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a89d282 | 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticContinuationEndpoint
namespace OAI
/-! Ordinary continuation derivatives pulled back through the actual
optimizing bias. The weighted second derivative has zero boundary limits. -/
noncomputable section
open Filter Set
open scoped Topology
namespace InvariantIsing
def magneticContinuationPullback (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ :=
J.value (magneticBias h s)
def magneticContinuationPullbackFirst (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ :=
J.first (magneticBias h s) / fieldBiasCurvature h (magneticBias h s)
def magneticContinuationPullbackSecond (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) : ℝ :=
J.second (magneticBias h s) / (fieldBiasCurvature h (magneticBias h s)) ^ 2 -
J.first (magneticBias h s) * fieldBiasThird h (magneticBias h s) /
(fieldBiasCurvature h (magneticBias h s)) ^ 3
lemma magneticContinuationPullback_hasDerivAt (h : FieldStep) (J : MagneticContinuationJet)
{s : ℝ} (hs : |s| < 1) :
HasDerivAt (magneticContinuationPullback h J)
(magneticContinuationPullbackFirst h J s) s := by
convert (J.dValue (magneticBias h s)).comp s (hasDerivAt_magneticBias h hs) using 1
· rfl
· simp only [magneticContinuationPullbackFirst, div_eq_mul_inv]
lemma magneticContinuationPullbackFirst_hasDerivAt (h : FieldStep) (J : MagneticContinuationJet)
{s : ℝ} (hs : |s| < 1) :
HasDerivAt (magneticContinuationPullbackFirst h J)
(magneticContinuationPullbackSecond h J s) s := by
have hZ : HasDerivAt (magneticBias h) (1 / fieldBiasCurvature h (magneticBias h s)) s := by
simpa only [one_div] using hasDerivAt_magneticBias h hs
exact inverse_mean_pullback_quotient (fieldBiasCurvature_pos h (magneticBias h s)).ne'
hZ rfl rfl (J.dFirst (magneticBias h s)) (hasDerivAt_fieldBiasCurvature h (magneticBias h s))
lemma magneticContinuationPullback_second_deriv (h : FieldStep) (J : MagneticContinuationJet)
{s : ℝ} (hs : |s| < 1) :
deriv (deriv (magneticContinuationPullback h J)) s =
magneticContinuationPullbackSecond h J s := by
have hh := magneticContinuationPullbackFirst_hasDerivAt h J hs
apply HasDerivAt.deriv
apply hh.congr_of_eventuallyEq
filter_upwards [Ioo_mem_nhds (abs_lt.mp hs).1 (abs_lt.mp hs).2] with t ht
exact (magneticContinuationPullback_hasDerivAt h J (abs_lt.mpr ht)).deriv
lemma magneticContinuation_weighted_second (h : FieldStep) (J : MagneticContinuationJet) (s : ℝ) :
(magneticCurvature h s) ^ 2 * magneticContinuationPullbackSecond h J s =
J.second (magneticBias h s) - J.first (magneticBias h s) *
(fieldBiasThird h (magneticBias h s) / fieldBiasCurvature h (magneticBias h s)) := by
unfold magneticCurvature magneticContinuationPullbackSecond
field_simp [(fieldBiasCurvature_pos h (magneticBias h s)).ne']
lemma magneticFieldLevel_weighted_second_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) :
Tendsto (fun s => (magneticCurvature h s) ^ 2 *
deriv (deriv (fun t => magneticFieldLevel h t i)) s)
(𝓝[<] (1 : ℝ)) (𝓝 0) := by
obtain ⟨J, hJ⟩ := magneticLevelAtBias_has_bounded_jet h i
have hvalue : magneticContinuationPullback h J = fun t => magneticFieldLevel h t i := by
funext t
change J.value (magneticBias h t) = magneticLevelAtBias h (magneticBias h t) i
exact congrFun hJ (magneticBias h t)
apply (magneticContinuation_weighted_endpoint h i J hJ).congr'
have hN : ∀ᶠ s : ℝ in 𝓝[<] (1 : ℝ), -1 < s :=
(show ∀ᶠ s : ℝ in 𝓝 (1 : ℝ), -1 < s from Ioi_mem_nhds (by norm_num)).filter_mono
nhdsWithin_le_nhds
filter_upwards [hN, self_mem_nhdsWithin] with s hs hs1
rw [← hvalue, magneticContinuationPullback_second_deriv h J (abs_lt.mpr ⟨hs, hs1⟩)]
exact (magneticContinuation_weighted_second h J s).symm
lemma magneticFieldLevel_weighted_second_neg_endpoint (h : FieldStep) (i : Fin (h.depth + 1)) :
Tendsto (fun s => (magneticCurvature h s) ^ 2 *
deriv (deriv (fun t => magneticFieldLevel h t i)) s)
(𝓝[>] (-1 : ℝ)) (𝓝 0) := by
obtain ⟨J, hJ⟩ := magneticLevelAtBias_has_bounded_jet h i
have hvalue : magneticContinuationPullback h J = fun t => magneticFieldLevel h t i := by
funext t
change J.value (magneticBias h t) = magneticLevelAtBias h (magneticBias h t) i
exact congrFun hJ (magneticBias h t)
apply (magneticContinuation_weighted_neg_endpoint h i J hJ).congr'
have hN : ∀ᶠ s : ℝ in 𝓝[>] (-1 : ℝ), s < 1 :=
(show ∀ᶠ s : ℝ in 𝓝 (-1 : ℝ), s < 1 from Iio_mem_nhds (by norm_num)).filter_mono
nhdsWithin_le_nhds
filter_upwards [hN, self_mem_nhdsWithin] with s hs hs1
rw [← hvalue, magneticContinuationPullback_second_deriv h J (abs_lt.mpr ⟨hs1, hs⟩)]
exact (magneticContinuation_weighted_second h J s).symm
end InvariantIsing
end
end OAI
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