Manuscript III-B@1 · III-B-v1.tex · section · d2c98f62

A strictly positive compact pair

Compact coupled Ward observables › A strictly positive compact pair · lines 1432–1457

III-B@1#eq:positive-pair

Choose

0 < s_{1} < s_{2} < min {a, ρ /2, π / (2 m)},

and nonnegative nonzero

b \in C_{c}^{\infty} (B_{ρ /4}), a_{0} \in C_{c}^{\infty} ((s_{1}, s_{2})) .

Set

J^{00} = a_{0} (s) b (x), J^{0 i} = J^{ij} = 0,

and

q (s, x) = \frac{2 a _{0}^{'} ( s ) b ( x )}{A _{*} m sin ( m s )} on (s_{1}, s_{2}), q = 0 elsewhere .

This is smooth and compact: the denominator is nonzero on the support and the numerator vanishes near the interval endpoints.

Displayed formulas in this passage

III-B@1#eq:positive-pair · line 1448
$q (s, x) = \frac{2 a _{0}^{'} ( s ) b ( x )}{A _{*} m sin ( m s )} on (s_{1}, s_{2}), q = 0 elsewhere .$
unlabeled · line 1435
$0 < s_{1} < s_{2} < min {a, ρ /2, π / (2 m)},$
unlabeled · line 1439
$b \in C_{c}^{\infty} (B_{ρ /4}), a_{0} \in C_{c}^{\infty} ((s_{1}, s_{2})) .$
unlabeled · line 1444
$J^{00} = a_{0} (s) b (x), J^{0 i} = J^{ij} = 0,$