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import OAI.Analysis.BiholderTransport.Model
namespace OAI
section
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
section ModelMixedConorm
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteModelMixed (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end ModelMixedConorm
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff
namespace WeakMTWTransport
section UniformMixedConorm
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteUniformConorm (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformMixedConorm
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff NNReal
namespace WeakMTWTransport
section UniformRadialProfile
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteUniformProfile (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformRadialProfile
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff
namespace WeakMTWTransport
section AffineMTW
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteAffine (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end AffineMTW
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff NNReal
namespace WeakMTWTransport
section UniformJoinSpectrum
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteJoinBound (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformJoinSpectrum
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff
namespace WeakMTWTransport
section UniformAffineProfile
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteAffineProfile (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformAffineProfile
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff
namespace WeakMTWTransport
section AffineSpectrumComparison
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteAffineSpectrum (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end AffineSpectrumComparison
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff
namespace WeakMTWTransport
section FixedJoinDeterminant
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
[IsRiemannianManifold 𝓘(ℝ,Model n) M]
local instance tangentFiniteJoinDet (x : M) :
FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
inferInstanceAs (FiniteDimensional ℝ (Model n))
end FixedJoinDeterminant
end WeakMTWTransport
end
end
section
noncomputable section
open Module
namespace WeakMTWTransport
section NormDetBound
variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[FiniteDimensional ℝ F]
lemma norm_eigenvector_image_eq_singularValue (A : E →ₗ[ℝ] F)
{n : ℕ} (hn : finrank ℝ E=n) (i : Fin n) :
‖A (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i)‖=A.singularValues i := by
let v := A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i
have hv : ‖v‖=1 := (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn).norm_eq_one i
have he : (A.adjoint.comp A) v=A.isSymmetric_adjoint_comp_self.eigenvalues hn i • v :=
A.isSymmetric_adjoint_comp_self.apply_eigenvectorBasis hn i
have hsq : ‖A v‖^2=A.isSymmetric_adjoint_comp_self.eigenvalues hn i := by
rw [←real_inner_self_eq_norm_sq,←A.adjoint_inner_left v (A v)]
change inner ℝ ((A.adjoint.comp A) v) v=_
rw [he,real_inner_smul_left,real_inner_self_eq_norm_sq,hv]
ring
apply (sq_eq_sq₀ (norm_nonneg _) (A.singularValues_nonneg i)).mp
rw [hsq,A.sq_singularValues_fin hn i]
end NormDetBound
end WeakMTWTransport
end
end
end
end OAI
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