Manuscript Em02-A@6 · Em02-A-v6.tex · section · cb2f0a9a

Characteristic chart and exact translated equations

Characteristic chart and exact translated equations · lines 1832–1899

Em02-A@6#eq:translated-cones

For $s = b - a$ and $d_{0} = y - a$ , the chart inverse is

n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2},

α = \frac{d _{0} ^{0} + n \cdot d _{0}}{2}, v = \frac{d _{0} ^{0} - n \cdot d _{0}}{2}, ξ^{A} = e_{A} \cdot d_{0}, r = z - a .

The Jacobian in $(β, u)$ is two; spatial polar coordinates contribute $(β - u)^{2}$ . At fixed $n$ , the $(α, v, ξ)$ Jacobian is two. Block triangularity proves (eq:density).

The aligned locus and response incidence are

N = {r = u = v = ξ = 0}, S_{resp} = {r = 0, u = 0} .

Along $N$ ,

d q_{R} = 4 β d u, d q_{1} = 4 α d v, d q_{2} = 4 (β - α) (d u - d v) .

Their rank is two. On $T S_{resp}$ along $N$ , the crossed pair has rank one. This is not a rank statement throughout the incidence.

For bounded translation variables $p, q$ ,

U = α ℓ + v k + ξ^{A} E_{A}, V = (α - β) ℓ + (v - u) k + ξ^{A} E_{A} .

Under $u = δ u_{*}$ , $v = δ v_{*}$ , $ξ = δ X$ ,

(U - δ p)^{2} = (V - δ q)^{2} = δ (4 α v_{*} - ∣ X ∣^{2} - 2 α ℓ \cdot p) - 2 δ^{3/2} X^{A} E_{A} \cdot p + δ^{2} (p^{2} - 2 v_{*} k \cdot p), δ (4 (α - β) (v_{*} - u_{*}) - ∣ X ∣^{2} - 2 (α - β) ℓ \cdot q) - 2 δ^{3/2} X^{A} E_{A} \cdot q + δ^{2} (q^{2} - 2 (v_{*} - u_{*}) k \cdot q) .

The normal Jacobian is $δ^{3}$ .

If $q_{R} = O (δ)$ and both translated squares are $O (δ)$ , compactness and bounded translations first give $q_{1}, q_{2} = O (δ)$ . The quadratic defect then gives $u, v = O (δ)$ and $ξ = O (δ)$ . This proves the translated core estimate without dropping the higher-order terms.

Displayed formulas in this passage

Em02-A@6#eq:translated-cones · line 1873
$(U - δ p)^{2} = (V - δ q)^{2} = δ (4 α v_{*} - ∣ X ∣^{2} - 2 α ℓ \cdot p) - 2 δ^{3/2} X^{A} E_{A} \cdot p + δ^{2} (p^{2} - 2 v_{*} k \cdot p), δ (4 (α - β) (v_{*} - u_{*}) - ∣ X ∣^{2} - 2 (α - β) ℓ \cdot q) - 2 δ^{3/2} X^{A} E_{A} \cdot q + δ^{2} (q^{2} - 2 (v_{*} - u_{*}) k \cdot q) .$
unlabeled · line 1836
$n = \frac{s}{∣ s ∣}, β = \frac{s ^{0} + ∣ s ∣}{2}, u = \frac{s ^{0} - ∣ s ∣}{2},$
unlabeled · line 1841
$α = \frac{d _{0} ^{0} + n \cdot d _{0}}{2}, v = \frac{d _{0} ^{0} - n \cdot d _{0}}{2}, ξ^{A} = e_{A} \cdot d_{0}, r = z - a .$
unlabeled · line 1852
$N = {r = u = v = ξ = 0}, S_{resp} = {r = 0, u = 0} .$
unlabeled · line 1857
$d q_{R} = 4 β d u, d q_{1} = 4 α d v, d q_{2} = 4 (β - α) (d u - d v) .$
unlabeled · line 1867
$U = α ℓ + v k + ξ^{A} E_{A}, V = (α - β) ℓ + (v - u) k + ξ^{A} E_{A} .$

Nearby in this section

proposition: Proposition: Failure of automatic internal-core localization