关于矩阵张量积的谱

<正> 设 A=(a_(jk))_(m_1×m_2),B=(b_(jk))_(n_1×n_2),则 m_1n_1×m_2n_2矩阵(?)称为 A 与 B 的张量积(也称直积或 Kronecker 积).矩阵的张量积是多重线性代数的重要工具之一,在群表示论中也有重要应用.本文的主要目的是在作者过去工作的基础上给出矩阵张量积的一些谱性质.     

n次对称群S_n的元素的阶的集合

<正> 我们试图给出 S_n 的元素的阶所构成的集合,为此,要用到下列的基本事实:1°任一个 n 个文字的置换可以分解为不相连的(即彼此无公共文字)循环置换的乘积.2°两个不相连的循环置换可交换.3°k 个文字的循环置换(a_1,a_2,…a_k)的阶为 k.4°群 G 的元素 g_1,g_2,…,g_s 的阶分别为 m_1,m_2,…,m_s,这些元素两两可换,这些阶数两两互质,则积 g_1g_2…g_s 真的阶为 m_1m_2…m_s.5°群 G 的元素 g_1,g_2…,g_s 的阶分别为 t_1,t_2,…,t_s,这些元素两两可换,则积g_1g_2…g_s 真的阶为 t_1,t_2,…t_s 的最小公倍数的因数.     

n维空间中方体表面的最短路

1983年在TYCMJ杂志上有人提示了3维空间中方体表面最短路的计数问题。1985年李慰萱和王子侠解决了这一问题。本文将这一问题推广到n维空间(n≥1)。设m_1,m_2,…,m_n为正整数,n维空间中的点集Ω={(x_1,x_2,…,x_n)|0≤x_i≤m_i,i=1,2,…,n}称为是一个m_1×m_2×…×m_n方体。先约定n≥2。     

具有超前和滞后的泛函微分方程的周期解

考虑具有超前和滞后的泛函微分方程的ω-周期解的存在性问题,其中L_i,R_j,φ_k,ψ_k:R→R(i=1,…,m_1,j=1,…1,…,m_2,k=1,…,m_3)是连续的ω周期函数,D_i:R~2→R~(n×n)连续,关于t以ω为周期;f:R×R~n×…×R~n→R~n连续,关于t以ω为周期;m_1,m_2,m_3为正整数,ω为正常数。 近些年来,人们利用Liapunov第二方法研究常微分方程和具有有限滞后或无限滞     

一个递推公式及其应用

本文介绍一个递推公式及其在解题中的广泛应用。1 递推公式设F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥0,n∈Z),构造以x_1,x_2,…,x_k为根的方程: x~k+m_1x~(k-1)+m_2x~(k-2)+…+m_k=0 我们称这个方程为F(n)的特征方程,则F(n)=a_1x_1~n+a_2x_2~n+…+a_kx_k~n(n≥k,x∈Z)满足下列递推公式:     

关于乘法分拆的计数函数

本文讨论了乘法分拆的计数函数 g(n)并对 g(n)的均值作了下界的估值。一 引言考虑集合 T(n)={(m_1,m_2,…,m_s);n=m_1m_2…m_s,m_i>1,1≤i≤s},此处不计m_1,m_2,…,m_s 的次序。我们定义 g(n)=|T(n)|并且 g(1)=1。例如 g(24)=7,因为24=3·8=3·4·2=3·2·2·2=6·4=6·2·2=12·2.在1983年,John F.Hughes 和 J.O.Shallit 证明了 g(n)≤2n 2~(1/2)     

圆内接六边形相对顶点联线共点的充要条件及应用

命题:圆内接六边形ABCDEF的边长AB、BC、CD、DE、EF、FA依次记为a_1、a_2、a_3、a_4、a_5、a_6,则其相对顶点联线AD、BE、CF相交于一点的充要条件为a_1a_3a_5=a_2a_4a_6。证明:1°必要性如图(1)若AD、BE、CF相交于O,依立于同弧的圆周角有△OAB∽△OED(?) a_1/a_4=OA/OE ①同理 a_5/a_2=OE/OC ② a_3/a_6=OC/OA ③①×②×③得 a_1a_3a_5=a_2a_4a_6。 2°充分性若 a_1a_3a_5=a_2a_4a_6。     

多重Toeplitz矩阵与多重Hankel矩阵相乘的复杂度

1.二重Toeplitz矩阵相乘的快速算法nm阶方阵 称为nm型2重Toeplitz矩阵,其中A_i(i=-n+1,…,n-1)为m阶Toeplitz矩阵. 定义.设p_1×p_2矩阵A=(a_(ij))_(p_1×p_2),B为q_1×q_2矩阵.称p_1q_1×p_2q_2矩阵     

有限域上Carlitz方程的解数

设F_q是含有q个元素的有限域,其中q=p~t,t≥1,p是一个奇素数.研究了Carlitz方程的推广形式(a_1x_1~(m_1)+…+a_nx_n~(m_n)+a_(n+1)x_(n+1)~(m_(n+1))+…+a_(n+s)x_(n+s)~(m_(n+s)))~k=bx_1~(k_1)…x_n~(k_n),其中ai,b∈F_q~*,s≥1,n≥1.当方程变量的指数满足一定条件时,得到了方程的解数公式.     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

Powers of general digital sums

Let m0,m1,m2,…be positive integers with mi〉 2 for all i. It is well known that each nonnegative integer n can be uniquely represented as n= a0 ＋ a1m0＋a2m0m1＋…＋atm0m1m2…mt-1,where 0≤ai≤mi-1 for all i and at≠0.let each fi be a function defined on {0,1,2…,mi-1} with fi（0）=0.write S（n）=i=0∑tfi（ai）.In this paper, we give the asymptotic formula for x^-1∑n≤xS（n）^k,where k is a positive integer.     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

Constructing 2 × 2 Bricks from Unitary Numerical Semigroups

This paper investigates numerical semigroups that yield 2 × 2 bricks. We demonstrate the existence of an infinite family of 2 × 2 bricks that includes all of the perfect 2 × 2 bricks. We provide a formula for the Frobenius numbers of these semigroups as well as a necessary and sufficient condition for the semigroups to be symmetric.     

Onq-projectively recurrent spaces

In questa nota sono studiati gli spazi proiettivamenteq-ricorrenti, definiti dalla relazione % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb \(W_{ijk, m_1 m_2 \ldots m_q }^p = a_{m_1 m_2 \ldots m_q } W_{ijk}^p \) ; doveW _ijk ^p sono le componenti del tensore proiettivo di Weyl. Nei teoremi 4 e 7 è data una forma canonica del tensore di curvatura di questi spazi. È quindi dimostrato che uno spazio di questo tipo non può essere uno spazio di Einstein. Sono pure studiati gli spazi proiettivamenteq-ricorrenti che nello stesso tempo sono conformemente Euclidei e il casoq=2.     

複合形在歐氏空間中的實現問题Ⅰ

<正> 在拓撲發展之初很早就知道一個抽象的n維單純複合形(有限或無限)必可在2n+1維歐氏空間及R~(2n+1)中得到實現,它的證明也很簡單(例如見[1]§2或[2]第Ⅲ章§2).從這一定理知道2n+1維的歐氏空間實際上已包括了所有想像得到的n維複合形,可是是否有不能在R~m中實現但能在R~(m+1)中實現的     

On the regular summability of multiple function series and Lebesgue functions

Основной целью работ ы является обобщение одного результата Кратца и Т раутнера [4], известного для одном ерных функциональны х рядов, на кратные ряды. Этот рез ультат касается суммируемо сти функционального ряда почти всюду при слабых пред положениях. В частности, он примен им к суммируемости по Чезаро и по Риссу. Мы рассматриваемd-кр атный ряд $$\mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x), \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty c_{k_1 ,...,k_d }^2< \infty $$ и предполагается, что функции \(f_{k_1 ,...,k_d } (x)\) интегрируе мы по пространству с полож ительной мерой и имеют почти вс юду ограниченные фун кции Лебега для метода суммирова ния Т. Метод Т определяетсяd-мерной матрицей \(T = \{ a_{m_1 ,...,m_d ;k_1 ,...,k_d } \} \) сл едующим образом: $$t_{m_1 ,...,m_d } (x) = \mathop \sum \limits_{k_1 = 0}^\infty \cdots \mathop \sum \limits_{k_d = 0}^\infty a_{m_1 ,...,m_d ;k_1 ,...,k_d } c_{k_1 ,...,k_d } f_{k_1 ,...,k_d } (x).$$ Эти средние существу ют, поскольку мы предп олагаем, что \(a_{m_1 ,...,m_d ;k_1 ,...,k_d } = 0\) ,если max(k ₁,...,k _d) достаточно вели к (в зависимости, конеч но, отm ₁,...,m _d). При некоторых дополнительных усло виях на матрицуТ (см. (7)– (9) в разделе 3) устанавлива ется почти всюду регулярная схо димость средних \(t_{m_1 ,...,m_d } (x) \user2{} \user2{(}m_1 \user2{,}...\user2{,}m_d \user2{)} \to \infty \) . Как вспомогательный результат, в работе об общается теорема Алексича [1] о сх одимости почти всюду некоторы х подпоследовательн остей частных сумм функцио нального ряда.     

基于最小二乘法的张量积Said-Ball曲面降多阶逼近

给出张量积Said-Ball曲面降多阶逼近的一种方法.该方法根据原张量积Said-Ball曲面Pn,m(u,v)与降多阶张量积Said-Ball曲面Qn1,m1(u,v)(n1≤n-1,m1≤m-1)在最小二乘范数下的距离函数在单位正方形[0,1]×[0,1]上取最小值,从而得到了用矩阵表示的降多阶张量积Said-Ball曲面Qn1,m1(u,v)的控制顶点{qij}in1=,0,m1j=0的显示表示式.在降多阶过程中,分别考虑了带角点高阶插值条件和不带角点插值条件的情形.文末附有数值例子,并将本文方法与参考文献(9)的方法做了比较.     

论数论函数ψ(n),σ(n)及d(n)的一些性质

<正> 命 f(n)为一数论函数.关于函数比值(?)的分布问题,Soma-yajulu,Sierpi(?)ski 及 Schinzel 曾用算术的方法,对于ω(n),σ(n)及 d(n)加以处理.华罗庚教授首先指出用 Brun 节法处理这一类问题的途径.按这一方向,作者与     

Packing boxes with harmonic bricks

Given a box of integral dimensions and a supply of bricks all having the same integral dimensions, what is the largest number of bricks that can be packed in the box with the sides of the bricks parallel to the sides of the box? In this general form the question is very difficult. We can think of the box as being made up of unit cells and say that a set R of cells is a representing set for a brick of given dimensions provided every brick that can be placed in the box, sides parallel to the box, contains at least one of the cells in R. The maximum number of bricks that can be placed in the box is then less than or equal to the minimum cardinality of a representing set. In general, there is not equality. In the case of two dimensions and a harmonic brick, we prove there is equality always and exhibit a best packing. For three-dimensional boxes and a harmonic brick, there need not be equality. We derive several results which are of the nature that if certain inequalities relating the dimensions of the box and the brick are satisfied, then equality occurs. Our results are strong enough to imply, for example, that if the smallest face of the brick packs each face of the box perfectly, then there is equality. For a 1 × 2 × 4 brick, there is always equality if one of the dimensions of the box is even.