Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Frozen rows and complete seed transpositions

Frozen rows and complete seed transpositions · lines 2250–2318

Define

L^{μ, I} (s, a) = A^{α μ, I} \partial_{a^{α}} R (s - a),

Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),

T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .

The source incidence is the submersion $(v, a, z) \mapsto (v - z, v - a)$ . The rows are

K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = - \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)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .

The internal $y$ -slot is the smooth constant one.

For $D_{0} = R, B$ ,

\int D_{0} (b - v) Q^{L, I, J} (v; a, z) d^{4} v = α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β}^{L, I, J} D_{0} (b - a) δ_{D}^{(4)} (a - z)] .

The incidence makes this pushforward locally proper.

The scalar seeds are

R (b - a) R (b - z), R (b - a) R (a - z), R (b - z) R (z - a), R (b - a) δ_{D}^{(4)} (a - z) .

They are tensor-product pullbacks under invertible coordinate changes, including the residual tips. Their row derivative orders are at most $4, 4, 4, 2, 3, 3, 2$ .

Displayed formulas in this passage

unlabeled · line 2270
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9 d}^{I, J} K_{9 m}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
unlabeled · line 2254
$L^{μ, I} (s, a) = A^{α μ, I} \partial_{a^{α}} R (s - a),$
unlabeled · line 2257
$Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),$
unlabeled · line 2263
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$
unlabeled · line 2302
$\int D_{0} (b - v) Q^{L, I, J} (v; a, z) d^{4} v = α, β \sum (- 1)^{∣ α ∣ + ∣ β ∣} \partial_{z}^{α} \partial_{a}^{β} [q_{α β}^{L, I, J} D_{0} (b - a) δ_{D}^{(4)} (a - z)] .$
unlabeled · line 2312
$R (b - a) R (b - z), R (b - a) R (a - z), R (b - z) R (z - a), R (b - a) δ_{D}^{(4)} (a - z) .$

Nearby in this section

section: Frozen rows and complete seed transpositions