一类试验次数为4p2的非对称正交表的构造

Nowadays orthogonal arrays play important roles in statistics,computer science, coding theory and cryptography.The usual difference matrices are essential for the con- struction of many mixed orthogonal arrays.But there are also many orthogonal arrays, especially mixed-level or asymmetrical which can not be obtained by the usual difference matrices.In order to construct these asymmetrical orthogonal arrays,a class of special matrices,so-called generalized difference matrices,were discovered by Zhang(1989,1990, 1993) by the orthogonal decompositions of projective matrices.In this article,an interesting equivalent relationship between the orthogonal arrays and the generalized difference matri- ces is presented.As an application,a family of orthogonal arrays of run sizes 4p~2,such as L_(36)(6~13~42~(10)),are constructed.     

Orthogonal arrays obtained by generalized difference matrices with g levels

Nowadays orthogonal arrays play important roles in statistics, computer science, coding theory and cryptography. The usual difference matrices are essential for the construction for many mixed orthogonal arrays. But there are also orthogonal arrays which cannot be obtained by the usual difference matrices, such as mixed orthogonal arrays of run size 60. In order to construct these mixed orthogonal arrays, a class of special so-called generalized difference matrices were discovered by Zhang (1989,1990,1993,2...     

Orthogonal arrays constructed by generalized Kronecker product

In this paper, we propose a new general approach to construct asymmetrical orthogonal arrays, namely generalized Kronecker product. The operation is not usual Kronecker product in the theory of matrices, but it is interesting since the interaction of two columns of asymmetrical orthogonal arrays can be often written out by the generalized Kronecker product. As an application of the method, some new mixed-level orthogonal arrays of run sizes 72 and 96 are constructed.     

Orthogonal arrays obtained by repeating-column difference matrices

In this paper, by using the repeating-column difference matrices and orthogonal decompositions of projection matrices, we propose a new general approach to construct asymmetrical orthogonal arrays. As an application of the method, some new orthogonal arrays with run sizes 72 and 96 are constructed.     

Generalized latin matrix and construction of orthogonal arrays

In this paper, generalized Latin matrix and orthogonal generalized Latin matrices are proposed. By using the property of orthogonal array, some methods for checking orthogonal generalized Latin matrices are presented. We study the relation between orthogonal array and orthogonal generalized Latin matrices and obtain some useful theorems for their construction. An example is given to illustrate applications of main theorems and a new class of mixed orthogonal arrays are obtained.     

正交平衡区组设计统计模型中的方差分析

正交平衡区组设计(或广义正交表)的数据分析类似于正交拉丁方(或正交表)的数据分析.利用类似于正交表数据分析中的投影矩阵的正交分解技术,研究正交平衡区组设计的统计分析模型,给出了方差分析中的二次型以及各因子的二次型的分布性质,从而给出正交平衡区组设计统计模型中的方差分析方法.     

Sequences and arrays involving free variables

Abstact: Sequences in free variables are introduced and used to construct arrays in free variables which are suitable for circulant matrices. Most of the arrays found are maximal in the number of free variables. Applications include many new Goethals‐Seidel type arrays and complex orthogonal designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 17–27, 2001     

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.     

Generalized Riordan arrays

In this paper, we generalize the concept of Riordan array. A generalized Riordan array with respect to c_n is an infinite, lower triangular array determined by the pair (g(t),f(t)) and has the generic element d_n,k=[tⁿ/c_n]g(t)(f(t))^k/c_k, where c_n is a fixed sequence of non-zero constants with c₀=1.We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.     

广义Nekrasov矩阵的细分迭代判别法

利用细分矩阵下标集合的思想,构造递进系数和特殊的正对角矩阵,结合不等式的放缩,给出广义Nekrasov矩阵的迭代判别法,推广和改进了已有相关结果,并用数值实例说明了所得结果的优越性.     

一类正交投影矩阵及其相关正交表

本文给出了一类正交投影矩阵及其相关的强度2正交表.使用这些正交投影矩阵和正交表,我们提供了一种构造正交表的方法,并且构造了一些混合水平正交表.     

k-广义Hermite矩阵

本文给出了k-广义(反)Hermite矩阵的概念,研究了它的性质及其与k-广义酉矩阵之间的联系,推广了酉矩阵和(反)Hermite矩阵的相应结果.     

非负广义幂等矩阵

非负幂等矩阵在［１］中已有详细地讨论和重要地应用。［２］给出了Ａ≥０，Ａｋ＋１＝Ａ，ｋ是正整数，Ａｋ是零对称幂等矩阵的充要条件。本文给出了一般非负广义幂等矩阵的充要条件。并讨论了广义幂等矩阵的秩与其广义幂等指数的关系，推广了［２］中的部分结论。     

局部环上幂等矩阵线性组合广义逆之间的关系

设R是2为单位的局部环.研究了R上三个两两可换的n阶非零幂等矩阵的线性组合广义逆之间的包含关系,确定了R上一类特殊矩阵广义逆的列表算法.利用这种列表算法和相关的矩阵理论,得到了这些矩阵线性组合广义逆之间的包含关系的充要条件,推广了矩阵自反广义逆的逆反律的相关结果.     

Generalized orthogonal arrays: Constructions and related graphs

Generalized orthogonal arrays were first defined to provide a combinatorial characterization of (t, m, s)-nets. In this article we describe three new constructions for generalized orthogonal arrays. Two of these constructions are straightforward generalizations of constructions for orthogonal arrays and one employs new techniques. We construct families of generalized orthogonal arrays using orthogonal arrays and provide net parameters obtained from our constructions. In addition, we define a set of graphs associated with a generalized orthogonal array which provide information about the structure of the array. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 31–39, 1999     

New orthogonal designs of order 56

Amicable sets of eight matrices are very useful in the construction of orthogonal designs using the Kharaghani array. In this article we use a simple procedure to construct many new amicable sets of eight matrices of order 7 and then new orthogonal designs of order 56. Some of these are restricted to be short amicable sets of two or four circulant matrices.. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 387–393, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10030     

Out-of-band quasiseparable matrices

We study structured matrices which consist of a band part and quasiseparable parts below and upper the band. We extend algorithms known for quasiseparable matrices, i.e. for the case when the band consists of the main diagonal only, to a wider class of matrices. The matrices which we consider may be treated as an usual quasiseparable matrices with larger orders of generators. Hence one can apply the methods developed for usual quasiseparable matrices and obtain various linear complexity O(N) algorithms. However in this case the coefficients in N in the complexity estimates turns out to be quite large. In this paper we use the structure more accurately by division of the matrix into three parts in which the middle part is the band instead of diagonal as it is used for usual quasiseparable matrices. This approach allows to use better the structure of the matrix in order to improve the coefficients in N in the complexity estimates for the algorithms. This method works for algorithms which keep invariant the structure.     

α-对角占优与广义严格对角占优矩阵的判定

1 引言与记号 广义严格对角占优矩阵在数学、物理、控制论及经济学等许多领域有着重要的研究价值和实用价值.广义严格对角占优矩阵就是非奇异日一矩阵,它是一类范围很广的特殊矩阵,熟知的严格对角占优矩阵,不可约对角占优矩阵,非奇异M-矩阵等都是其特殊情形.如何在实际应用中简便地判别一个矩阵是否是日一矩阵,一直是人们关注的问题.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

实正定矩阵与Minkowski不等式的再推广

本文给出实广义正定矩阵概念的新推广及其基本性质,讨论它及常见几种定义下广义正定矩阵的代数结构,得到非对称正定矩阵乘积的一个新刻画,并利用所获广义正定矩阵的性质,拓广了Minkowski,OstrowskiTaussky等矩阵不等式的取值范围.