Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Ward reduction and the target function

The frozen response problem › Ward reduction and the target function · lines 704–734

II-B@1#eq:qfactor, II-B@1#eq:response-factorization

Define

q (d) = d_{112}^{2} + d_{122}^{2} - d_{111} d_{122} - d_{112} d_{222},

and let $d_{*}$ have $d_{112} = 1$ and all other independent components zero. Then $q (d_{*}) = 1$ . Put $m = F (d_{*})$ .

For species $1122$ or $2211$ ,

D_{s} = d_{111} d_{122} + d_{112} d_{222}, D_{t} = D_{u} = d_{112}^{2} + d_{122}^{2},

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

Other two-and-two placements permute the channel coefficients. For $1111, 2222, 1112, 1222$ , up to permutation, the three channel coefficients agree, so $F (d) = 0$ . Consequently the complete response has the global factorization

r (d, θ; β) = q (d) m (β) + ℓ \sum θ_{ℓ} n_{ℓ} (β), β \in X .

This factorization is not a global contact identity.

Displayed formulas in this passage

II-B@1#eq:qfactor · line 719
$F (d) = q (d) \frac{- 2 W _{2} + W _{3} + W _{4}}{3} .$
II-B@1#eq:response-factorization · line 727
$r (d, θ; β) = q (d) m (β) + ℓ \sum θ_{ℓ} n_{ℓ} (β), β \in X .$
unlabeled · line 707
$q (d) = d_{112}^{2} + d_{122}^{2} - d_{111} d_{122} - d_{112} d_{222},$
unlabeled · line 714
$D_{s} = d_{111} d_{122} + d_{112} d_{222}, D_{t} = D_{u} = d_{112}^{2} + d_{122}^{2},$

Ward reduction and the target function

Displayed formulas in this passage

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