| import OAI.MathematicalPhysics.Transonic.Phase.RadialFlux |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SepticProfile |
| open Set |
| open scoped ContDiff BigOperators |
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| abbrev PhysicalFunction := ℝ → PhysicalSpace → ℝ |
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| def partialTime (s : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ := |
| deriv (fun q => s q X) t |
| def partialSpace (s : PhysicalFunction) (j : Fin 4) (t : ℝ) (X : PhysicalSpace) : ℝ := |
| deriv (fun r => s t (coordinateLine X j r)) 0 |
| def minkowskiPair (s f : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ := |
| partialTime s t X*partialTime f t X- |
| ∑ j : Fin 4, partialSpace s j t X*partialSpace f j t X |
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| def phaseTransport (s : PhysicalFunction) (t : ℝ) (X : PhysicalSpace) : ℝ := |
| partialTime (fun q Z => (minkowskiSquare s q Z)^((1:ℝ)/3)*partialTime s q Z) t X- |
| ∑ j : Fin 4, partialSpace (fun q Z => |
| (minkowskiSquare s q Z)^((1:ℝ)/3)*partialSpace s j q Z) j t X |
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| def GlobalProfile.w0 (P : GlobalProfile) (t : ℝ) (X : PhysicalSpace) : ℝ := |
| (minkowskiSquare P.s0 t X)^((1:ℝ)/6) |
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| def GlobalProfile.linearizedPhase (P : GlobalProfile) (f : PhysicalFunction) |
| (t : ℝ) (X : PhysicalSpace) : ℝ := |
| partialTime (fun q Z => (P.w0 q Z)^2*partialTime f q Z+ |
| (2/3:ℝ)*(P.w0 q Z)^(-4:ℤ)*minkowskiPair P.s0 f q Z*partialTime P.s0 q Z) t X- |
| ∑ j : Fin 4, partialSpace (fun q Z => (P.w0 q Z)^2*partialSpace f j q Z+ |
| (2/3:ℝ)*(P.w0 q Z)^(-4:ℤ)*minkowskiPair P.s0 f q Z*partialSpace P.s0 j q Z) j t X |
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| def InProfileSpace (a : ℝ) (f : PhysicalFunction) : Prop := |
| ∃ (N : ℕ) (F : ℕ → ℝ → ℝ), |
| (∀ j≤N, ContDiffOn ℝ ∞ (F j) (Ici 0)) ∧ |
| ∀ t : ℝ, 0<t → ∀ X : PhysicalSpace, |
| f t X=t^a*∑ j∈Finset.range (N+1), (Real.log t)^j*F j (radiusSq X/t^2) |
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| def ProfileSurjectivity (P : GlobalProfile) : Prop := |
| ∀ (a : ℝ) (f : PhysicalFunction), InProfileSpace (a-2-2*P.beta/3) f → |
| ∃ phi : PhysicalFunction, InProfileSpace a phi ∧ |
| ∀ t : ℝ, 0<t → ∀ X : PhysicalSpace, P.linearizedPhase phi t X=f t X |
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| def FullProfileInputExists : Prop := |
| ∃ P : GlobalProfile, (∀ t : ℝ, 0<t → ∀ X : PhysicalSpace, |
| 0 < minkowskiSquare P.s0 t X ∧ phaseTransport P.s0 t X=0) ∧ ProfileSurjectivity P |
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| end SepticProfile |
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| end |
| end |
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| end OAI |
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