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content/Math4211/Math4211_L1.md

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+# Math4211 Lecture 1
+

content/Math4211/_meta.js

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+export default {
+    index: "Course Description",
+    "---":{
+        type: 'separator'
+    },
+    Math4211_L1: "Topology I (Lecture 1)",
+}

content/Math4211/index.md

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+# Math4211
+
+Topology I

content/Math4501/Math4501_L1.md

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+# 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)

content/_meta.js

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         timestamp: true,
       }
     },
+    Math4211: {
+      type: 'page',
+      theme:{
+        timestamp: true,
+      }
+    },
     Math416: {
       type: 'page',
       theme:{