| import OAI.Geometry.RuledCovering.Model |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace RuledApproximation |
|
|
| def comparisonOperator : Plane →L[ℝ] Plane := |
| (2 • EuclideanSpace.proj 1 : Plane →L[ℝ] ℝ).smulRight (vec2 1 0) + |
| ((1 / 8 : ℝ) • EuclideanSpace.proj 0 : Plane →L[ℝ] ℝ).smulRight (vec2 0 1) |
|
|
| def SpectralNormComparisonStatement : Prop := |
| (∀ lam : ℝ, Module.End.HasEigenvalue comparisonOperator.toLinearMap lam ↔ |
| lam = 1 / 2 ∨ lam = -(1 / 2)) ∧ |
| ‖comparisonOperator‖ = 2 ∧ |
| (∀ p : Plane, HyperbolicAt 1 comparisonOperator p) ∧ |
| ¬ ∃ kappa : ℝ, kappa < 1 ∧ ‖comparisonOperator‖ ≤ kappa |
|
|
| def transverseScaling (h : ℝ) (hh : h ≠ 0) : Plane ≃ₗ[ℝ] Plane where |
| toFun p := vec2 (p 0) (h * p 1) |
| invFun p := vec2 (p 0) (p 1 / h) |
| left_inv p := by |
| ext i |
| fin_cases i <;> simp [vec2, hh] |
| right_inv p := by |
| ext i |
| fin_cases i <;> simp [vec2] |
| field_simp |
| map_add' p q := by |
| ext i |
| fin_cases i <;> simp [vec2, mul_add] |
| map_smul' c p := by |
| ext i |
| fin_cases i <;> simp [vec2, mul_left_comm] |
|
|
| def PhysicalScalingInvarianceStatement : Prop := |
| ∀ (h : ℝ) (hh : 0 < h) (A : Plane →ₗ[ℝ] Plane), |
| (∀ lam : ℝ, Module.End.HasEigenvalue ((transverseScaling h hh.ne').conj A) lam ↔ |
| Module.End.HasEigenvalue A lam) ∧ |
| (((transverseScaling h hh.ne').conj A).comp ((transverseScaling h hh.ne').conj A) = 0 ↔ |
| A.comp A = 0) |
|
|
| def rescaledComparisonOperator : Plane →L[ℝ] Plane := |
| LinearMap.toContinuousLinearMap |
| ((transverseScaling 4 (by norm_num)).conj comparisonOperator.toLinearMap) |
|
|
| def PhysicalNormChangeStatement : Prop := |
| ‖comparisonOperator‖ = 2 ∧ ‖rescaledComparisonOperator‖ = 1 / 2 |
|
|
| end RuledApproximation |
|
|
| end |
|
|
| end OAI |
|
|
