广义超度量矩阵的封闭性质

本文研究了广义超度量矩阵的封闭性质.证明了若A为非奇异的广义超度量矩阵,则A与A的转置的Hadamard积仍然是一个广义超度量矩阵,并且它的逆矩阵是一个对角占优的M矩阵.给出了两个广义超度量矩阵Hadamard积封闭的一个充分条件.最后,讨论了广义超度量矩阵的Perron补与和的封闭条件.     

On Circulant Complex Hadamard Matrices

We show that a circulant complex Hadamard matrix of order n is equivalent to a relative difference set in the group C ₄×C _n where the forbidden subgroup is the unique subgroup of order two which is contained in the C ₄ component. We obtain non-existence results for these relative difference sets. Our results are sufficient to prove there are no circulant complex Hadamard matrices for many orders.     

广义循环矩阵的一个性质

可表为非奇异对角矩阵和循环矩阵乘积的矩阵,我们称其为广义循环矩阵.本文给出了单位矩阵与广义循环矩阵的和矩阵的非奇异的充要条件,得到了这样和矩阵的相对增益阵列的显示表达式.     

A Series of Regular Hadamard Matrices

Let p and 2p−1 be prime powers and p ≡ 3 (mod 4). Then there exists a symmetric design with parameters (4p², 2p² − p, p² − p). Thus there exists a regular Hadamard matrix of order 4p².     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

Cocyclic Generalised Hadamard Matrices and Central Relative Difference Sets

Cocyclic matrices have the form where G is a finite group, C is a finite abelian group and : G × G C is a (two-dimensional) cocycle; that is,

关于Hadamard乘积矩阵的一些性质的注记

对文 [1 ]的主要结论作了说明 ,给出 Hadamard乘积矩阵有关性质的更一般的结果 .     

Power Hadamard matrices

We introduce power Hadamard matrices, in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and perhaps as useful, as the objects that motivated them.We develop a basic theory of power Hadamard matrices, explore these relationships, and offer some new insights into old results. For example, we show that all 4×4 Butson Hadamard matrices are equivalent to circulant ones, and how to move between equivalence classes.We provide, among other new things, an infinite family of circulant Butson Hadamard matrices that extends a known class to include one of each positive integer order.Dedication: In 1974 Jennifer Seberry (Wallis) introduced what was then a totally new structure, orthogonal designs, in order to study the existence and construction of Hadamard matrices. They have proved their worth for this purpose, and have also become an object of interest for their own sake and in applications (e.g., [H.J.V. Tarok, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (1999) 1456-1467. [26]]). Since then many other generalizations of Hadamard matrices have been introduced, including some discussed herein. In the same spirit we introduce a new object showing this kind of promise.Seberry's contributions to this field are not limited to her own work, of which orthogonal designs are but one example—she has mentored many young mathematicians who have expanded her legacy by making their own marks in this field. It is fitting, therefore, that our contribution to this volume is a collaboration between one who has worked in this field for over a decade and an undergraduate student who had just completed his third year of study at the time of the work.     

Zero-Term Ranks of Real Matrices and Their Preservers

Zero-term rank of a matrix is the minimum number of lines (rows or columns) needed to cover all the zero entries of the given matrix. We characterize the linear operators that preserve zero-term rank of the m× nreal matrices. We also obtain combinatorial equivalent condition for the zero-term rank of a real matrix.     

Hadamard Matrices and Dihedral Groups

Let D _2p be a dihedral group of order 2p, where p is an odd integer. Let ZD _2p be the group ring of D _2p over the ring Z of integers. We identify elements of ZD _2p and their matrices of the regular representation of ZD _2p . Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD ₆. We discuss relation of ZD _2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.     

Hadamard and Conference Matrices

We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative -difference set where n – 1 is not a prime power.     

广义Nekrasov矩阵的充分条件

对任意给定的矩阵,从矩阵元素出发,利用定义构造特殊的正对角矩阵,结合不等式的放缩和数学归纳法,给出广义Nekrasov矩阵判定的三个充分条件,并用数值实例说明了所得结果的有效性.     

Necessary and Sufficient Condition for Generalized Diagonal Dominance Matrices

ＮｅｃｅｓｓａｒｙａｎｄＳｕｆｆｉｃｉｅｎｔＣｏｎｄｉｔｉｏｎｆｏｒＧｅｎｅｒａｌｉｚｅｄＤｉａｇｏｎａｌＤｏｍｉｎａｎｃｅＭａｔｒｉｃｅｓＹａｎｇＹｉｍｉｎ（杨益民）（ＡｎｈｕｉＭｅｃｈａｎｉｃａｌａｎｄＥｌｅｃｔｒｏｎｉｃＣｏｌｌｅｇｅ，Ｗｕｈｕ...     

广义Nekrasov矩阵的新判据

广义Nekrasov矩阵是一类应用广泛的特殊矩阵,通过构造不同的系数因子,结合不等式的放缩技巧,给出广义Nekrasov矩阵两个新的充分判据,改进和推广了近期文献的已有结果.     

Regular Hadamard matrices constructed from Hadamard 2-designs and conference graphs

Suppose there exists a Hadamard 2-$(m,\frac{m?1}{2},\frac{m?3}{4})$ design having skew incidence matrix. If there exists a conference graph on $2m?1$ vertices, then there exists a regular Hadamard matrix of order $4m^2$. A conference graph on $2m+3$ vertices yields a regular Hadamard matrix of order $4{(m+1)}^2$.     

Multilevel and multidimensional Hadamard matrices

Multilevel Hadamard matrices (MHMs), whose entries are integers as opposed to the traditional restriction to {±1}, were introduced by Trinh, Fan, and Gabidulin in 2006 as a way to construct multilevel zero-correlation zone sequences, which have been studied for use in approximately synchronized code division multiple access systems. We answer the open question concerning the maximum number of distinct elements permissible in an order n MHM by proving the existence of an order n MHM with n elements of distinct absolute value for all n. We also define multidimensional MHMs and prove an analogous existence result.      

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

四元数矩阵的加权广义逆

利用四元数矩阵的加权奇异值分解 ,给出了四元数矩阵加权广义逆的显式表达 ,简化了应瓦金给出的相应结果 ,同时解答了有关的公开问题 .