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[微分拓樸] 淺論 Manifold (2) - Manifold 的 Boundary 與 Regular point

回憶前篇,我們說一個 Manifold with boundary 定義如下:

==========================
Definition: Manifold with Boundary
集合 $M \subset \mathbb{R}^n$ 為 $k$-dimensional manifold of class $C^r$ 若下列條件成立:
對任意點 $p \in M$,存在鄰域 $U_p \subset \mathbb{R}^k$ open 或者 $U_p \subset \mathbb{H}^k$ open in $\mathbb{H}^k$ 且 $V_p \subset M$;與 coordinate patch $\alpha: U_p \to V_p$ 滿足
 (1) $\alpha \in C^r$
 (2) $\alpha^{-1} \in C^0$
 (3) $D \alpha$ 有 rank $k$
==========================


接著我們可以介紹 Manifold 的 Interior Point 與 Boundary point:

========================== Definition: Interior Point and Boundary Point of a Manifold
令 $M \subset \mathbb{R}^n$ 為 $k$-manifold 且 $p \in M$
我們說 $p$ 為 Manifold $M$ 中的 interior point 若下列條件成立:
對上述的 $p$ 而言,存在 coordinate patch $\alpha : U_p \to V_p$ 使得 $U_p$ 為 open in $\mathbb{R}^k$

反之,我們則稱此點 $p$ 為 boundary point
==========================
Comments:
上述的定義的 interior/boundary point 與 一般的 topology 中定義的 interior/ boundary point 不盡相同! 讀者須小心分辨


以下我們有個更好的判斷法來辨別是否為 interior point 或者 boundary point : =====…

[微分拓樸] 淺論 Manifold (1)

基本想法:
Manifold 一般譯為 "流形" ,本質上是作為  $\mathbb{R}^n$ 空間中 曲線 or 曲面 的進一步推廣。故我們可將 Manifold 視為 $\mathbb{R}^n$ 空間中的子集,這樣一來,整個 $\mathbb{R}^n$ 之上定義的概念 (極限、微分、積分) 都可以在 manifold上面做處理。

==============
Definition: k-dimensional manifold of class $C^r$ without Boundary
一個子集合 $M \subset \mathbb{R}^n$ 為 k-dimensional manifold of class $C^r$ without Boundary 若下列條件成立:對任意 $p \in M$ 存在兩個 open sets: $U_p \subset \mathbb{R}^k$ 與 $V_p \subset M$ ($V_p$ open in $M$) 且 存在 連續 bijection 函數 $\alpha:U_p \to V_p$ 滿足
 (i) $\alpha \in C^1$
 (ii) $\alpha^{-1}$ 為 連續
 (iii) $D \alpha$ 具備 rank $k$
==============

Comment: 
$\alpha: U_p \to V_p$ 一般稱為 coordinate patch。

以下我們看幾個例子:
-----------
Example 1: 1-manifold example
考慮 $M:=\{(x,y) \in \mathbb{R}^2: x^2 + y^2 = 1 \}$,試問此集合是否為 1-dimensional manifold of class $C^1$?
-----------

Proof:
要證明上述集合為 manifold 我們需要證明:
對任意 $p \in M$ 存在兩個 open sets $U_p \subset \mathbb{R}^k$ 與 $V_p \subset M$ open in $M$ 且 存在 一連續 bijection 函數 $\alpha:U_p \to V_p$ 且
 (i) $\alpha \in C^1$
 (ii) $\…

[整理] 多元智能理論(英)

最近涉獵了一些 多元智能理論 (Multiple Intelligence theory),個人覺得頗為有趣。 整理的資料如下 
Multiple Intelligence theory, Edited by Chung-Han Hsieh

The Multiple Intelligence theory, or so called MI theory, which is proposed by Professor Howard Gardner. He believed that the human beings can have a number of relatively different but interacted intelligence not just only one general intelligence. (from Harvard Graduate school of Education)
Two Definitions of Intelligence 1. From Merriam-Webster Dictionary
Definition: the ability to learn or understand or to deal with new or trying (difficult) situations. (Merriam-Webster Dict.)2. From Professor Howard Gardner:
Howard Gardner viewed intelligence as 'the capacity to solve problems or to fashion products that are valued in one or more cultural setting' (Gardner & Hatch, 1989)
The main idea of Multiple Intelligence theory, MI theory
According to this MI theory, human beings have a number of relatively different but interacted intelligences. This theory is about un…

[隨機分析] Ito Isometry Property in H^2 Space

已知 $f \in \cal{H}^2[0,t]$,我們有 Ito Isometry 如下:
\[
E\left[ \left( \int_0^t f(s) dB_s \right)^2 \right] = E \left [ \int_0^t f(s)^2 ds \right]
\]
現在我們看看 cross term 會怎麼樣?

考慮 $f,g \in \cal{H}^2[0,t]$
\[E\left[ {\int_0^t f (s)d{B_s} \cdot \int_0^t g (s)d{B_s}} \right] = E\left[ {\int_0^t f (s)g\left( s \right)ds} \right]
\]
Proof:
觀察下式:
\[\begin{array}{l}
E\left[ {{{\left( {\int_0^t {f(s)} d{B_s} + \int_0^t {g\left( s \right)} d{B_s}} \right)}^2}} \right]\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} = E\left[ {{{\left( {\int_0^t {f(s)} d{B_s}} \right)}^2} + 2\int_0^t {f(s)} d{B_s}\int_0^t {g\left( s \right)} d{B_s} + {{\left( {\int_0^t {g(s)} d{B_s}} \right)}^2}} \right]\\
\begin{array}{*{20}{c}}
{}&{}
\end{array} = E\left[ {{{\left( {\int_0^t {f(s)} d{B_s}} \right)}^2}} \right] + E\left[ {2\int_0^t {f(s)} d{B_s}\int_0^t {g\left( s \right)} d{B_s}} \right]\\
\begin{array}{*{20}{c}}
{}&{}&{}&{}&{}&{}&{}&{}
\end{array} + E\left[ {{{\left( {\int_0^t {g(s)} d{…