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Merge branch 'main' of https://git.trance-0.com/Trance-0/NoteNextra-origin

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Zheyuan Wu
2026-03-25 18:35:23 -05:00
parent 37707302bb dbdc8f528b
commit b94eef4848
5 changed files with 239 additions and 4 deletions
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content/Math4202/Math4202_L23.md
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@@ -29,6 +29,8 @@ $f|_{B^2}$ is a continuous map from $B^2\to \mathbb{R}^2-\{0\}$.
$f|_{S^1=\partial B^2}:S^1\to \mathbb{R}-\{0\}$ **is nulhomotopic**.
> Recall that: Any map $g:S^1\to Y$ is nulhomotopic whenever it extends to a continuous map $G:B^2\to Y$.
Construct a homotopy between $f|_{S^1}$ and $g$
$$
@@ -57,10 +59,9 @@ Therefore $f$ must have a root in $B^2$.
<details>
<summary>Proof: part 2</summary>
If \|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|< R$ has a root in the disk $B^2_R$. (and $R\geq 1$, otherwise follows part 1)
Consider $\tilde{f}(x)=f(Rx)$.
If $\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|< R$ has a root in the disk $B^2_R$. (and $R\geq 1$, otherwise follows part 1)
Consider $\tilde{f}(x)=f(Rx)$.
$$
\begin{aligned}
\tilde{f}(x)
@@ -71,7 +72,7 @@ $$
$$
\begin{aligned}
\|\frac{a_{n-1}}{R}\|+\|\frac{a_{n-2}}{R^2}\|+\cdots+\|\frac{a_0}{R^n}\|&=\frac{1}{R}\|a_{n-1}\|+\frac{1}{R^2}\|a_{n-2}\|+\cdots+\frac{1}{R^n}\|a_0\|\\
\left\|\frac{a_{n-1}}{R}\right\|+\left\|\frac{a_{n-2}}{R^2}\right\|+\cdots+\left\|\frac{a_0}{R^n}\right\|&=\frac{1}{R}\|a_{n-1}\|+\frac{1}{R^2}\|a_{n-2}\|+\cdots+\frac{1}{R^n}\|a_0\|\\
&<\frac{1}{R}\left(\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|\right)\\
&<\frac{1}{R}<1
\end{aligned}