| import OAI.Geometry.CylinderCovering.Physical.CoordinatesDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.CapsDefinitions |
|
|
| namespace OAI |
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|
| noncomputable section |
| open Filter |
| open scoped Topology |
| namespace RadialPhysical |
|
|
| def RelativeCostExpansion (f : ℝ → ℝ) (reference coefficient : ℝ) : Prop := |
| (fun τ => f τ - reference * (1 + coefficient * τ ^ 2)) |
| =O[𝓝[>] (0 : ℝ)] (fun τ => τ ^ 4) |
|
|
| def PhysicalCostTableStatement : Prop := |
| RelativeCostExpansion (fun τ => scaledCost 1 τ (linearCap 1 τ (1/120))) |
| (height 1 / 2) (-1/60) ∧ |
| RelativeCostExpansion (fun τ => scaledCost (1/2) (τ/2) (linearCap (1/2) (τ/2) (1/120))) |
| (height (1/2) / 4) (-1/3840) ∧ |
| RelativeCostExpansion (fun τ => scaledCost (1/Real.sqrt 2) τ |
| (linearCap (1/Real.sqrt 2) τ (1/125))) |
| ((1/Real.sqrt 2) * height (1/Real.sqrt 2) / 2) (-13/3000) ∧ |
| RelativeCostExpansion (fun τ => scaledCost (1/Real.sqrt 2) τ |
| (squareCap (1/Real.sqrt 2) τ (1/125))) |
| ((1/Real.sqrt 2) * height (1/Real.sqrt 2) / 2) (-13/3000) ∧ |
| RelativeCostExpansion (fun τ => scaledCost (1/2) τ (linearCap (1/2) τ (4/625))) |
| (height (1/2) / 4) (-77/60000) ∧ |
| RelativeCostExpansion (fun τ => scaledCost (1/2) (2*τ) (squareCap (1/2) (2*τ) (1/120))) |
| (height (1/2) / 4) (-1/240) ∧ |
| RelativeCostExpansion (fun τ => scaledCost 1 (2*τ) (linearCap 1 (2*τ) (1/200))) |
| (height 1 / 2) (-7/75) |
|
|
| def centeredFractionalCap (b η s : ℝ) : ℝ := |
| 1 + η ^ 2 * s ^ 2 * (b ^ 2 + s ^ 2) / 2 + η ^ (5/2 : ℝ) |
|
|
| def nonorthCost (ε : ℝ) (R : ℝ → ℝ) : ℝ := |
| scaledCost (1/Real.sqrt 2) (2*ε) (fun s => R (Real.sqrt 2 * s)) |
|
|
| structure PhysicalCostVariantsStatement : Prop where |
| linearExpansion : ∀ b β a : ℝ, 0 < b → 0 ≤ β → |
| (fun τ => scaledCost b (a*τ) (linearCap b (a*τ) β) - quadraticCost b (a*τ) β) |
| =O[𝓝[>] (0:ℝ)] (fun τ => τ^4) |
| squareExpansion : ∀ b β a : ℝ, 0 < b → 0 ≤ β → |
| (fun τ => scaledCost b (a*τ) (squareCap b (a*τ) β) - quadraticCost b (a*τ) β) |
| =O[𝓝[>] (0:ℝ)] (fun τ => τ^4) |
| linearSaving : ∀ b β : ℝ, 0 < b → 0 < β → β < 1/60 → |
| ∀ᶠ η : ℝ in 𝓝[>] 0, scaledCost b η (linearCap b η β) < b*height b/2 |
| squareSaving : ∀ b β : ℝ, 0 < b → 0 < β → β < 1/60 → |
| ∀ᶠ η : ℝ in 𝓝[>] 0, scaledCost b η (squareCap b η β) < b*height b/2 |
| table : PhysicalCostTableStatement |
| centeredLinearExpansion : ∀ b β a : ℝ, 0 < b → 0 ≤ β → |
| (fun τ => scaledCost b (a*τ) (centeredLinearCap b (a*τ) β) - |
| 2*b*height b*(1+relativeQuadraticTerm b (a*τ) β)) |
| =O[𝓝[>] (0:ℝ)] (fun τ => τ^4) |
| centeredLinearSaving : ∀ b β : ℝ, 0 < b → 0 < β → β < 1/60 → |
| ∀ᶠ η : ℝ in 𝓝[>] 0, scaledCost b η (centeredLinearCap b η β) < 2*b*height b |
| centeredFractionalDictionary : ∀ η : ℝ, 0 < η → ∀ r : ℝ, |
| centeredFractionalCap (1/Real.sqrt 2) η ((1/Real.sqrt 2)*r) = |
| CylinderCovering.RadialCaps.fractionalCap (4*Real.sqrt 2) (η/2) r |
| centeredFractionalExpansion : |
| (fun η => scaledCost (1/Real.sqrt 2) η (centeredFractionalCap (1/Real.sqrt 2) η) - |
| (1/Real.sqrt 2)*(2-η^2/60+4*η^(5/2:ℝ))) |
| =O[𝓝[>] (0:ℝ)] (fun η => η^4) |
| centeredFractionalLittleO : |
| (fun η => scaledCost (1/Real.sqrt 2) η (centeredFractionalCap (1/Real.sqrt 2) η) - |
| (1/Real.sqrt 2)*(2-η^2/60)) =o[𝓝[>] (0:ℝ)] (fun η => η^2) |
| centeredFractionalSaving : |
| ∀ᶠ η : ℝ in 𝓝[>] 0, scaledCost (1/Real.sqrt 2) η |
| (centeredFractionalCap (1/Real.sqrt 2) η) < 2/Real.sqrt 2 |
| nonorthDictionary : ∀ ε : ℝ, ∀ R : ℝ → ℝ, |
| nonorthCost ε R = Real.sqrt 2 * CylinderCovering.RadialCaps.Q R ε |
| nonorthCubic : |
| (fun ε => nonorthCost ε (CylinderCovering.RadialCaps.cubicCap 5 ε) - |
| Real.sqrt 2*(1-ε^2/30+10*ε^3)) =O[𝓝[>] (0:ℝ)] (fun ε => ε^4) |
| nonorthFractional : |
| (fun ε => nonorthCost ε (CylinderCovering.RadialCaps.fractionalCap 1 ε) - |
| Real.sqrt 2*(1-ε^2/30+2*ε^(5/2:ℝ))) =O[𝓝[>] (0:ℝ)] (fun ε => ε^4) |
| end RadialPhysical |
|
|
| end |
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|
| end OAI |
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|
