Manuscript Hawk05-C@1 · Hawk05-C.tex · theorem · ba0719f1

Theorem: Fixed tensor-release compatibility

Replacing the clock: compatibility, crossover, and cutoff › The constant-capacity-sum requirement · lines 906–932 · proof included or linked

Hawk05-C@1#thm:compatibility, Hawk05-C@1#eq:sumconstraint, Hawk05-C@1#eq:sumdifference

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For the proxy of Section sec:nonuniform,

lo g_{2} r_{k} + lo g_{2} b_{k} = lo g_{2} D

at every cut. If target capacities $A (t_{k}), B (t_{k})$ satisfy the rounding bounds of Definition def:comparison, then

∣ A (t_{k}) + B (t_{k}) - lo g_{2} D ∣ \leq e_{R, k} + e_{B, k} .

In particular, for any two cuts $j, k$ ,

∣ [A + B] (t_{k}) - [A + B] (t_{j}) ∣ \leq e_{R, k} + e_{B, k} + e_{R, j} + e_{B, j} .

This is a necessary condition for that fixed-dimensional model, not for every possible evaporation model.

Proof. Equation (eq:dimensions) gives $r_{k} b_{k} = D$ . Subtract the two target capacities from the corresponding logarithms and use the triangle inequality.

Displayed formulas in this passage

Hawk05-C@1#eq:sumconstraint · line 914
$∣ A (t_{k}) + B (t_{k}) - lo g_{2} D ∣ \leq e_{R, k} + e_{B, k} .$
Hawk05-C@1#eq:sumdifference · line 919
$∣ [A + B] (t_{k}) - [A + B] (t_{j}) ∣ \leq e_{R, k} + e_{B, k} + e_{R, j} + e_{B, j} .$
unlabeled · line 909
$lo g_{2} r_{k} + lo g_{2} b_{k} = lo g_{2} D$

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section: The constant-capacity-sum requirement