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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.SpinParameterMoments
import OAI.Probability.SKBarriers.Hierarchy.AverageMeanVariance
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
attribute [local instance 2000] parameterNormedGroup parameterNormedSpace
def blockAdaptiveOverlap {D N k : ℕ} (I : Fin D → Finset (Fin N)) (a : Fin D → ℝ)
(β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) (j : Fin (k+1)) : ℝ :=
hierarchyMeanOverlap (blockDimension D N k) (blockMass D N k)
(blockExponent I (fun i => p.1*a i) (fun b => β*p.2 b))
(fun i s => spin (s i)) (blockLevel D N k j)
theorem blockCoefficients_contDiff {D N k : ℕ} (a : Fin D → ℝ) (β : ℝ) :
ContDiff ℝ 2 (fun p : ℝ × (Fin (k+1) → ℝ) =>
blockCoefficients (N := N) (fun i => p.1*a i) (fun b => β*p.2 b)) := by
apply contDiff_pi.mpr
intro t
refine Fin.addCases (fun i => ?_) (fun i => ?_) t
· simp only [blockCoefficients,Fin.addCases_left]
exact contDiff_fst.mul contDiff_const
· simp only [blockCoefficients,Fin.addCases_right]
exact contDiff_const.mul ((contDiff_apply ℝ ℝ _).comp contDiff_snd)
theorem blockAdaptiveOverlap_contDiff {D N k : ℕ} (I : Fin D → Finset (Fin N))
(a : Fin D → ℝ) (β : ℝ) : ContDiff ℝ 2 (blockAdaptiveOverlap (k := k) I a β) := by
apply contDiff_pi.mpr
intro j
exact (spinHierarchyOverlap_contDiff (blockDimension D N k) (blockMass D N k)
(blockInteraction I) (fun i s => spin (s i)) (fun i s => by cases s i <;> norm_num [spin])
(blockLevel D N k j)).comp (blockCoefficients_contDiff a β)
theorem blockAdaptiveOverlap_bounds {D N k : ℕ} (I : Fin D → Finset (Fin N))
(a : Fin D → ℝ) (β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) (j : Fin (k+1)) :
blockAdaptiveOverlap I a β p j ∈ Set.Icc (0:ℝ) 1 :=
hierarchyMeanOverlap_bounds _ _ _ _ (fun i s => by cases s i <;> norm_num [spin]) _
theorem blockAdaptiveOverlap_monotone {D N k : ℕ} (I : Fin D → Finset (Fin N))
(a : Fin D → ℝ) (β : ℝ) (p : ℝ × (Fin (k+1) → ℝ)) :
Monotone (blockAdaptiveOverlap I a β p) :=
(hierarchyMeanOverlap_mono _ _ _ _ (fun i s => by cases s i <;> norm_num [spin])).comp
(blockLevel_mono D N k)
end SK.Analytic
end
end
end
end OAI
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