Manuscript VIII-C@3 · VIII-C-v3.tex · section · 3b070436

A separately installed collar module

Derived fixed-packet flow › A separately installed collar module · lines 1097–1124

VIII-C@3#eq:collar-flow, VIII-C@3#eq:collar-lengths

Fix $μ \in I$ and a regular $s = s_{a}$ . In an already refined collar representation, let

E_{i} (t_{col}) = R_{col} (t_{col}) e_{i}, T (E_{i}) = 2 + s_{a}, R_{col} (t_{col}) = \frac{15}{4} + \frac{t _{col}}{8}, - 1 \leq t_{col} \leq 1.

Replace each $Q_{i}$ , in this installed representation, by

Q_{i}^{in} = [A_{i}, E_{i}, E_{i + 1}, A_{i + 1}], Q_{i}^{out} = [E_{i}, D_{i}, D_{i + 1}, E_{i + 1}] .

For $t_{col} = t_{col, a} + v_{col} (τ - τ_{a})$ ,

edge [A_{i}, E_{i}] [E_{i}, D_{i}] [E_{i}, E_{i + 1}] ℓ_{e} R_{col} - r 4 - R_{col} R_{col} L_{α} d ℓ_{e} / d τ v_{col} /8 - v_{col} /8 v_{col} L_{α} /8

with multiplicity three per row. All other edges are frozen.

Displayed formulas in this passage

VIII-C@3#eq:collar-flow · line 1101
$E_{i} (t_{col}) = R_{col} (t_{col}) e_{i}, T (E_{i}) = 2 + s_{a}, R_{col} (t_{col}) = \frac{15}{4} + \frac{t _{col}}{8}, - 1 \leq t_{col} \leq 1.$
VIII-C@3#eq:collar-lengths · line 1114
$edge [A_{i}, E_{i}] [E_{i}, D_{i}] [E_{i}, E_{i + 1}] ℓ_{e} R_{col} - r 4 - R_{col} R_{col} L_{α} d ℓ_{e} / d τ v_{col} /8 - v_{col} /8 v_{col} L_{α} /8$
unlabeled · line 1109
$Q_{i}^{in} = [A_{i}, E_{i}, E_{i + 1}, A_{i + 1}], Q_{i}^{out} = [E_{i}, D_{i}, D_{i + 1}, E_{i + 1}] .$