Commit a2bf17be authored by Felix Hilsky's avatar Felix Hilsky
Browse files

fix unicode to latex issues

parent d553fc2c
......@@ -168,7 +168,7 @@ From boundedness we get a weakly convergent subsequence $`n^{(k_l)} ⇀ n`$, whi
---
*Lemma 2*: non-orientability is a stable property with respect to the $`W^{1,p}(Ω, ℝ^9)`$ norm:
Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there exists $`ε > 0`$, depending on $`Q`$, so that for all $`Q̃ ∈ W^{1,p}(Ω, 𝒬)`$ with $`\lVert Q̃ − Q \rVert_{W^{1,p}(M,ℝ^9)} < ε`$ the line field Q̃ is also non-orientable.
Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there exists $`ε > 0`$, depending on $`Q`$, so that for all $`\tilde Q ∈ W^{1,p}(Ω, 𝒬)`$ with $`\lVert \tilde Q − Q \rVert_{W^{1,p}(M,ℝ^9)} < ε`$ the line field Q̃ is also non-orientable.
*Proof of Lemma 2*: obvious, given Proposition 4
......@@ -222,7 +222,7 @@ Given that $`Q`$ stays close to some $`\overline{Q}`$, calculation shows that $`
*Theorem 1*: If $`Q: G := Ω \setminus \bigcup_{i = 1}^N \overline{ω_i} → 𝒬_2`$ is orientable at all $`∂ω_i`$, then $`Q`$ is orientable.
*Proof sketch*:
- For $`f:[0,1] → \overline{Ω}`$ define $`f^*`$ by replacing parts of $`f`$ in ω_i by (shorter) parts on $`∂ω_i`$. $`f^*`$ is continous (with proof in Lemma 4, but really?).
- For $`f:[0,1] → \overline{Ω}`$ define $`f^*`$ by replacing parts of $`f`$ in $`ω_i`$ by (shorter) parts on $`∂ω_i`$. $`f^*`$ is continous (with proof in Lemma 4, but really?).
- *Lemma 5*:
1) $`\overline{Ω}`$ is path-connected and simply connected. Proof: Schoenflies theorem: Jordan curve creates homeomorphism to unit disk.
2) $`\overline{G}`$ is path-connected. Proof: see previous bullet point.
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......@@ -13,5 +13,6 @@
\newcommand{\mathsymbol}[2]{\newunicodechar{#1}{\ifmmode#2\else#1\fi}}
\mathsymbol{ê}{\hat{e}}
\mathsymbol{ñ}{\tilde{n}}
\mathsymbol{𝕊}{\mathbb{S}}
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