Manuscript En03-A@1 · En03-A.tex · section · 5398614e

Four affine faces

External conical geometry and certified local moves › Four affine faces · lines 1495–1520

En03-A@1#eq:fourplanes, En03-A@1#eq:rational

Let $s = 0$ be the event. In a common chart $z = (T, x, y)$ , choose

0 < q < p < 1, a = 1 - p^{2}, b = 1 - q^{2}, Δ = b - a > 0,

and define

f_{E} f_{N} = p x + a s, = q y + b s, f_{W} f_{S} = - p x + a s, = - q y + b s, F_{s} = max {f_{E}, f_{W}, f_{N}, f_{S}} .

An active face attains the maximum; its interior is the strict-maximum region. The concrete certificate uses

p = \frac{3}{5}, q = \frac{5}{13}, a = \frac{4}{5}, b = \frac{12}{13}, Δ = \frac{8}{65} .

Displayed formulas in this passage

En03-A@1#eq:fourplanes · line 1503
$f_{E} f_{N} = p x + a s, = q y + b s, f_{W} f_{S} = - p x + a s, = - q y + b s, F_{s} = max {f_{E}, f_{W}, f_{N}, f_{S}} .$
En03-A@1#eq:rational · line 1514
$p = \frac{3}{5}, q = \frac{5}{13}, a = \frac{4}{5}, b = \frac{12}{13}, Δ = \frac{8}{65} .$
unlabeled · line 1498
$0 < q < p < 1, a = 1 - p^{2}, b = 1 - q^{2}, Δ = b - a > 0,$