From cbebc5893546bf9c6ec729378f66d09b726c9041 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 25 Aug 2025 22:10:12 -0500
Subject: [PATCH] updates

---
 content/Math4211/Math4211_L1.md |  2 ++
 content/Math4211/_meta.js       |  7 ++++
 content/Math4211/index.md       |  3 ++
 content/Math4501/Math4501_L1.md | 60 +++++++++++++++++++++++++++++++++
 content/_meta.js                |  6 ++++
 5 files changed, 78 insertions(+)
 create mode 100644 content/Math4211/Math4211_L1.md
 create mode 100644 content/Math4211/_meta.js
 create mode 100644 content/Math4211/index.md
 create mode 100644 content/Math4501/Math4501_L1.md

diff --git a/content/Math4211/Math4211_L1.md b/content/Math4211/Math4211_L1.md
new file mode 100644
index 0000000..8953927
--- /dev/null
+++ b/content/Math4211/Math4211_L1.md
@@ -0,0 +1,2 @@
+# Math4211 Lecture 1
+
diff --git a/content/Math4211/_meta.js b/content/Math4211/_meta.js
new file mode 100644
index 0000000..ee9a445
--- /dev/null
+++ b/content/Math4211/_meta.js
@@ -0,0 +1,7 @@
+export default {
+    index: "Course Description",
+    "---":{
+        type: 'separator'
+    },
+    Math4211_L1: "Topology I (Lecture 1)",
+}
diff --git a/content/Math4211/index.md b/content/Math4211/index.md
new file mode 100644
index 0000000..304af29
--- /dev/null
+++ b/content/Math4211/index.md
@@ -0,0 +1,3 @@
+# Math4211
+
+Topology I
\ No newline at end of file
diff --git a/content/Math4501/Math4501_L1.md b/content/Math4501/Math4501_L1.md
new file mode 100644
index 0000000..4e9ebe0
--- /dev/null
+++ b/content/Math4501/Math4501_L1.md
@@ -0,0 +1,60 @@
+# Math4501 Lecture 1
+
+In many practical problems (ODEs, PdEs, Sys of eqn)
+
+closed-form analytical solutions are unknown.
+
+-> resort ot computational algorithms (approximation)
+
+For example,
+
+Deep learning classifiers
+
+**Root finding**
+
+$$
+f(x)=\sum_{i=1}^n a_i x^i
+$$
+
+for $n\geq 5$.
+
+find all roots $x\in \mathbb{R}$ of $f(x)=0$.
+
+**Investment**
+
+Invest a dollars every month return with the rate $r$.
+
+$g(r)=a\sum_{i=1}^n (1+r)^i=a\left[\frac{(1+r)^{n+1}-(1+r)}{r}\right]$
+
+Say want $g(r)=b$ for some $b$.
+
+$f(r)=a(1+n)^{n+1}-a(1+n)-br=0$
+
+use Newton's method to find $r$ such that $f(r)=0$.
+
+Since $f$ is non-linear, that is $f(x+y)\neq f(x)+f(y)$.
+
+Let
+
+$$
+f_1(x_1,\dots, x_m)=0\\
+\vdots\\
+f_m(x_1,\dots, x_m)=0
+$$
+
+be a system of $m$ equations $\vec{f} \mathbb{R}^m \to \mathbb{R}^m$. and $f_1(\vec{x})=\vec{0}$.
+
+If $\vec{f}$ is linear, note that
+
+$$
+\begin{aligned}
+\vec{f}(\vec{x})&=\vec{f}(\begin{bmatrix}x_1\\ \vdots\\ x_m\end{bmatrix})\\
+&=\vec{f}(x_1\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix}+x_2\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix}+\cdots+x_m\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
+&=x_1\vec{f}(\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix})+x_2\vec{f}(\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix})+\cdots+x_m\vec{f}(\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
+&=A\vec{x}
+\end{aligned}
+$$
+
+where $\vec{e}_i$ is the $i$-th standard basis vector.
+
+Gaussian elimination (LU factorization)
\ No newline at end of file
diff --git a/content/_meta.js b/content/_meta.js
index e1b7520..b6a0a67 100644
--- a/content/_meta.js
+++ b/content/_meta.js
@@ -41,6 +41,12 @@ export default {
         timestamp: true,
       }
     },
+    Math4211: {
+      type: 'page',
+      theme:{
+        timestamp: true,
+      }
+    },
     Math416: {
       type: 'page',
       theme:{