Manuscript Em02-D@4 · Em02-D-v4.tex · section · fcf2fac7

Rows in independent seed variables

Smoothing, vertex, and exact rows › Rows in independent seed variables · lines 557–623

Em02-D@4#eq:K47, Em02-D@4#eq:K5a, Em02-D@4#eq:K5z, Em02-D@4#eq:K6, Em02-D@4#eq:K8, Em02-D@4#eq:K9d, Em02-D@4#eq:K9m

Temporal primes on $R$ mean derivatives of its propagation argument. On the retained neighborhood, the two-line kernels are equivalently

K_{47}^{22} K_{5 a}^{22} K_{5 z}^{22} = (\partial_{s^{0}} + \partial_{t^{0}}) [R^{'} (s) R^{'} (t)], = \partial_{s^{0}} [f (a + s) R^{'} (s) R^{''} (t)], = \partial_{s^{0}} [f (c + s) R^{'} (s) R^{''} (t)], s = b - a, t = b - z, s = b - a, t = a - z, c = z, s = b - z, t = z - a .

Every seed product is a tensor-product pullback, including at its inner vertex.

For comparison with endpoint notation, set $L^{0, 2} (q, a) = \partial_{a^{0}} R (q - a)$ . Then $K_{5 a}$ and $K_{5 z}$ are the two terms

\partial_{b^{0}} [f (b) \partial_{a^{0}} R (b - a) \partial_{a^{0}} L^{0, 2} (a, z)],

\partial_{b^{0}} [f (b) \partial_{z^{0}} R (b - z) \partial_{z^{0}} L^{0, 2} (z, a)] .

For $47$ , the exact bracket cancellation gives

K_{47}^{22} = \partial_{b^{0}} [(\partial_{0} f) (b) \partial_{z^{0}} R (b - z) \partial_{a^{0}} R (b - a)] .

The plateau value $\partial_{0} f = 1$ gives (eq:K47).

In relative variables, put $D = \partial_{S^{0}}$ . The response rows are

K_{6}^{22} K_{8}^{22} K_{9 d}^{22} K_{9 m}^{j j} = - D (f D R) δ^{(4)} (r), = - D [f (D B) H_{r}], = - D R H_{r}, = - f R H_{r}, j = 1, 2.

Here $f = f (a + S)$ . The endpoint derivative in row $6$ is line-only. It is not redistributed onto $δ^{(4)} (r)$ .

All undisplayed components vanish, and

K_{ret} = K_{47} + K_{5 a} + K_{5 z} + K_{6} + K_{8} + K_{9 d} + K_{9 m} .

Displayed formulas in this passage

Em02-D@4#eq:K47 · line 562
$K_{47}^{22} K_{5 a}^{22} K_{5 z}^{22} = (\partial_{s^{0}} + \partial_{t^{0}}) [R^{'} (s) R^{'} (t)], = \partial_{s^{0}} [f (a + s) R^{'} (s) R^{''} (t)], = \partial_{s^{0}} [f (c + s) R^{'} (s) R^{''} (t)], s = b - a, t = b - z, s = b - a, t = a - z, c = z, s = b - z, t = z - a .$
Em02-D@4#eq:K5a · line 562
$K_{47}^{22} K_{5 a}^{22} K_{5 z}^{22} = (\partial_{s^{0}} + \partial_{t^{0}}) [R^{'} (s) R^{'} (t)], = \partial_{s^{0}} [f (a + s) R^{'} (s) R^{''} (t)], = \partial_{s^{0}} [f (c + s) R^{'} (s) R^{''} (t)], s = b - a, t = b - z, s = b - a, t = a - z, c = z, s = b - z, t = z - a .$
Em02-D@4#eq:K5z · line 562
$K_{47}^{22} K_{5 a}^{22} K_{5 z}^{22} = (\partial_{s^{0}} + \partial_{t^{0}}) [R^{'} (s) R^{'} (t)], = \partial_{s^{0}} [f (a + s) R^{'} (s) R^{''} (t)], = \partial_{s^{0}} [f (c + s) R^{'} (s) R^{''} (t)], s = b - a, t = b - z, s = b - a, t = a - z, c = z, s = b - z, t = z - a .$
Em02-D@4#eq:K6 · line 605
$K_{6}^{22} K_{8}^{22} K_{9 d}^{22} K_{9 m}^{j j} = - D (f D R) δ^{(4)} (r), = - D [f (D B) H_{r}], = - D R H_{r}, = - f R H_{r}, j = 1, 2.$
Em02-D@4#eq:K8 · line 605
$K_{6}^{22} K_{8}^{22} K_{9 d}^{22} K_{9 m}^{j j} = - D (f D R) δ^{(4)} (r), = - D [f (D B) H_{r}], = - D R H_{r}, = - f R H_{r}, j = 1, 2.$
Em02-D@4#eq:K9d · line 605
$K_{6}^{22} K_{8}^{22} K_{9 d}^{22} K_{9 m}^{j j} = - D (f D R) δ^{(4)} (r), = - D [f (D B) H_{r}], = - D R H_{r}, = - f R H_{r}, j = 1, 2.$
Em02-D@4#eq:K9m · line 605
$K_{6}^{22} K_{8}^{22} K_{9 d}^{22} K_{9 m}^{j j} = - D (f D R) δ^{(4)} (r), = - D [f (D B) H_{r}], = - D R H_{r}, = - f R H_{r}, j = 1, 2.$
unlabeled · line 585
$\partial_{b^{0}} [f (b) \partial_{a^{0}} R (b - a) \partial_{a^{0}} L^{0, 2} (a, z)],$
unlabeled · line 589
$\partial_{b^{0}} [f (b) \partial_{z^{0}} R (b - z) \partial_{z^{0}} L^{0, 2} (z, a)] .$
unlabeled · line 594
$K_{47}^{22} = \partial_{b^{0}} [(\partial_{0} f) (b) \partial_{z^{0}} R (b - z) \partial_{a^{0}} R (b - a)] .$
unlabeled · line 619
$K_{ret} = K_{47} + K_{5 a} + K_{5 z} + K_{6} + K_{8} + K_{9 d} + K_{9 m} .$