Manuscript XIV-J@1 · XIV-J.tex · input · 1c0e72ca

Input statement: Precomposition presentation

J0: copied premises and their domains › XIV-I's precomposition sum · lines 528–564

XIV-J@1#input:presum, XIV-J@1#eq:K47, XIV-J@1#eq:K6, XIV-J@1#eq:presum

The regrouped $K_{4} + K_{7}$ kernel is

K_{47}^{I, J} (a, b, z) = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)] .

The local density piece is

K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .

The exact retained sum is

K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9} .

Its reconstructed value agrees with the canonical composition on the inherited proved domain $U_{can}$ .

No map composing this sum with the grouped cross-lines on tests meeting $N$ was constructed XIVI.

Displayed formulas in this passage

XIV-J@1#eq:K47 · line 531
$K_{47}^{I, J} (a, b, z) = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)] .$
XIV-J@1#eq:K6 · line 545
$K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XIV-J@1#eq:presum · line 555
$K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9} .$