Manuscript II@2 · II.tex · section · 50d64def

Factorization on the whole locked domain

Ward reduction and coupling factorization › Factorization on the whole locked domain · lines 869–900

II@2#eq:m1122

For species $1122$ ,

D_{s} = a c + b e, D_{t} = D_{u} = b^{2} + c^{2},

and therefore

F_{1122} (d) = q (d) \frac{- 2 W _{2} + W _{3} + W _{4}}{3} .

The same formula holds for $2211$ . Other two-and-two placements permute the three coefficients.

For the remaining species patterns, all three $D$ 's agree:

pattern, up to permutation 1111222211121222 D_{s} = D_{t} = D_{u} a^{2} + b^{2} c^{2} + e^{2} ab + b c b c + ce .

Their reference response is zero.

Define

d_{*} = (a, b, c, e) = (0, 1, 0, 0), q (d_{*}) = 1, m = F (d_{*}) .

Displayed formulas in this passage

II@2#eq:m1122 · line 876
$F_{1122} (d) = q (d) \frac{- 2 W _{2} + W _{3} + W _{4}}{3} .$
unlabeled · line 872
$D_{s} = a c + b e, D_{t} = D_{u} = b^{2} + c^{2},$
unlabeled · line 884
$pattern, up to permutation 1111222211121222 D_{s} = D_{t} = D_{u} a^{2} + b^{2} c^{2} + e^{2} ab + b c b c + ce .$
unlabeled · line 897
$d_{*} = (a, b, c, e) = (0, 1, 0, 0), q (d_{*}) = 1, m = F (d_{*}) .$