$$U \subset \mathbb{R}^d$$

$$\xi: U \rightarrow \mathbb{R}^d$$

$$x: U \rightarrow \mathbb{R}$$

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

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

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

$$s$$

$$u$$

$$s+\xi(s)$$

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

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

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

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

$$\|A\|_\infty = \|USV\|_\infty = \|S\|_\infty = \max_i S_i$$

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

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

$$u$$

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

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

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

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

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

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

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

$$\Leftrightarrow$$

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

$$\Leftrightarrow$$

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

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

By Mario Geiger

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