Manuscript En02-D@4 · En02-D-v4.tex · section · e19ec871

Absorption and drain-free turnover

Absorption and drain-free turnover · lines 863–886

For time-homogeneous stage A, $r_{n} = P_{n} (1) > 0$ ,

E T_{abs} = n = 1 \sum^{K} r_{n}^{- 1}, E z^{T_{abs}} = n = 1 \prod^{K} \frac{r _{n} z}{1 - ( 1 - r _{n} ) z} .

For the synthetic law $r_{n} = (2^{n} + 1)^{- 1}$ , $K = 5$ , the uniform bound is $165$ opportunities and the exact mean is $67$ .

For a pure input and $R_{t} = D_{\leq t} F_{\leq t}$ ,

S (R_{t}) = S (C_{t}) \leq h_{2} (s_{t}) + s_{t} lo g_{2} (2^{K + 1} - 2) .

The endpoint block is one-dimensional and the complement has dimension at most $2^{K + 1} - 2$ . Also

∣ S (D_{\leq t}) - S (F_{\leq t}) ∣ \leq S (C_{t}) .

Permanent inaccessible loss need not give zero accessible entropy.

Displayed formulas in this passage

unlabeled · line 866
$E T_{abs} = n = 1 \sum^{K} r_{n}^{- 1}, E z^{T_{abs}} = n = 1 \prod^{K} \frac{r _{n} z}{1 - ( 1 - r _{n} ) z} .$
unlabeled · line 875
$S (R_{t}) = S (C_{t}) \leq h_{2} (s_{t}) + s_{t} lo g_{2} (2^{K + 1} - 2) .$
unlabeled · line 882
$∣ S (D_{\leq t}) - S (F_{\leq t}) ∣ \leq S (C_{t}) .$