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

Proposition: Independent-variable tree products

The retained target and the available inherited formulas › Internal lines and tensor formulas · lines 1822–1871 · proof included or linked

XV-B@2#prop:tree, XV-B@2#eq:treeproducts

For any $R \in D^{'} (R^{4})$ , the products

R (b - a) R (a - z), R (b - z) R (z - a), R (b - a) R (b - z)

are defined distributions in $(a, b, z)$ by diffeomorphic pullback of tensor products. The product

\partial^{ν} R (b - a) δ^{(4)} (a - z)

is defined by the same independent-variable construction. Passive residual variables may be retained.

Proof. For the first product use

F_{1} (a, b, z) = (a, b - a, a - z), F_{1}^{- 1} (a, r, u) = (a, a + r, a - u) .

It is a diffeomorphism, and the product is $F_{1}^{*} (1 \otimes R \otimes R)$ .

For the second use

F_{2} (a, b, z) = (z, b - z, z - a), F_{2}^{- 1} (z, r, u) = (z - u, z + r, z) .

For the third use

F_{3} (a, b, z) = (b, b - a, b - z), F_{3}^{- 1} (b, r, u) = (b - r, b, b - u) .

The respective tensor products exist, and all pullbacks are by diffeomorphisms.

Finally,

F_{1}^{*} (1 \otimes \partial^{ν} R \otimes δ_{0})

gives the supported product. Adding passive variables corresponds to adjoining a smooth factor $1$ .

Displayed formulas in this passage

XV-B@2#eq:treeproducts · line 1825
$R (b - a) R (a - z), R (b - z) R (z - a), R (b - a) R (b - z)$
unlabeled · line 1833
$\partial^{ν} R (b - a) δ^{(4)} (a - z)$
unlabeled · line 1842
$F_{1} (a, b, z) = (a, b - a, a - z), F_{1}^{- 1} (a, r, u) = (a, a + r, a - u) .$
unlabeled · line 1851
$F_{2} (a, b, z) = (z, b - z, z - a), F_{2}^{- 1} (z, r, u) = (z - u, z + r, z) .$
unlabeled · line 1857
$F_{3} (a, b, z) = (b, b - a, b - z), F_{3}^{- 1} (b, r, u) = (b - r, b, b - u) .$
unlabeled · line 1866
$F_{1}^{*} (1 \otimes \partial^{ν} R \otimes δ_{0})$