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Figure 8
Splitting of the optimal branches as plotted against Σ instead of σ in the rectangle system for the cases [\rho = \sqrt{10/7} ] and [\rho = \sqrt{21/16} ]. The red dots and lines correspond to [\Sigma_{k,1}] and the blue dots and lines to [\Sigma_{1,k^{\prime}}]. In the first case, p = 2×5 the red branch splits into four subbranches in the ratio 1, 2, 5 and 10, and since q = 7, the blue branch splits into two subbranches of ratio 1 and 7. In the second case, p = 21 = 3×7, the red branch splits into four subbranches in the ratio 1, 3, 7 and 21, whereas for q = 24, the branch splits into five subbranches in the ratio 1, 2, 4, 8 and 16. In both cases, the red and blue branches superimpose for k = pl and [k^{\prime} = q\ell,\ \ell\in{\bb Z}].

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