Manuscript En08-B@1 · En08-B.tex · section · 23cac52a

Null targets and future-directed rounding

The connecting theorem › Null targets and future-directed rounding · lines 681–716

En08-B@1#eq:null-construction, En08-B@1#eq:B, En08-B@1#eq:H

Let $(T, x) \in K_{c}$ be null:

∥ x ∥_{2} = c T .

Define

x_{j} = Q_{j} x, N = N_{j} (x_{j}), n_{j} (T, x) = max {N, ⌈ \frac{T}{δ _{j}} ⌉}, T_{j} = n_{j} (T, x) δ_{j} .

First execute the route to $x_{j}$ . If necessary, wait there until tick $n_{j} (T, x)$ . Thus $T_{j} \geq T$ .

This final waiting convention repairs a possible sign issue: rounding $x$ inward can make the unpadded route arrive slightly before the original continuum target time. Spatial rounding alone does not prove a nonnegative delay relative to $T$ .

For $L > 0$ , define

B_{j} (L) H_{j} (L) = \frac{ε _{j}}{c} + \frac{κ ( L + ε _{j} )}{c ( j - κ )} + δ_{j}, = ⌈ \frac{L + ε _{j}}{h _{j} ( j - κ )} ⌉ + 1.

Displayed formulas in this passage

En08-B@1#eq:null-construction · line 688
$x_{j} = Q_{j} x, N = N_{j} (x_{j}), n_{j} (T, x) = max {N, ⌈ \frac{T}{δ _{j}} ⌉}, T_{j} = n_{j} (T, x) δ_{j} .$
En08-B@1#eq:B · line 706
$B_{j} (L) H_{j} (L) = \frac{ε _{j}}{c} + \frac{κ ( L + ε _{j} )}{c ( j - κ )} + δ_{j}, = ⌈ \frac{L + ε _{j}}{h _{j} ( j - κ )} ⌉ + 1.$
En08-B@1#eq:H · line 706
$B_{j} (L) H_{j} (L) = \frac{ε _{j}}{c} + \frac{κ ( L + ε _{j} )}{c ( j - κ )} + δ_{j}, = ⌈ \frac{L + ε _{j}}{h _{j} ( j - κ )} ⌉ + 1.$
unlabeled · line 684
$∥ x ∥_{2} = c T .$

Nearby in this section

lemma: Lemma: Endpoint, waiting, and uniform resource bounds