File size: 3,782 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 | import OAI.Geometry.CylinderCovering.Ruled.LocalDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.ProductLocal
open MeasureTheory RuledApproximation
def LogarithmicLocalStatement : Prop :=
∀ (D : Set Plane) (v : Plane → Plane) (h M : ℝ),
IsCompact D → D.Nonempty → 0 < h → 0 < M → SmoothNear 1 D v →
(∀ p ∈ D, HyperbolicAt M v p) →
∀ (eta m gamma : ℝ), 0 < eta → 1 < m → 0 < gamma →
(1+gamma)*(1/m^2) < 1 → ProductCells.areaRatio (1/m^2) gamma ≤ 1+eta →
∃ t0 : ℝ, 0 < t0 ∧ ∀ p ∈ D,
∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane),
IsUnitEigenframe (fderiv ℝ v p) lam L → 0 < M*lam → M*lam < 1 →
∀ t : ℝ, 0 < t → t < t0 →
affineCell p t L (ProductCells.logarithmicCell m) ⊆ D →
∃ C : Cylinder,
C.axis = ‖vec3 1 (v p 0) (h*v p 1)‖⁻¹ • vec3 1 (v p 0) (h*v p 1) ∧
C.carrier = firstPhysicalMap h '' auxiliaryCylinder
(affineCell p t L (ProductCells.logarithmicIntercept m gamma)) (v p) ∧
C.openBounded ∧
physicalRuled h (affineCell p t L (ProductCells.logarithmicCell m)) v M ⊆ C.carrier ∧
C.area ≤ h*(1+eta)^2 *
(∫ q in affineCell p t L (ProductCells.logarithmicCell m), physicalWeight h (v q))
def SourceLogarithmicLocalStatement : Prop :=
∀ (X k : ℝ) (D : Set Plane) (v : Plane → Plane),
0 < X → 0 < k → IsCompact D → D.Nonempty → OpenRectangleC2 D v →
(∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧
LinearMap.det (fderiv ℝ v p).toLinearMap < 0 ∧
k*X*Real.sqrt (-LinearMap.det (fderiv ℝ v p).toLinearMap) < 1) →
∀ eta : ℝ, 0 < eta →
∃ m : ℝ, 1 < m ∧ ∃ gamma : ℝ, 0 < gamma ∧
(1+gamma)*(1/m^2) < 1 ∧ ProductCells.areaRatio (1/m^2) gamma < 1+eta ∧
∃ t0 : ℝ, 0 < t0 ∧ ∀ p ∈ D,
∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane),
0 < lam → IsUnitEigenframe (fderiv ℝ v p) lam L →
∀ t : ℝ, 0 < t → t < t0 →
affineCell p t L (ProductCells.logarithmicCell m) ⊆ D →
∃ C : Cylinder,
C.axis = ‖vec3 1 (k*v p 0) (k*v p 1)‖⁻¹ • vec3 1 (k*v p 0) (k*v p 1) ∧
C.carrier = firstPhysicalMap 1 '' auxiliaryCylinder
(affineCell p t L (ProductCells.logarithmicIntercept m gamma)) (k • v p) ∧
C.openBounded ∧
physicalRuled 1 (affineCell p t L (ProductCells.logarithmicCell m))
(fun q => k • v q) X ⊆ C.carrier ∧
C.area ≤ (1+eta)^2 *
(∫ q in affineCell p t L (ProductCells.logarithmicCell m),
1 / Real.sqrt (1+k^2*‖v q‖^2))
def LogarithmicRemainderStatement : Prop :=
∀ (X k : ℝ) (D : Set Plane) (v : Plane → Plane),
0 < X → 0 < k → IsCompact D → OpenRectangleC2 D v →
(∀ p ∈ D, LinearMap.trace ℝ Plane (fderiv ℝ v p).toLinearMap = 0 ∧
LinearMap.det (fderiv ℝ v p).toLinearMap < 0) →
∀ m : ℝ, 1 < m → ∃ t0 B : ℝ, 0 < t0 ∧ 0 < B ∧
∀ p ∈ D, ∀ (lam : ℝ) (L : (ℝ × ℝ) ≃L[ℝ] Plane),
0 < lam → IsUnitEigenframe (fderiv ℝ v p) lam L →
∀ t : ℝ, 0 < t → t < t0 →
∀ s ∈ ProductCells.logarithmicCell m, ∀ x : ℝ, |x| ≤ X →
∃ E : ℝ × ℝ,
(let q := p + t • L s
q + (x*k) • (v q-v p) =
p + t • L (((1+x*k*lam)*s.1+E.1,(1-x*k*lam)*s.2+E.2))) ∧
‖vec2 E.1 E.2‖ ≤ B*t
def CompleteLocalStatement : Prop :=
ProductCells.GeometryStatement ∧ LocalStatement ∧
LogarithmicLocalStatement ∧ SourceLogarithmicLocalStatement ∧ LogarithmicRemainderStatement
end CylinderCovering.ProductLocal
end
end OAI
|