Manuscript XIX-D@1 · XIX-D.tex · proposition · 1f47e33b

Proposition: A compact preparation difference can leave a late response

Retarded memory, formal order, and gauge closure › Late source agreement does not determine retarded response · lines 846–872 · proof included or linked

XIX-D@1#prop:memory

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There are two smooth sources agreeing identically after a fixed preparation interval whose equal-zero-data retarded responses differ at every later time.

Proof. Fix $0 < ϵ < T$ . Let $b \in C_{c}^{\infty} ((0, ϵ))$ be nonnegative and nonzero. Compare the sources $F_{1} = b$ and $F_{2} = 0$ for

u^{''} = F, u (0) = u^{'} (0) = 0.

Their difference is

d (t) = \int_{0}^{t} (t - s) b (s) d s .

For $t \geq ϵ$ ,

d (t) = t M_{0} - M_{1}, M_{0} = \int_{0}^{ϵ} b (s) d s > 0, M_{1} = \int_{0}^{ϵ} s b (s) d s < ϵ M_{0} .

Hence $d (t) > 0$ for $t \geq ϵ$ , although the sources agree there.

Displayed formulas in this passage

unlabeled · line 857
$u^{''} = F, u (0) = u^{'} (0) = 0.$
unlabeled · line 861
$d (t) = \int_{0}^{t} (t - s) b (s) d s .$
unlabeled · line 865
$d (t) = t M_{0} - M_{1}, M_{0} = \int_{0}^{ϵ} b (s) d s > 0, M_{1} = \int_{0}^{ϵ} s b (s) d s < ϵ M_{0} .$

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section: Late source agreement does not determine retarded response