If every fixed point of f is a interior point of T, then either every point of T is a periodic point of f with period r < n or fr has a turbulence for some natural number r < n.

In order to transform n-dimensions data into one-dimension data. The author first considers two sets: the natural number set N and the cartesian product N × N.

为了实现多维数据到一维数据的转换，本文首先考虑两个集合：自然数集N和笛卡尔积N×N。

On Sum of Four Prime Squares and k-TH Powr of Prime in Short Intervals

表自然数为四个素数的平方与一个素数的k次方之和的小区间问题

Researches on Stirling number appearing in power sum of natural numbers

自然数方幂和中的Stirling数研究

After the rotor passes its resonant zone, the phase angle does not increase any more, and it begins to fluctuate around a constant value ( 2n+ 1)π rad.

在远离临界转速的高速区，转子的相位角不再增大，而是围绕某一定值波动，此定值为（2n+1）πrad（n为自然数）；

This paper proves one of the two important limits with continuous function, which is different from the mathematicians' demonstration by treating x as natural number.

本文是对两个重要极限之一的证明，以往数学家们都是对x为自然数的证明，这里是以连续函数来进行证明。

It introduces the trigeminal liked list storage structure of binary tree for encoding, while binary tree traversal-based coding is used formerly. The natural numbers are expressed as the serial numbers of the node encoding.

该编码模式利用二叉树的三叉链表结构来存储XML文档树的结点，用自然数作为结点的编码序号。

Two extended Petri net models implementing the involution and the extraction of natural numbers

实现自然数M次乘方和开M次方计算的增广PETRI网模型

Probability relation between the natural number sequence and the Fibonacci sequence

Fibonacci序列与自然数序列的概率联系

Stirling's formula is that n, n!

Stirling公式为对于任意自然数n，n！

On Additive Partitions of Type ( a, b, k) of the Set of Natural Numbers

自然数集的（a，b，k）型可加划分

An edge-labelling f of a graph G is a 1-1 map from E ( G) into the natural numbers.

图G的边的一个标号f是指边集E（G）到自然数的子集的一个一一映射。

A recurrence formula of the coefficient of the sum of natural numbers power and the calculating formula of Bernoulli Number

自然数幂和公式系数的递推公式和有关Bernoulli数的计算公式

Some properties of the Fibonacci sequence are discussed and some probability relations between the Fibonacci sequence and the natural number sequence are obtained.

讨论了Fibonacci序列的一些性质，得到Fibonacci序列与自然数序列的某些概率联系。

Secondly, by using of the medium axiomatic set theory ( MS), a natural number system in MS is constructed, and it is proved that five axioms of Peano ′ s natural number system are theorems is MS.

其次，利用中介公理集合论MS的相关理论，构造了MS中的自然数系统，证明了Peano5条公理为MS中的定理。

Deduces the operational formula of the function between the third power Cartesian product of the natural number set N& N × N × N and N, and further researches into complex computation of countably infinite sets.

文中推导出了自然数集合N的三次笛卡尔积运算&N×N×N与自然数集合N之间的双射函数运算公式，对可数无穷集合的复杂计算作了进一步研究。

A Generalization of Preceding N-trmed Power Sum for Natural Number

自然数前n项方幂和推广

All of these indicate that the system of Peano ′ s natural number can be generated in MS and a theoretical foundation provided to prove that MS is a foundation of classical mathematics.

这表明Peano自然数系统能在MS中产生，为最终证明精确性经典数学能奠基于MS提供了理论基础。

Several precise expressions involving Stirling numbers as coefficients for the power sum of natural numbers are given. Some new properties of the Stirling number and counting formulae of the Bernoulli number are obtained.

给出了自然数方幂和的包含Stirling数的几种精确表示式，得到了Stirling数的一些新的性质和Bernouli数的计数式。

The Stirling Numbers of the Second Kind and Multiple Sums of Powers of Natural Numbers

第二类Stirling数与多重自然数方幂和

K Difference of k Power of Arbitrary k+ 1 Adjacent Natural Numbers Equal to k Factorial

任意k+1个相邻自然数k次方的k次差等于k阶乘