Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Exact backwards support

Prescribed kinematics and schedule dependence › Exact backwards support · lines 627–655

VII-E@1#eq:schedule-past

Let $(X, d)$ be a finite metric space, with site Hilbert spaces $H_{x}$ . For $Y \subseteq X$ , set

H_{X} = x \in X ⨂ H_{x}, A (Y) = L (x \in Y ⨂ H_{x}) \otimes I_{X ∖ Y} .

At tick $t$ , let $B_{t}$ be a partition of $X$ , and suppose

diam B \leq Δ, Φ_{t} = B \in B_{t} ⨂ Φ_{t, B},

where every block map is a channel. If ticks have duration $τ > 0$ , the resulting scalar schedule bound is $Δ/ τ$ .

Define the backwards sets recursively by

N_{t, t} (Y) N_{s, t} (Y) = Y, = ⋃ {B \in B_{s + 1} : B \cap N_{s + 1, t} (Y) \neq = \emptyset}, s < t .

Displayed formulas in this passage

VII-E@1#eq:schedule-past · line 646
$N_{t, t} (Y) N_{s, t} (Y) = Y, = ⋃ {B \in B_{s + 1} : B \cap N_{s + 1, t} (Y) \neq = \emptyset}, s < t .$
unlabeled · line 631
$H_{X} = x \in X ⨂ H_{x}, A (Y) = L (x \in Y ⨂ H_{x}) \otimes I_{X ∖ Y} .$
unlabeled · line 637
$diam B \leq Δ, Φ_{t} = B \in B_{t} ⨂ Φ_{t, B},$