QEtaSpecificationRing C¶
qetaspecring.spad line 91 [edit on github]
QEtaSpecificationRing is the domain of integer linear combinations of eta-quotients.
- 0: %
from AbelianMonoid
- 1: %
from MagmaWithUnit
- *: (%, %) -> %
from Magma
- *: (%, C) -> %
from RightModule C
- *: (C, %) -> %
from LeftModule C
- *: (Integer, %) -> %
from AbelianGroup
- *: (NonNegativeInteger, %) -> %
from AbelianMonoid
- *: (PositiveInteger, %) -> %
from AbelianSemiGroup
- +: (%, %) -> %
from AbelianSemiGroup
- -: % -> %
from AbelianGroup
- -: (%, %) -> %
from AbelianGroup
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- associator: (%, %, %) -> %
from NonAssociativeRng
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if C has CharacteristicNonZero
- coefficient: (%, QEtaSpecification) -> C
from FreeModuleCategory(C, QEtaSpecification)
- coefficients: % -> List C
from FreeModuleCategory(C, QEtaSpecification)
- coerce: % -> %
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: C -> %
from Algebra C
- coerce: Fraction Polynomial C -> % if C has IntegralDomain
coerce(p)
returns specificationPolynomial(p
,e
) wheree
is the first letter of the first variable ofp
. This, of course, assumes that all variables ofp
are of the form ed or ed_g whered
is a positive integer and 0<g<=d/2- coerce: Integer -> %
from NonAssociativeRing
- coerce: List Record(k: QEtaSpecification, c: C) -> %
from MonoidRingCategory(C, QEtaSpecification)
- coerce: Polynomial C -> %
coerce(p)
returns specificationPolynomial(p
,e
) wheree
is the first letter of the first variable ofp
. This, of course, assumes that all variables ofp
are of the form ed or ed_g whered
is a positive integer and 0<g<=d/2- coerce: QEtaSpecification -> %
- commutator: (%, %) -> %
from NonAssociativeRng
- construct: List Record(k: QEtaSpecification, c: C) -> %
from IndexedProductCategory(C, QEtaSpecification)
- constructOrdered: List Record(k: QEtaSpecification, c: C) -> %
from IndexedProductCategory(C, QEtaSpecification)
- convert: % -> InputForm if C has Finite and QEtaSpecification has Finite
from ConvertibleTo InputForm
- etaPolynomial: % -> Polynomial C
etaPolynomial(x)
returns etaPolynomial(x
,”E”,"Y"
).
- etaPolynomial: (%, String, String) -> Polynomial C
etaPolynomial(x,e,y)
replaces in each term ofx
the specification spec by monomial(spec,e
,y
).
- etaRationalFunction: % -> Fraction Polynomial C if C has IntegralDomain
etaRationalFunction(x)
returns etaRationalFunction(x
,”E”).
- etaRationalFunction: (%, String) -> Fraction Polynomial C if C has IntegralDomain
etaRationalFunction(x,e)
replaces in each term ofx
the specification spec by quotient(spec,e
).
- hash: % -> SingleInteger if C has Finite and QEtaSpecification has Finite
from Hashable
- hashUpdate!: (HashState, %) -> HashState if C has Finite and QEtaSpecification has Finite
from Hashable
- index: PositiveInteger -> % if C has Finite and QEtaSpecification has Finite
from Finite
- latex: % -> String
from SetCategory
- leadingCoefficient: % -> C
from IndexedProductCategory(C, QEtaSpecification)
- leadingMonomial: % -> %
from IndexedProductCategory(C, QEtaSpecification)
- leadingTerm: % -> Record(k: QEtaSpecification, c: C)
from IndexedProductCategory(C, QEtaSpecification)
- leftPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- leftPower: (%, PositiveInteger) -> %
from Magma
- leftRecip: % -> Union(%, failed)
from MagmaWithUnit
- level: % -> PositiveInteger
level(x)
returns the least common multiple of the levels of all terms.
- linearExtend: (QEtaSpecification -> C, %) -> C
from FreeModuleCategory(C, QEtaSpecification)
- listOfTerms: % -> List Record(k: QEtaSpecification, c: C)
- lookup: % -> PositiveInteger if C has Finite and QEtaSpecification has Finite
from Finite
- map: (C -> C, %) -> %
from IndexedProductCategory(C, QEtaSpecification)
- modularGamma0?: % -> Boolean
modularGamma0?(x)
returnstrue
if the specification in each term ofx
is modular with respect to $Gamma_0(N
)$ where $N
$ is thelcm
of all levels of the specifications of the terms ofx
.
- modularGamma1?: % -> Boolean
modularGamma1?(x)
returnstrue
if the specification in each term ofx
is modular with respect to $Gamma_0(N
)$ where $N
$ is thelcm
of all levels of the specifications of the terms ofx
.
- monomial?: % -> Boolean
from IndexedProductCategory(C, QEtaSpecification)
- monomial: (C, QEtaSpecification) -> %
from IndexedProductCategory(C, QEtaSpecification)
- monomials: % -> List %
from FreeModuleCategory(C, QEtaSpecification)
- one?: % -> Boolean
from MagmaWithUnit
- opposite?: (%, %) -> Boolean
from AbelianMonoid
- plenaryPower: (%, PositiveInteger) -> %
from NonAssociativeAlgebra C
- pretty: (%, Integer) -> OutputForm
pretty(x,fmt)
returns pretty(x
,fmt
,0).
- pretty: (%, Integer, (Integer, Integer) -> OutputForm, Fraction Integer) -> OutputForm
pretty(x,fmt,v,e)
multipliesx
by q^e and transforms the relation into a nicely looking output form using the variable functionv
deciding via format whether it must print a factor ofq
.
- pretty: (%, Integer, Fraction Integer) -> OutputForm
pretty(x,fmt,e)
chooses an appropriate variable functionv
according tofmt
and returns pretty(x
,fmt
,v
,e
).
- purify: % -> %
purify(x)
applies purify(spec) to each term c*spec ofx
.
- random: () -> % if C has Finite and QEtaSpecification has Finite
from Finite
- recip: % -> Union(%, failed)
from MagmaWithUnit
- reductum: % -> %
from IndexedProductCategory(C, QEtaSpecification)
- retract: % -> C
from RetractableTo C
- retract: % -> QEtaSpecification
- retractIfCan: % -> Union(C, failed)
from RetractableTo C
- retractIfCan: % -> Union(QEtaSpecification, failed)
- rightPower: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- rightPower: (%, PositiveInteger) -> %
from Magma
- rightRecip: % -> Union(%, failed)
from MagmaWithUnit
- sample: %
from AbelianMonoid
- size: () -> NonNegativeInteger if C has Finite and QEtaSpecification has Finite
from Finite
- smaller?: (%, %) -> Boolean if C has Comparable
from Comparable
- specificationPolynomial: (Fraction Polynomial C, String) -> % if C has IntegralDomain
specificationPolynomial(p,e)
replaces each variableei
by the respective eta-specification and thus turnsp
into an element of this domain. Note that the specifications of each monomial may live in different levels. This function can be seen as a coercion.
- specificationPolynomial: (Polynomial C, String) -> %
specificationPolynomial(p,e)
replaces each variableei
by the respective eta-specification and thus turnsp
into an element of this domain. Note that the specifications of each monomial may live in different levels. This function can be seen as a coercion.
- specificationPolynomial: (Polynomial C, String, List QEtaSpecification) -> %
specificationPolynomial(p,m,specs)
replaces each of the variablesmi
inp
by the respective elementspecs
.i
and thus creates a formal linear combination of eta-quotients.
- specificationPolynomial: (PositiveInteger, Fraction Polynomial C, String) -> % if C has IntegralDomain
specificationPolynomial(nn,p,e)
replaces each variableei
by the respective eta-specification (of levelnn
) and thus turnsp
into an element of this domain. It can be seen as a coercion function.
- specificationPolynomial: (PositiveInteger, Polynomial C, String) -> %
specificationPolynomial(nn,p,e)
replaces each variableei
by the respective eta-specification (of levelnn
) and thus turnsp
into an element of this domain. It can be seen as a coercion function.
- subtractIfCan: (%, %) -> Union(%, failed)
- support: % -> List QEtaSpecification
from FreeModuleCategory(C, QEtaSpecification)
- terms: % -> List Record(k: QEtaSpecification, c: C)
from MonoidRingCategory(C, QEtaSpecification)
- zero?: % -> Boolean
from AbelianMonoid
Algebra %
Algebra C
BiModule(%, %)
BiModule(C, C)
CharacteristicNonZero if C has CharacteristicNonZero
CharacteristicZero if C has CharacteristicZero
CoercibleFrom QEtaSpecification
Comparable if C has Comparable
ConvertibleTo InputForm if C has Finite and QEtaSpecification has Finite
Finite if C has Finite and QEtaSpecification has Finite
FreeModuleCategory(C, QEtaSpecification)
Hashable if C has Finite and QEtaSpecification has Finite
IndexedDirectProductCategory(C, QEtaSpecification)
IndexedProductCategory(C, QEtaSpecification)
Module %
Module C
MonoidRingCategory(C, QEtaSpecification)