Manuscript IV@1 · IV.tex · section · 351cb2d7

Edge error and measured limit

Cutoff ledger: Model 1 › Edge error and measured limit · lines 3490–3521

The only removed regulator is $ε$ :

∥ u_{a, f, ε} - u_{a, f} ∥ \leq B_{a, f} ε, a = e, r .

For $2 C ∣ τ ∣ < 1$ ,

∥ (U_{ε} - U) Ψ_{\leq n} ∥ \leq \frac{2 D _{ε} ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}} ∥ Ψ_{\leq n} ∥ .

The coherent-state remainder is

E_{ε} (μ; τ) = n in f [2 Pois (μ) Pr (N > n) + \frac{2 D _{ε} ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}}] .

It tends to zero by choosing $n$ first and then $ε$ . Strong convergence for arbitrary fixed time follows by subdivision, not by extrapolating the small-time bound.

Figures fig:edge-removal and fig:model1-record summarize the removed regulator and the associated record control.

Displayed formulas in this passage

unlabeled · line 3493
$∥ u_{a, f, ε} - u_{a, f} ∥ \leq B_{a, f} ε, a = e, r .$
unlabeled · line 3498
$∥ (U_{ε} - U) Ψ_{\leq n} ∥ \leq \frac{2 D _{ε} ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}} ∥ Ψ_{\leq n} ∥ .$
unlabeled · line 3505
$E_{ε} (μ; τ) = n in f [2 Pois (μ) Pr (N > n) + \frac{2 D _{ε} ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}}] .$