### 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. ... ...
 ... ... @@ -13,5 +13,6 @@ \newcommand{\mathsymbol}{\newunicodechar{#1}{\ifmmode#2\else#1\fi}} \mathsymbol{ê}{\hat{e}} \mathsymbol{ñ}{\tilde{n}} \mathsymbol{𝕊}{\mathbb{S}}
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