Manuscript Hawk02-C@2 · Hawk02-C.tex · remark · bb332d66

Remark: A stable packet-distance formula

Prescribed packet access and the complete residual › Accessible and residual spaces · lines 1000–1019

Hawk02-C@2#rem:stable, Hawk02-C@2#eq:stabledefect, Hawk02-C@2#eq:stabledistance

For $p > 0$ , define

d_{N} = \frac{p ℓ _{N}}{p + p p _{N}} = \frac{ℓ _{N}}{1 + p _{N} / p} .

Then $h_{N} = 1 - d_{N}$ , and

d_{⋄} (Ad_{W_{N}}, Ad_{W_{N}}) = d_{N} (2 - d_{N}), W_{N} - W_{N}_{\infty} = 2 d_{N} .

These are exact algebraic identities. They avoid subtracting nearly equal numbers in $1 - h_{N}^{2}$ . For $p = 0$ , both distances are zero.

Displayed formulas in this passage

Hawk02-C@2#eq:stabledefect · line 1003
$d_{N} = \frac{p ℓ _{N}}{p + p p _{N}} = \frac{ℓ _{N}}{1 + p _{N} / p} .$
Hawk02-C@2#eq:stabledistance · line 1010
$d_{⋄} (Ad_{W_{N}}, Ad_{W_{N}}) = d_{N} (2 - d_{N}), W_{N} - W_{N}_{\infty} = 2 d_{N} .$