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双语推荐：秩

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分析了传统线性代数教材中秩的概念引入顺序的不足.在教学过程中,抛开矩阵的秩的行列式定义,直接从向量组的秩出发定义矩阵的秩,这样推导更易帮助学生理解向量组的秩和矩阵的秩的本质联系.
Analyzed the shortcoming of traditional linear algebra teaching on how to introduce the concept of rank. Put aside the determinant of the rank of matrix by k order determinant,directly gave the the concept of matrix rank from the concept of rank of vector.Practice shown that it helps students to understand the essence of the rank of matrix and vector set.

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矩阵的秩是线性代数中一个基本的概念，它在矩阵理论和线性空间理论中占有基本的重要性。本文从Syl-vester关于矩阵秩的定义出发，结合秩的几何意义，探讨矩阵秩的定义的教学，以便学生更好地理解这一较为抽象的概念，从而提高教学效果。
In linear algebra, the rank of a matrix is a measure of the system of linear equations and linear transformation enco-ded by matrix.The rank is one of the fundamental pieces of data associated with a matrix , and thus it holds an important position in the theory of matrix and linear space.In this paper, we mainly focus on the several definitions of the rank of a matrix so as to facilitate students’ understanding of this abstract definition.

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在随机化临床试验中，针对一般非参数Behrens －Fisher问题，经常采用 O＇Brien秩和检验法和Huang等人的秩和检验法。为此，对O＇Brien秩和检验法进行改进，讨论新检验方法的统计性质，并模拟三个检验法犯第一类错误的概率和检验功效。结果显示：新检验方法犯第一类错误的概率往往低于O＇Brien秩和检验法及 H uang等人的秩和检验法，而且新检验方法的功效高于其它两种检验。
In randomized clinical trials ,O''Brien (1984) rank sum procedure and Huang et al (2005) rank sum procedure are used to solve the generalized Nonparametric Behrens‐Fisher problem .This paper improves O''Brien rank sum procedure , simulates the Type I error probability and power of the new procedure .The simulation results show that the Type I error probability of the new procedure is often lower than O''Brien (1984) rank sum procedure and Huang et al (2005) rank sum procedure .Moreover , power of the new test is higher than both of them .

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在经典测量平差中,按间接平差法列出的误差方程系数阵为列满秩,所得到的法方程系数阵为一个对称的满秩矩阵,法方程组具有唯一解。而当控制网中没有必要的起算数据时,则按间接平差法所列出的误差方程系数阵是非列满秩的,其对应的法方程系数阵也是一个秩亏的奇异阵。探讨了秩亏自由网的法方程系数阵的一种解算方法。
In classical adjustment, the normal equations have a unique solution if the error equation coefficient matrix achieved by indirect adjustment is a full column rank, and the normal equation coefficient matrix achieved by the same method is a sym-metrical full rank matrix. When there is no necessary initial data in control network, the error equation coefficient matrix by indi-rect adjustment is a non-full column rank and the corresponding normal equation is also a rank deficient singular matrix. The paper discusses the solution for normal equation coefficient matrix in the rank defect free network.

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本文运用矩阵分块，矩阵满秩分解，线性空间维数，以及广义矩阵初等变换四种方法证明矩阵秩Frobenius不等式。
In this paper,the Frobenius Inequality of Matrix Rank is proved in the following four ways:partitioned matrix,full rank de-composition,linear space dimension,and elementary transformation of generalized matrix.

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设RW n是有限链[n]上的正则保序且压缩奇异变换半群。对任意的r(2≤r≤n-1),考虑半群W(n,r)={α∈RW n:|Imα|≤r}的非群元秩和非幂等元秩。证明了:W(n,r)是由秩为r的元素生成的;确定了当1≤l≤r时,半群W(n,r)关于其理想W(n,l)的相关秩。
Let RWn be the semigroup of all regular order-preserving and compressing singular transformations on a finite-chain [n].For an arbitrary integer r(2≤r≤n-1), the non-group rank and non-idempotent rank of the semig-roup W(n,r)={α∈RWn:|Imα|≤r} were studied.The semigroup W(n,r) generated by elements of rank r is proved.Furthermore, it is shown that for 1≤l≤r, the relative rank of the semigroup W( n,r) with respect to itself each ideal W( n,l) .

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提出了三维装配约束求解中雅克比矩阵近似更新的方法。该方法通过对迭代过程中满秩以及行秩秩亏雅克比矩阵进行近似更新,提高了约束求解的效率。首先在非线性迭代求解过程中添加雅克比矩阵及其逆矩阵近似更新的公式;然后给出使用近似更新公式需要满足的限制条件;最后通过对奇异点扰动算法的描述介绍迭代求解过程中雅克比矩阵发生行秩秩亏的处理办法。文中提出的策略与算法已在三维装配约束求解引擎CBABench中实现,给出的实例表明本文提出的方法效果显著。
A new method of approximately updating Jacobian matrix during 3D assembly constraint solving is proposed in this paper. This method principally improves the efficiency of constraints solving based on the approximate update of Jacobian matrix. First, an approximate update formula of Jacobian matrix and its inverse matrix are inserted to the non-linear iterative solution process. After that, the indispensable constraint is put forward, which must be satisfied when using the formulas above. At last, a solution handling row rank defect of Jacobian matrix is introduced via disturbance algorithm description. The methodology presented is implemented in a 3D assembly constraint solving engine, named CBABench. An example given at the end of this paper shows that the method has achieved a considerable effect.

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将鲁棒主成分分析、矩阵补全和低秩表示统称为低秩矩阵恢复, 并对近年来出现的低秩矩阵恢复算法进行了简要的综述。讨论了鲁棒主成分分析的各种优化模型及相应的迭代算法, 分析了矩阵补全问题及求解它的不精确增广拉格朗日乘子算法, 介绍了低秩表示的优化模型及求解算法。最后指出了有待进一步研究的问题。
This paper collectively referred robust principal component analysis, matrix completion and low-rank representation to as low-rank matrix recovery, and made a brief survey on the existing algorithms of low-rank matrix recovery. Firstly, it discussed various optimization models and the corresponding iterative algorithms for robust principal component analysis. Next, it analyzed the matrix completion problem and proposed the inexact augmented Lagrange multipliers algorithm to solve the problem. In addition, it introduced the optimization models for the low-rank representation problem and presented the iterative algorithm. Finally, this paper discussed several problems which need further research.

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基于布尔函数与形态算子关系,用秩函数取代结构化映射中的布尔函数,通过选取不同秩函数阈值的方法,对二值形态变换进行扩展和推广。提出了具有调节秩函数阈值的广义二值形态变换理论,以期为形态算子的应用以及新算法的研究提供新的思路。
Based on the relationship between the Boolean functions and morphological operators ,the Boolean functions in the structural mapping are replaced by the rank function . By choosing the threshold of the rank function , the binary morphological transforms are extended . We hope the generalized morphological transform theory with the adjustable rank function threshold can provide an innovative idea for the researches of morphological operators and algorithm .

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分析了初等变换方法求矩阵的秩、利用初等变换求矩阵的秩与高斯消元法解线性方程组，向量组的线性表示．向量组的线性相关性的相通性原理，将初等变换求秩应用在以上方面，既解决了三个问题的求解判断，更将知识融会贯通．紧密联系在一起，为以后相关知识的学习奠定基础。
This paper analyzes the method of calculating matrix rank with elementary transformation, utilizes the method and Gauss elimination to solve linear equations, analyzes the linear representation of vector group, the similarity principle of the linear relevance of vector group, and applies the method of calculating matrix rank with elementary transformation to the above three aspects. It can not only settle the solving and judgment of the three problems, but also integrate the learned knowledge, so as to lay a foundation for future learning.

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