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

Representative born lengths

Exact algebraic and resource checks › Representative born lengths · lines 2347–2370

For $v_{i} = [P, C_{i}]$ ,

Δ T = b_{μ} C s, Δ x = C s (e_{i} + e_{i + 1}),

ℓ (v_{i})^{2} = C^{2} s^{2} (2 (1 + z_{α}) - b_{μ}^{2}) .

For $w_{i}^{-} = [B_{i}, C_{i}]$ ,

Δ T = b_{μ} C s, Δ x = s ((C - B) e_{i} + C e_{i + 1}),

giving $W_{α}^{2} s^{2}$ . The plus edge has the same length by reflection.

The corresponding displacements are nonzero for $s > 0$ and lie in faces of gradient norm below one. Hence the born coefficients are strictly positive. The persistent $e_{i}$ rate is negative for positive flow speed; positivity of a length does not require positivity of its rate.

Displayed formulas in this passage

unlabeled · line 2350
$Δ T = b_{μ} C s, Δ x = C s (e_{i} + e_{i + 1}),$
unlabeled · line 2355
$ℓ (v_{i})^{2} = C^{2} s^{2} (2 (1 + z_{α}) - b_{μ}^{2}) .$
unlabeled · line 2359
$Δ T = b_{μ} C s, Δ x = s ((C - B) e_{i} + C e_{i + 1}),$