【例1】

## 数学内蕴于事物

１，给每只羊起一个名字，并因为某个个性特征而永远不会把名字给搞混淆了，那么我们晚上把羊赶回羊圈时，可以站在门口，一只一只地把回来的羊与记忆当中的那些具有个性特征的羊对上号，最后，我们可以知道，是不是丢失了羊；
２，数数，记住这群羊一共有多少只，然后在羊圈里面，再点一次数，就知道是不是丢失了羊。

## 流形的基本概念|Basic Notions of Manifold

Basic Notions of Manifold

When we mention Euclid space, maybe it induct excess meanings. So,
we can try to construct an object that its meaning is less than
Euclid space but however, is meaningful and analyzable.

At least, we need some points, then, if we want a point is
analyzable, it must has its neighborhood, moreover, to make it
computable or realizable, let its neighborhood homeomorphism with
an open set of Euclid space.

Such points constitutes a set, then make this set is a separated
set, a Hausdorff set, we definite such a set a manifold.

The point x\in M, M is a manifold. U is a neighborhood of x, and
there is a homeomorphism \phi:U\mapsto \phi(U). then, Any such a
$(U,\phi(U))$ means a realization of any point of U: $\phi(y)$ is

## 最基本的代数观念-Module

[t]a\equiv b(mod c)[/t]

$$modc:a\mapsto b$$

$$modc: xy\mapsto bd.$$

=>任何多项式在此映射下是不变的.

Fermat Theorem:[t]$a^{p-1}-1\equiv0(mod p)$[/t]

Euler Theorem:$$a^{k}-1\equiv0(mod b)$$