Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · section · 8ca47088

Relational expansion and insertion sign

Strict positivity and the evaluated classical obstruction › Relational expansion and insertion sign · lines 659–690

X-A-conv-ST@1#eq:Ocoefficients

The formal inverse of the coordinate map is

Y^{μ} (z) = z^{μ} - κ w^{μ} (z) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (z) (mod κ^{3}) .

Define

O_{f, g, ε} [h, k, ϕ] = \int f (z) ϕ (Y (z)) d^{4} z .

Here $f$ is fixed in relational-label space. No numerical global inverse is assumed.

Set

J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .

Taylor expansion gives

O^{(0)} = \int f ϕ, O^{(1)} = - \int f X_{w} ϕ, O^{(2)} = \int f (J (w) ϕ - X_{v} ϕ) .

The negative first insertion sign is essential XConv (:Ocoeff).

Displayed formulas in this passage

X-A-conv-ST@1#eq:Ocoefficients · line 682
$O^{(0)} = \int f ϕ, O^{(1)} = - \int f X_{w} ϕ, O^{(2)} = \int f (J (w) ϕ - X_{v} ϕ) .$
unlabeled · line 662
$Y^{μ} (z) = z^{μ} - κ w^{μ} (z) + κ^{2} (w^{ν} \partial_{ν} w^{μ} - v^{μ}) (z) (mod κ^{3}) .$
unlabeled · line 668
$O_{f, g, ε} [h, k, ϕ] = \int f (z) ϕ (Y (z)) d^{4} z .$
unlabeled · line 676
$J (w) = (w^{ν} \partial_{ν} w^{μ}) \partial_{μ} + \frac{1}{2} w^{μ} w^{ν} \partial_{μ} \partial_{ν} .$