| import Mathlib | |
| namespace OAI | |
| open Filter Topology Asymptotics | |
| noncomputable section | |
| namespace CylinderCovering.SingularCutoff | |
| def Admissible (p : ℝ) (ρ q : ℝ → ℝ) : Prop := | |
| (∀ᶠ c in 𝓝[>] (0 : ℝ), 0 < ρ c ∧ 0 < q c) ∧ | |
| ρ =o[𝓝[>] (0 : ℝ)] (fun c => c) ∧ | |
| Tendsto (fun c => c * (ρ c) ^ (p - 2)) (𝓝[>] (0 : ℝ)) (𝓝 0) ∧ | |
| q =o[𝓝[>] (0 : ℝ)] (fun c => c ^ (2 : ℕ)) ∧ | |
| (fun c => c ^ (3 : ℕ)) =o[𝓝[>] (0 : ℝ)] q | |
| def powerCutoff (A β : ℝ) (c : ℝ) : ℝ := (A * c) ^ β | |
| def powerPadding (A B η : ℝ) (c : ℝ) : ℝ := B * (A * c) ^ η | |
| def edgeOneCoefficient (p : ℝ) : ℝ := | |
| (2 * p ^ 2 + 10 * p - 13) / (15360 * p) | |
| def boundaryCoefficient (p : ℝ) : ℝ := (2 - p) / 2560 + p / 1536 | |
| def energyCoefficient (p : ℝ) : ℝ := | |
| (1 / (16 * p)) * | |
| ((((2 - p) ^ 2 / 80 + p * (2 - p) / 24 + p ^ 2 / 16) / 16) + 1 / 96) | |
| end CylinderCovering.SingularCutoff | |
| end | |
| end OAI | |