Chapter 10 — Worked Examples

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1. #1

   $$g_t^h(i,j)=e^{i({\alpha }_t(i,j)+{\phi }_i-{\phi }_j)}.$$
2. #2

   For triangle :

   $${\Omega }_t(i,j,k)=e^{i({\alpha }_{ij}+{\alpha }_{jk}+{\alpha }_{ki})}.$$
3. #3

   For triangle : Discrete curvature:

   $${\omega }_t(i,j,k)=2(1-\cos ({\alpha }_{ij}+{\alpha }_{jk}+{\alpha }_{ki})).$$
4. #4

   $${\mathcal{E}}_{F^{\circ }}(\phi )=\sum _{\begin{array}{c}i,j\in F^{\circ }\\ W_t(i,j)>0\end{array}}{W}_t(i,j)\cdot 2(1-\cos ({\alpha }_{ij}+{\phi }_i-{\phi }_j)).$$
5. #5

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

   $$L{\phi }^{\ast }=-B\mathrm{diag}(W)\alpha ,{\phi }^{\ast }=-L^{†}B\mathrm{diag}(W)\alpha +c\cdot 1$$
6. #6

   $${\tilde{\alpha }}_{ij}={\alpha }_{ij}+{\phi }_i^{\ast }-{\phi }_j^{\ast },\rho (i)=\sum _{j\sim i}{W}_t(i,j){\tilde{\alpha }}_{ij}^2.$$
7. #7

   Incidence matrix for on (edges ordered as ):

   $$B=\left(\begin{array}{ccc}-1& 1& 0\\ -1& 0& 1\\ 0& -1& 1\end{array}\right),W=\mathrm{diag}(8,8,8)=8I.$$
8. #8

   Existence of Fruit B:

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

   with:

   $$g(1,2)=e^{i\pi {\sigma }_x/4},g(2,3)=e^{i\pi {\sigma }_y/4},g(3,1)=e^{i\pi {\sigma }_z/4}$$