Manuscript En05-B@1 · En05-B.tex · section · 8b626460

Parameters and the stencil

The substrate: increasing directions and separated ports › Parameters and the stencil · lines 390–413

En05-B@1#eq:parameters

Work in calibrated length units. For integers $n \geq 4$ , define

K_{n} = n, N_{n} = 2 n + 1, ℓ_{n} = n^{- 8}, δ_{n} = \frac{ℓ _{n}}{K _{n}}, a_{n} = \frac{ℓ _{n}}{N _{n}} (1 + \frac{1}{N _{n}}), ξ_{n} = \frac{ℓ _{n}}{10 N _{n} ^{2}}, h_{n} = \frac{a _{n}}{c} .

Here $δ_{n}$ is the hub mesh, $ℓ_{n}$ the maximum nominal lane length, $a_{n}$ the physical allowance for one graph edge, and $ξ_{n}$ a small separation allowance.

Let

P_{n} = {p \in Z^{2} : 0 < ∥ p ∥_{2} \leq K_{n}}, S_{n} = ∣ P_{n} ∣ \leq (2 K_{n} + 1)^{2} .

Both orientations occur. The vectors $e_{1}, e_{2}$ occur, ensuring connectivity.

Displayed formulas in this passage

En05-B@1#eq:parameters · line 393
$K_{n} = n, N_{n} = 2 n + 1, ℓ_{n} = n^{- 8}, δ_{n} = \frac{ℓ _{n}}{K _{n}}, a_{n} = \frac{ℓ _{n}}{N _{n}} (1 + \frac{1}{N _{n}}), ξ_{n} = \frac{ℓ _{n}}{10 N _{n} ^{2}}, h_{n} = \frac{a _{n}}{c} .$
unlabeled · line 407
$P_{n} = {p \in Z^{2} : 0 < ∥ p ∥_{2} \leq K_{n}}, S_{n} = ∣ P_{n} ∣ \leq (2 K_{n} + 1)^{2} .$

Nearby in this section

lemma: Lemma: Directional resolution