矩阵幂和问题的进一步讨论

本文证明了;(1)F_p~m上p~m次幂矩阵的充要条件;(2)F_p~m上任一方阵都可表示为2个其最小多项式均无重因式的q次幂矩阵之和;(3)任一整数方阵可表示成不超过7个平方次幂整数矩阵之和,从而推广和改进了文[1,2]的结果.     

关于矩阵幂和的Newman问题

M.Newman[2]提出以下几个未解决的问题:(1)在 F_2上,确定全体 n 阶平方次幂矩阵的数目。(2)在整数环上,对任意的 n,确定最小的整正数 M(n),使任一 n 阶方阵都可表示成 M(n)个平方次幂矩阵之和。(3)把以上问题推广到高次幂。本文分别讨论上述问题,得到如下结果:(1)给出全体平方矩阵计数公式。(2)对任一整数矩阵,若它可以有理标准化,则可表示成4个平方次矩阵之和。这与数论中著名的 Lagrange 定理[4]相吻合。(3)在域 F_p 上,任一 n 阶方阵都可表示2个 p 次幂矩阵之和。     

两个素数平方、四个素数立方和2的整数幂*

本文中, 作者考虑了 Linnik 型的非齐次幂的Waring-Goldbach问题.具体地说, 作者证明了所有充分大的偶数都可以表示成两个素数的平方、四个素数的立方和18个2的正整数幂之和的形式.这改进了Zhao的结果, 即需要43个2的正整数幂.     

一些特殊整数的分拆问题

本文的主要目的是利用初等及解析方法证明对任意满足条件P≡1(mod 3)的素数P,6p可以表示成三个整数的平方之和;18p~((2))可以表示成三个整数的四次方之和.即存在整数x,y及z使得6p=x~((2))+y~((2))+z~((2));存在整数u,υ及w使得18p~((2))=u~((4))+v~((4))+w~((4)).同时借助于Legendre符号在多项式上的求和给出了x,y,z以及u,v,w的具体表示形式.     

整数方阵的Waring问题

本文证明了:对任意给定的整数 m≥2和 n≥2,存在正整数 f(n,m),使得任何 n 阶整数方阵均可表示为 f(n,m)个整数方阵的 m 次幂之和.并对 f(n,m)作了估计,从而推广、改进了 M.Newman 1985年的结果.     

保持二次算子的可加映射

设X是具有无限重复度的无限维或维数不小于3的有限维复Banach空间,B(X)是X上全体有界线性算子组成的Banach代数.首先证明了单位算子不能表示成3个平方幂零算子之和,利用算子分块矩阵技巧获得了平方幂零算子的本质特征.以此特征为基础,刻画了B(X)上双边保持二次算子可加满射的结构.     

证明非平方数的若干方法

如果某数是一个整数的平方,那么称某数是完全平方数,简称平方数。否则称为非平方数。本文介绍证明一个整数是非平方数的若干方法,不当之处,请批评指出, 一、间隔法根据“在两个相邻整数的平方之间的任一整数都不是平方数”。要证明整数M是非平方数,只须证明M在两个相邻整数的平方之间。     

应用(a±b)~2=a~2±2ab+b~2速算整数的平方

熟记11～25各整数的平方,是速算整数平方的基础。因而本文是在能熟记11～25各整数平方的基础上展开讨论的。一 25～100各整数的平方 1° 25～75各整数的平方 25～50的二位数可表示为50-a(a∈Z,且a≤25),a叫做该二位数对于50的补数; 50～75的二位数可表示为50+a(a∈Z,且a≤25),a叫做该二位数对于50的过数。 (50±a)~2,=50~2±2×50·a+a~2=2500±100a+a~2=(25±a)×100+a~2=〔(50±a)-25〕×100+a~2。上式说明:求25～75各整数的平方,可先求该数与25之差的100倍,再加上补数或过数的平方。     

柯召、孙琦—单墫定理的拓广

一、引言人们容易证明任意3个整数中必有两个整数之和为2整除,任意5个整数中必有3个整数之和为3整除,柯老和孙琦教授在[1]中证明了任意7个整数中必有4个整数之和为4整除,并猜测任意2n-1(n>1)个整数中必有n个整数之和能为n整除。1983年单墫     

整数矩阵除数函数在无平方因子数集上的均值估计

利用解析数论的经典方法,研究了整数矩阵除数函数在无平方因子数集上的均值,并得到了渐近公式,推广了相关结果.     

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.     

Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order

A magic square is a square matrix whereby the sum of any row, column, or any one of the two principal diagonals is equal. A surrogate of this abstract mathematical construct, introduced in 2012 by Fahimi and Jaleh, is the “electrostatic potential (ESP)” that results from treating the matrix elements of the magic square as electric charges. The overarching idea is to characterize patterns associated with these matrices that can possibly be used, in the future, in reverse to generate these squares. This study focuses on squares of order 4 and 5 with 880 and 275,305,224 distinct (irreducible/unique) realizations, respectively. It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares. The electrostatic potentials for natural magic squares exhibit a striking pattern of maxima and minima in all distinct 880 of the 4th order and all distinct 275,305,224 of the 5th order matrices. The minimum values of ESP of Dudeney groups are discussed. Equipotential points and certain constants are found among the ESP sums along horizontal and vertical lines on the square lattice. These findings may help to open a new perspective regarding magic squares unsolved problems. While mathematics often leads discovery in physics, the latter (physics) is used here to detect otherwise invisible patterns in a mathematical object such as magic squares.     

論矩陣的一種變換

<正> 在本文中,數域限定為複數域.我們要來研究如下的變換:(1)(它將方陣A變成方陣B),式中P表示任意正則陣,P表示P的元素的共軛救構成的陣.所有的變换(1)顯然成羣.這種變換現在姑稱之為種變換.如果二方陣A與B可由一個種變換變此成彼,我們就說,A與B是對相似的.     

Complexité et circuits eulériens dans les sommes tensorielles de graphes

The paper starts from two general remarks about the spectra of tensorial products and tensorial sums of square matrices. Since some classical results about spanning trees and Eulerian circuits may be reformulated in terms of eigenvalues of (what is called here) the second associated matrix, the initial remark about tensorial sums of matrices leads to natural applications to the enumeration of spanning trees and Eulerian circuits in tensorial sums (sometimes simply called “products”) of graphs. Examples and numerical results are given, in particular for the complexity of tensorial sums of chains and/or cycles and for the enumeration of the Eulerian circuits of the tensorial sum of two circuits.     

Sums of nilpotent matrices

We study matrices over general rings which are sums of nilpotent matrices. We show that over commutative rings all matrices with nilpotent trace are sums of three nilpotent matrices. We characterize 2-by-2 matrices with integer entries which are sums of two nilpotents via the solvability of a quadratic Diophantine equation. Some exemples in the case of matrices over noncommutative rings are given.     

Maximum and minimum weighted diagonal sums of certain non-negative matrices

This paper extends the notion of diagonal sums of a square matrix to “weighted diagonal sums”. Using simple probabilistic arguments, most of the results of Wang [5] concerning the maximum and minimum diagonal sums of doubly stochastic matrices are extended to maximum and minimum weighted diagonal sums of stochastic matrices (Sec. 3). Two stronger versions of one of Wang's conjectures are also proven (Theorems 4.1 and 5.1), of which the latter easily generalizes to the case of non-negative matrices (Theorem 5.2). The paper ends with a few open questions and counter-examples.     

The centro-invertible matrix: A new type of matrix arising in pseudo-random number generation

This paper defines a new type of matrix (which will be called a centro-invertible matrix) with the property that the inverse can be found by simply rotating all the elements of the matrix through 180 degrees about the mid-point of the matrix. Centro-invertible matrices have been demonstrated in a previous paper to arise in the analysis of a particular algorithm used for the generation of uniformly-distributed pseudo-random numbers.An involutory matrix is one for which the square of the matrix is equal to the identity. It is shown that there is a one-to-one correspondence between the centro-invertible matrices and the involutory matrices. When working in modular arithmetic this result allows all possible k by k centro-invertible matrices with integer entries modulo M to be enumerated by drawing on existing theoretical results for involutory matrices.Consider the k by k matrices over the integers modulo M. If M takes any specified finite integer value greater than or equal to two then there are only a finite number of such matrices and it is valid to consider the likelihood of such a matrix arising by chance. It is possible to derive both exact expressions and order-of-magnitude estimates for the number of k by k centro-invertible matrices that exist over the integers modulo M. It is shown that order (√N) of the N=M^(k2) different k by k matrices modulo M are centro-invertible, so that the proportion of these matrices that are centro-invertible is order (1/√N).     

Trace forms, kronecker sums, and the shuffle matrix

The trace map on the ring of square matrices with entries in a field can be used to define various quadratic forms on this ring. This paper makes a study of some of these forms and in particular the "scaled trace forms" are shown to have a symmetric matrix representation involving both Kronecker sums and the shuffle matrix.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.