Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Non-tip derivative placement and cubic cancellation

Non-tip derivative placement and cubic cancellation · lines 1404–1437

Em01-G@5#eq:positive-density

Write $r = A ℓ + B k + Z E$ . Then

r^{2} = 4 A B - ∣ Z ∣^{2}, (b - z)^{2} = 4 (β - A) (u - B) - ∣ Z ∣^{2}, d^{4} r = 2 d A d B d^{2} Z .

The $47$ graph charts are

B = u - \frac{∣ Z ∣ ^{2}}{4 ( β - A )}, β - A > 0, A = β - \frac{∣ Z ∣ ^{2}}{4 ( u - B )}, u - B > 0.

They cover compact nonzero future residual cones after finite localization. On the first graph, with $u = 0$ , the seed density is

\frac{β}{8 π ^{2} ( β - A )} d ϑ d A d^{2} Z d v d^{2} ξ .

For $5 +$ , put

- r = ϵ j_{ω}, j_{ω} = (1, ω), ϵ > 0.

The seed density is

\frac{β ϵ}{8 π ^{2}} d ϑ d ϵ d Ω_{ω} d v d^{2} ξ, u = 0.

Displayed formulas in this passage

Em01-G@5#eq:positive-density · line 1431
$\frac{β ϵ}{8 π ^{2}} d ϑ d ϵ d Ω_{ω} d v d^{2} ξ, u = 0.$
unlabeled · line 1408
$r^{2} = 4 A B - ∣ Z ∣^{2}, (b - z)^{2} = 4 (β - A) (u - B) - ∣ Z ∣^{2}, d^{4} r = 2 d A d B d^{2} Z .$
unlabeled · line 1414
$B = u - \frac{∣ Z ∣ ^{2}}{4 ( β - A )}, β - A > 0, A = β - \frac{∣ Z ∣ ^{2}}{4 ( u - B )}, u - B > 0.$
unlabeled · line 1421
$\frac{β}{8 π ^{2} ( β - A )} d ϑ d A d^{2} Z d v d^{2} ξ .$
unlabeled · line 1426
$- r = ϵ j_{ω}, j_{ω} = (1, ω), ϵ > 0.$