| import Mathlib |
| import OAI.Analysis.Conductivity.Fourier.GeneralModeFiniteEndNormal |
|
|
| namespace OAI |
|
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| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
|
|
| lemma angularLiftMatrix_background {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) |
| (s : Fin 3 → ℝ) : |
| angularLiftMatrix A*flatBackgroundTensor (normalizedAngularTensor s A)*(angularLiftMatrix A)ᵀ= |
| angularNormalization A • flatBackgroundTensor s := by |
| have hd := congrArg (fun z : ℝ => z^2) (real_angular_det_one hA) |
| norm_num only [one_pow] at hd |
| have hm := Real.sq_sqrt (angularNormalization_pos A).le |
| ext i j |
| fin_cases i <;> fin_cases j <;> |
| simp [Matrix.mul_apply,Fin.sum_univ_three,Matrix.transpose_apply, |
| angularLiftMatrix,flatBackgroundTensor,normalizedAngularTensor] |
| · nlinarith [hm] |
| · linear_combination angularNormalization A * s 0 * hd |
| · linear_combination angularNormalization A * s 1 * hd |
| · linear_combination angularNormalization A * s 1 * hd |
| · linear_combination angularNormalization A * s 2 * hd |
|
|
| lemma angularInverseLift_shift (A : Matrix (Fin 2) (Fin 2) ℤ) |
| (T : ℝ) (n : Fin 2 → ℤ) (x : Coord3) : |
| angularInverseLift A (x+angularShift T n)=angularInverseLift A x+ |
| angularShift T ![A 1 1*n 0-A 0 1*n 1,-A 1 0*n 0+A 0 0*n 1] := by |
| ext i |
| fin_cases i <;> simp [angularInverseLift,angularShift] <;> ring |
|
|
| lemma AngularPeriodic.angularInverseLift {E : Type*} {T : ℝ} {f : Coord3 → E} |
| (hf : AngularPeriodic T f) (A : Matrix (Fin 2) (Fin 2) ℤ) : |
| AngularPeriodic T (f ∘ angularInverseLift A) := by |
| intro n x |
| simp only [Function.comp_apply,angularInverseLift_shift] |
| exact hf _ _ |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
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| section |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Set Filter Topology MeasureTheory Matrix Real |
| open scoped Matrix.Norms.Elementwise |
|
|
| def undoAngularPoint (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) |
| (d : ℝ) (x : Coord3) : Coord3 := |
| angularInverseLift A (x-![d,0,0])-![0,θ 0,θ 1] |
|
|
| lemma undoAngularPoint_zero (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) |
| (d : ℝ) (x : Coord3) : |
| undoAngularPoint A θ d x 0=(x 0-d)/sqrt (angularNormalization A) := by |
| simp [undoAngularPoint,angularInverseLift] |
|
|
| lemma undoAngularPoint_forward {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) |
| (θ : Fin 2 → ℝ) (d : ℝ) (x : Coord3) : |
| angularNormalizedLift A (angularRealTranslation θ (undoAngularPoint A θ d x))=x-![d,0,0] := by |
| have he : angularRealTranslation θ (undoAngularPoint A θ d x)= |
| angularInverseLift A (x-![d,0,0]) := by |
| ext i |
| fin_cases i <;> simp [angularRealTranslation,undoAngularPoint] |
| rw [he,angularNormalizedLift_inverse hA] |
|
|
| lemma undoAngularPoint_contDiff {A : Matrix (Fin 2) (Fin 2) ℤ} (hA : A.det=1) |
| (θ : Fin 2 → ℝ) (d : ℝ) : ContDiff ℝ (↑(⊤ : ℕ∞)) (undoAngularPoint A θ d) := |
| (((angularNormalizedEquiv A hA).symm.contDiff).comp |
| (contDiff_id.sub contDiff_const)).sub contDiff_const |
|
|
| lemma AngularPeriodic.undoAngularPoint {F : Type*} {T : ℝ} {f : Coord3 → F} |
| (hf : AngularPeriodic T f) (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) (d : ℝ) : |
| AngularPeriodic T (f ∘ undoAngularPoint A θ d) := by |
| intro n x |
| simp only [Function.comp_apply,ScalarConductivity.undoAngularPoint] |
| rw [show x+angularShift T n-![d,0,0]=(x-![d,0,0])+angularShift T n by abel, |
| angularInverseLift_shift] |
| rw [show ScalarConductivity.angularInverseLift A (x-![d,0,0])+ |
| angularShift T ![A 1 1*n 0-A 0 1*n 1,-A 1 0*n 0+A 0 0*n 1]-![0,θ 0,θ 1]= |
| (ScalarConductivity.angularInverseLift A (x-![d,0,0])-![0,θ 0,θ 1])+ |
| angularShift T ![A 1 1*n 0-A 0 1*n 1,-A 1 0*n 0+A 0 0*n 1] by abel] |
| exact hf _ _ |
|
|
| def undoAngularValue (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) |
| (d lam : ℝ) (v : Coord3 → Fin 2 → ℝ) (x : Coord3) : Fin 2 → ℝ := |
| ![sqrt (angularNormalization A)*v (undoAngularPoint A θ d x) 0+d, |
| exp (-lam*d)*v (undoAngularPoint A θ d x) 1] |
|
|
| def undoAngularTensor (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) |
| (d : ℝ) (E : Coord3 → Mat3) (x : Coord3) : Mat3 := |
| (angularNormalization A)⁻¹ • |
| (angularLiftMatrix A*E (undoAngularPoint A θ d x)*(angularLiftMatrix A)ᵀ) |
|
|
| lemma undoAngularValue_matching {s : Fin 3 → ℝ} {A : Matrix (Fin 2) (Fin 2) ℤ} |
| (hA : A.det=1) (θ : Fin 2 → ℝ) (d : ℝ) (h₀ : Fin 2 → ℤ) |
| (a b pa pb : (Fin 2 → ℤ) → ℝ) {v : Coord3 → Fin 2 → ℝ} |
| (hv : ∀ y,y 0∈Icc 0 (1/2) → v y= |
| ![y 0+flatFourier s (normalizedFourierCoefficients s 0 d a) pa |
| (angularNormalizedLift A (angularRealTranslation θ y))/sqrt (angularNormalization A), |
| flatPhaseMode s h₀ (pb h₀) (angularNormalizedLift A (angularRealTranslation θ y))+ |
| flatFourier s (normalizedFourierCoefficients s (torusRate s h₀) d b) pb |
| (angularNormalizedLift A (angularRealTranslation θ y))]) |
| {x : Coord3} (hx : (x 0-d)/sqrt (angularNormalization A)∈Icc 0 (1/2)) : |
| undoAngularValue A θ d (torusRate s h₀) v x= |
| ![x 0+flatFourier s a pa x,flatPhaseMode s h₀ (pb h₀) x+flatFourier s b pb x] := by |
| have hm : sqrt (angularNormalization A)≠0 := (sqrt_pos.mpr (angularNormalization_pos A)).ne' |
| have hxy : (x-![d,0,0])+![d,0,0]=x := sub_add_cancel _ _ |
| have he : exp (-torusRate s h₀*d)*exp (torusRate s h₀*d)=1 := by |
| rw [←exp_add,show -torusRate s h₀*d+torusRate s h₀*d=0 by ring,exp_zero] |
| have hmode := flatPhaseMode_translation s h₀ (pb h₀) d (x-![d,0,0]) |
| rw [hxy] at hmode |
| unfold undoAngularValue |
| rw [hv _ (by simpa only [undoAngularPoint_zero] using hx),undoAngularPoint_forward hA, |
| flatFourier_normalized_translation,flatFourier_normalized_translation,hxy] |
| ext j |
| fin_cases j |
| · change sqrt (angularNormalization A)*(undoAngularPoint A θ d x 0+ |
| exp (0*d)*flatFourier s a pa x/sqrt (angularNormalization A))+d=x 0+flatFourier s a pa x |
| rw [undoAngularPoint_zero] |
| simp only [zero_mul,exp_zero,one_mul] |
| field_simp [hm] |
| ring |
| · change exp (-torusRate s h₀*d)*(flatPhaseMode s h₀ (pb h₀) (x-![d,0,0])+ |
| exp (torusRate s h₀*d)*flatFourier s b pb x)=_ |
| rw [mul_add,←mul_assoc,he,one_mul,←hmode] |
| rfl |
|
|
| lemma undoAngularTensor_matching {s : Fin 3 → ℝ} {A : Matrix (Fin 2) (Fin 2) ℤ} |
| (hA : A.det=1) (θ : Fin 2 → ℝ) (d : ℝ) {E : Coord3 → Mat3} |
| (hE : ∀ y,y 0∈Icc 0 (1/2) → E y=flatBackgroundTensor (normalizedAngularTensor s A)) |
| {x : Coord3} (hx : (x 0-d)/sqrt (angularNormalization A)∈Icc 0 (1/2)) : |
| undoAngularTensor A θ d E x=flatBackgroundTensor s := by |
| rw [undoAngularTensor,hE _ (by simpa only [undoAngularPoint_zero] using hx), |
| angularLiftMatrix_background hA,smul_smul,inv_mul_cancel₀ (angularNormalization_pos A).ne',one_smul] |
|
|
| lemma undoAngularValue_terminal (A : Matrix (Fin 2) (Fin 2) ℤ) (θ : Fin 2 → ℝ) |
| (d lam T : ℝ) {v : Coord3 → Fin 2 → ℝ} (hv : ∀ y,T≤y 0 → v y=![y 0,0]) |
| {x : Coord3} (hx : d+sqrt (angularNormalization A)*T≤x 0) : |
| undoAngularValue A θ d lam v x=![x 0,0] := by |
| have hm : 0<sqrt (angularNormalization A) := sqrt_pos.mpr (angularNormalization_pos A) |
| have hy : T≤undoAngularPoint A θ d x 0 := by |
| rw [undoAngularPoint_zero,le_div_iff₀ hm] |
| nlinarith |
| unfold undoAngularValue |
| rw [hv _ hy] |
| ext j |
| fin_cases j |
| · change sqrt (angularNormalization A)*undoAngularPoint A θ d x 0+d=x 0 |
| rw [undoAngularPoint_zero] |
| field_simp [hm.ne'] |
| ring |
| · simp |
|
|
| end ScalarConductivity |
|
|
| end |
| end |
|
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| end OAI |
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|
