Manuscript IX-E-2@1 · IX-E-2.tex · section · e70d9077

An explicit representative

The IX-E witness class: exact zero › An explicit representative · lines 1004–1038

IX-E-2@1#eq:bumps, IX-E-2@1#eq:explicit-zero-tests

Define smooth nonnegative bumps

β (a) = {exp (- 1/ (1 - a^{2})), 0, ∣ a ∣ < 1, ∣ a ∣ \geq 1, χ (v) = {exp (- 1/ (1 - ∣ v ∣^{2})), 0, ∣ v ∣ < 1, ∣ v ∣ \geq 1.

Let $e_{1} = (1, 0, 0)$ , and take

f (t, x) g (s, y) = β (8 (t - \frac{1}{2})) χ (10 x), = β (8 (s - \frac{1}{2})) χ (10 (y - 3 e_{1})) .

Their time supports lie in $[3/8, 5/8] \subset (1/4, 3/4)$ , their spatial radii are $1/10$ , and their center distance is $3$ . Taking $T = 1$ gives

3 > 2 + \frac{1}{10} + \frac{1}{10} .

This is a retained two-way pair with exact zero commutator. No numerical integration is involved.

Displayed formulas in this passage

IX-E-2@1#eq:explicit-zero-tests · line 1022
$f (t, x) g (s, y) = β (8 (t - \frac{1}{2})) χ (10 x), = β (8 (s - \frac{1}{2})) χ (10 (y - 3 e_{1})) .$
IX-E-2@1#eq:bumps · line 1007
$β (a) = {exp (- 1/ (1 - a^{2})), 0, ∣ a ∣ < 1, ∣ a ∣ \geq 1, χ (v) = {exp (- 1/ (1 - ∣ v ∣^{2})), 0, ∣ v ∣ < 1, ∣ v ∣ \geq 1.$
unlabeled · line 1033
$3 > 2 + \frac{1}{10} + \frac{1}{10} .$