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Homological Algebra Programming

HAP

Version 1.62

01 Feb 2024

Graham Ellis
Email: Graham.Ellis@nuigalway.ie

Address:
School of Mathematics
National University of Irelnd, Galway
Galway
(Ireland)

Abstract

HAP is a homological algebra library for use with the GAP computer algebra system, and is still under development. The current version 1.62 was released on 01 Feb 2024 .
The initial focus of the library was on computations related to the cohomology of finite and infinite groups, with particular emphasis on integral coefficients. The focus has since broadened to include Steenrod algebras of finite groups, Bredon homology, cohomology of simplicial groups, and general computations in algebraic topology relating to finite CW-complexes, covering spaces, knots, knotted surfaces, and topics such as persitent homology arising in topological data analysis.
This document describes the functions available in HAP. Examples illustrating these functions are available in the HAP tutorial.

Copyright

© 2005-2019 by Graham Ellis

Acknowledgements

The HAP project has been supported by the School of Maths at NUI Galway, various PhD students and postdoctoral researchers at NUI Galway, the Irish Research Council, Science Foundation Ireland, and the EU Marie Curie programme.

Contents

1 Basic functionality for cellular complexes, fundamental groups and homology

1.1 Data \(\longrightarrow\) Cellular Complexes

  1.1-1 RegularCWPolytope
  1.1-2 CubicalComplex
  1.1-3 PureCubicalComplex
  1.1-4 PureCubicalKnot
  1.1-5 PurePermutahedralKnot
  1.1-6 PurePermutahedralComplex
  1.1-7 CayleyGraphOfGroup
  1.1-8 EquivariantEuclideanSpace
  1.1-9 EquivariantOrbitPolytope
  1.1-10 EquivariantTwoComplex
  1.1-11 QuillenComplex
  1.1-12 RestrictedEquivariantCWComplex
  1.1-13 RandomSimplicialGraph
  1.1-14 RandomSimplicialTwoComplex
  1.1-15 ReadCSVfileAsPureCubicalKnot
  1.1-16 ReadImageAsPureCubicalComplex
  1.1-17 ReadImageAsFilteredPureCubicalComplex
  1.1-18 ReadImageAsWeightFunction
  1.1-19 ReadPDBfileAsPureCubicalComplex
  1.1-20 ReadPDBfileAsPurepermutahedralComplex
  1.1-21 RegularCWPolytope
  1.1-22 SimplicialComplex
  1.1-23 SymmetricMatrixToFilteredGraph
  1.1-24 SymmetricMatrixToGraph

1.2 Metric Spaces

  1.2-1 CayleyMetric
  1.2-2 EuclideanMetric
  1.2-3 EuclideanSquaredMetric
  1.2-4 HammingMetric
  1.2-5 KendallMetric
  1.2-6 ManhattanMetric
  1.2-7 VectorsToSymmetricMatrix

1.3 Cellular Complexes \(\longrightarrow\) Cellular Complexes

  1.3-1 BoundaryMap
  1.3-2 CliqueComplex
  1.3-3 ConcentricFiltration
  1.3-4 DirectProduct
  1.3-5 FiltrationTerm
  1.3-6 Graph
  1.3-7 HomotopyGraph
  1.3-8 Nerve
  1.3-9 RegularCWComplex
  1.3-10 RegularCWMap
  1.3-11 ThickeningFiltration

1.4 Cellular Complexes \(\longrightarrow\) Cellular Complexes (Preserving Data Types)

  1.4-1 ContractedComplex
  1.4-2 ContractibleSubcomplex
  1.4-3 KnotReflection
  1.4-4 KnotSum
  1.4-5 OrientRegularCWComplex
  1.4-6 PathComponent
  1.4-7 PureComplexBoundary
  1.4-8 PureComplexComplement
  1.4-9 PureComplexDifference
  1.4-10 PureComplexInterstection
  1.4-11 PureComplexThickened
  1.4-12 PureComplexUnion
  1.4-13 SimplifiedComplex
  1.4-14 ZigZagContractedComplex

1.5 Cellular Complexes \(\longrightarrow\) Homotopy Invariants

  1.5-1 AlexanderPolynomial
  1.5-2 BettiNumber
  1.5-3 EulerCharacteristic
  1.5-4 EulerIntegral
  1.5-5 FundamentalGroup
  1.5-6 FundamentalGroupOfQuotient
  1.5-7 IsAspherical
  1.5-8 KnotGroup
  1.5-9 PiZero
  1.5-10 PersistentBettiNumbers

1.6 Data \(\longrightarrow\) Homotopy Invariants

1.6-1 DendrogramMat

1.7 Cellular Complexes \(\longrightarrow\) Non Homotopy Invariants

  1.7-1 ChainComplex
  1.7-2 ChainComplexEquivalence
  1.7-3 ChainComplexOfQuotient
  1.7-4 ChainMap
  1.7-5 CochainComplex
  1.7-6 CriticalCells
  1.7-7 DiagonalApproximation
  1.7-8 Size

1.8 (Co)chain Complexes \(\longrightarrow \) (Co)chain Complexes

  1.8-1 FilteredTensorWithIntegers
  1.8-2 FilteredTensorWithIntegersModP
  1.8-3 HomToIntegers
  1.8-4 TensorWithIntegersModP

1.9 (Co)chain Complexes \(\longrightarrow \) Homotopy Invariants

  1.9-1 Cohomology
  1.9-2 CupProduct
  1.9-3 Homology

1.10 Visualization

  1.10-1 BarCodeDisplay
  1.10-2 BarCodeCompactDisplay
  1.10-3 CayleyGraphOfGroup
  1.10-4 Display
  1.10-5 DisplayArcPresentation
  1.10-6 DisplayCSVKnotFile
  1.10-7 DisplayDendrogram
  1.10-8 DisplayDendrogramMat
  1.10-9 DisplayPDBfile
  1.10-10 OrbitPolytope
  1.10-11 ScatterPlot

2 Basic functionality for \(\mathbb ZG\)-resolutions and group cohomology

2.1 Resolutions

  2.1-1 EquivariantChainMap
  2.1-2 FreeGResolution
  2.1-3 ResolutionBieberbachGroup
  2.1-4 ResolutionCubicalCrystGroup
  2.1-5 ResolutionFiniteGroup
  2.1-6 ResolutionNilpotentGroup
  2.1-7 ResolutionNormalSeries
  2.1-8 ResolutionPrimePowerGroup
  2.1-9 ResolutionSL2Z
  2.1-10 ResolutionSmallGroup
  2.1-11 ResolutionSubgroup

2.2 Algebras \(\longrightarrow \) (Co)chain Complexes

2.2-1 LeibnizComplex

2.3 Resolutions \(\longrightarrow \) (Co)chain Complexes

  2.3-1 HomToIntegers
  2.3-2 HomToIntegralModule
  2.3-3 TensorWithIntegers
  2.3-4 TensorWithIntegersModP

2.4 Cohomology rings

  2.4-1 AreIsomorphicGradedAlgebras
  2.4-2 HAPDerivation
  2.4-3 HilbertPoincareSeries
  2.4-4 HomologyOfDerivation
  2.4-5 IntegralCohomologyGenerators
  2.4-6 LHSSpectralSequence
  2.4-7 LHSSpectralSequenceLastSheet
  2.4-8 ModPCohomologyGenerators
  2.4-9 ModPCohomologyRing
  2.4-10 Mod2CohomologyRingPresentation

2.5 Group Invariants

  2.5-1 GroupCohomology
  2.5-2 GroupHomology
  2.5-3 PrimePartDerivedFunctor
  2.5-4 PoincareSeries
  2.5-5 PoincareSeries
  2.5-6 RankHomologyPGroup

2.6 \(\mathbb F_p\)-modules

  2.6-1 GroupAlgebraAsFpGModule
  2.6-2 Radical
  2.6-3 RadicalSeries

3 Basic functionality for homological group theory

  3.1-1 CcGroup
  3.1-2 CocycleCondition
  3.1-3 StandardCocycle

3.2 G-Outer Groups

  3.2-1 ActedGroup
  3.2-2 ActingGroup
  3.2-3 Centre
  3.2-4 GOuterGroup

3.3 \(G\)-cocomplexes

3.3-1 CohomologyModule
3.3-2 HomToGModule

4 Basic functionality for parallel computation

4.1 Six Core Functions

4.1-1 ChildCreate
4.1-2 ChildCreate

5 Resolutions of the ground ring

  5.1-1 TietzeReducedResolution
  5.1-2 ResolutionArithmeticGroup
  5.1-3 FreeGResolution
  5.1-4 ResolutionGTree
  5.1-5 ResolutionSL2Z
  5.1-6 ResolutionAbelianGroup
  5.1-7 ResolutionAlmostCrystalGroup
  5.1-8 ResolutionAlmostCrystalQuotient
  5.1-9 ResolutionArtinGroup
  5.1-10 ResolutionAsphericalPresentation
  5.1-11 ResolutionBieberbachGroup
  5.1-12 ResolutionCoxeterGroup
  5.1-13 ResolutionDirectProduct
  5.1-14 ResolutionExtension
  5.1-15 ResolutionFiniteDirectProduct
  5.1-16 ResolutionFiniteExtension
  5.1-17 ResolutionFiniteGroup
  5.1-18 ResolutionFiniteSubgroup
  5.1-19 ResolutionGraphOfGroups
  5.1-20 ResolutionNilpotentGroup
  5.1-21 ResolutionNormalSeries
  5.1-22 ResolutionPrimePowerGroup
  5.1-23 ResolutionSmallFpGroup
  5.1-24 ResolutionSubgroup
  5.1-25 ResolutionSubnormalSeries
  5.1-26 TwistedTensorProduct
  5.1-27 ConjugatedResolution
  5.1-28 RecalculateIncidenceNumbers

6 Resolutions of modules

6.1-1 ResolutionFpGModule

7 Induced equivariant chain maps

7.1-1 EquivariantChainMap

  8.1-1 ExtendScalars
  8.1-2 HomToIntegers
  8.1-3 HomToIntegersModP
  8.1-4 HomToIntegralModule
  8.1-5 TensorWithIntegralModule
  8.1-6 HomToGModule
  8.1-7 InduceScalars
  8.1-8 LowerCentralSeriesLieAlgebra
  8.1-9 TensorWithIntegers
  8.1-10 FilteredTensorWithIntegers
  8.1-11 TensorWithTwistedIntegers
  8.1-12 TensorWithIntegersModP
  8.1-13 TensorWithTwistedIntegersModP
  8.1-14 TensorWithRationals

9 Chain complexes

  9.1-1 ChainComplex
  9.1-2 ChainComplexOfPair
  9.1-3 ChevalleyEilenbergComplex
  9.1-4 LeibnizComplex
  9.1-5 SuspendedChainComplex
  9.1-6 ReducedSuspendedChainComplex
  9.1-7 CoreducedChainComplex
  9.1-8 TensorProductOfChainComplexes
  9.1-9 LefschetzNumber

10 Sparse Chain complexes

  10.1-1 SparseMat
  10.1-2 TransposeOfSparseMat
  10.1-3 ReverseSparseMat
  10.1-4 SparseRowMult
  10.1-5 SparseRowInterchange
  10.1-6 SparseRowAdd
  10.1-7 SparseSemiEchelon
  10.1-8 RankMatDestructive
  10.1-9 RankMat
  10.1-10 SparseChainComplex
  10.1-11 SparseChainComplexOfRegularCWComplex
  10.1-12 SparseBoundaryMatrix
  10.1-13 Bettinumbers

11 Homology and cohomology groups

  11.1-1 Cohomology
  11.1-2 CohomologyModule
  11.1-3 CohomologyPrimePart
  11.1-4 GroupCohomology
  11.1-5 GroupHomology
  11.1-6 PersistentHomologyOfQuotientGroupSeries
  11.1-7 PersistentCohomologyOfQuotientGroupSeries
  11.1-8 UniversalBarCode
  11.1-9 PersistentHomologyOfSubGroupSeries
  11.1-10 PersistentHomologyOfFilteredChainComplex
  11.1-11 PersistentHomologyOfCommutativeDiagramOfPGroups
  11.1-12 PersistentHomologyOfFilteredPureCubicalComplex
  11.1-13 PersistentHomologyOfPureCubicalComplex
  11.1-14 ZZPersistentHomologyOfPureCubicalComplex
  11.1-15 RipsHomology
  11.1-16 BarCode
  11.1-17 BarCodeDisplay
  11.1-18 Homology
  11.1-19 HomologyPb
  11.1-20 HomologyVectorSpace
  11.1-21 HomologyPrimePart
  11.1-22 LeibnizAlgebraHomology
  11.1-23 LieAlgebraHomology
  11.1-24 PrimePartDerivedFunctor
  11.1-25 RankHomologyPGroup
  11.1-26 RankPrimeHomology

12 Poincare series

  12.1-1 EfficientNormalSubgroups
  12.1-2 ExpansionOfRationalFunction
  12.1-3 PoincareSeries
  12.1-4 PoincareSeriesPrimePart
  12.1-5 PoincareSeriesLHS
  12.1-6 Prank

13 Cohomology ring structure

  13.1-1 IntegralCupProduct
  13.1-2 IntegralRingGenerators
  13.1-3 ModPCohomologyGenerators
  13.1-4 ModPCohomologyRing
  13.1-5 ModPRingGenerators
  13.1-6 Mod2CohomologyRingPresentation

14 Cohomology rings of \(p\)-groups (mainly \(p=2)\)

14.1-1 Mod2CohomologyRingPresentation
14.1-2 PoincareSeriesLHS

15 Commutator and nonabelian tensor computations

  15.1-1 BaerInvariant
  15.1-2 BogomolovMultiplier
  15.1-3 Bogomology
  15.1-4 Coclass
  15.1-5 EpiCentre
  15.1-6 NonabelianExteriorProduct
  15.1-7 NonabelianSymmetricKernel
  15.1-8 NonabelianSymmetricSquare
  15.1-9 NonabelianTensorProduct
  15.1-10 NonabelianTensorSquare
  15.1-11 RelativeSchurMultiplier
  15.1-12 TensorCentre
  15.1-13 ThirdHomotopyGroupOfSuspensionB
  15.1-14 UpperEpicentralSeries

16 Lie commutators and nonabelian Lie tensors

  16.1-1 LieCoveringHomomorphism
  16.1-2 LeibnizQuasiCoveringHomomorphism
  16.1-3 LieEpiCentre
  16.1-4 LieExteriorSquare
  16.1-5 LieTensorSquare
  16.1-6 LieTensorCentre

17 Generators and relators of groups

  17.1-1 CayleyGraphOfGroupDisplay
  17.1-2 IdentityAmongRelatorsDisplay
  17.1-3 IsAspherical
  17.1-4 PresentationOfResolution
  17.1-5 TorsionGeneratorsAbelianGroup

18 Orbit polytopes and fundamental domains

  18.1-1 CoxeterComplex
  18.1-2 ContractibleGcomplex
  18.1-3 QuotientOfContractibleGcomplex
  18.1-4 TruncatedGComplex
  18.1-5 FundamentalDomainStandardSpaceGroup
  18.1-6 OrbitPolytope
  18.1-7 PolytopalComplex
  18.1-8 PolytopalGenerators
  18.1-9 VectorStabilizer

  19.1-1 CcGroup
  19.1-2 CocycleCondition
  19.1-3 StandardCocycle
  19.1-4 Syzygy

20 Words in free \(ZG\)-modules

  20.1-1 AddFreeWords
  20.1-2 AddFreeWordsModP
  20.1-3 AlgebraicReduction
  20.1-4 MultiplyWord
  20.1-5 Negate
  20.1-6 NegateWord
  20.1-7 PrintZGword
  20.1-8 TietzeReduction
  20.1-9 ResolutionBoundaryOfWord

22 Meataxe modules

  22.1-1 DesuspensionMtxModule
  22.1-2 FpG_to_MtxModule
  22.1-3 GeneratorsOfMtxModule

23 G-Outer Groups

  23.1-1 GOuterGroup
  23.1-2 GOuterGroupHomomorphismNC
  23.1-3 GOuterHomomorphismTester
  23.1-4 Centre
  23.1-5 DirectProductGog

24 Cat-1-groups

  24.1-1 AutomorphismGroupAsCatOneGroup
  24.1-2 HomotopyGroup
  24.1-3 HomotopyModule
  24.1-4 QuasiIsomorph
  24.1-5 ModuleAsCatOneGroup
  24.1-6 MooreComplex
  24.1-7 NormalSubgroupAsCatOneGroup
  24.1-8 XmodToHAP

25 Simplicial groups

  25.1-1 NerveOfCatOneGroup
  25.1-2 EilenbergMacLaneSimplicialGroup
  25.1-3 EilenbergMacLaneSimplicialGroupMap
  25.1-4 MooreComplex
  25.1-5 ChainComplexOfSimplicialGroup
  25.1-6 SimplicialGroupMap
  25.1-7 HomotopyGroup
  25.1-8 Representation of elements in the bar resolution
  25.1-9 BarResolutionBoundary
  25.1-10 BarResolutionHomotopy
  25.1-11 Representation of elements in the bar complex
  25.1-12 BarComplexBoundary
  25.1-13 BarResolutionEquivalence
  25.1-14 BarComplexEquivalence
  25.1-15 Representation of elements in the bar cocomplex
  25.1-16 BarCocomplexCoboundary

26 Coxeter diagrams and graphs of groups

  26.1-1 CoxeterDiagramComponents
  26.1-2 CoxeterDiagramDegree
  26.1-3 CoxeterDiagramDisplay
  26.1-4 CoxeterDiagramFpArtinGroup
  26.1-5 CoxeterDiagramFpCoxeterGroup
  26.1-6 CoxeterDiagramIsSpherical
  26.1-7 CoxeterDiagramMatrix
  26.1-8 CoxeterSubDiagram
  26.1-9 CoxeterDiagramVertices
  26.1-10 EvenSubgroup
  26.1-11 GraphOfGroupsDisplay
  26.1-12 GraphOfResolutions
  26.1-13 GraphOfGroups
  26.1-14 GraphOfResolutionsDisplay
  26.1-15 GraphOfGroupsTest
  26.1-16 TreeOfGroupsToContractibleGcomplex
  26.1-17 TreeOfResolutionsToContractibleGcomplex

27 Torsion Subcomplexes

  27.1-1 RigidFacetsSubdivision
  27.1-2 IsPNormal
  27.1-3 TorsionSubcomplex
  27.1-4 DisplayAvailableCellComplexes
  27.1-5 VisualizeTorsionSkeleton
  27.1-6 ReduceTorsionSubcomplex
  27.1-7 EquivariantEulerCharacteristic
  27.1-8 CountingCellsOfACellComplex
  27.1-9 CountingControlledSubdividedCells
  27.1-10 CountingBaryCentricSubdividedCells
  27.1-11 EquivariantSpectralSequencePage
  27.1-12 ExportHapCellcomplexToDisk

28 Simplicial Complexes

29 Cubical Complexes

  29.1-1 ArrayToPureCubicalComplex
  29.1-2 PureCubicalComplex
  29.1-3 FramedPureCubicalComplex
  29.1-4 RandomCubeOfPureCubicalComplex
  29.1-5 PureCubicalComplexIntersection
  29.1-6 PureCubicalComplexUnion
  29.1-7 PureCubicalComplexDifference
  29.1-8 ReadImageAsPureCubicalComplex
  29.1-9 ReadLinkImageAsPureCubicalComplex
  29.1-10 ReadImageSequenceAsPureCubicalComplex
  29.1-11 Size
  29.1-12 Dimension
  29.1-13 WritePureCubicalComplexAsImage
  29.1-14 ViewPureCubicalComplex
  29.1-15 Homology
  29.1-16 Bettinumbers
  29.1-17 DirectProductOfPureCubicalComplexes
  29.1-18 SuspensionOfPureCubicalComplex
  29.1-19 EulerCharacteristic
  29.1-20 PathComponentOfPureCubicalComplex
  29.1-21 ChainComplex
  29.1-22 ChainComplexOfPair
  29.1-23 ExcisedPureCubicalPair
  29.1-24 ChainInclusionOfPureCubicalPair
  29.1-25 ChainMapOfPureCubicalPairs
  29.1-26 ContractPureCubicalComplex
  29.1-27 ContractedComplex
  29.1-28 ZigZagContractedPureCubicalComplex
  29.1-29 ContractCubicalComplex
  29.1-30 DVFReducedCubicalComplex
  29.1-31 SkeletonOfCubicalComplex
  29.1-32 ContractibleSubomplexOfPureCubicalComplex
  29.1-33 AcyclicSubomplexOfPureCubicalComplex
  29.1-34 HomotopyEquivalentMaximalPureCubicalSubcomplex
  29.1-35 HomotopyEquivalentMinimalPureCubicalSubcomplex
  29.1-36 BoundaryOfPureCubicalComplex
  29.1-37 SingularitiesOfPureCubicalComplex
  29.1-38 ThickenedPureCubicalComplex
  29.1-39 CropPureCubicalComplex
  29.1-40 BoundingPureCubicalComplex
  29.1-41 MorseFiltration
  29.1-42 ComplementOfPureCubicalComplex
  29.1-43 PureCubicalComplexToTextFile
  29.1-44 ThickeningFiltration
  29.1-45 Dendrogram
  29.1-46 DendrogramDisplay
  29.1-47 DendrogramToPersistenceMat
  29.1-48 ReadImageAsFilteredPureCubicalComplex
  29.1-49 ComplementOfFilteredPureCubicalComplex
  29.1-50 PersistentHomologyOfFilteredPureCubicalComplex

30 Regular CW-Complexes

  30.1-1 SimplicialComplexToRegularCWComplex
  30.1-2 CubicalComplexToRegularCWComplex
  30.1-3 CriticalCellsOfRegularCWComplex
  30.1-4 ChainComplex
  30.1-5 ChainComplexOfRegularCWComplex
  30.1-6 FundamentalGroup

31 Knots and Links

  31.1-1 PureCubicalKnot
  31.1-2 ViewPureCubicalKnot
  31.1-3 KnotSum
  31.1-4 KnotGroup
  31.1-5 AlexanderMatrix
  31.1-6 AlexanderPolynomial
  31.1-7 ProjectionOfPureCubicalComplex
  31.1-8 ReadPDBfileAsPureCubicalComplex

33 Finite metric spaces and their filtered complexes

  33.1-1 CayleyMetric
  33.1-2 HammingMetric
  33.1-3 KendallMetric
  33.1-4 EuclideanSquaredMetric
  33.1-5 EuclideanApproximatedMetric
  33.1-6 ManhattanMetric
  33.1-7 VectorsToSymmetricMatrix
  33.1-8 SymmetricMatDisplay
  33.1-9 SymmetricMatrixToFilteredGraph
  33.1-10 PermGroupToFilteredGraph

35 Arrays and Pseudo lists

  35.1-1 Array
  35.1-2 PermuteArray
  35.1-3 ArrayDimension
  35.1-4 ArrayDimensions
  35.1-5 ArraySum
  35.1-6 ArrayValue
  35.1-7 ArrayValueFunctions
  35.1-8 ArrayAssign
  35.1-9 ArrayAssignFunctions
  35.1-10 ArrayIterate
  35.1-11 BinaryArrayToTextFile
  35.1-12 FrameArray
  35.1-13 UnframeArray
  35.1-14 Add
  35.1-15 Append
  35.1-16 ListToPseudoList

36 Parallel Computation - Core Functions

  36.1-1 ChildProcess
  36.1-2 ChildClose
  36.1-3 ChildCommand
  36.1-4 NextAvailableChild
  36.1-5 IsAvailableChild
  36.1-6 ChildPut
  36.1-7 ChildGet
  36.1-8 HAPPrintTo
  36.1-9 HAPRead

37 Parallel Computation - Extra Functions

  37.1-1 ChildFunction
  37.1-2 ChildRead
  37.1-3 ChildReadEval
  37.1-4 ParallelList

38 Some functions for accessing basic data

  38.1-1 BoundaryMap
  38.1-2 BoundaryMatrix
  38.1-3 Dimension
  38.1-4 EvaluateProperty
  38.1-5 GroupOfResolution
  38.1-6 Length
  38.1-7 Map
  38.1-8 Source
  38.1-9 Target

39 Miscellaneous

  39.1-1 SL2Z
  39.1-2 BigStepLCS
  39.1-3 Classify
  39.1-4 RefineClassification
  39.1-5 Compose
  39.1-6 HAPcopyright
  39.1-7 IsLieAlgebraHomomorphism
  39.1-8 IsSuperperfect
  39.1-9 MakeHAPManual
  39.1-10 PermToMatrixGroup
  39.1-11 SolutionsMatDestructive
  39.1-12 LinearHomomorphismsPersistenceMat
  39.1-13 NormalSeriesToQuotientHomomorphisms
  39.1-14 TestHap

40 HAP variables that are not yet documented

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