chap0.txt

  
  
               numericalsgps-- a package for numerical semigroups
  
  
                                 Version 1.1.5
  
  
                                 Manuel Delgado
  
                            Pedro A. García-Sánchez
  
                                José João Morais
  
  
  
  Manuel Delgado
      Email:    mailto:[email protected]
      Homepage: http://www.fc.up.pt/cmup/mdelgado
  Pedro A. García-Sánchez
      Email:    mailto:[email protected]
      Homepage: http://www.ugr.es/~pedro
  
  -------------------------------------------------------
  Copyright
  ©  2005--2015  Centro  de  Matemática da Universidade do Porto, Portugal and
  Universidad de Granada, Spain
  
  Numericalsgps  is  free  software;  you can redistribute it and/or modify it
  under     the     terms     of    the    GNU    General    Public    License
  (http://www.fsf.org/licenses/gpl.html)  as  published  by  the Free Software
  Foundation;  either  version 2 of the License, or (at your option) any later
  version.
  
  
  -------------------------------------------------------
  Acknowledgements
  The  first  author's  work  was  (partially)  supported  by  the  Centro  de
  Matemática  da  Universidade  do  Porto  (CMUP),  financed by FCT (Portugal)
  through  the  programs  POCTI  (Programa  Operacional  "Ciência, Tecnologia,
  Inovação")  and  POSI  (Programa  Operacional Sociedade da Informação), with
  national  and  European Community structural funds and a sabbatical grant of
  FCT.
  
  The second author was supported by the projects MTM2004-01446, MTM2007-62346
  and MTM2010-15595, the Junta de Andalucía group FQM-343, and FEDER founds.
  
  The third author acknowledges financial support of FCT and the POCTI program
  through  a  scholarship  given  by  Centro  de Matemática da Universidade do
  Porto.
  
  The authors wish to thank J. I. García-García and Alfredo Sánchez-R. Navarro
  for  many  helpful  discussions  and  for  helping  in  the  programming  of
  preliminary  versions  of  some  functions,  and  also  to  C.  O'Neill,  A.
  Sammartano,  I.  Ojeda, C. M. Moreno Ávila, A. Herrera-Poyatos and K. Stokes
  for  their  contributions  (see  Contributions Chapter). We are also in debt
  with  S. Gutsche, M. Horn, H. Schönemann, C. Söeger and M. Barakat for their
  fruitful  advices  concerning  4ti2Interface,  SingularInterface,  Singular,
  Normaliz, NormalizInterface and GradedModules.
  
  The  first  and second authors warmly thank María Burgos for her support and
  help.
  
  Concerning the mantainment:
  
  The   first   author   was   (partially)   supported   by  the  FCT  project
  PTDC/MAT/65481/2006  and also by the Centro de Matemática da Universidade do
  Porto  (CMUP),  funded by the European Regional Development Fund through the
  programme  COMPETE  and  by  the  Portuguese  Government  through  the FCT -
  Fundação    para    a   Ciência   e   a   Tecnologia   under   the   project
  PEst-C/MAT/UI0144/2011.
  
  The  second author was/is supported by the project MTM2014-55367-P, which is
  funded  by  Ministerio de Economía y Competitividad and the Fondo Europeo de
  Desarrollo Regional FEDER.
  
  Both   maintainers   want   to   acknowledge   partial   support   by   CMUP
  (UID/MAT/00144/2013),  which is funded by FCT (Portugal) with national (MEC)
  and  European  structural  funds  through  the  programs  FEDER,  under  the
  partnership agreement PT2020.
  
  Both maintainers are also supported by the project MTM2014-55367-P, which is
  funded  by  Ministerio  de  Economía  y  Competitividad and Fondo Europeo de
  Desarrollo Regional FEDER.
  
  The  maintainers  want  to  thank the organizers of GAPDays in their several
  editions.
  
  The  authors  also  thank  the Centro de Servicios de Informática y Redes de
  Comunicaciones  (CSIRC), Universidad de Granada, for providing the computing
  time, specially Rafael Arco Arredondo for installing everything this package
  and the extra software needed in alhambra.ugr.es.
  
  
  -------------------------------------------------------
  Colophon
  This  work started when (in 2004) the first author visited the University of
  Granada in part of a sabbatical year. Since Version 0.96 (released in 2008),
  the package is maintained by the first two authors. Bug reports, suggestions
  and comments are, of course, welcome. Please use our email addresses to this
  effect.
  
  If  you have benefited from the use of the numerigalsgps GAP package in your
  research,  please  cite  it  in addition to GAP itself, following the scheme
  proposed in http://www.gap-system.org/Contacts/cite.html.
  
  If you have predominantly used the functions in the Appendix, contributed by
  other  authors,  please  cite in addition these authors, referring "software
  implementations available in the GAP package NumericalSgps".
  
  
  -------------------------------------------------------
  
  
  Contents (NumericalSgps)
  
  1 Introduction
  2 Numerical Semigroups
    2.1 Generating Numerical Semigroups
      2.1-1 NumericalSemigroupByGenerators
      2.1-2 NumericalSemigroupBySubAdditiveFunction
      2.1-3 NumericalSemigroupByAperyList
      2.1-4 NumericalSemigroupBySmallElements
      2.1-5 NumericalSemigroupByGaps
      2.1-6 NumericalSemigroupByFundamentalGaps
      2.1-7 NumericalSemigroupByAffineMap
      2.1-8 ModularNumericalSemigroup
      2.1-9 ProportionallyModularNumericalSemigroup
      2.1-10 NumericalSemigroupByInterval
      2.1-11 NumericalSemigroupByOpenInterval
    2.2 Some basic tests
      2.2-1 IsNumericalSemigroup
      2.2-2 RepresentsSmallElementsOfNumericalSemigroup
      2.2-3 RepresentsGapsOfNumericalSemigroup
      2.2-4 IsAperyListOfNumericalSemigroup
      2.2-5 IsSubsemigroupOfNumericalSemigroup
      2.2-6 IsSubset
      2.2-7 BelongsToNumericalSemigroup
  3 Basic operations with numerical semigroups
    3.1 Invariants
      3.1-1 Multiplicity
      3.1-2 GeneratorsOfNumericalSemigroup
      3.1-3 EmbeddingDimension
      3.1-4 SmallElements
      3.1-5 FirstElementsOfNumericalSemigroup
      3.1-6 RthElementOfNumericalSemigroup
      3.1-7 AperyList
      3.1-8 AperyList
      3.1-9 AperyList
      3.1-10 AperyListOfNumericalSemigroupAsGraph
      3.1-11 KunzCoordinatesOfNumericalSemigroup
      3.1-12 KunzPolytope
      3.1-13 CocycleOfNumericalSemigroupWRTElement
      3.1-14 FrobeniusNumber
      3.1-15 Conductor
      3.1-16 PseudoFrobeniusOfNumericalSemigroup
      3.1-17 TypeOfNumericalSemigroup
      3.1-18 Gaps
      3.1-19 DesertsOfNumericalSemigroup
      3.1-20 IsOrdinaryNumericalSemigroup
      3.1-21 IsAcuteNumericalSemigroup
      3.1-22 Holes
      3.1-23 LatticePathAssociatedToNumericalSemigroup
      3.1-24 Genus
      3.1-25 FundamentalGaps
      3.1-26 SpecialGaps
    3.2 Wilf's conjecture
      3.2-1 WilfNumber
      3.2-2 EliahouNumber
      3.2-3 ProfileOfNumericalSemigroup
      3.2-4 EliahouSlicesOfNumericalSemigroup
  4 Presentations of Numerical Semigroups
    4.1 Presentations of Numerical Semigroups
      4.1-1 MinimalPresentationOfNumericalSemigroup
      4.1-2 GraphAssociatedToElementInNumericalSemigroup
      4.1-3 BettiElementsOfNumericalSemigroup
      4.1-4 PrimitiveElementsOfNumericalSemigroup
      4.1-5 ShadedSetOfElementInNumericalSemigroup
    4.2 Uniquely Presented Numerical Semigroups
      4.2-1 IsUniquelyPresented
      4.2-2 IsGeneric
  5 Constructing numerical semigroups from others
    5.1 Adding and removing elements of a numerical semigroup
      5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup
      5.1-2 AddSpecialGapOfNumericalSemigroup
    5.2 Intersections, and quotients and multiples by integers
      5.2-1 Intersection
      5.2-2 QuotientOfNumericalSemigroup
      5.2-3 MultipleOfNumericalSemigroup
      5.2-4 Difference
      5.2-5 NumericalDuplication
      5.2-6 InductiveNumericalSemigroup
    5.3 Constructing the set of all numerical semigroups containing a given
    numerical semigroup
      5.3-1 OverSemigroupsNumericalSemigroup
    5.4 Constructing the set of numerical semigroup with given Frobenius
    number
      5.4-1 NumericalSemigroupsWithFrobeniusNumber
    5.5 Constructing the set of numerical semigroups with genus g, that is,
    numerical semigroups with exactly g gaps
      5.5-1 NumericalSemigroupsWithGenus
    5.6 Constructing the set of numerical semigroups with a given set of
    pseudo-Frobenius numbers
      5.6-1 ForcedIntegersForPseudoFrobenius
      5.6-2 SimpleForcedIntegersForPseudoFrobenius
      5.6-3 NumericalSemigroupsWithPseudoFrobeniusNumbers
      5.6-4 ANumericalSemigroupWithPseudoFrobeniusNumbers
  6 Irreducible numerical semigroups
    6.1 Irreducible numerical semigroups
      6.1-1 IsIrreducibleNumericalSemigroup
      6.1-2 IsSymmetricNumericalSemigroup
      6.1-3 IsPseudoSymmetric
      6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber
      6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber
      6.1-6 DecomposeIntoIrreducibles
    6.2 Complete intersection numerical semigroups
      6.2-1 AsGluingOfNumericalSemigroups
      6.2-2 IsCompleteIntersection
      6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber
      6.2-4 IsFree
      6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber
      6.2-6 IsTelescopic
      6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber
      6.2-8 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity
      6.2-9 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber
      6.2-10 IsAperySetGammaRectangular
      6.2-11 IsAperySetBetaRectangular
      6.2-12 IsAperySetAlphaRectangular
    6.3 Almost-symmetric numerical semigroups
      6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible
      6.3-2 IsAlmostSymmetric
      6.3-3 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber
  7 Ideals of numerical semigroups
    7.1 Definitions and basic operations
      7.1-1 IdealOfNumericalSemigroup
      7.1-2 IsIdealOfNumericalSemigroup
      7.1-3 MinimalGenerators
      7.1-4 Generators
      7.1-5 AmbientNumericalSemigroupOfIdeal
      7.1-6 IsIntegral
      7.1-7 SmallElements
      7.1-8 Conductor
      7.1-9 Minimum
      7.1-10 BelongsToIdealOfNumericalSemigroup
      7.1-11 SumIdealsOfNumericalSemigroup
      7.1-12 MultipleOfIdealOfNumericalSemigroup
      7.1-13 SubtractIdealsOfNumericalSemigroup
      7.1-14 Difference
      7.1-15 TranslationOfIdealOfNumericalSemigroup
      7.1-16 Intersection
      7.1-17 MaximalIdealOfNumericalSemigroup
      7.1-18 CanonicalIdealOfNumericalSemigroup
      7.1-19 IsCanonicalIdeal
      7.1-20 TypeSequenceOfNumericalSemigroup
    7.2 Blow ups and closures
      7.2-1 HilbertFunctionOfIdealOfNumericalSemigroup
      7.2-2 BlowUpIdealOfNumericalSemigroup
      7.2-3 ReductionNumber
      7.2-4 BlowUpOfNumericalSemigroup
      7.2-5 LipmanSemigroup
      7.2-6 RatliffRushNumberOfIdealOfNumericalSemigroup
      7.2-7 RatliffRushClosureOfIdealOfNumericalSemigroup
      7.2-8 AsymptoticRatliffRushNumberOfIdealOfNumericalSemigroup
      7.2-9 MultiplicitySequenceOfNumericalSemigroup
      7.2-10 MicroInvariantsOfNumericalSemigroup
      7.2-11 AperyListOfIdealOfNumericalSemigroupWRTElement
      7.2-12 AperyTableOfNumericalSemigroup
      7.2-13 StarClosureOfIdealOfNumericalSemigroup
    7.3 Patterns for ideals
      7.3-1 IsAdmissiblePattern
      7.3-2 IsStronglyAdmissiblePattern
      7.3-3 AsIdealOfNumericalSemigroup
      7.3-4 BoundForConductorOfImageOfPattern
      7.3-5 ApplyPatternToIdeal
      7.3-6 ApplyPatternToNumericalSemigroup
      7.3-7 IsAdmittedPatternByIdeal
      7.3-8 IsAdmittedPatternByNumericalSemigroup
    7.4 Graded associated ring of numerical semigroup
      7.4-1 IsGradedAssociatedRingNumericalSemigroupCM
      7.4-2 IsGradedAssociatedRingNumericalSemigroupBuchsbaum
      7.4-3 TorsionOfAssociatedGradedRingNumericalSemigroup
      7.4-4 BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup
      7.4-5 IsMpure
      7.4-6 IsPure
      7.4-7 IsGradedAssociatedRingNumericalSemigroupGorenstein
      7.4-8 IsGradedAssociatedRingNumericalSemigroupCI
  8 Numerical semigroups with maximal embedding dimension
    8.1 Numerical semigroups with maximal embedding dimension
      8.1-1 IsMED
      8.1-2 MEDNumericalSemigroupClosure
      8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup
    8.2 Numerical semigroups with the Arf property and Arf closures
      8.2-1 IsArf
      8.2-2 ArfNumericalSemigroupClosure
      8.2-3 ArfCharactersOfArfNumericalSemigroup
      8.2-4 ArfNumericalSemigroupsWithFrobeniusNumber
      8.2-5 ArfNumericalSemigroupsWithFrobeniusNumberUpTo
      8.2-6 ArfNumericalSemigroupsWithGenus
      8.2-7 ArfNumericalSemigroupsWithGenusUpTo
      8.2-8 ArfNumericalSemigroupsWithGenusAndFrobeniusNumber
    8.3 Saturated numerical semigroups
      8.3-1 IsSaturated
      8.3-2 SaturatedNumericalSemigroupClosure
      8.3-3 SaturatedNumericalSemigroupsWithFrobeniusNumber
  9 Nonunique invariants for factorizations in numerical semigroups
    9.1 Factorizations in Numerical Semigroups
      9.1-1 FactorizationsIntegerWRTList
      9.1-2 FactorizationsElementWRTNumericalSemigroup
      9.1-3 FactorizationsElementListWRTNumericalSemigroup
      9.1-4 RClassesOfSetOfFactorizations
      9.1-5 LShapesOfNumericalSemigroup
      9.1-6 DenumerantOfElementInNumericalSemigroup
    9.2 Invariants based on lengths
      9.2-1 LengthsOfFactorizationsIntegerWRTList
      9.2-2 LengthsOfFactorizationsElementWRTNumericalSemigroup
      9.2-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup
      9.2-4 ElasticityOfNumericalSemigroup
      9.2-5 DeltaSetOfSetOfIntegers
      9.2-6 DeltaSetOfFactorizationsElementWRTNumericalSemigroup
      9.2-7 DeltaSetPeriodicityBoundForNumericalSemigroup
      9.2-8 DeltaSetPeriodicityStartForNumericalSemigroup
      9.2-9 DeltaSetListUpToElementWRTNumericalSemigroup
      9.2-10 DeltaSetUnionUpToElementWRTNumericalSemigroup
      9.2-11 DeltaSetOfNumericalSemigroup
      9.2-12 MaximumDegreeOfElementWRTNumericalSemigroup
      9.2-13 MaximalDenumerantOfElementInNumericalSemigroup
      9.2-14 MaximalDenumerantOfSetOfFactorizations
      9.2-15 MaximalDenumerantOfNumericalSemigroup
      9.2-16 AdjustmentOfNumericalSemigroup
      9.2-17 IsAdditiveNumericalSemigroup
      9.2-18 IsSuperSymmetricNumericalSemigroup
    9.3 Invariants based on distances
      9.3-1 CatenaryDegreeOfSetOfFactorizations
      9.3-2 AdjacentCatenaryDegreeOfSetOfFactorizations
      9.3-3 EqualCatenaryDegreeOfSetOfFactorizations
      9.3-4 MonotoneCatenaryDegreeOfSetOfFactorizations
      9.3-5 CatenaryDegreeOfElementInNumericalSemigroup
      9.3-6 TameDegreeOfSetOfFactorizations
      9.3-7 CatenaryDegreeOfNumericalSemigroup
      9.3-8 EqualPrimitiveElementsOfNumericalSemigroup
      9.3-9 EqualCatenaryDegreeOfNumericalSemigroup
      9.3-10 MonotonePrimitiveElementsOfNumericalSemigroup
      9.3-11 MonotoneCatenaryDegreeOfNumericalSemigroup
      9.3-12 TameDegreeOfNumericalSemigroup
      9.3-13 TameDegreeOfElementInNumericalSemigroup
    9.4 Primality
      9.4-1 OmegaPrimalityOfElementInNumericalSemigroup
      9.4-2 OmegaPrimalityOfElementListInNumericalSemigroup
      9.4-3 OmegaPrimalityOfNumericalSemigroup
    9.5 Homogenization of Numerical Semigroups
      9.5-1 BelongsToHomogenizationOfNumericalSemigroup
      9.5-2 FactorizationsInHomogenizationOfNumericalSemigroup
      9.5-3 HomogeneousBettiElementsOfNumericalSemigroup
      9.5-4 HomogeneousCatenaryDegreeOfNumericalSemigroup
    9.6 Divisors, posets
      9.6-1 MoebiusFunctionAssociatedToNumericalSemigroup
      9.6-2 DivisorsOfElementInNumericalSemigroup
    9.7 Feng-Rao distances and numbers
      9.7-1 FengRaoDistance
      9.7-2 FengRaoNumber
  10 Polynomials and numerical semigroups
    10.1 Generating functions or Hilbert series
      10.1-1 NumericalSemigroupPolynomial
      10.1-2 IsNumericalSemigroupPolynomial
      10.1-3 NumericalSemigroupFromNumericalSemigroupPolynomial
      10.1-4 HilbertSeriesOfNumericalSemigroup
      10.1-5 GraeffePolynomial
      10.1-6 IsCyclotomicPolynomial
      10.1-7 IsKroneckerPolynomial
      10.1-8 IsCyclotomicNumericalSemigroup
      10.1-9 IsSelfReciprocalUnivariatePolynomial
    10.2 Semigroup of values of algebraic curves
      10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity
      10.2-2 IsDeltaSequence
      10.2-3 DeltaSequencesWithFrobeniusNumber
      10.2-4 CurveAssociatedToDeltaSequence
      10.2-5 SemigroupOfValuesOfPlaneCurve
      10.2-6 SemigroupOfValuesOfCurve_Local
      10.2-7 SemigroupOfValuesOfCurve_Global
      10.2-8 GeneratorsModule_Global
      10.2-9 GeneratorsKahlerDifferentials
      10.2-10 IsMonomialNumericalSemigroup
  11 Affine semigroups
    11.1 Defining affine semigroups
      11.1-1 AffineSemigroupByGenerators
      11.1-2 AffineSemigroupByEquations
      11.1-3 AffineSemigroupByInequalities
      11.1-4 Generators
      11.1-5 MinimalGenerators
      11.1-6 AsAffineSemigroup
      11.1-7 IsAffineSemigroup
      11.1-8 BelongsToAffineSemigroup
      11.1-9 IsFull
      11.1-10 HilbertBasisOfSystemOfHomogeneousEquations
      11.1-11 HilbertBasisOfSystemOfHomogeneousInequalities
      11.1-12 EquationsOfGroupGeneratedBy
      11.1-13 BasisOfGroupGivenByEquations
    11.2 Gluings of affine semigroups
      11.2-1 GluingOfAffineSemigroups
    11.3 Presentations of affine semigroups
      11.3-1 GeneratorsOfKernelCongruence
      11.3-2 CanonicalBasisOfKernelCongruence
      11.3-3 GraverBasis
      11.3-4 MinimalPresentationOfAffineSemigroup
      11.3-5 BettiElementsOfAffineSemigroup
      11.3-6 ShadedSetOfElementInAffineSemigroup
      11.3-7 IsGeneric
      11.3-8 IsUniquelyPresentedAffineSemigroup
      11.3-9 PrimitiveElementsOfAffineSemigroup
    11.4 Factorizations in affine semigroups
      11.4-1 FactorizationsVectorWRTList
      11.4-2 ElasticityOfAffineSemigroup
      11.4-3 DeltaSetOfAffineSemigroup
      11.4-4 CatenaryDegreeOfAffineSemigroup
      11.4-5 EqualCatenaryDegreeOfAffineSemigroup
      11.4-6 HomogeneousCatenaryDegreeOfAffineSemigroup
      11.4-7 MonotoneCatenaryDegreeOfAffineSemigroup
      11.4-8 TameDegreeOfAffineSemigroup
      11.4-9 OmegaPrimalityOfElementInAffineSemigroup
      11.4-10 OmegaPrimalityOfAffineSemigroup
  12 Good semigroups
    12.1 Defining good semigroups
      12.1-1 IsGoodSemigroup
      12.1-2 NumericalSemigroupDuplication
      12.1-3 AmalgamationOfNumericalSemigroups
      12.1-4 CartesianProductOfNumericalSemigroups
      12.1-5 GoodSemigroup
    12.2 Notable elements
      12.2-1 BelongsToGoodSemigroup
      12.2-2 Conductor
      12.2-3 SmallElements
      12.2-4 RepresentsSmallElementsOfGoodSemigroup
      12.2-5 GoodSemigroupBySmallElements
      12.2-6 MaximalElementsOfGoodSemigroup
      12.2-7 IrreducibleMaximalElementsOfGoodSemigroup
      12.2-8 GoodSemigroupByMaximalElements
      12.2-9 MinimalGoodGeneratingSystemOfGoodSemigroup
      12.2-10 MinimalGenerators
    12.3 Symmetric semigroups
      12.3-1 IsSymmetricGoodSemigroup
      12.3-2 ArfGoodSemigroupClosure
    12.4 Good ideals
      12.4-1 GoodIdeal
      12.4-2 GoodGeneratingSystemOfGoodIdeal
      12.4-3 AmbientGoodSemigroupOfGoodIdeal
      12.4-4 MinimalGoodGeneratingSystemOfGoodIdeal
      12.4-5 BelongsToGoodIdeal
      12.4-6 SmallElementsOfGoodIdeal
      12.4-7 CanonicalIdealOfGoodSemigroup
  13 External packages
    13.1 Using external packages
      13.1-1 NumSgpsUse4ti2
      13.1-2 NumSgpsUse4ti2gap
      13.1-3 NumSgpsUseNormalize
      13.1-4 NumSgpsUseSingular
      13.1-5 NumSgpsUseSingularInterface
      13.1-6 NumSgpsUseSingularGradedModules
  A Generalities
    A.1 Bézout sequences
      A.1-1 BezoutSequence
      A.1-2 IsBezoutSequence
      A.1-3 CeilingOfRational
    A.2 Periodic subadditive functions
      A.2-1 RepresentsPeriodicSubAdditiveFunction
      A.2-2 IsListOfIntegersNS
  B "Random" functions
    B.1 Random functions
      B.1-1 RandomNumericalSemigroup
      B.1-2 RandomListForNS
      B.1-3 RandomModularNumericalSemigroup
      B.1-4 RandomProportionallyModularNumericalSemigroup
      B.1-5 RandomListRepresentingSubAdditiveFunction
      B.1-6 NumericalSemigroupWithRandomElementsAndFrobenius
  C Contributions
    C.1 Functions implemented by A. Sammartano
    C.2 Functions implemented by C. O'Neill
    C.3 Functions implemented by K. Stokes
    C.4 Functions implemented by I. Ojeda and C. J. Moreno Ávila
    C.5 Functions implemented by A. Sánchez-R. Navarro
    C.6 Functions implemented by G. Zito
    C.7 Functions implemented by A. Herrera-Poyatos
    C.8 Functions implemented by Benjamin Heredia
    C.9 Functions implemented by Juan Ignacio García-García