迭代公式如下x=x+p,(2)其中x为前次迭代近似,x为紧接着x后的迭代近似,p=-[F'(x)]~(-1)F(x)为牛顿修正,F'(x)为x处的雅可比矩阵.     

基于样条的高阶矩阵微分方程近似解

矩阵微分方程经常出现在许多物理模型和工程技术模型中.利用矩阵样条构造形如{y(p)(x)=Ap-1(x)y(p-1)(x)+Ap-2(x)y(p-2)(x)+…+A1(x)y(1)(x)+A0(x)y(x)+B0(x),y(a)=ya,…,y(p-1)(a)=y(p-1)a,x∈[a,b];Ai(x),B0(x)∈C4[a,b],0≤i≤p-烅烄烆1的高阶矩阵线性微分方程初值问题的数值解.给出实现算法和数值解的近似误差估计以及数值实例.先将高阶矩阵微分方程转化为一阶矩阵微分方程,然后利用三次矩阵样条求出一阶矩阵线性微分方程的数值解,从而解决高阶微分方程问题.     

一类插值多项式的导数逼近(英文)

本文给出了以雅可比多项式的零点作为插值节点的一类插值多项式 Bn( f ;x)的导数逼近具有一阶连续导数的函数的收敛阶 .并且指出 limn→∞ Bn′( f;-1 )≠f′( -1 ) .     

厄米特矩阵的迹的几点性质

应用矩阵Ａ＝（ａｉｊ）∈Ｃｎ×ｎ的弗罗伯尼范数ＡＦ和谱范数ＡＳ，研究厄米特矩阵的迹的性质，得到几个结论：Ｔｒ（ＡＢ）＝∑ｎｉ＝１λｉ∑ｎｊ＝１ｔｉｊμｊ（λｉ，μｊ分别为Ａ，Ｂ的特征值，０≤ｔｉｊ≤１，且∑ｎｉ＝１ｔｉｊ＝１，ｊ＝１，２，…，ｎ）；Ｔｒ（ＡＢ）≤Ｔｒ（Ａ）ＢＳ；Ｔｒ（ＡＢ）Ｈ（ＡＢ）］≤Ｔｒ（ＡＨＡ）［ｍａｘ１≤ｉ≤ｎλｉ］２（λｉ是Ｂ的特征值）等．     

矩阵Kronecker积的推广

由x,y的p次多项式f(x,y)=∑pi,j=0aijxiyj给出f(x,y)的广义Kronecker积f(A,B)=∑pi,j=0aijAi Bj,得到f(A,B)的特征值的分布,推广了已知的一些结果.     

一个关于非负矩阵的不等式

证明:若(xij)是一个元素不全为零的m×n非负矩阵,则当0相似文献   

The Stability of Systems of Neutral Type with Small Time Lag

In this paper, we have obtained the equivalence theorems of stability between the system of differential equations $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t)} (i = 1,2, \cdots ,n)\]$ and the system of differential-difference equations of neutral type $[{\dot x_i}(t) = \sum\limits_{j = 1}^n {{a_{ij}}{x_j}(t)} + \sum\limits_{j = 1}^n {{b_{ij}}{x_j}(t - {\Delta _{ij}})} + \sum\limits_{j = 1}^n {{c_{ij}}{{\dot x}_j}(t - {\Delta _{ij}})} (i = 1,2, \cdots ,n)\]$ where a_ij, b_ij, c_ij are given constants, and \Delta_ij are non-negative real constants.     

自然数幂求和公式的计算机实现

自然数幂的求和问题 ,一直受到人们的关注 .著名数学家陈景润对此就有过较好的研究 ,更多结果散见其他许多文献 .但都比较烦琐 .本文借助 Mathematica软件 ,利用高阶等差数列的一个结论 :m阶等差数列的充要条件是其前 n项和为 n的 m+ 1次多项式 .给出了一种求自然数幂前 n项和的一种简单方法 .利用此方法还可实现小于 m的自然数幂前 n项和的同时实现 .     

Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

Sparse quasi-Newton updates with positive definite matrix completion

Quasi-Newton methods are powerful techniques for solving unconstrained minimization problems. Variable metric methods, which include the BFGS and DFP methods, generate dense positive definite approximations and, therefore, are not applicable to large-scale problems. To overcome this difficulty, a sparse quasi-Newton update with positive definite matrix completion that exploits the sparsity pattern of the Hessian is proposed. The proposed method first calculates a partial approximate Hessian , where , using an existing quasi-Newton update formula such as the BFGS or DFP methods. Next, a full matrix H _k+1, which is a maximum-determinant positive definite matrix completion of , is obtained. If the sparsity pattern E (or its extension F) has a property related to a chordal graph, then the matrix H _k+1 can be expressed as products of some sparse matrices. The time and space requirements of the proposed method are lower than those of the BFGS or the DFP methods. In particular, when the Hessian matrix is tridiagonal, the complexities become O(n). The proposed method is shown to have superlinear convergence under the usual assumptions.      

MW-分形集的分离性质

令（ａｉｊ）ｎ×ｎ为０ １不可约矩阵．对每一ａｉｊ＝１，取Ｒｄ中具有相似率０＜ｒｉｊ＜１的相似压缩映射φｉｊ．则对应地存在Ｒｄ中唯一紧集族Ｆ１，…，Ｆｎ满足：Ｆｉ＝∪ｎｊ＝１ａｉｊ＝１φｉｊ（Ｆｊ）．我们证明开集条件成立当且仅当强开集条件成立当且仅当对某个１ｉｎ，Ｆｉ为一ｓ－集，此处ｓ为使得矩阵ｒｓｉｊｎ×ｎ的谱半径为１的唯一非负实数．     

解一般矩阵方程的GPBiCG(m,l)法

The generalized product bi-conjugate gradient(GPBiCG(m,l))method has been recently proposed as a hybrid variant of the GPBi CG and the Bi CGSTAB methods to solve the linear system Ax=b with non-symmetric coefficient matrix,and its attractive convergence behavior has been authenticated in many numerical experiments.By means of the Kronecker product and the vectorization operator,this paper aims to develop the GPBi CG(m,l)method to solve the general matrix equation■ and the general discrete-time periodic matrix equations■ which include the well-known Lyapunov,Stein,and Sylvester matrix equations that arise in a wide variety of applications in engineering,communications and scientific computations.The accuracy and efficiency of the extended GPBi CG(m,l)method assessed against some existing iterative methods are illustrated by several numerical experiments.