Manuscript XV-D@1 · XV-D.tex · section · 967136eb

Chart, insertions, and two reconstructions

The null defect in the one source-complete module › Chart, insertions, and two reconstructions · lines 637–671

XV-D@1#eq:insert, XV-D@1#eq:beta

Let $X \subset R^{d}$ and $V_{q} \subset R^{m}$ be open with $0 \in V_{q}$ , let $Z \subset X$ be relatively closed, and put

X^{\circ} = X ∖ Z, U = X \times V_{q}, U^{\circ} = X^{\circ} \times V_{q}, S = X \times {0} .

For $b \in D^{'} (X)$ and $ϕ \in D (U)$ the insertion is

⟨ E_{γ} b, ϕ ⟩ = (- 1)^{∣ γ ∣} ⟨ b, \partial_{q}^{γ} ϕ (\cdot, 0)⟩ .

The same formula with $X^{\circ}$ , $U^{\circ}$ in place of $X$ , $U$ defines $E^{\circ}_{γ} : D^{'} (X^{\circ}) \to D^{'} (U^{\circ})$ .

Fix $r \geq 0$ and the presentation space

P_{r}^{jet} = ∣ α ∣ \leq r ⨁ C^{\infty} (U), a = (a_{α})_{∣ α ∣ \leq r} .

For $a \in P_{r}^{jet}$ and $∣ γ ∣ \leq r$ define the smooth coefficient

β_{γ} (a) (x) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0) \in C^{\infty} (X) .

Displayed formulas in this passage

XV-D@1#eq:insert · line 648
$⟨ E_{γ} b, ϕ ⟩ = (- 1)^{∣ γ ∣} ⟨ b, \partial_{q}^{γ} ϕ (\cdot, 0)⟩ .$
XV-D@1#eq:beta · line 662
$β_{γ} (a) (x) = α \geq γ ∣ α ∣ \leq r \sum (- 1)^{∣ α - γ ∣} (γ α) (\partial_{q}^{α - γ} a_{α}) (x, 0) \in C^{\infty} (X) .$
unlabeled · line 641
$X^{\circ} = X ∖ Z, U = X \times V_{q}, U^{\circ} = X^{\circ} \times V_{q}, S = X \times {0} .$
unlabeled · line 657
$P_{r}^{jet} = ∣ α ∣ \leq r ⨁ C^{\infty} (U), a = (a_{α})_{∣ α ∣ \leq r} .$