Commit ce4bb1ef authored by Felix Hilsky's avatar Felix Hilsky
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move question with answer to appropriate section

parent 91ad41e5
......@@ -35,6 +35,7 @@
- otherwise *biaxial* (more complicated)
- Constraint on $`s`$:
- $`s`$ defined by limiting process $`L → 0`$ and the constants $`a, b, c`$
- To make spectral representation of $`Q`$ fit to $`Q = s(n ⊗ n - \tfrac13 \operatorname{Id})`$, we need $`s = -3 λ_1 = -3 λ_2, n = ê_3`$. With $`s`$ depending on $`a, b, c`$ (see formula (3) on page 3 (=495)), $`Q`$ is further reduced to the choice of $`n`$. That is what we assumed as our model at the beginning.
- but also $`Q`$ has an original definition via $`μ`$. Together:
```math
Qn · n
......@@ -172,11 +173,6 @@ Let $`Q ∈ W^{1,p}(Ω, 𝒬), 1 ≤ p ≤ ∞`$ be non-orientable. Then there e
- conversion between energy functionals possible
## Questions
- What is the third index $`k`$ for $`Q`$ on page 3 (= 495)?
- Probably it's a derivative of $`Q_{ij}`$ in direction $`k`$.
- And why is it written as a product instead of a square$`^2`$?
- To make spectral representation of $`Q`$ fit to $`Q = s(n \otimes n - \tfrac13 \operatorname{Id})`$, we need $`s = -3 λ_1 = -3 λ_2, n = ê_3`$. How can $`s`$ then depend on $`a, b, c`$ (see formula (3) on page 3 (=495))? If that's the case, $`Q`$ is further reduced to the choice of $`n`$ but that's what we assumed as our model at the beginning. So apparently that's fine.
- in Proposition 1 it should be $`\det Q = +\frac{2s^3}{27}`$ since the $`-`$ exists in exactly two eigenvalues: $`\det Q = λ_1 λ_2 λ_3 = (-\frac s3)(-\frac s3)(+\frac {2s}3) = +\frac{2s^3}{27}`$. Correct?
## Errors in the paper
p.5 (=497) Proposition 1 (iii): sign of $`\operatorname{det} Q`$ is wrong. Must be positive, since exactly two of the eigenvalues are negative
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