从(n 1)/n×(n 1)=(n 1)/n (n 1)谈起

一问题的提出本刊2003年第5期刊载了《运用发现法解题》(以下简称《解题》)一文,文章在谈到“归纳发现法”时,提到这样一个例子: 1.观察下列各式     

The n �� n KdV hierarchy

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of sl (n, \mathbbC){sl (n, \mathbb{C})} . The flows in the hierarchy include systems of coupled nonlinear Schr?dinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and B?cklund transformations for these hierarchies.     

Balanced path decomposition of λ K n,n and λ K n,n *

Let P _k denote a path with k edges and λ K _n,n denote the λ-fold complete bipartite graph with both parts of size n. In this paper, we obtain the necessary and sufficient conditions for λ K _n,n to have a balanced P _k -decomposition. We also obtain the directed version of this result.     

数列{n√n!/n}极限的简便求法

给出了数列{^n√n!/n}l的极限的一种简便求法。     

数列{n/n／n!}极限的教学价值

给出数列(n)/(n)/(n!)极限的八种灵活新颖的求解方法,进一步讨论了它的教学价值.     

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

The infinity of the rank of ergodic symmetric powers of automorphisms of the Lebesgue space is proved, and sharp upper bounds for their local rank are found.     

Minima of decomposable forms of degree n in n variables for n ≥ 3

The following theorem is proved: if for all (X0)one has ¦ F(x) ¦ >0, where F(x) is a decomposable form of degree n of n variables, then, for n 3, F(x) is proportional to an integral form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 142–154, 1990.     

级数∞∑n=1nαsin(nβ)的收敛性

通过推导给出了使级数∞∑n=1nαsin(nβ)收敛的α与β的值.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

二阶矩阵的n次方和n次方根

利用Chebyshev多项式讨论了二阶矩阵的n次方的各元素的表达式和其n次方根的存在条件.     

n王后问题

n 王后问题是高斯(Gauss)提出(n=8)的一个著名问题: 在 n×n 的国际象棋棋盘上摆上 n 个王后,使得每一个王后都不能攻击另外一个(王后可以横走、竖走,还可以沿着棋盘的两条对角线中任何一条的方向走)。这个问题是有意义的,它等价于求一个单纯图 G=(X,E)的极大稳固集,其中 G 有n~2个顶点并且如果顶点 x 和 y 处于同一行或同一列或处于与对角线平行的同一条直线上,则 y∈F_G(x)。(见文[1])。这个问题也等价于求满足一定条件的特殊置换类的问题。此外,有趣的是,1977年,Loren C.Larson 利用 n 王后     

n元等比级数

定义了n元等比数列、n元等比级数,给出了它们的通项公式及前n项和.并解决了多元等比级数的敛散性问题,求出了多元等比级数的和.指出了多元等比数列及多元等比级数在特殊情况下与一元等比数列及一元等比级数的一致性.     

n维立方体

文[1]介绍了盖尔范德（I．M．Gelfand）所著《代数》一书中如何深入浅出地数出4维立方体的顶点数、边数和面数的方法．读后倍受启发,将这种思想方法推而广之,我们可以研究任意n维空间中的n维立方体,并得到一些较为一般性的结论．1n维空间的m维面．设有序实数（x1,x2,...,xn）表示n维空间中的一个点,则n维空间即为集合Vn=｛（x1,x2,...,x。）|其中xiεR,i=1,2,...,n）．若（x1,...,xn）中n个坐标均为常数,则表示n维空间中的一个点,可称为0维面．若（x1,...,xn）中有（n-1）个坐标为常数,只有一个坐标为变量,则表示n…     

从求lim(n→ ∞)ln(((n!)~(1/n)/n))说起

求 limn→ ∞lnn n!n 是一个老题 ,同济大学数学教研室所编高等数学总习题 5中收入了该题 ,并将它与另外两个可作定积分定义来求的极限问题放在一起。各色各样题解之类的书也都用定积分定义来求这个极限 ,使我感到有必要说几句话。1 952年我开始工作 ,有个学生就曾以此题问过我。我学的熊庆来的《高等算学分析》书中有过用定积分定义求极限的问题 ,但要求所考虑函数在闭区间上连续。这个题所考虑的函数在闭区间上不连续 ,甚至还是无界的。不好办 !我告诉他 ,假设 lnx在 [0 ,1 ]上是连续的 ,倒可以由定积分的定义知道所求极限就是∫10lnxdx …     

Global Behavior of Solutions of x n + 1 = max {x n , A}/x n x n – 1

We investigate the asymptotic behavior, the oscillatory character, and theperiodic nature of solutions of the difference equation