Manuscript VIII-ST@1 · VIII-ST.tex · section · fe979f3b

Cells and exact finite metric data

An explicit persistent transverse cellulation › Cells and exact finite metric data · lines 590–624

Put

z_{ϑ} = cos ϑ, S_{ϑ} = sin ϑ, L_{ϑ} = 2 (1 - z_{ϑ}),

and

W_{ϑ} = (χ - β)^{2} + χ^{2} + 2 z_{ϑ} χ (χ - β) - b^{2} χ^{2} .

Normalized transverse edge lengths are:

\toprule slot \midrule r_{i}^{rad} r_{i}^{crest} = [A_{i}, A_{i + 1}] r_{i}^{conn} = [A_{i}, D_{i}] r_{i}^{outer} = [D_{i}, D_{i + 1}] u_{i} = [P, B_{i}] v_{i} = [P, C_{i}] w_{i}^{-} = [B_{i}, C_{i}] w_{i}^{+} = [C_{i}, B_{i + 1}] \bottomrule s \leq 0 r 1 - a^{2} r L_{ϑ} 4 - r 4 L_{ϑ} inactive inactive inactive inactive s > 0 (r - β s) 1 - a^{2} r L_{ϑ} 4 - r 4 L_{ϑ} β s χs 2 (1 + z_{ϑ}) - b^{2} W_{ϑ} s W_{ϑ} s

The radial slot means $[P, A_{i}]$ before the event and $[B_{i}, A_{i}]$ afterwards. There are 24 finite slots.

Displayed formulas in this passage

unlabeled · line 593
$z_{ϑ} = cos ϑ, S_{ϑ} = sin ϑ, L_{ϑ} = 2 (1 - z_{ϑ}),$
unlabeled · line 598
$W_{ϑ} = (χ - β)^{2} + χ^{2} + 2 z_{ϑ} χ (χ - β) - b^{2} χ^{2} .$
unlabeled · line 604
$\toprule slot \midrule r_{i}^{rad} r_{i}^{crest} = [A_{i}, A_{i + 1}] r_{i}^{conn} = [A_{i}, D_{i}] r_{i}^{outer} = [D_{i}, D_{i + 1}] u_{i} = [P, B_{i}] v_{i} = [P, C_{i}] w_{i}^{-} = [B_{i}, C_{i}] w_{i}^{+} = [C_{i}, B_{i + 1}] \bottomrule s \leq 0 r 1 - a^{2} r L_{ϑ} 4 - r 4 L_{ϑ} inactive inactive inactive inactive s > 0 (r - β s) 1 - a^{2} r L_{ϑ} 4 - r 4 L_{ϑ} β s χs 2 (1 + z_{ϑ}) - b^{2} W_{ϑ} s W_{ϑ} s$