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Felix Hilsky
masterthesis
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2186dcaa
Commit
2186dcaa
authored
Jan 19, 2022
by
Felix Hilsky
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move compatibility section
parent
ce4bb1ef
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notes/papernotes/q4OrientiationEquivalence.md
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2186dcaa
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@@ 75,11 +75,6 @@ minimizers of $`ℱ_{\text{LG}}`$ for $`L → 0`$ are "suitably approximated" by
Q = s(n ⊗ n  \tfrac13 \operatorname{Id}) \quad (s = 3 λ_1 = 3 λ_2, n = ê_3)
```
## Compatibility

OseenFrank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)

otherwise OseenFrank might miss a global minimizer because it is not orientable

Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)
### Notation

$
`P : 𝕊^2 → 𝒬`
$ removes orientation. Orientable = in the image of $
`P`
$. Same for $
`Q`
$ only defined on $
`∂Ω`
$

$
`𝒬 := \{Q = s(n ⊗ n  \frac13 \operatorname{ID})  n ∈ 𝕊^2\}`
$
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@@ 172,6 +167,16 @@ Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be nonorientable. Then there e

specific energy functionals are regarded (p. 11/503) because they were looked at before and OseenFrank was successful for them

conversion between energy functionals possible
## Compatibility

OseenFrank and de Gennes are compatible if the oriented line field (= unit vector field) can be oriented (without changing regularity)

otherwise OseenFrank might miss a global minimizer because it is not orientable

Question if orientable calculable with integer programming problem (p. 4 = 496) (= (linear) optimization problem with only integer coefficiants)

In chapter 3, orienting a line field is translated into lifting $
`Q`
$ from $
`ℝP^2`
$ to covering space $
`𝕊^2`
$.

Classical algebraic topology tools only consider continous maps, but we have $
`W^{1,p}(Ω)`
$ maps and want to preserve this regularity
## Questions
## Errors in the paper
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