Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Exact characteristic normal coordinates

Exact characteristic normal coordinates · lines 1043–1075

These rank statements are localized to $N$ ; rank may increase away from it. The flag has codimensions

codim_{W} S_{resp} = 5, codim_{S_{resp}} N = 3, codim_{W} N = 8.

It is not a replacement numerical extension problem.

The dilation

D_{λ} (τ; r, u, v, ξ) = (τ; λ r, λ^{2} u, λ^{2} v, λ ξ)

satisfies

q_{j} \circ D_{λ} = λ^{2} q_{j}, j = R, 1, 2,

and has normal coordinate Jacobian $λ^{10}$ . The weight of $r$ is declared; the cone equations do not determine it. The ambient density has the additional smooth ratio

\frac{( β - λ ^{2} u ) ^{2}}{( β - u ) ^{2}} .

This geometric dilation is distinct from the isotropic smoothing in (eq:regularization). No physical scaling degree is assigned by this calculation alone. Extension frameworks require their additional growth and microlocal hypotheses Dang,BF.

Displayed formulas in this passage

unlabeled · line 1047
$codim_{W} S_{resp} = 5, codim_{S_{resp}} N = 3, codim_{W} N = 8.$
unlabeled · line 1055
$D_{λ} (τ; r, u, v, ξ) = (τ; λ r, λ^{2} u, λ^{2} v, λ ξ)$
unlabeled · line 1061
$q_{j} \circ D_{λ} = λ^{2} q_{j}, j = R, 1, 2,$
unlabeled · line 1067
$\frac{( β - λ ^{2} u ) ^{2}}{( β - u ) ^{2}} .$

Exact characteristic normal coordinates

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