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f205175 | 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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import Mathlib
import OAI.Probability.CriticalPercolation.Harmonic.GeneralizedGramSpectrum
import OAI.Probability.CriticalPercolation.Harmonic.HarmonicFunctionSpace
namespace OAI
section
noncomputable section
open scoped Classical BigOperators Matrix
namespace CriticalPercolation.HarmonicAnalysis
variable {ι X : Type*} [Fintype ι]
def functionGram (F : Finset X) (v : ι → X → ℝ) : Matrix ι ι ℝ :=
fun i j => ∑ x∈F, v i x*v j x
def functionCombination (v : ι → X → ℝ) (a : ι → ℝ) : X → ℝ :=
fun x => ∑ i, a i*v i x
omit [Fintype ι] in
lemma functionGram_hermitian (F : Finset X) (v : ι → X → ℝ) :
(functionGram F v).IsHermitian := by
apply Matrix.IsHermitian.ext
intro i j
simp only [star_trivial,functionGram,mul_comm]
lemma functionGram_quadratic (F : Finset X) (v : ι → X → ℝ) (a : ι → ℝ) :
matrixQuadratic (functionGram F v) a = squareMass F (functionCombination v a) := by
simp only [matrixQuadratic,functionGram,Matrix.mulVec,dotProduct,
squareMass,functionCombination,pow_two,Finset.mul_sum,Finset.sum_mul]
calc
_ = ∑ i, ∑ x∈F, ∑ j, a i*(v i x*v j x*a j) := by
apply Finset.sum_congr rfl
intro i _
exact Finset.sum_comm
_ = ∑ x∈F, ∑ i, ∑ j, a i*(v i x*v j x*a j) := Finset.sum_comm
_ = _ := by
apply Finset.sum_congr rfl
intro x _
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
lemma functionGram_posSemidef (F : Finset X) (v : ι → X → ℝ) :
(functionGram F v).PosSemidef := by
rw [Matrix.posSemidef_iff_dotProduct_mulVec]
refine ⟨functionGram_hermitian F v,?_⟩
intro a
simp only [star_trivial]
change 0 ≤ matrixQuadratic (functionGram F v) a
rw [functionGram_quadratic]
exact squareMass_nonneg _ _
lemma functionGram_posDef (F : Finset X) (v : ι → X → ℝ)
(hdetect : ∀ a, (∀ x∈F, functionCombination v a x=0) → a=0) :
(functionGram F v).PosDef := by
rw [Matrix.posDef_iff_dotProduct_mulVec]
refine ⟨functionGram_hermitian F v,?_⟩
intro a ha
simp only [star_trivial]
change 0 < matrixQuadratic (functionGram F v) a
rw [functionGram_quadratic]
apply lt_of_le_of_ne (squareMass_nonneg _ _)
intro hz
exact ha (hdetect a ((squareMass_eq_zero_iff _ _).mp hz.symm))
lemma functionGram_quadratic_mono {F H : Finset X} (hFH : F⊆H)
(v : ι → X → ℝ) (a : ι → ℝ) :
matrixQuadratic (functionGram F v) a ≤ matrixQuadratic (functionGram H v) a := by
rw [functionGram_quadratic,functionGram_quadratic]
exact squareMass_mono hFH _
lemma functionGram_det_mono {F H : Finset X} (hFH : F⊆H)
(v : ι → X → ℝ) (hF : (functionGram F v).PosDef) :
(functionGram F v).det ≤ (functionGram H v).det := by
obtain ⟨μ,b,hμ,hdet,_,_⟩ := generalized_gram_spectrum
(functionGram F v) (functionGram H v) hF (functionGram_hermitian H v)
(functionGram_quadratic_mono hFH v)
rw [hdet]
have hp : (1:ℝ) ≤ ∏ i, μ i := Finset.one_le_prod₀ (fun index _ => hμ index)
exact le_mul_of_one_le_right hF.det_pos.le hp
omit [Fintype ι] in
lemma functionGram_entry_bound (F : Finset X) (v : ι → X → ℝ) (M : ℝ)
(hM : 0 ≤ M) (hv : ∀ i, ∀ x∈F, |v i x| ≤ M) (i j : ι) :
|functionGram F v i j| ≤ (F.card:ℝ)*M^2 := by
calc
_ ≤ ∑ x∈F, |v i x*v j x| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ x∈F, M^2 := by
apply Finset.sum_le_sum
intro x hx
rw [abs_mul,pow_two]
exact mul_le_mul (hv i x hx) (hv j x hx) (abs_nonneg _) hM
_ = _ := by simp
lemma functionGram_det_bound (F : Finset X) (v : ι → X → ℝ) (M : ℝ)
(hM : 0 ≤ M) (hv : ∀ i, ∀ x∈F, |v i x| ≤ M) :
|(functionGram F v).det| ≤ ((Fintype.card ι).factorial:ℝ)*((F.card:ℝ)*M^2)^(Fintype.card ι) := by
have hh := Matrix.det_le (abv := AbsoluteValue.abs) (functionGram_entry_bound F v M hM hv)
simpa only [AbsoluteValue.abs_apply,nsmul_eq_mul] using hh
end CriticalPercolation.HarmonicAnalysis
end
end
end OAI
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