Chapter 10 — Worked Examples

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1. #1

   $$g_t^h(i,j)=e^{i(\alpha_t(i,j)+\varphi_i-\varphi_j)}.$$
2. #2

   For triangle :

   $$\Omega_t(i,j,k)=e^{i(\alpha_{ij}+\alpha_{jk}+\alpha_{ki})}.$$
3. #3

   For triangle : Discrete curvature:

   $$\omega_t(i,j,k)=2\bigl(1-\cos(\alpha_{ij}+\alpha_{jk}+\alpha_{ki})\bigr).$$
4. #4

   $$\mathcal{E}_{F^\circ}(\varphi)=\sum_{\substack{i,j\in F^\circ\\W_t(i,j)>0}}W_t(i,j)\cdot 2\bigl(1-\cos(\alpha_{ij}+\varphi_i-\varphi_j)\bigr).$$
5. #5

   In the small-curvature regime, the normal equations give:

   $$L\,\varphi^*=-B\,\mathrm{diag}(W)\,\alpha,\qquad\varphi^*=-L^\dagger B\,\mathrm{diag}(W)\,\alpha+c\cdot\mathbf{1}$$
6. #6

   $$\tilde\alpha_{ij}=\alpha_{ij}+\varphi_i^*-\varphi_j^*,\qquad\rho(i)=\sum_{j\sim i}W_t(i,j)\,\tilde\alpha_{ij}^2.$$
7. #7

   Incidence matrix for on (edges ordered as ):

   $$B=\begin{pmatrix}-1&1&0\\-1&0&1\\0&-1&1\end{pmatrix},\quad W=\mathrm{diag}(8,8,8)=8I.$$
8. #8

   Existence of Fruit B:

   $$\mathrm{Existence}(\{7,8,9,10\},t)=\bigl([\tilde g|_{B^\circ}]_{\mathcal{G}_{\mathrm{const}}},\;\{8\},\;\{0.6\}\bigr)$$
9. #9

   with:

   $$g(1,2)=e^{i\pi\sigma_x/4},\quad g(2,3)=e^{i\pi\sigma_y/4},\quad g(3,1)=e^{i\pi\sigma_z/4}$$