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

Geometry and refinement

The frozen model › Geometry and refinement · lines 186–219

En08-B@1#eq:scales, En08-B@1#eq:stencil

Fix

d \geq 2, D \geq 2, ℓ > 0, c > 0, κ = d .

The restriction $D \geq 2$ makes unknown-state transport and signalling nontrivial. The geometric estimates also hold for $D = 1$ , but a one-dimensional payload does not witness nontrivial influence.

For each integer $j \geq 1$ , declare

h_{j} = \frac{ℓ}{j ^{2}}, R_{j} = j, a_{j} = h_{j} R_{j} = \frac{ℓ}{j}, δ_{j} = \frac{a _{j}}{c} = \frac{ℓ}{c j} .

The spatial permission lattice is

Λ_{j} = h_{j} Z^{d},

and the displacement stencil is

S_{j} = {s \in Z^{d} : ∥ s ∥_{2} \leq j} .

In particular, $0 \in S_{j}$ .

The stencil is finite at each $j$ . Its physical radius $a_{j}$ tends to zero, whereas its radius in lattice units grows. No fixed-stencil or fixed-degree substrate is asserted.

Displayed formulas in this passage

En08-B@1#eq:scales · line 198
$h_{j} = \frac{ℓ}{j ^{2}}, R_{j} = j, a_{j} = h_{j} R_{j} = \frac{ℓ}{j}, δ_{j} = \frac{a _{j}}{c} = \frac{ℓ}{c j} .$
En08-B@1#eq:stencil · line 210
$S_{j} = {s \in Z^{d} : ∥ s ∥_{2} \leq j} .$
unlabeled · line 189
$d \geq 2, D \geq 2, ℓ > 0, c > 0, κ = d .$
unlabeled · line 206
$Λ_{j} = h_{j} Z^{d},$