Manuscript X-A-conv@1 · X-A-conv.tex · section · c76fffa3

An explicit switch and plateau

Convention sheet and frozen data › An explicit switch and plateau · lines 390–427

X-A-conv@1#eq:switch, X-A-conv@1#eq:UO

Let $x_{*} = (4, 0, 0, 0)$ , and use the Euclidean norm only to specify compact supports and the regulator. Put

r (x) = ∣ x - x_{*} ∣_{E} .

Define

u (s) = {e^{- 1/ s}, 0, s > 0, s \leq 0, c (s) = \frac{u ( s )}{u ( s ) + u ( 1 - s )} .

The denominator is everywhere positive. Then $c = 0$ for $s \leq 0$ and $c = 1$ for $s \geq 1$ .

Freeze

g (x) = 1 - c (\frac{r ( x ) ^{2} - 4}{5}) .

Thus $0 \leq g \leq 1$ , $g = 1$ on $r \leq 2$ , and

supp g \subset {r \leq 3} ⋐ (0, 8) \times R^{3} .

In particular $g = 0$ near $Σ$ and near $t = 8$ .

Freeze the open sets

U = {r < 1}, O = {r < 1/4} .

The switch is background data and is not transformed.

Displayed formulas in this passage

X-A-conv@1#eq:switch · line 411
$g (x) = 1 - c (\frac{r ( x ) ^{2} - 4}{5}) .$
X-A-conv@1#eq:UO · line 422
$U = {r < 1}, O = {r < 1/4} .$
unlabeled · line 394
$r (x) = ∣ x - x_{*} ∣_{E} .$
unlabeled · line 398
$u (s) = {e^{- 1/ s}, 0, s > 0, s \leq 0, c (s) = \frac{u ( s )}{u ( s ) + u ( 1 - s )} .$
unlabeled · line 416
$supp g \subset {r \leq 3} ⋐ (0, 8) \times R^{3} .$