摘要
建立3^2阶和(2k+1)^2(3■(2k+1))阶等内积性对角拉丁方正交偶的一般公式,引入矩阵的函数型Kronecker积的概念,以此为基础解决(2m+1)^2(m≥1)阶等内积性对角拉丁方正交偶的存在性和构造性问题.进而,(2m+1)^2(m≥1)阶二次幻方、加乘幻方的存在性和构造性问题得到彻底解决,且将构成两类幻方的数集拓广至二维等差矩阵.
For the cases of orders 3^2 and(2k+1)^2(3■(2k+1)),the general formulae for a pair of orthogonal diagonal Latin squares with the property of equi-scalar product have been established directly.We introduce a number of concepts of the functional Kronecker products which are used to obtain the existence and construction of orthogonal pair of diagonal Latin squares with the property of equi-scalar of order(2m+1)^2(m≥1).As an application,the existence and construction problems of quadratic magic squares and addition-multiplication magic squares of order(2m+1)^2 have been solved and the number sets constructing such kinds of magic squares can be extended to the arithmetic matrices in each direction.
作者
张世德
李程
张迪
ZHANG Shi-de;LI Cheng;ZHANG Di(College Mathematical and Information Science,Henan Normal University,Xinxiang 453007,China;Daxiang Press Co.Ltd.Zhengzhou 450044,China;Civil Safe Insurance Pic,Phnom Penh,Cambodia)
出处
《数学的实践与认识》
北大核心
2020年第24期215-230,共16页
Mathematics in Practice and Theory
关键词
等度方阵
矩阵的函数型Kronecker积
等内积对角拉丁方正交偶
二维等差矩阵
二次幻方
加乘幻方
equal multiplicity matrix
functional Kronecker product of matrices
orthogonal pair of diagonal Latin squares with the property of equi-scalar product
arithmetic matrix in each direction
quadratic magic square
addition-multiplication magic square
