File size: 4,127 Bytes
785c727 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 | import OAI.Geometry.CylinderCovering.Physical.CoordinatesDefinitions
import OAI.Geometry.CylinderCovering.Radial.CapsDefinitions
namespace OAI
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
end OAI
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