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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 | 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
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