Manuscript I-B-G@2 · I-B-G-v2.tex · lemma · 3b4d700f

Lemma: Uniform lower bound

Supported replacement of flat-completion operators › The geometric arrival window · lines 1666–1690 · proof included or linked

I-B-G@2#lem:window, I-B-G@2#eq:uniform-window

In an angularly small chart,

T_{arr} (x, x^{'}) \geq L_{U} := 2 r_{U} sin (\frac{π α}{2}) > 0.

Proof. For $k \neq = 0$ in (eq:Tarr),

π α \leq ∣ ψ - 2 π α k ∣ \leq π .

Consequently,

R_{k}^{2} = (r - r^{'})^{2} + 4 r r^{'} sin^{2} (\frac{ψ - 2 π α k}{2}) + Δ z^{2} \geq 4 r_{U}^{2} sin^{2} (π α /2) .

Also $ρ_{0} \geq r + r^{'} > 2 r_{U}$ .

Displayed formulas in this passage

I-B-G@2#eq:uniform-window · line 1669
$T_{arr} (x, x^{'}) \geq L_{U} := 2 r_{U} sin (\frac{π α}{2}) > 0.$
unlabeled · line 1678
$π α \leq ∣ ψ - 2 π α k ∣ \leq π .$
unlabeled · line 1682
$R_{k}^{2} = (r - r^{'})^{2} + 4 r r^{'} sin^{2} (\frac{ψ - 2 π α k}{2}) + Δ z^{2} \geq 4 r_{U}^{2} sin^{2} (π α /2) .$