Manuscript XV-B@2 · XV-B.tex · proposition · 4d3f12e8

Proposition: Existence of the independent grouped pullback

The retained target and the available inherited formulas › A grouped cross-line is not its product with the target · lines 1947–1978 · proof included or linked

XV-B@2#prop:crosspullback

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Given distributions $Δ_{1}^{T}, Δ_{2}^{T}, Δ_{1}, Δ_{2}$ in their respective four-dimensional relative variables, the distribution

Δ_{1}^{T} (y - a) Δ_{2}^{T} (y - b) - Δ_{1} (y - a) Δ_{2} (y - b)

is defined on the attachment variables by independent-variable tensor-product pullback. This assertion does not define its product with $K_{ret}$ .

Proof. The map

(a, b, z, y) ⟼ (y - a, y - b, z, y)

is a diffeomorphism, with inverse

(u, v, z, y) ⟼ (y - u, y - v, z, y) .

Pull back

(Δ_{1}^{T} \otimes Δ_{2}^{T} - Δ_{1} \otimes Δ_{2}) \otimes 1 \otimes 1.

This defines the grouped distribution. An additional product with $K_{ret}$ is not part of this pullback construction.

Displayed formulas in this passage

unlabeled · line 1952
$Δ_{1}^{T} (y - a) Δ_{2}^{T} (y - b) - Δ_{1} (y - a) Δ_{2} (y - b)$
unlabeled · line 1964
$(a, b, z, y) ⟼ (y - a, y - b, z, y)$
unlabeled · line 1968
$(u, v, z, y) ⟼ (y - u, y - v, z, y) .$
unlabeled · line 1972
$(Δ_{1}^{T} \otimes Δ_{2}^{T} - Δ_{1} \otimes Δ_{2}) \otimes 1 \otimes 1.$

Proposition: Existence of the independent grouped pullback

Displayed formulas in this passage

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