File size: 1,337 Bytes
d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Geometry.FarNormalization
namespace OAI
/-! # Exact conversion from positive collision to far-shell normalization -/
namespace Falconer
theorem profile_far_collision_normalization {N : ℕ} (hN : 0 < N)
(d : ℕ) (m : ℝ → ℝ) (a b : ℕ) (v lam : ℝ) :
((2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N a)⁻¹^4*
(2:ℝ)^(-((d:ℝ)-1)*(v-b)) /
((2:ℝ)^(v-b)*(profileModelMass m N b)⁻¹^4)) *
((2:ℝ)^N)^(2*(m ((b:ℝ)/N)-m ((a:ℝ)/N))+4*((d:ℝ)+1)*lam) =
((2:ℝ)^N)^(-2*((m ((b:ℝ)/N)-((d:ℝ)/2)*(b:ℝ)/N)-
(m ((a:ℝ)/N)-((d:ℝ)/2)*(a:ℝ)/N))+4*((d:ℝ)+1)*lam) := by
have ha : profileModelMass m N a ≠ 0 := by unfold profileModelMass; positivity
have hb : profileModelMass m N b ≠ 0 := by unfold profileModelMass; positivity
have hv : (2:ℝ)^(v-b) ≠ 0 := by positivity
have hi : ((2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N a)⁻¹^4*
(2:ℝ)^(-((d:ℝ)-1)*(v-b)) /
((2:ℝ)^(v-b)*(profileModelMass m N b)⁻¹^4)) =
(2:ℝ)^((d:ℝ)*(v-a))*(profileModelMass m N b/profileModelMass m N a)^4*
(2:ℝ)^(-((d:ℝ)-1)*(v-b))*(2:ℝ)^(-(v-b)) := by
rw [Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2)]
field_simp
rw [hi]
exact profile_far_prefactor hN d m a b v lam
end Falconer
end OAI
|