\(U \subset \mathbb{R}^d\)

\(\xi: U \rightarrow \mathbb{R}^d\)

\(x: U \rightarrow \mathbb{R}\)

 

\([\tau x](s) = x(s')\)          where      \(s' + \xi(s') = s\)

\(s\mapsto s + \xi(s)\) is bijective and continuous

\(\forall s\in D, u\neq0\in \mathbb{R}^d \quad \|u + (u\cdot\nabla) \xi(s) \| \neq 0\)

\(s\)

\(u\)

\(s+\xi(s)\)

\(u + (u\cdot \nabla) \xi(s)\)

\(u\cdot(u + (u\cdot\nabla) \xi(s) ) > 0\)

\( \frac12 \left( \sqrt{\|\nabla \xi\|^2 - 2 \det(\nabla \xi)} - \mathrm{Tr}(\nabla\xi) \right) < 1 \)

\(\|A\|_\infty \equiv \sup\{\|Au\|: u \text{ unitary} \}\)

\(\|A\|_\infty = \|USV\|_\infty = \|S\|_\infty = \max_i S_i\)

\( \| A \|_\infty^2 = \frac12\left( \|A\|^2 + \sqrt{\|A\|^4-4\det(A)^2} \right)\)

\(\| \nabla \xi(s) \|_\infty < 1\)

\(u\)

\(\|(u\cdot\nabla) \xi(s)\|_\infty < 1\)

\(\frac12\left( \|\nabla \xi\|^2 + \sqrt{\|\nabla \xi\|^4-4\det(\nabla \xi)^2} \right) < 1\)

\(\|u + (u\cdot\nabla) \xi(s) \| \neq 0\)

\((u\cdot\nabla) \xi(s) \neq -u\)

\(u + (u\cdot\nabla) \xi(s)\)

\((u\cdot\nabla) \xi(s)\)

\(u\cdot(u + (u\cdot\nabla) \xi(s) ) > 0 \quad \forall u\)

 

\(\Leftrightarrow\)

 

\(\frac12 \left(\sqrt{(\partial_x \xi_x - \partial_y \xi_y)^2 + (\partial_x \xi_y + \partial_y \xi_x)^2} - \partial_x \xi_x - \partial_y \xi_y\right) < 1\)

 

\(\Leftrightarrow\)

 

 

\( \frac12 \left( \sqrt{\|\nabla \xi\|^2 - 2 \det(\nabla \xi)} - \mathrm{Tr}(\nabla\xi) \right) < 1 \)

\(\max_s \frac12 \left( \sqrt{\|\nabla \xi\|^2 - 2 \det(\nabla \xi)} - \mathrm{Tr}(\nabla\xi) \right) \approx 2.5 \frac{c^{1.1}}{n} \sqrt{T}\)

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By Mario Geiger

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