| import OAI.MeasureTheory.Falconer.Angular.AngularEnvelope |
| import OAI.MeasureTheory.Falconer.Graphs.ProfileGraphEstimate |
|
|
| namespace OAI |
|
|
| /-! # The actual profile graph estimate on the admissible energy envelope |
|
|
| Restriction to a child is implemented by the existing spatial cell symbol. |
| The original restricted endpoint integral is unchanged, and the support |
| condition is propagated explicitly to the child's good set. |
| -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
| open scoped SchwartzMap |
|
|
| noncomputable def finiteEnvelopeMaximum {ι : Type*} (S : Finset ι) (F : ι → ℝ) : ℝ := |
| sSup (insert 0 (F '' (S : Set ι))) |
|
|
| theorem finiteEnvelopeMaximum_nonneg {ι : Type*} (S : Finset ι) (F : ι → ℝ) : |
| 0 ≤ finiteEnvelopeMaximum S F := |
| le_csSup ((S.finite_toSet.image F).insert 0).bddAbove (Or.inl rfl) |
|
|
| theorem le_finiteEnvelopeMaximum {ι : Type*} (S : Finset ι) (F : ι → ℝ) |
| {i : ι} (hi : i ∈ S) : F i ≤ finiteEnvelopeMaximum S F := |
| le_csSup ((S.finite_toSet.image F).insert 0).bddAbove (Or.inr ⟨i,hi,rfl⟩) |
|
|
| theorem finiteEnvelopeMaximum_le {ι : Type*} (S : Finset ι) (F : ι → ℝ) |
| {H : ℝ} (hH : 0 ≤ H) (h : ∀ i ∈ S, F i ≤ H) : |
| finiteEnvelopeMaximum S F ≤ H := by |
| change sSup (insert (0:ℝ) (F '' (S : Set ι))) ≤ H |
| apply csSup_le (show (insert (0:ℝ) (F '' (S : Set ι))).Nonempty from ⟨0,Or.inl rfl⟩) |
| intro x hx |
| rcases hx with rfl | ⟨i,hi,rfl⟩ |
| · exact hH |
| · exact h i hi |
|
|
| theorem angularEnvelope_le_of_admissible {n : ℕ} |
| (μ ν : Measure (Space (n+2))) (M P : ℝ) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) |
| {H : ℝ} (hH : 0 ≤ H) |
| (h : ∀ b c, AngularAdmissible k D GoodLeft b → AngularAdmissible k D GoodRight c → |
| independentAngularEnergy a v Ψ |
| (fun z => endpointTransform μ M b z.1 z.2) |
| (fun z => endpointTransform ν P c z.1 z.2) ≤ H) : |
| angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight ≤ H := by |
| apply csSup_le (angularEnergyValues_nonempty μ ν M P k D a v Ψ GoodLeft GoodRight) |
| intro E hE |
| rcases hE with rfl | ⟨b,c,hb,hc,rfl⟩ |
| · exact hH |
| · exact h b c hb hc |
|
|
| theorem AngularAdmissible.cell {n k : ℕ} {D : ℝ} |
| {Good Good' : Set (Space (n+2) × Sphere (n+2))} |
| {b : Space (n+2) → Sphere (n+2) → ℂ} |
| (h : AngularAdmissible k D Good b) (hD : 0 ≤ D) (Q : DyadicCube (n+2)) |
| (hGood : ∀ x ∈ Q.carrier, ∀ w, (x,w) ∈ Good → (x,w) ∈ Good') : |
| AngularAdmissible k D Good' (cellSymbol Q b) := by |
| refine ⟨h.1.cell (by norm_num) hD Q,?_⟩ |
| intro x w hw |
| have hx := cellSymbol_active Q b hw |
| have hb : b x w ≠ 0 := by simpa only [cellSymbol,ite_eq_left hx] using hw |
| exact hGood x hx w (h.2 x w hb) |
|
|
| theorem child_energy_le_angularEnvelope {n k : ℕ} {D : ℝ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| (Q₁ Q₂ : DyadicCube (n+2)) {M P : ℝ} (hM : 0 < M) (hP : 0 < P) |
| (hD : 0 ≤ D) (a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight ChildLeft ChildRight : Set (Space (n+2) × Sphere (n+2))) |
| (hLeft : ∀ x ∈ Q₁.carrier, ∀ w, (x,w) ∈ GoodLeft → (x,w) ∈ ChildLeft) |
| (hRight : ∀ x ∈ Q₂.carrier, ∀ w, (x,w) ∈ GoodRight → (x,w) ∈ ChildRight) |
| (b c : Space (n+2) → Sphere (n+2) → ℂ) |
| (hb : AngularAdmissible k D GoodLeft b) (hc : AngularAdmissible k D GoodRight c) : |
| independentAngularEnergy a v Ψ |
| (fun z => endpointTransform (μ.restrict Q₁.carrier) M b z.1 z.2) |
| (fun z => endpointTransform (ν.restrict Q₂.carrier) P c z.1 z.2) ≤ |
| angularEnvelope (μ.restrict Q₁.carrier) (ν.restrict Q₂.carrier) |
| M P k D a v Ψ ChildLeft ChildRight := by |
| have h := admissible_energy_le_angularEnvelope (μ.restrict Q₁.carrier) (ν.restrict Q₂.carrier) |
| hM hP k D a v Ψ ChildLeft ChildRight (cellSymbol Q₁ b) (cellSymbol Q₂ c) |
| (hb.cell hD Q₁ hLeft) (hc.cell hD Q₂ hRight) |
| simpa only [endpointTransform_cellSymbol] using h |
|
|
| theorem profile_angular_envelope_terminal {n : ℕ} (k : ℕ) (D : ℝ) |
| (Q : DyadicCube (n+2)) (ρ : Measure (Space (n+2))) |
| {m : ℝ → ℝ} {N : ℕ} {s lam : ℝ} |
| (hreg : Regularization.RegularizedProfileData Q ρ N s lam m) (hN : 0 < N) |
| {a q : ℕ} (ha : a ≤ N) (X Y : List (Fin (n+2) → Bool)) |
| (hX : X.length = a) (hY : Y.length = a) (v : ℝ) |
| (hv : v ≤ (a:ℝ)+100*q) (Ψ : ℝ → ℝ) {W : ℝ} |
| (hW : 0 ≤ W) (hΨ : ∀ t, Ψ t ≤ W) (hsupp : ∀ t, 4 < t → Ψ t = 0) |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| angularEnvelope (ρ.restrict (Q.descendant X).carrier) (ρ.restrict (Q.descendant Y).carrier) |
| (profileModelMass m N a) (profileModelMass m N a) k D a v Ψ GoodLeft GoodRight ≤ |
| (4*W*(sphereMeasure (n+2)).real univ^2)* |
| ((2:ℝ)^N)^(100*((n+2:ℕ):ℝ)*q/N+4*(((n+2:ℕ):ℝ)+1)*lam) := by |
| apply angularEnvelope_le_of_admissible _ _ _ _ _ _ _ _ _ _ _ (by positivity) |
| intro b c hb hc |
| simpa only [one_pow,mul_one] using profile_angular_terminal Q ρ hreg hN ha X Y |
| hX hY v hv Ψ hW hΨ hsupp (by norm_num : (0:ℝ) ≤ 1) b c hb.1.bound hc.1.bound |
|
|
| / |
| noncomputable def childAngularEnvelopeMaximum {n : ℕ} |
| (Q₁ Q₂ : DyadicCube (n+2)) (ρ₁ ρ₂ : Measure (Space (n+2))) (j : ℕ) |
| (M P : ℝ) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| (GoodLeft GoodRight : List (Fin (n+2) → Bool) → |
| Set (Space (n+2) × Sphere (n+2))) : ℝ := |
| finiteEnvelopeMaximum (Q₁.occupiedPaths ρ₁ j ×ˢ Q₂.occupiedPaths ρ₂ j) |
| (fun row => angularEnvelope |
| (ρ₁.restrict (Q₁.descendant row.1).carrier) |
| (ρ₂.restrict (Q₂.descendant row.2).carrier) |
| M P k D a v Ψ (GoodLeft row.1) (GoodRight row.2)) |
|
|
| / |
| theorem profile_near_le_child_angular_envelope {n k : ℕ} (D : ℝ) (hD : 0 ≤ D) |
| (Q : DyadicCube (n+2)) (hQ : Q.side = 1) (ρ : Measure (Space (n+2))) |
| {m : ℝ → ℝ} {N : ℕ} {s lam : ℝ} |
| (hreg : Regularization.RegularizedProfileData Q ρ N s lam m) (hN : 0 < N) |
| {a p : ℕ} (hap : a ≤ p) (hp : p ≤ N) |
| (X Y : List (Fin (n+2) → Bool)) (hX : X.length = a) (hY : Y.length = a) |
| (v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) |
| (ChildLeft ChildRight : List (Fin (n+2) → Bool) → |
| Set (Space (n+2) × Sphere (n+2))) |
| (hLeft : ∀ q ∈ (Q.descendant X).occupiedPaths ρ (p-a), |
| ∀ x ∈ ((Q.descendant X).descendant q).carrier, ∀ z, |
| (x,z) ∈ GoodLeft → (x,z) ∈ ChildLeft q) |
| (hRight : ∀ q ∈ (Q.descendant Y).occupiedPaths ρ (p-a), |
| ∀ x ∈ ((Q.descendant Y).descendant q).carrier, ∀ z, |
| (x,z) ∈ GoodRight → (x,z) ∈ ChildRight q) |
| (b c : Space (n+2) → Sphere (n+2) → ℂ) |
| (hb : AngularAdmissible k D GoodLeft b) (hc : AngularAdmissible k D GoodRight c) : |
| profileNearEnergy (Q.descendant X) (Q.descendant Y) ρ m N a p v Ψ b c ≤ |
| ((2*(10*((n+2)+1))+1)^(2*(n+2)):ℕ) * |
| ((2:ℝ)^N)^(-2*((m ((p:ℝ)/N)-(((n+2:ℕ):ℝ)/2)*(p:ℝ)/N)- |
| (m ((a:ℝ)/N)-(((n+2:ℕ):ℝ)/2)*(a:ℝ)/N))+4*(((n+2:ℕ):ℝ)+1)*lam)* |
| childAngularEnvelopeMaximum (Q.descendant X) (Q.descendant Y) ρ ρ (p-a) |
| (profileModelMass m N p) (profileModelMass m N p) k D p v Ψ |
| ChildLeft ChildRight := by |
| let := hreg.probability |
| have hM : 0 < profileModelMass m N p := by unfold profileModelMass; positivity |
| apply profile_near_reduction Q hQ ρ hreg hN hap hp X Y hX hY v Ψ b c |
| hb.1.measurable hc.1.measurable (by norm_num) (by norm_num) |
| (finiteEnvelopeMaximum_nonneg _ _) hb.1.bound hc.1.bound |
| intro row hrow |
| have hr := Finset.mem_product.mp hrow |
| apply (child_energy_le_angularEnvelope ρ ρ |
| ((Q.descendant X).descendant row.1) ((Q.descendant Y).descendant row.2) |
| hM hM hD p v Ψ GoodLeft GoodRight (ChildLeft row.1) (ChildRight row.2) |
| (hLeft row.1 hr.1) (hRight row.2 hr.2) b c hb hc).trans |
| exact le_finiteEnvelopeMaximum _ (fun row => |
| angularEnvelope (ρ.restrict ((Q.descendant X).descendant row.1).carrier) |
| (ρ.restrict ((Q.descendant Y).descendant row.2).carrier) |
| (profileModelMass m N p) (profileModelMass m N p) k D p v Ψ |
| (ChildLeft row.1) (ChildRight row.2)) hrow |
|
|
| / |
| suprema. Only the localization estimate uses the enlarged analytic budget. -/ |
| theorem dyadic_profile_angular_envelope_step |
| (hpublished : PublishedGnomonicSurface) (n k : ℕ) (hn : 1 ≤ n) |
| (φ : 𝓢(ℝ, ℂ)) (Ψ : ℝ → ℝ) |
| (hφ : ∀ u, φ u = (Ψ u : ℂ)) (hΨ : Measurable Ψ) |
| (hpos : ∀ u, 0 ≤ Ψ u) (hzero : ∀ u ≤ 0, Ψ u = 0) |
| {W : ℝ} (hW : 0 ≤ W) (hΨW : ∀ t, Ψ t ≤ W) |
| (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4 : ℝ) 4) |
| {α : ℝ} (hα : 0 < α) (T : ℝ) : |
| ∃ A C : ℝ, 0 ≤ A ∧ 0 < C ∧ |
| ∀ (Q₁ Q₂ : DyadicCube (n+2)), Q₁.side = 1 → Q₂.side = 1 → |
| ∀ (ρ₁ ρ₂ : Measure (Space (n+2))) (N : ℕ) (s₁ s₂ lam : ℝ) |
| (m₁ m₂ : ℝ → ℝ), |
| Regularization.RegularizedProfileData Q₁ ρ₁ N s₁ lam m₁ → |
| Regularization.RegularizedProfileData Q₂ ρ₂ N s₂ lam m₂ → 0 < N → |
| ∀ (a p g₁ g₂ : ℕ) (σ : ℝ), a ≤ p → p ≤ N → 0 ≤ σ → |
| a ≤ g₁ → g₁ ≤ p → (g₁:ℝ)-a ≤ 2*σ*N → |
| a ≤ g₂ → g₂ ≤ p → (g₂:ℝ)-a ≤ 2*σ*N → |
| ∀ (X Y : List (Fin (n+2) → Bool)), X.length = a → Y.length = a → |
| ∀ (v τ δ H Dsym Dgraph : ℝ), 0 < τ → 0 < δ → δ ≤ 1/2 → |
| 0 ≤ H → 0 ≤ Dsym → Dsym ≤ Dgraph → 1 ≤ Dgraph → δ⁻¹ ≤ Dgraph → |
| Real.sqrt (n+2 : ℕ)*(Q₁.descendant X).side ≤ (τ/4)*Dgraph → |
| 2*Real.sqrt (n+2 : ℕ)*((Q₁.descendant X).side/(2:ℝ)^(p-a))+ |
| 8*δ*(Real.sqrt (n+2 : ℕ)*(Q₁.descendant X).side) ≤ τ/4 → |
| 4*Real.pi*((2:ℝ)^N)^α ≤ (2:ℝ)^v*τ → |
| 4*τ*((2:ℝ)^(p-a)/(Q₁.descendant X).side) ≤ |
| (⌈4*((2:ℝ)^N)^(30*σ)⌉₊:ℝ) → |
| ∀ {ι : Type*} (K : Finset ι) (w : ι → Sphere (n+2)) |
| (η : ι → Sphere (n+2) → ℝ), |
| (∀ j ∈ K, ChartCapControl k (η j) (w j : Space (n+2)) δ H) → |
| (∀ j ∈ K, ∀ z, 0 ≤ η j z) → (∀ z, ∑ j ∈ K, η j z = 1) → |
| ∀ (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) |
| (ChildLeft ChildRight : List (Fin (n+2) → Bool) → |
| Set (Space (n+2) × Sphere (n+2))), |
| (∀ q ∈ (Q₁.descendant X).occupiedPaths ρ₁ (p-a), |
| ∀ x ∈ ((Q₁.descendant X).descendant q).carrier, ∀ z, |
| (x,z) ∈ GoodLeft → (x,z) ∈ ChildLeft q) → |
| (∀ q ∈ (Q₂.descendant Y).occupiedPaths ρ₂ (p-a), |
| ∀ x ∈ ((Q₂.descendant Y).descendant q).carrier, ∀ z, |
| (x,z) ∈ GoodRight → (x,z) ∈ ChildRight q) → |
| ∀ (W₁ W₂ : ℝ), |
| 2*τ+2*δ*(Real.sqrt (n+2 : ℕ)*((Q₁.descendant X).side/(2:ℝ)^(g₁-a))) ≤ W₁ → |
| 2*τ+2*δ*(Real.sqrt (n+2 : ℕ)*((Q₂.descendant Y).side/(2:ℝ)^(g₂-a))) ≤ W₂ → |
| (∀ q ∈ (Q₁.descendant X).occupiedPaths ρ₁ (p-a), |
| ∀ x ∈ ((Q₁.descendant X).descendant q).carrier, |
| ∀ z : Sphere (n+2), (x,z) ∈ GoodLeft → |
| DyadicParallelTest (Q₁.descendant X) ((Q₁.descendant X).occupiedPaths ρ₁ (p-a)) |
| (g₁-a) q z W₁ |
| ⌈((2:ℝ)^N)^(PlanarFalconer.height |
| (fun j => m₁ ((j:ℝ)/N)-((n+2:ℕ):ℝ)/2*j/N) g₁ p+ |
| 1000*(((n+2:ℕ):ℝ)+1)*σ)⌉₊) → |
| (∀ q ∈ (Q₂.descendant Y).occupiedPaths ρ₂ (p-a), |
| ∀ x ∈ ((Q₂.descendant Y).descendant q).carrier, |
| ∀ z : Sphere (n+2), (x,z) ∈ GoodRight → |
| DyadicParallelTest (Q₂.descendant Y) ((Q₂.descendant Y).occupiedPaths ρ₂ (p-a)) |
| (g₂-a) q z W₂ |
| ⌈((2:ℝ)^N)^(PlanarFalconer.height |
| (fun j => m₂ ((j:ℝ)/N)-((n+2:ℕ):ℝ)/2*j/N) g₂ p+ |
| 1000*(((n+2:ℕ):ℝ)+1)*σ)⌉₊) → |
| ∀ [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)], |
| let E := childAngularEnvelopeMaximum (Q₁.descendant X) (Q₂.descendant Y) |
| ρ₁ ρ₂ (p-a) (profileModelMass m₁ N p) (profileModelMass m₂ N p) |
| k Dsym p v Ψ ChildLeft ChildRight |
| angularEnvelope (ρ₁.restrict (Q₁.descendant X).carrier) |
| (ρ₂.restrict (Q₂.descendant Y).carrier) |
| (profileModelMass m₁ N a) (profileModelMass m₂ N a) |
| k Dsym a v Ψ GoodLeft GoodRight ≤ |
| 2*11^(n+2)*((2:ℝ)^N)^( |
| -((m₁ ((p:ℝ)/N)-((n+2:ℕ):ℝ)/2*p/N)- |
| (m₁ ((a:ℝ)/N)-((n+2:ℕ):ℝ)/2*a/N))- |
| ((m₂ ((p:ℝ)/N)-((n+2:ℕ):ℝ)/2*p/N)- |
| (m₂ ((a:ℝ)/N)-((n+2:ℕ):ℝ)/2*a/N))+ |
| (min (PlanarFalconer.height (fun j => m₁ ((j:ℝ)/N)-((n+2:ℕ):ℝ)/2*j/N) a p) |
| (PlanarFalconer.height (fun j => m₂ ((j:ℝ)/N)-((n+2:ℕ):ℝ)/2*j/N) a p)+ |
| (1000*(((n+2:ℕ):ℝ)+1)+32*(n+2:ℕ))*σ)+4*(((n+2:ℕ):ℝ)+1)*lam)*E + |
| (2:ℝ)^(((n+2:ℕ):ℝ)*(v-a)) * |
| ((profileModelMass m₁ N p/profileModelMass m₁ N a)* |
| (profileModelMass m₂ N p/profileModelMass m₂ N a))^2 * |
| capGraphError (n+2) k A C W ((2:ℝ)^N) T |
| (profileModelMass m₁ N p) (profileModelMass m₂ N p) H 1 Dgraph v τ K.card |
| ((Q₁.descendant X).occupiedPaths ρ₁ (p-a) ×ˢ |
| (Q₂.descendant Y).occupiedPaths ρ₂ (p-a)).card := by |
| obtain ⟨A,C,hA,hC,hgraph⟩ := dyadic_profile_graph_energy_step hpublished n k hn φ Ψ |
| hφ hΨ hpos hzero hW hΨW hsupp hα T |
| refine ⟨A,C,hA,hC,?_⟩ |
| intro Q₁ Q₂ hQ₁ hQ₂ ρ₁ ρ₂ N s₁ s₂ lam m₁ m₂ hreg₁ hreg₂ hN |
| a p g₁ g₂ σ hap hp hσ hag₁ hgp₁ hgap₁ hag₂ hgp₂ hgap₂ X Y hX hY |
| v τ δ H Dsym Dgraph hτ hδ hδsmall hH hDsym hsymgraph hD hδD hnorm herr hfreq hL |
| ι K w η hcap hη hpart GoodLeft GoodRight ChildLeft ChildRight |
| hleft hright W₁ W₂ hw₁ hw₂ htest₁ htest₂ _ |
| let := hreg₁.probability |
| let := hreg₂.probability |
| let E := childAngularEnvelopeMaximum (Q₁.descendant X) (Q₂.descendant Y) |
| ρ₁ ρ₂ (p-a) (profileModelMass m₁ N p) (profileModelMass m₂ N p) |
| k Dsym p v Ψ ChildLeft ChildRight |
| have hE : 0 ≤ E := finiteEnvelopeMaximum_nonneg _ _ |
| have hmass (m : ℝ → ℝ) (j : ℕ) : 0 < profileModelMass m N j := by |
| unfold profileModelMass |
| positivity |
| have herror : 0 ≤ capGraphError (n+2) k A C W ((2:ℝ)^N) T |
| (profileModelMass m₁ N p) (profileModelMass m₂ N p) H 1 Dgraph v τ K.card |
| ((Q₁.descendant X).occupiedPaths ρ₁ (p-a) ×ˢ |
| (Q₂.descendant Y).occupiedPaths ρ₂ (p-a)).card := by |
| unfold capGraphError |
| positivity |
| apply angularEnvelope_le_of_admissible _ _ _ _ _ _ _ _ _ _ _ (by positivity) |
| intro b c hb hc |
| apply hgraph Q₁ Q₂ hQ₁ hQ₂ ρ₁ ρ₂ N s₁ s₂ lam m₁ m₂ hreg₁ hreg₂ hN |
| a p g₁ g₂ σ hap hp hσ hag₁ hgp₁ hgap₁ hag₂ hgp₂ hgap₂ X Y hX hY |
| v τ δ H 1 Dgraph hτ hδ hδsmall hH (by norm_num) hD hδD hnorm herr hfreq hL |
| K w η hcap hη hpart b c |
| (hb.mono hDsym hsymgraph subset_rfl).1 |
| (hc.mono hDsym hsymgraph subset_rfl).1 W₁ W₂ hw₁ hw₂ |
| · intro q hq |
| filter_upwards [ae_restrict_mem ((Q₁.descendant X).descendant q).measurable] with x hx |
| intro z hz |
| exact htest₁ q hq x hx z (hb.2 x z hz) |
| · intro q hq |
| filter_upwards [ae_restrict_mem ((Q₂.descendant Y).descendant q).measurable] with x hx |
| intro z hz |
| exact htest₂ q hq x hx z (hc.2 x z hz) |
| · exact hE |
| · intro row hrow |
| have hr := Finset.mem_product.mp hrow |
| apply (child_energy_le_angularEnvelope ρ₁ ρ₂ |
| ((Q₁.descendant X).descendant row.1) ((Q₂.descendant Y).descendant row.2) |
| (hmass m₁ p) (hmass m₂ p) hDsym p v Ψ GoodLeft GoodRight |
| (ChildLeft row.1) (ChildRight row.2) |
| (hleft row.1 hr.1) (hright row.2 hr.2) b c hb hc).trans |
| exact le_finiteEnvelopeMaximum _ (fun row => |
| angularEnvelope (ρ₁.restrict ((Q₁.descendant X).descendant row.1).carrier) |
| (ρ₂.restrict ((Q₂.descendant Y).descendant row.2).carrier) |
| (profileModelMass m₁ N p) (profileModelMass m₂ N p) k Dsym p v Ψ |
| (ChildLeft row.1) (ChildRight row.2)) hrow |
|
|
| end Falconer |
|
|
| end OAI |
|
|
