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Diff — Schwarzian derivative

Revision #2415 → #3562 · back to history

modifiedSchwarzian derivative (complex)b543d81ff517
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noteNo Schwarzian derivative definition appears in Mathlib4 (grep for 'Schwarzian' returns no files).Grep for 'Schwarzian' in Mathlib returns no files, so no such definition exists.
modifiedSchwarzian derivative (real C^3)ff8f67d0ef87
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noteNo real-variable Schwarzian derivative is defined in Mathlib4.No real-variable Schwarzian derivative appears in Mathlib (grep 'Schwarzian' empty).
modifiedMöbius transformations characterized by vanishing Schwarzian664823e9e135
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noteNo Schwarzian characterization of Möbius maps in Mathlib4.No Schwarzian-based characterization of Möbius maps in Mathlib.
modifiedChain rule for Schwarzian derivative830caf9bcf31
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noteNo Schwarzian derivative and so no chain rule for it in Mathlib4.No Schwarzian derivative, so no chain rule for it in Mathlib.
modifiedSign of Schwarzian preserved under iteration31922a6e9714
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noteNot present in Mathlib4.Not present in Mathlib (no Schwarzian defined).
modifiedAuxiliary function of two complex variablesd21bf809eb8c
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noteThis Schwarzian-related auxiliary function is not defined in Mathlib4.This Schwarzian-related auxiliary function is not defined in Mathlib.
addedSchwarzian via second mixed partial derivativea9d384a92e53
modifiedInversion formula for the Schwarzian502732f5ab01
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noteNot formalized; no Schwarzian in Mathlib4.Not formalized; no Schwarzian in Mathlib.
modifiedThurston's geometric interpretation433f7a39383b
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noteThurston's interpretation is not in Mathlib4.Thurston's interpretation is not in Mathlib.
modifiedImage of circle under conformal map92b4de4eb6a0
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noteNot present in Mathlib4.Not present in Mathlib.
modifiedCross-ratio transformation and Möbius invariance87df3c65f3f8
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noteNo cross-ratio definition or Möbius invariance lemma found in Mathlib4.No cross-ratio definition or Möbius invariance lemma found in Mathlib (grep 'crossRatio' empty).
modifiedLinear second-order ODE associated to Schwarzian555849fdad46
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noteThis specific ODE setup is not formalized in Mathlib4.This specific ODE setup is not formalized in Mathlib.
modifiedSchwarzian of evaluation kernel equals coefficient411fcd2b1fa4
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedEqual Schwarzians implies fractional linear relationd23d2df64178
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noteNot in Mathlib4.Not in Mathlib.
modifiedRatio of solutions satisfies Schwarzian equatione94258ac0666
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedConverse: holomorphic g determines pair of solutionsb2690b77d50e
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noteNot in Mathlib4.Not in Mathlib.
modifiedQ-value of a second-order ODEe2d4692c47f0
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noteQ-value terminology is not defined in Mathlib4.Q-value terminology is not defined in Mathlib.
modifiedGaussian hypergeometric ODEf8b49c8a45e4
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noteMathlib has the hypergeometric series (`ordinaryHypergeometric`) but not the associated ODE in Schwarzian form.Mathlib has the hypergeometric series (`ordinaryHypergeometric`, verified) but not the associated ODE in Schwarzian form.
modifiedKraus–Nehari necessary condition for univalenceb67988483f25
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noteNo Kraus–Nehari univalence bound in Mathlib4.No Kraus–Nehari univalence bound in Mathlib (grep 'Nehari' empty).
modifiedNehari sufficient condition for univalence6bb41a066ce5
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noteNehari's sufficient univalence condition is not formalized in Mathlib4.Nehari's sufficient univalence condition is not formalized in Mathlib.
modifiedParticular sufficient condition for univalence3f03a31cc998
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noteNot in Mathlib4.Not in Mathlib.
modifiedConformal map from Schwarzian to circular-arc polygon77002d220d3d
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noteNot in Mathlib4.Not in Mathlib.
modifiedSchwarzian extends to rational function with double polescb72d56babae
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedAccessory parameters8ff1f1cf2010
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noteAccessory parameters are not defined in Mathlib4.Accessory parameters are not defined in Mathlib.
modifiedTriangle case (Schwarz triangle function)a82fce92321c
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noteSchwarz triangle function is not in Mathlib4.Schwarz triangle function is not in Mathlib.
modifiedQuadrilateral case as Sturm–Liouvillec3d68c9fa6bf
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noteNot in Mathlib4; no Sturm–Liouville theory either.Not in Mathlib; no Sturm–Liouville theory either.
modifiedLowest-eigenvalue condition via Sturm separation9be6d953995c
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noteSturm separation theorem is not in Mathlib4.Sturm separation theorem is not in Mathlib.
modifiedUniversal Teichmüller spaceb4074767096d
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noteMathlib's 'Teichmuller' files concern Witt vectors, not Teichmüller spaces.Mathlib's 'Teichmuller' files concern Witt vectors and Teichmüller–Tukey, not Teichmüller spaces of Riemann surfaces.
addedBeltrami differential equation6a4f4c607771
modifiedBers embedding via Schwarziana09b03796319
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noteBers embedding is not in Mathlib4.Bers embedding is not in Mathlib.
modifiedGehring's characterization of the Bers image88b28aad4e21
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noteGehring's theorem is not formalized in Mathlib4.Gehring's theorem is not formalized in Mathlib.
modifiedTeichmüller space of compact surface as quadratic differentials4d06ef5e850f
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noteNot in Mathlib4.Not in Mathlib.
modifiedSchwarzian as 1-cocycle of Diff(S^1)3084b7982d83
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noteNot in Mathlib4.Not in Mathlib.
modifiedVanishing of higher cohomology for compact group action6e3429cf373e
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noteContinuous cohomology vanishing for compact groups on topological vector spaces is not in Mathlib4.Continuous cohomology vanishing for compact groups on topological vector spaces is not in Mathlib.
modifiedInfinitesimal 1-cocycle on Vect(S^1)081140f96ec2
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedUniqueness of non-zero 1-cocycle solution7fe2f1b9e008
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noteNot in Mathlib4.Not in Mathlib.
modifiedVirasoro algebra as central extension042f6c56de4c
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noteVirasoro algebra is not in Mathlib4.Virasoro algebra is not in Mathlib (grep 'Virasoro' empty).
modifiedQuasisymmetric homeomorphisms from quasiconformal self-maps73bce0c2fd89
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noteQuasiconformal/quasisymmetric theory is absent from Mathlib4.Quasiconformal/quasisymmetric theory is absent from Mathlib (grep empty).
modifiedDual of Lie algebra as Hill's operatorsb2abee7296dc
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedSchwarzian in coadjoint action of Diff(S^1)fd532277b666
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noteNot in Mathlib4.Not in Mathlib.
modifiedHolomorphic pseudogroup3ea60eee2d2b
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noteHolomorphic pseudogroups are not defined in Mathlib4.Holomorphic pseudogroups are not defined in Mathlib.
modifiedPseudogroup defined by differential equations7e91228095f6
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noteNot in Mathlib4.Not in Mathlib.
modifiedClassification of flat pseudogroups via affine/Möbiusaf96d2b3dc27
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedSchwarzian as 1-cocycle for biholomorphism pseudogroupb7aa439c8a75
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noteNot in Mathlib4.Not in Mathlib.
modifiedAffine and projective structures on Riemann surfaces394c3aed4671
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noteAffine/projective structures on Riemann surfaces are not formalized in Mathlib4.Affine/projective structures on Riemann surfaces are not formalized in Mathlib.
modifiedGunning's projective connection result1761b9c7460e
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noteNot in Mathlib4.Not in Mathlib.
modifiedOsgood–Stowe Schwarzian on conformal manifoldsbec6b07d2077
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedCocycle law for conformal-manifold Schwarzianac4392b24183
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noteNot in Mathlib4.Not in Mathlib.
modifiedMöbius transformations have vanishing Schwarzian conformal factor04ac46d651e8
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noteNot formalized in Mathlib4.Not formalized in Mathlib.
modifiedSolutions of Schwarzian equation on Euclidean space16680a1f79d7
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noteNot in Mathlib4.Not in Mathlib.
modifiedLagrangian Schwarzian for positive curvesd905acda20b3
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noteLagrangian Grassmannian and Lagrangian Schwarzian are not defined in Mathlib4.Lagrangian Grassmannian and Lagrangian Schwarzian are not defined in Mathlib.
modifiedVanishing Lagrangian Schwarzian characterizes symplectic equivalencec50fbfc62574
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noteNot in Mathlib4.Not in Mathlib.
modifiedLagrangian Schwarzian via second-order ODE0fdc992c40b9
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noteNot formalized in Mathlib4.Not formalized in Mathlib.