Manuscript Oai119-A@1 · Oai119-A.tex · lemma · fdff7292

Lemma: The two capacities are an algebraic allocation

Stage 1: the five proof interfaces › The mean-dependent production profile · lines 441–477 · proof included or linked

Oai119-A@1#lem:capacities, Oai119-A@1#eq:capacities

Put

α = 1 - m^{2}, c_{m} = h (m) - α ℓ .

Then $c_{m} \geq 0$ , and

B_{m} (I) = u + v = I 0 \leq u \leq c_{m} 0 \leq v < α ℓ in f {2 u + α F (v / α)} .

The infimum is attained at $u = min {I, c_{m}}$ .

Proof. Integrating the power series for $atanh r$ gives

ψ (r) = k \geq 1 \sum \frac{r ^{2 k}}{2 k ( 2 k - 1 )} .

Its positive coefficients sum to $ℓ$ , by continuity at $r = 1$ . Hence $ψ (m) \leq m^{2} ℓ$ , proving $c_{m} \geq 0$ .

At fixed $u + v = I$ , increasing $u$ cannot increase the objective, since its derivative is $2 - F^{'} ((I - u) / α) \leq 0$ . The largest feasible $u$ is $min {I, c_{m}}$ . Substitution gives $2 I$ when $I \leq c_{m}$ , and

2 c_{m} + α F ((I - c_{m}) / α)

otherwise. These are exactly the two branches of (eq:Bprofile).

Displayed formulas in this passage

Oai119-A@1#eq:capacities · line 448
$B_{m} (I) = u + v = I 0 \leq u \leq c_{m} 0 \leq v < α ℓ in f {2 u + α F (v / α)} .$
unlabeled · line 444
$α = 1 - m^{2}, c_{m} = h (m) - α ℓ .$
unlabeled · line 461
$ψ (r) = k \geq 1 \sum \frac{r ^{2 k}}{2 k ( 2 k - 1 )} .$
unlabeled · line 472
$2 c_{m} + α F ((I - c_{m}) / α)$

Lemma: The two capacities are an algebraic allocation

Displayed formulas in this passage

Nearby in this section