FunctionSpace RΒΆ

fspace.spad line 388

A space of formal functions with arguments in an arbitrary ordered set.

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
*: (%, R) -> % if R has CommutativeRing
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 IntegralDomain or R has Group
from Group
/: (SparseMultivariatePolynomial(R, Kernel %), SparseMultivariatePolynomial(R, Kernel %)) -> % if R has IntegralDomain
p1/p2 returns the quotient of p1 and p2 as an element of %.
=: (%, %) -> Boolean
from BasicType
^: (%, Integer) -> % if R has IntegralDomain or R has Group
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
algtower(f) is algtower([f])
algtower: List % -> List Kernel % if R has IntegralDomain
algtower([f1, ..., fn]) returns list of kernels [ak1, ..., akl] such that each toplevel algebraic kernel in one of f1, ..., fn or in arguments of ak1, ..., akl is one of ak1, ..., akl.
annihilate?: (%, %) -> Boolean if R has Ring
from Rng
antiCommutator: (%, %) -> % if R has Ring
from NonAssociativeSemiRng
applyQuote: (Symbol, %) -> %
applyQuote(foo, x) returns 'foo(x).
applyQuote: (Symbol, %, %) -> %
applyQuote(foo, x, y) returns 'foo(x, y).
applyQuote: (Symbol, %, %, %) -> %
applyQuote(foo, x, y, z) returns 'foo(x, y, z).
applyQuote: (Symbol, %, %, %, %) -> %
applyQuote(foo, x, y, z, t) returns 'foo(x, y, z, t).
applyQuote: (Symbol, List %) -> %
applyQuote(foo, [x1, ..., xn]) returns 'foo(x1, ..., xn).
associates?: (%, %) -> Boolean if R has IntegralDomain
from EntireRing
associator: (%, %, %) -> % if R has Ring
from NonAssociativeRng
belong?: BasicOperator -> Boolean
from ExpressionSpace
box: % -> %
from ExpressionSpace
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 IntegralDomain or R has RetractableTo Fraction Integer
from Algebra Fraction Integer
coerce: Fraction Polynomial Fraction R -> % if R has IntegralDomain
coerce(f) returns f as an element of %.
coerce: Fraction Polynomial R -> % if R has IntegralDomain
from RetractableTo Fraction Polynomial R
coerce: Fraction R -> % if R has IntegralDomain
coerce(q) returns q as an element of %.
coerce: Integer -> % if R has RetractableTo Integer or R has Ring
from NonAssociativeRing
coerce: Kernel % -> %
from RetractableTo Kernel %
coerce: Polynomial Fraction R -> % if R has IntegralDomain
coerce(p) returns p as an element of %.
coerce: Polynomial R -> % if R has Ring
from RetractableTo Polynomial R
coerce: R -> %
from RetractableTo R
coerce: SparseMultivariatePolynomial(R, Kernel %) -> % if R has Ring
coerce(p) returns p as an element of %.
coerce: Symbol -> %
from RetractableTo Symbol
commutator: (%, %) -> % if R has Ring or R has Group
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
convert(f1\^e1 ... fm\^em) returns (f1)\^e1 ... (fm)\^em as an element of %, using formal kernels created using a paren.
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 ExpressionSpace
denom: % -> SparseMultivariatePolynomial(R, Kernel %) if R has IntegralDomain
denom(f) returns the denominator of f viewed as a polynomial in the kernels over R.
denominator: % -> % if R has IntegralDomain
denominator(f) returns the denominator of f converted to %.
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 ExpressionSpace
distribute: (%, %) -> %
from ExpressionSpace
divide: (%, %) -> Record(quotient: %, remainder: %) if R has IntegralDomain
from EuclideanDomain
elt: (BasicOperator, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, %, %, %, %, %, %, %, %, %) -> %
from ExpressionSpace
elt: (BasicOperator, List %) -> %
from ExpressionSpace
euclideanSize: % -> NonNegativeInteger if R has IntegralDomain
from EuclideanDomain
eval: % -> % if R has ConvertibleTo InputForm
eval(f) unquotes all the quoted operators in f.
eval: (%, %, %) -> %
from InnerEvalable(%, %)
eval: (%, BasicOperator, % -> %) -> %
from ExpressionSpace
eval: (%, BasicOperator, %, Symbol) -> % if R has ConvertibleTo InputForm
eval(x, s, f, y) replaces every s(a) in x by f(y) with y replaced by a for any a.
eval: (%, BasicOperator, List % -> %) -> %
from ExpressionSpace
eval: (%, Equation %) -> %
from Evalable %
eval: (%, Kernel %, %) -> %
from InnerEvalable(Kernel %, %)
eval: (%, List %, List %) -> %
from InnerEvalable(%, %)
eval: (%, List BasicOperator, List % -> %) -> %
from ExpressionSpace
eval: (%, List BasicOperator, List %, Symbol) -> % if R has ConvertibleTo InputForm
eval(x, [s1, ..., sm], [f1, ..., fm], y) replaces every si(a) in x by fi(y) with y replaced by a for any a.
eval: (%, List BasicOperator, List List % -> %) -> %
from ExpressionSpace
eval: (%, List Equation %) -> %
from Evalable %
eval: (%, List Kernel %, List %) -> %
from InnerEvalable(Kernel %, %)
eval: (%, List Symbol) -> % if R has ConvertibleTo InputForm
eval(f, [foo1, ..., foon]) unquotes all the fooi's in f.
eval: (%, List Symbol, List % -> %) -> %
from ExpressionSpace
eval: (%, List Symbol, List List % -> %) -> %
from ExpressionSpace
eval: (%, List Symbol, List NonNegativeInteger, List % -> %) -> % if R has Ring
eval(x, [s1, ..., sm], [n1, ..., nm], [f1, ..., fm]) replaces every si(a)^ni in x by fi(a) for any a.
eval: (%, List Symbol, List NonNegativeInteger, List List % -> %) -> % if R has Ring
eval(x, [s1, ..., sm], [n1, ..., nm], [f1, ..., fm]) replaces every si(a1, ..., an)^ni in x by fi(a1, ..., an) for any a1, ..., am.
eval: (%, Symbol) -> % if R has ConvertibleTo InputForm
eval(f, foo) unquotes all the foo's in f.
eval: (%, Symbol, % -> %) -> %
from ExpressionSpace
eval: (%, Symbol, List % -> %) -> %
from ExpressionSpace
eval: (%, Symbol, NonNegativeInteger, % -> %) -> % if R has Ring
eval(x, s, n, f) replaces every s(a)^n in x by f(a) for any a.
eval: (%, Symbol, NonNegativeInteger, List % -> %) -> % if R has Ring
eval(x, s, n, f) replaces every s(a1, ..., am)^n in x by f(a1, ..., am) for any a1, ..., am.
even?: % -> Boolean if % has RetractableTo Integer
from ExpressionSpace
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 ExpressionSpace
freeOf?: (%, Symbol) -> Boolean
from ExpressionSpace
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: % -> R
ground(f) returns f as an element of R. An error occurs if f is not an element of R.
ground?: % -> Boolean
ground?(f) tests if f is an element of R.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState, %) -> HashState
from SetCategory
height: % -> NonNegativeInteger
from ExpressionSpace
inv: % -> % if R has IntegralDomain or R has Group
from Group
is?: (%, BasicOperator) -> Boolean
from ExpressionSpace
is?: (%, Symbol) -> Boolean
from ExpressionSpace
isExpt: % -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has SemiGroup
isExpt(p) returns [x, n] if p = x^n and n ~= 0.
isExpt: (%, BasicOperator) -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has Ring
isExpt(p, op) returns [x, n] if p = x^n and n ~= 0 and x = op(a).
isExpt: (%, Symbol) -> Union(Record(var: Kernel %, exponent: Integer), failed) if R has Ring
isExpt(p, f) returns [x, n] if p = x^n and n ~= 0 and x = f(a).
isMult: % -> Union(Record(coef: Integer, var: Kernel %), failed) if R has AbelianSemiGroup
isMult(p) returns [n, x] if p = n * x and n ~= 0.
isPlus: % -> Union(List %, failed) if R has AbelianSemiGroup
isPlus(p) returns [m1, ..., mn] if p = m1 +...+ mn and n > 1.
isPower: % -> Union(Record(val: %, exponent: Integer), failed) if R has Ring
isPower(p) returns [x, n] if p = x^n and n ~= 0.
isTimes: % -> Union(List %, failed) if R has SemiGroup
isTimes(p) returns [a1, ..., an] if p = a1*...*an and n > 1.
kernel: (BasicOperator, %) -> %
from ExpressionSpace
kernel: (BasicOperator, List %) -> %
from ExpressionSpace
kernels: % -> List Kernel %
from ExpressionSpace
kernels: List % -> List Kernel %
from ExpressionSpace
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 ExpressionSpace
map: (% -> %, Kernel %) -> %
from ExpressionSpace
minPoly: Kernel % -> SparseUnivariatePolynomial % if % has Ring
from ExpressionSpace
multiEuclidean: (List %, %) -> Union(List %, failed) if R has IntegralDomain
from EuclideanDomain
numer: % -> SparseMultivariatePolynomial(R, Kernel %) if R has Ring
numer(f) returns the numerator of f viewed as a polynomial in the kernels over R if R is an integral domain. If not, then numer(f) = f viewed as a polynomial in the kernels over R.
numerator: % -> % if R has Ring
numerator(f) returns the numerator of f converted to %.
odd?: % -> Boolean if % has RetractableTo Integer
from ExpressionSpace
one?: % -> Boolean if R has SemiGroup
from MagmaWithUnit
operator: BasicOperator -> BasicOperator
from ExpressionSpace
operators: % -> List BasicOperator
from ExpressionSpace
opposite?: (%, %) -> Boolean if R has AbelianSemiGroup
from AbelianMonoid
paren: % -> %
from ExpressionSpace
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
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 Integer and R has IntegralDomain or R has RetractableTo Fraction Integer
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 Integer and R has IntegralDomain or R has RetractableTo Fraction Integer
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 MagmaWithUnit
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 ExpressionSpace
subst: (%, List Equation %) -> %
from ExpressionSpace
subst: (%, List Kernel %, List %) -> %
from ExpressionSpace
subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup
from CancellationAbelianMonoid
tower: % -> List Kernel %
from ExpressionSpace
tower: List % -> List Kernel %
from ExpressionSpace
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
univariate(f, k) returns f viewed as a univariate fraction in k.
variables: % -> List Symbol
variables(f) returns the list of all the variables of f.
variables: List % -> List Symbol
variables([f1, ..., fn]) returns the list of all the variables of f1, ..., fn.
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

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

Field if R has IntegralDomain

FullyLinearlyExplicitOver R if R has Ring

FullyPatternMatchable R

FullyRetractableTo R

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

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 Fraction Integer or R has RetractableTo Integer and R has IntegralDomain

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 R if R has CommutativeRing

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

UniqueFactorizationDomain if R has IntegralDomain

unitsKnown if R has Ring or R has Group