FunctionSpace R

fspace.spad line 1158 [edit on github]

• R: Comparable

undocumented

0: % if R has AbelianSemiGroup

from AbelianMonoid

1: % if R has SemiGroup

from MagmaWithUnit

*: (%, %) -> % if R has SemiGroup

from Magma

*: (%, Fraction Integer) -> % if R has IntegralDomain

from RightModule Fraction Integer

*: (%, Integer) -> % if R has Ring and R has LinearlyExplicitOver Integer

from RightModule Integer

*: (%, R) -> % if R has Ring

from RightModule R

*: (Fraction Integer, %) -> % if R has IntegralDomain

from LeftModule Fraction Integer

*: (Integer, %) -> % if R has AbelianGroup

from AbelianGroup

*: (NonNegativeInteger, %) -> % if R has AbelianSemiGroup

from AbelianMonoid

*: (PositiveInteger, %) -> % if R has AbelianSemiGroup

from AbelianSemiGroup

*: (R, %) -> % if R has CommutativeRing

from LeftModule R

+: (%, %) -> % if R has AbelianSemiGroup

from AbelianSemiGroup

-: % -> % if R has AbelianGroup

from AbelianGroup

-: (%, %) -> % if R has AbelianGroup

from AbelianGroup

/: (%, %) -> % if R has Group or R has IntegralDomain

from Group

/: (SparseMultivariatePolynomial(R, Kernel %), SparseMultivariatePolynomial(R, Kernel %)) -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

=: (%, %) -> Boolean

from BasicType

^: (%, Integer) -> % if R has Group or R has IntegralDomain

from Group

^: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

^: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

~=: (%, %) -> Boolean

from BasicType

algtower: % -> List Kernel % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

algtower: List % -> List Kernel % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

annihilate?: (%, %) -> Boolean if R has Ring

from Rng

antiCommutator: (%, %) -> % if R has Ring

from NonAssociativeSemiRng

applyQuote: (Symbol, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, %, %, %, %) -> %

from FunctionSpace2(R, Kernel %)

applyQuote: (Symbol, List %) -> %

from FunctionSpace2(R, Kernel %)

associates?: (%, %) -> Boolean if R has IntegralDomain

from EntireRing

associator: (%, %, %) -> % if R has Ring

from NonAssociativeRng

belong?: BasicOperator -> Boolean

from ExpressionSpace2 Kernel %

box: % -> %

from ExpressionSpace2 Kernel %

characteristic: () -> NonNegativeInteger if R has Ring

from NonAssociativeRing

charthRoot: % -> Union(%, failed) if R has CharacteristicNonZero

from CharacteristicNonZero

coerce: % -> % if R has IntegralDomain

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: Fraction Integer -> % if R has RetractableTo Fraction Integer or R has IntegralDomain

from CoercibleFrom Fraction Integer

coerce: Fraction Polynomial Fraction R -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

coerce: Fraction Polynomial R -> % if R has IntegralDomain

from CoercibleFrom Fraction Polynomial R

coerce: Fraction R -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

coerce: Integer -> % if R has Ring or R has RetractableTo Integer

from NonAssociativeRing

coerce: Kernel % -> %

from CoercibleFrom Kernel %

coerce: Polynomial Fraction R -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

coerce: Polynomial R -> % if R has Ring

from CoercibleFrom Polynomial R

coerce: R -> %

from Algebra R

coerce: SparseMultivariatePolynomial(R, Kernel %) -> % if R has Ring

from FunctionSpace2(R, Kernel %)

coerce: Symbol -> %

from CoercibleFrom Symbol

commutator: (%, %) -> % if R has Group or R has Ring

from NonAssociativeRng

conjugate: (%, %) -> % if R has Group

from Group

convert: % -> InputForm if R has ConvertibleTo InputForm

from ConvertibleTo InputForm

convert: % -> Pattern Float if R has ConvertibleTo Pattern Float

from ConvertibleTo Pattern Float

convert: % -> Pattern Integer if R has ConvertibleTo Pattern Integer

from ConvertibleTo Pattern Integer

convert: Factored % -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

D: (%, List Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

D: (%, Symbol, NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

definingPolynomial: % -> % if % has Ring

from ExpressionSpace2 Kernel %

denom: % -> SparseMultivariatePolynomial(R, Kernel %) if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

denominator: % -> % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

differentiate: (%, List Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, List Symbol, List NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, Symbol) -> % if R has Ring

from PartialDifferentialRing Symbol

differentiate: (%, Symbol, NonNegativeInteger) -> % if R has Ring

from PartialDifferentialRing Symbol

distribute: % -> %

from ExpressionSpace2 Kernel %

distribute: (%, %) -> %

from ExpressionSpace2 Kernel %

divide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain

from EuclideanDomain

elt: (BasicOperator, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, %, %, %, %, %, %, %, %, %) -> %

from ExpressionSpace2 Kernel %

elt: (BasicOperator, List %) -> %

from ExpressionSpace2 Kernel %

euclideanSize: % -> NonNegativeInteger if R has IntegralDomain

from EuclideanDomain

eval: (%, %, %) -> %

from InnerEvalable(%, %)

eval: (%, BasicOperator, % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, BasicOperator, %, Symbol) -> % if R has ConvertibleTo InputForm

from FunctionSpace2(R, Kernel %)

eval: (%, BasicOperator, List % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Equation %) -> %

from Evalable %

eval: (%, Kernel %, %) -> %

from InnerEvalable(Kernel %, %)

eval: (%, List %, List %) -> %

from InnerEvalable(%, %)

eval: (%, List BasicOperator, List %, Symbol) -> % if R has ConvertibleTo InputForm

from FunctionSpace2(R, Kernel %)

eval: (%, List BasicOperator, List(% -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List BasicOperator, List(List % -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List Equation %) -> %

from Evalable %

eval: (%, List Kernel %, List %) -> %

from InnerEvalable(Kernel %, %)

eval: (%, List Symbol, List NonNegativeInteger, List(% -> %)) -> % if R has Ring

from FunctionSpace2(R, Kernel %)

eval: (%, List Symbol, List NonNegativeInteger, List(List % -> %)) -> % if R has Ring

from FunctionSpace2(R, Kernel %)

eval: (%, List Symbol, List(% -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, List Symbol, List(List % -> %)) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, List % -> %) -> %

from ExpressionSpace2 Kernel %

eval: (%, Symbol, NonNegativeInteger, % -> %) -> % if R has Ring

from FunctionSpace2(R, Kernel %)

eval: (%, Symbol, NonNegativeInteger, List % -> %) -> % if R has Ring

from FunctionSpace2(R, Kernel %)

even?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 Kernel %

expressIdealMember: (List %, %) -> Union(List %, failed) if R has IntegralDomain

from PrincipalIdealDomain

exquo: (%, %) -> Union(%, failed) if R has IntegralDomain

from EntireRing

extendedEuclidean: (%, %) -> Record(coef1: %, coef2: %, generator: %) if R has IntegralDomain

from EuclideanDomain

extendedEuclidean: (%, %, %) -> Union(Record(coef1: %, coef2: %), failed) if R has IntegralDomain

from EuclideanDomain

factor: % -> Factored % if R has IntegralDomain

from UniqueFactorizationDomain

freeOf?: (%, %) -> Boolean

from ExpressionSpace2 Kernel %

freeOf?: (%, Symbol) -> Boolean

from ExpressionSpace2 Kernel %

gcd: (%, %) -> % if R has IntegralDomain

from GcdDomain

gcd: List % -> % if R has IntegralDomain

from GcdDomain

gcdPolynomial: (SparseUnivariatePolynomial %, SparseUnivariatePolynomial %) -> SparseUnivariatePolynomial % if R has IntegralDomain

from GcdDomain

ground?: % -> Boolean

from FunctionSpace2(R, Kernel %)

ground: % -> R

from FunctionSpace2(R, Kernel %)

height: % -> NonNegativeInteger

from ExpressionSpace2 Kernel %

inv: % -> % if R has Group or R has IntegralDomain

from Group

is?: (%, BasicOperator) -> Boolean

from ExpressionSpace2 Kernel %

is?: (%, Symbol) -> Boolean

from ExpressionSpace2 Kernel %

isExpt: % -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has SemiGroup

from FunctionSpace2(R, Kernel %)

isExpt: (%, BasicOperator) -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has Ring

from FunctionSpace2(R, Kernel %)

isExpt: (%, Symbol) -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has Ring

from FunctionSpace2(R, Kernel %)

isMult: % -> Union(Record(coef: Integer, var: Kernel %), failed) if R has AbelianSemiGroup

from FunctionSpace2(R, Kernel %)

isPlus: % -> Union(List %, failed) if R has AbelianSemiGroup

from FunctionSpace2(R, Kernel %)

isPower: % -> Union(Record(val: %, exponent: Integer), failed) if R has Ring

from FunctionSpace2(R, Kernel %)

isTimes: % -> Union(List %, failed) if R has SemiGroup

from FunctionSpace2(R, Kernel %)

kernel: (BasicOperator, %) -> %

from ExpressionSpace2 Kernel %

kernel: (BasicOperator, List %) -> %

from ExpressionSpace2 Kernel %

kernels: % -> List Kernel %

from ExpressionSpace2 Kernel %

kernels: List % -> List Kernel %

from ExpressionSpace2 Kernel %

latex: % -> String

from SetCategory

lcm: (%, %) -> % if R has IntegralDomain

from GcdDomain

lcm: List % -> % if R has IntegralDomain

from GcdDomain

lcmCoef: (%, %) -> Record(llcm_res: %, coeff1: %, coeff2: %) if R has IntegralDomain

from LeftOreRing

leftPower: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

leftRecip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

mainKernel: % -> Union(Kernel %, failed)

from ExpressionSpace2 Kernel %

map: (% -> %, Kernel %) -> %

from ExpressionSpace2 Kernel %

minPoly: Kernel % -> SparseUnivariatePolynomial % if % has Ring

from ExpressionSpace2 Kernel %

multiEuclidean: (List %, %) -> Union(List %, failed) if R has IntegralDomain

from EuclideanDomain

numer: % -> SparseMultivariatePolynomial(R, Kernel %) if R has Ring

from FunctionSpace2(R, Kernel %)

numerator: % -> % if R has Ring

from FunctionSpace2(R, Kernel %)

odd?: % -> Boolean if % has RetractableTo Integer

from ExpressionSpace2 Kernel %

one?: % -> Boolean if R has SemiGroup

from MagmaWithUnit

operator: BasicOperator -> BasicOperator

from ExpressionSpace2 Kernel %

operators: % -> List BasicOperator

from ExpressionSpace2 Kernel %

opposite?: (%, %) -> Boolean if R has AbelianSemiGroup

from AbelianMonoid

paren: % -> %

from ExpressionSpace2 Kernel %

patternMatch: (%, Pattern Float, PatternMatchResult(Float, %)) -> PatternMatchResult(Float, %) if R has PatternMatchable Float

from PatternMatchable Float

patternMatch: (%, Pattern Integer, PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %) if R has PatternMatchable Integer

from PatternMatchable Integer

plenaryPower: (%, PositiveInteger) -> % if R has CommutativeRing

from NonAssociativeAlgebra R

prime?: % -> Boolean if R has IntegralDomain

from UniqueFactorizationDomain

principalIdeal: List % -> Record(coef: List %, generator: %) if R has IntegralDomain

from PrincipalIdealDomain

quo: (%, %) -> % if R has IntegralDomain

from EuclideanDomain

recip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix Integer, vec: Vector Integer) if R has Ring and R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: (Matrix %, Vector %) -> Record(mat: Matrix R, vec: Vector R) if R has Ring

from LinearlyExplicitOver R

reducedSystem: Matrix % -> Matrix Integer if R has Ring and R has LinearlyExplicitOver Integer

from LinearlyExplicitOver Integer

reducedSystem: Matrix % -> Matrix R if R has Ring

from LinearlyExplicitOver R

rem: (%, %) -> % if R has IntegralDomain

from EuclideanDomain

retract: % -> Fraction Integer if R has RetractableTo Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

from RetractableTo Fraction Integer

retract: % -> Fraction Polynomial R if R has IntegralDomain

from RetractableTo Fraction Polynomial R

retract: % -> Integer if R has RetractableTo Integer

from RetractableTo Integer

retract: % -> Kernel %

from RetractableTo Kernel %

retract: % -> Polynomial R if R has Ring

from RetractableTo Polynomial R

retract: % -> R

from RetractableTo R

retract: % -> Symbol

from RetractableTo Symbol

retractIfCan: % -> Union(Fraction Integer, failed) if R has RetractableTo Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

from RetractableTo Fraction Integer

retractIfCan: % -> Union(Fraction Polynomial R, failed) if R has IntegralDomain

from RetractableTo Fraction Polynomial R

retractIfCan: % -> Union(Integer, failed) if R has RetractableTo Integer

from RetractableTo Integer

retractIfCan: % -> Union(Kernel %, failed)

from RetractableTo Kernel %

retractIfCan: % -> Union(Polynomial R, failed) if R has Ring

from RetractableTo Polynomial R

retractIfCan: % -> Union(R, failed)

from RetractableTo R

retractIfCan: % -> Union(Symbol, failed)

from RetractableTo Symbol

rightPower: (%, NonNegativeInteger) -> % if R has SemiGroup

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> % if R has SemiGroup

from Magma

rightRecip: % -> Union(%, failed) if R has SemiGroup

from MagmaWithUnit

sample: % if R has AbelianSemiGroup or R has SemiGroup

from AbelianMonoid

sizeLess?: (%, %) -> Boolean if R has IntegralDomain

from EuclideanDomain

smaller?: (%, %) -> Boolean

from Comparable

squareFree: % -> Factored % if R has IntegralDomain

from UniqueFactorizationDomain

squareFreePart: % -> % if R has IntegralDomain

from UniqueFactorizationDomain

subst: (%, Equation %) -> %

from ExpressionSpace2 Kernel %

subst: (%, List Equation %) -> %

from ExpressionSpace2 Kernel %

subst: (%, List Kernel %, List %) -> %

from ExpressionSpace2 Kernel %

subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup

from CancellationAbelianMonoid

tower: % -> List Kernel %

from ExpressionSpace2 Kernel %

tower: List % -> List Kernel %

from ExpressionSpace2 Kernel %

unit?: % -> Boolean if R has IntegralDomain

from EntireRing

unitCanonical: % -> % if R has IntegralDomain

from EntireRing

unitNormal: % -> Record(unit: %, canonical: %, associate: %) if R has IntegralDomain

from EntireRing

univariate: (%, Kernel %) -> Fraction SparseUnivariatePolynomial % if R has IntegralDomain

from FunctionSpace2(R, Kernel %)

variables: % -> List Symbol

from FunctionSpace2(R, Kernel %)

variables: List % -> List Symbol

from FunctionSpace2(R, Kernel %)

zero?: % -> Boolean if R has AbelianSemiGroup

from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid if R has AbelianSemiGroup

AbelianSemiGroup if R has AbelianSemiGroup

Algebra % if R has IntegralDomain

Algebra Fraction Integer if R has IntegralDomain

Algebra R if R has CommutativeRing

BasicType

BiModule(%, %) if R has Ring

BiModule(Fraction Integer, Fraction Integer) if R has IntegralDomain

BiModule(R, R) if R has CommutativeRing

CancellationAbelianMonoid if R has AbelianGroup

canonicalsClosed if R has IntegralDomain

canonicalUnitNormal if R has IntegralDomain

CharacteristicNonZero if R has CharacteristicNonZero

CharacteristicZero if R has CharacteristicZero

CoercibleFrom Fraction Integer if R has RetractableTo Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

CoercibleFrom Fraction Polynomial R if R has IntegralDomain

CoercibleFrom Integer if R has RetractableTo Integer

CoercibleFrom Kernel %

CoercibleFrom Polynomial R if R has Ring

CoercibleFrom R

CoercibleFrom Symbol

CoercibleTo OutputForm

CommutativeRing if R has IntegralDomain

CommutativeStar if R has IntegralDomain

Comparable

ConvertibleTo InputForm if R has ConvertibleTo InputForm

ConvertibleTo Pattern Float if R has ConvertibleTo Pattern Float

ConvertibleTo Pattern Integer if R has ConvertibleTo Pattern Integer

DivisionRing if R has IntegralDomain

EntireRing if R has IntegralDomain

EuclideanDomain if R has IntegralDomain

Evalable %

ExpressionSpace

ExpressionSpace2 Kernel %

Field if R has IntegralDomain

FullyLinearlyExplicitOver R if R has Ring

FullyPatternMatchable R

FullyRetractableTo R

FunctionSpace2(R, Kernel %)

GcdDomain if R has IntegralDomain

Group if R has Group

InnerEvalable(%, %)

InnerEvalable(Kernel %, %)

IntegralDomain if R has IntegralDomain

LeftModule % if R has Ring

LeftModule Fraction Integer if R has IntegralDomain

LeftModule R if R has CommutativeRing

LeftOreRing if R has IntegralDomain

LinearlyExplicitOver Integer if R has Ring and R has LinearlyExplicitOver Integer

LinearlyExplicitOver R if R has Ring

Magma if R has SemiGroup

MagmaWithUnit if R has SemiGroup

Module % if R has IntegralDomain

Module Fraction Integer if R has IntegralDomain

Module R if R has CommutativeRing

Monoid if R has SemiGroup

NonAssociativeAlgebra % if R has IntegralDomain

NonAssociativeAlgebra Fraction Integer if R has IntegralDomain

NonAssociativeAlgebra R if R has CommutativeRing

NonAssociativeRing if R has Ring

NonAssociativeRng if R has Ring

NonAssociativeSemiRing if R has Ring

NonAssociativeSemiRng if R has Ring

noZeroDivisors if R has IntegralDomain

PartialDifferentialRing Symbol if R has Ring

Patternable R

PatternMatchable Float if R has PatternMatchable Float

PatternMatchable Integer if R has PatternMatchable Integer

PrincipalIdealDomain if R has IntegralDomain

RetractableTo Fraction Integer if R has RetractableTo Integer and R has IntegralDomain or R has RetractableTo Fraction Integer

RetractableTo Fraction Polynomial R if R has IntegralDomain

RetractableTo Integer if R has RetractableTo Integer

RetractableTo Kernel %

RetractableTo Polynomial R if R has Ring

RetractableTo R

RetractableTo Symbol

RightModule % if R has Ring

RightModule Fraction Integer if R has IntegralDomain

RightModule Integer if R has Ring and R has LinearlyExplicitOver Integer

RightModule R if R has Ring

Ring if R has Ring

Rng if R has Ring

SemiGroup if R has SemiGroup

SemiRing if R has Ring

SemiRng if R has Ring

SetCategory

TwoSidedRecip if R has Group or R has IntegralDomain

UniqueFactorizationDomain if R has IntegralDomain

unitsKnown if R has Group or R has Ring