Manuscript III@2 · III.tex · section · 82dad3e5

An explicit nonzero compact pair

Compact coupled Ward observables › An explicit nonzero compact pair · lines 2147–2178

III@2#eq:nonzero-J, III@2#eq:nonzero-c

Choose

0 < τ_{1} < τ_{2} < min (\frac{R}{2}, \frac{π}{2 m})

and nonnegative nonzero

a \in C_{c}^{\infty} ((τ_{1}, τ_{2})), b \in C_{c}^{\infty} ({∣ x ∣ < R /4}) .

Their support cylinder lies strictly inside the plateau (eq:plateau).

Define

J^{00} (t, x) = a (τ) b (x), J^{0 i} = J^{ij} = 0,

and

c (t, x) = \frac{2 a ^{'} ( τ )}{A _{⋆} m sin ( m τ )} b (x) on (τ_{1}, τ_{2}), c = 0 otherwise .

The denominator is nonzero, and $a^{'}$ vanishes near the endpoints. Thus $c$ is smooth and compact.

Displayed formulas in this passage

III@2#eq:nonzero-J · line 2163
$J^{00} (t, x) = a (τ) b (x), J^{0 i} = J^{ij} = 0,$
III@2#eq:nonzero-c · line 2169
$c (t, x) = \frac{2 a ^{'} ( τ )}{A _{⋆} m sin ( m τ )} b (x) on (τ_{1}, τ_{2}), c = 0 otherwise .$
unlabeled · line 2150
$0 < τ_{1} < τ_{2} < min (\frac{R}{2}, \frac{π}{2 m})$
unlabeled · line 2155
$a \in C_{c}^{\infty} ((τ_{1}, τ_{2})), b \in C_{c}^{\infty} ({∣ x ∣ < R /4}) .$