曲线拐点充分条件证明中的常见错误

文[1]给出了判别曲线拐点的两个充分条件,文[2]给出了一个充分条件,但三个定理的证明都是错误的.同时,文[1]的两个推论也是错误的.本文通过反例分析了其错因,并给出了文[1]中一个拐点充分条件的正确证明.     

三角形的一个向量性质再探究

文[1]给出了三角形重心向量的一个性质,并进行了空间拓广.文[2]对三角形内任一点的向量性质进行了探究,并进行了空间拓广.文[3]对文[1]的性质进行再探究,本文类比文[3]对文[2]的性质进行再探究,得到了两个定理,现叙述如下.定理1如图1所示,已知△ABC及其内部一点P,若λ1PA λ2PB     

三角形与三棱锥的一个向量性质的逆定理

文[1]、文[2]、文[3]及文[4]对一个三角形重心向量性质进行拓广,文[5]证明了文[1]的逆定理也成立,文[6]将以上的重心性质进行了再推广得到了两个定理,我们可以将这两个定理加强为以下两个命题,证明类似文[6]在此不再证明.命题1如图1所示,设P为△ABC所在平面上任意一点,λ1PA λ     

一类有限环

本文指出了文 [5 ]的一些错误 ,给出了文 [5 ]定理 1的一个反例 .并给出了一类交换环的等价条件 .     

凸函数和凹函数的幂平均不等式

文 [1 ]获得了当 α≥ 1时的凸函数的幂平均不等式 (3)、(4 ) [1] .本文指出文 [1 ]中的一个错误 ,并且得到了 α≤ 1时的凹函数的幂平均不等式 .修正和充实了文 [1 ]的定理 .同时讨论了当 α取其它值时不等式的情况 .     

一个精彩平方和不等式的推广

宋庆先生在文[1]给出了几个精彩的平方和不等式,文[2]对其中三个不等式进行了加权推广,本文将对文[1]的另一个不等式进行加权推广. 　　……     

关于度量和的一个不等式的推广

本文首先证明一个矩阵不等式,然后利用它将文[3]中关于度量和的一个不等式作实质性的推广,同时解决了文[3]中所提出的猜想.     

“勾股定理的一个简洁证明”的姐妹篇

文[1],[2],[3]分别给出了勾股定理的“简短证明”,文[4]给出了“一个更为简短而且整洁的证明”.本文运用圆的基本知识再给出一个前所未见的简证.     

圆锥曲线简易判别补遗

文[1]提出了一个避开坐标变换的繁琐计算,直接判别二元二次方程所表示的圆锥曲线形状的简易方法.本文拟补充文[1]所给判别法则的理论证明,并补充文[1]遗漏的判别法则及共证明与应用.     

再看二次曲线的一个性质

文[1]证明了二次曲线的一个性质,文[2]、[3]分别给出了一个简单证明.文[2]并把它统一叙述为:如图1,设A、B、C、D均在二次曲线上,且直线AD与BC相交于点M,则直线AC、BD的交点N必在点M的极线上.     

Computing a Nearest P-Symmetric Nonnegative Definite Matrix Under Linear Restriction

Let $P$ be an $n\times n$ symmetric orthogonal matrix. A real $n\times n$ matrix $A$ is called P-symmetric nonnegative definite if $A$ is symmetric nonnegative definite and $(PA)^T=PA$. This paper is concerned with a kind of inverse problem for P-symmetric nonnegative definite matrices: Given a real $n\times n$ matrix $\widetilde{A}$, real $n\times m$ matrices $X$ and $B$, find an $n\times n$ P-symmetric nonnegative definite matrix $A$ minimizing $||A-\widetilde{A}||_F$ subject to $AX =B$. Necessary and sufficient conditions are presented for the solvability of the problem. The expression of the solution to the problem is given. These results are applied to solve an inverse eigenvalue problem for P-symmetric nonnegative definite matrices.     

单位球上星形映射在极值点处的偏差定理

利用多复变数的边界型Schwarz引理,建立了C~n中单位球上正规化双全纯星形映射在极值点处的行列式型偏差定理和矩阵型偏差定理.     

D-optimal Designs for Seven Objects and a Large Number of Weighings

Let m ? 7 be an integer and let M m ,7 (0,1) be the set of all m × 7 (0,1)-matrices. Define $$ G(m,7) = \max \{ \det X^T X|X \in M_{m,7} (0,1)\} . \eqno (1) $$ A matrix X for which the maximum is achieved is said to be D- optimal . We determine G ( m , 7) for sufficiently large m and exhibit D-optimal matrices. For each m , the Gram matrices, X T X for D-optimal matrices X ε M m ,7 (0,1), are shown to be essentially unique.     

Matrix norm inequalities and the relative Dixmier property

If x is a selfadjoint matrix with zero diagonal and non-negative entries, then there exists a decomposition of the identity into k diagonal orthogonal projections {p_m} for which $$\parallel \sum p_m xp_m \parallel \leqslant (1/k)\parallel x\parallel $$ From this follows that all bounded matrices with non-negative entries satisfy the relative Dixmier property or, equivalently, the Kadison Singer extension property. This inequality fails for large Hadamard matrices. However a similar inequality holds for all matrices with respect to the Hilbert-Schmidt norm with constant k^?1/2 and for Hadamard matrices with respect to the Schatten 4-norm with constant 2^1/4k^?1/2.     

线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题

本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.     

THE PRIMAL—DUAL POTENTIAL REDUCTION ALGORITHM FOR POSITIVE SEMI—DEFINITE PROGRAMMING

In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primal-dual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either O(nlog(X0 · S0/ε) or O(nlog(X0· S0/ε) depends on the value of ρ in the primal-dual potential function, where X0 and S0 is the initial interior matrices of the positive semi-definite programming.     

Decomposition of Matrices into Involutory Matrices and Symmetric Matrices

Let \Omega be a field, and let F denote the Frobenius matrix: $[F = \left( {\begin{array}{*{20}{c}} 0&{ - {\alpha _n}}\{{E_{n - 1}}}&\alpha \end{array}} \right)\]$ where \alpha is an n-1 dimentional vector over Q, and E_n- 1 is identity matrix over \Omega. Theorem 1. There hold two elementary decompositions of Frobenius matrix: (i) F=SJB, where S, J are two symmetric matrices, and B is an involutory matrix; (ii) F=CQD, where O is an involutory matrix, Q is an orthogonal matrix over \Omega, and D is a diagonal matrix. We use the decomposition (i) to deduce the following two theorems: Theorem 2. Every square matrix over \Omega is a product of twe symmetric matrices and one involutory matrix. Theorem 3. Every square matrix over \Omega is a product of not more than four symmetric matrices. By using the decomposition (ii), we easily verify the following Theorem 4(Wonenburger-Djokovic') . The necessary and sufficient condition that a square matrix A may be decomposed as a product of two involutory matrices is that A is nonsingular and similar to its inverse A^-1 over Q (See [2, 3]). We also use the decomosition (ii) to obtain Theorem 5. Every unimodular matrix is similar to the matrix CQB, where C, B are two involutory matrices, and Q is an orthogonal matrix over Q. As a consequence of Theorem 5. we deduce immediately the following Theorem 6 (Gustafson-Halmos-Radjavi). Every unimodular matrix may be decomposed as a product of not more than four involutory matrices (See [1] ). Finally, we use the decomposition (ii) to derive the following Thoerem 7. If the unimodular matrix A possesses one invariant factor which is not constant polynomial, or the determinant of the unimodular matrix A is I and A possesses two invariant factors with the same degree (>0), then A may be decomposed as a product of three involutory matrices. All of the proofs of the above theorems are constructive.     

Drazin谱和算子矩阵的Weyl定理

A∈B(H)称为是一个Drazin可逆的算子,若A有有限的升标和降标．用σ_D(A)={λ∈C：A-λI不是Drazin可逆的)表示Drazin谱集．本文证明了对于Hilbert空间上的一个2×2上三角算子矩阵M_C=■,从σ_D(A)∪σ_D(G)到σ_D(M_C)的道路需要从前面子集中移动σ_D(A)∩σ_D(B)中一定的开子集,即有等式：σ_D(A)∪σ_D(B)=σ_D(M_C)∪G,其中G为σ_D(M_C)中一定空洞的并,并且为σ_D(A)∪σ_D(B)的子集．2×2算子矩阵不一定满足Weyl定理,利用Drazin谱,我们研究了2×2上三角算子矩阵的Weyl定理,Browder定理,a-Weyl定理和a-Browder定理．     

An example on strong shift equivalence of positive integral matrices

The following matrices are considered $$A_k = \left( {\begin{array}{*{20}c} k \\ 1 \\ \end{array} \begin{array}{*{20}c} 2 \\ k \\ \end{array} } \right), B_k \left( {\begin{array}{*{20}c} {k - 1} \\ 1 \\ \end{array} \begin{array}{*{20}c} 1 \\ {k + 1} \\ \end{array} } \right),k \in \mathbb{N},$$ which are strong shift equivalent in the sense ofWilliams [7]. In case \(k + \sqrt 2 \) is a prime number of the algebraic field \(\mathbb{Q}(\sqrt 2 )\) matrices are defined which determine the possible choices of rank two matrices connectingA _k andB _k in the sense of strong shift equivalence. A complete list of all these matrices is given.