Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Data and actual convolution

Exact nonphysical eight-variable diagnostic › Data and actual convolution · lines 1860–1917

Em01-D@2#eq:diagnosticprofile, Em01-D@2#eq:diagnosticlines, Em01-D@2#eq:diagnosticD

Fix

ℓ = (1, 0, 0, 1), k = (1, 0, 0, - 1), κ = (1, 0, 0, - 1) \in (R^{4})^{*} .

Then

κ (x) = x^{0} - x^{3}, κ (ℓ) = 0, κ (k) = 2, κ^{♯} = ℓ .

Choose nonnegative $p \in C_{c}^{\infty} (R)$ of integral one, and compact smooth $ψ, ω$ , of integral one on $R, R^{2}$ , respectively. For $h = A ℓ + u k + X$ , put

ρ_{4} (h) = p (2 u) ψ (A) ω (X), ρ = ρ_{4} \otimes ρ_{4} .

Since $d^{4} h = 2 d A d u d^{2} X$ , the profile is normalized and its $κ$ -marginal is $p$ .

Choose one tensor-component label $I_{*}$ for both incoming and residual slots, and set

H_{T, γ}^{μν}_{I} (x) = δ_{I, I_{*}} η^{μν} δ_{D} (κ (x)), H_{T, ϕ} (x) = δ_{D} (κ (x)),

H_{γ} = 0, H_{ϕ} = 0.

Set $m = 0$ , take

f = 1 near B_{0}, g (y) = y^{0} near Y,

and keep the source and all remaining internal data fixed. Compare $D_{0} = 0$ with the array whose sole nonzero entry is

D_{1}^{00, I_{*}, I_{*}} = 1.

Using the same slot label also makes this diagnostic array symmetric in its two input slots.

Displayed formulas in this passage

unlabeled · line 1863
$ℓ = (1, 0, 0, 1), k = (1, 0, 0, - 1), κ = (1, 0, 0, - 1) \in (R^{4})^{*} .$
unlabeled · line 1869
$κ (x) = x^{0} - x^{3}, κ (ℓ) = 0, κ (k) = 2, κ^{♯} = ℓ .$
Em01-D@2#eq:diagnosticprofile · line 1880
$ρ_{4} (h) = p (2 u) ψ (A) ω (X), ρ = ρ_{4} \otimes ρ_{4} .$
unlabeled · line 1891
$H_{T, γ}^{μν}_{I} (x) = δ_{I, I_{*}} η^{μν} δ_{D} (κ (x)), H_{T, ϕ} (x) = δ_{D} (κ (x)),$
Em01-D@2#eq:diagnosticlines · line 1898
$H_{γ} = 0, H_{ϕ} = 0.$
unlabeled · line 1903
$f = 1 near B_{0}, g (y) = y^{0} near Y,$
Em01-D@2#eq:diagnosticD · line 1911
$D_{1}^{00, I_{*}, I_{*}} = 1.$

Nearby in this section

lemma: Lemma: Exact smoothed crossed array