PartialDifferentialOperator(R, Var)

skpol.spad line 357 [edit on github]

PartialDifferentialOperator(R, V) defines a ring of partial differential operators in variables from V and with coefficients in a partial differential ring R. Multiplication of operators corresponds to composition of operators.

* : (%, %) -> %
from Magma
* : (%, R) -> %
from RightModule(R)
* : (%, Fraction(Integer)) -> % if R has Algebra(Fraction(Integer))
from RightModule(Fraction(Integer))
* : (%, Integer) -> % if R has LinearlyExplicitOver(Integer)
from RightModule(Integer)
* : (R, %) -> %
from LeftModule(R)
* : (Fraction(Integer), %) -> % if R has Algebra(Fraction(Integer))
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, R) -> % if R has Field
from AbelianMonoidRing(R, IndexedExponents(Var))
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
D : Var -> %

D(v) returns the operator corresponding to derivative with respect to v in R.

^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
adjoint : % -> %

adjoint(p) returns the adjoint of operator p.

annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
associates? : (%, %) -> Boolean if R has EntireRing
from EntireRing
associator : (%, %, %) -> %
from NonAssociativeRng
binomThmExpt : (%, %, NonNegativeInteger) -> % if % has CommutativeRing
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if R has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, Var, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
coefficient : (%, List(Var), List(NonNegativeInteger)) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
coefficient : (%, IndexedExponents(Var)) -> R
from AbelianMonoidRing(R, IndexedExponents(Var))
coefficients : % -> List(R)
from FreeModuleCategory(R, IndexedExponents(Var))
coerce : % -> % if R has CommutativeRing and % has VariablesCommuteWithCoefficients or R has IntegralDomain and % has VariablesCommuteWithCoefficients
from Algebra(%)
coerce : R -> %
from Algebra(R)
coerce : Fraction(Integer) -> % if R has Algebra(Fraction(Integer)) or R has RetractableTo(Fraction(Integer))
from Algebra(Fraction(Integer))
coerce : Integer -> %
from NonAssociativeRing
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
construct : List(Record(k : IndexedExponents(Var), c : R)) -> %
from IndexedProductCategory(R, IndexedExponents(Var))
constructOrdered : List(Record(k : IndexedExponents(Var), c : R)) -> %
from IndexedProductCategory(R, IndexedExponents(Var))
content : % -> R if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
degree : % -> IndexedExponents(Var)
from SolvableSkewPolynomialCategory(R, IndexedExponents(Var))
degree : (%, List(Var)) -> List(NonNegativeInteger)
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
degree : (%, Var) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
exquo : (%, %) -> Union(%, "failed") if R has EntireRing
from EntireRing
exquo : (%, R) -> Union(%, "failed") if R has EntireRing
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
fmecg : (%, IndexedExponents(Var), R, %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
ground : % -> R
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
ground? : % -> Boolean
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
latex : % -> String
from SetCategory
leadingCoefficient : % -> R
from SolvableSkewPolynomialCategory(R, IndexedExponents(Var))
leadingMonomial : % -> %
from SolvableSkewPolynomialCategory(R, IndexedExponents(Var))
leadingSupport : % -> IndexedExponents(Var)
from IndexedProductCategory(R, IndexedExponents(Var))
leadingTerm : % -> Record(k : IndexedExponents(Var), c : R)
from IndexedProductCategory(R, IndexedExponents(Var))
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
linearExtend : (Mapping(R, IndexedExponents(Var)), %) -> R if R has CommutativeRing
from FreeModuleCategory(R, IndexedExponents(Var))
listOfTerms : % -> List(Record(k : IndexedExponents(Var), c : R))
from IndexedDirectProductCategory(R, IndexedExponents(Var))
mainVariable : % -> Union(Var, "failed")
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
map : (Mapping(R, R), %) -> %
from IndexedProductCategory(R, IndexedExponents(Var))
mapExponents : (Mapping(IndexedExponents(Var), IndexedExponents(Var)), %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
minimumDegree : % -> IndexedExponents(Var)
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
monomial : (%, Var, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
monomial : (%, List(Var), List(NonNegativeInteger)) -> %
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
monomial : (R, IndexedExponents(Var)) -> %
from SolvableSkewPolynomialCategory(R, IndexedExponents(Var))
monomial? : % -> Boolean
from IndexedProductCategory(R, IndexedExponents(Var))
monomials : % -> List(%)
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
numberOfMonomials : % -> NonNegativeInteger
from IndexedDirectProductCategory(R, IndexedExponents(Var))
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
plenaryPower : (%, PositiveInteger) -> % if R has IntegralDomain and % has VariablesCommuteWithCoefficients or R has Algebra(Fraction(Integer)) or R has CommutativeRing and % has VariablesCommuteWithCoefficients
from NonAssociativeAlgebra(%)
pomopo! : (%, R, IndexedExponents(Var), %) -> %
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
primitiveMonomials : % -> List(%)
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
primitivePart : % -> % if R has GcdDomain
from FiniteAbelianMonoidRing(R, IndexedExponents(Var))
recip : % -> Union(%, "failed")
from MagmaWithUnit
reducedSystem : Matrix(%) -> Matrix(R)
from LinearlyExplicitOver(R)
reducedSystem : Matrix(%) -> Matrix(Integer) if R has LinearlyExplicitOver(Integer)
from LinearlyExplicitOver(Integer)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(R), vec : Vector(R))
from LinearlyExplicitOver(R)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if R has LinearlyExplicitOver(Integer)
from LinearlyExplicitOver(Integer)
reductum : % -> %
from SolvableSkewPolynomialCategory(R, IndexedExponents(Var))
retract : % -> R
from RetractableTo(R)
retract : % -> Fraction(Integer) if R has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retract : % -> Integer if R has RetractableTo(Integer)
from RetractableTo(Integer)
retractIfCan : % -> Union(R, "failed")
from RetractableTo(R)
retractIfCan : % -> Union(Fraction(Integer), "failed") if R has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retractIfCan : % -> Union(Integer, "failed") if R has RetractableTo(Integer)
from RetractableTo(Integer)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
sample : () -> %
from AbelianMonoid
smaller? : (%, %) -> Boolean if R has Comparable
from Comparable
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
support : % -> List(IndexedExponents(Var))
from FreeModuleCategory(R, IndexedExponents(Var))
totalDegree : % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
totalDegree : (%, List(Var)) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
totalDegreeSorted : (%, List(Var)) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
unit? : % -> Boolean if R has EntireRing
from EntireRing
unitCanonical : % -> % if R has EntireRing
from EntireRing
unitNormal : % -> Record(unit : %, canonical : %, associate : %) if R has EntireRing
from EntireRing
variables : % -> List(Var)
from MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

Module(Fraction(Integer))

NonAssociativeSemiRing

LeftModule(R)

BiModule(%, %)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

FullyRetractableTo(R)

SolvableSkewPolynomialCategory(R, IndexedExponents(Var))

SemiRing

EntireRing

MultivariateSkewPolynomialCategory(R, IndexedExponents(Var), Var)

NonAssociativeAlgebra(Fraction(Integer))

FreeModuleCategory(R, IndexedExponents(Var))

unitsKnown

FullyLinearlyExplicitOver(R)

FiniteAbelianMonoidRing(R, IndexedExponents(Var))

CharacteristicNonZero

IndexedProductCategory(R, IndexedExponents(Var))

noZeroDivisors

Magma

SemiGroup

IntegralDomain

LeftModule(%)

NonAssociativeRing

CharacteristicZero

MaybeSkewPolynomialCategory(R, IndexedExponents(Var), Var)

Module(R)

CommutativeRing

Algebra(%)

BiModule(R, R)

RightModule(Fraction(Integer))

Algebra(R)

RightModule(R)

NonAssociativeSemiRng

CancellationAbelianMonoid

Comparable

RetractableTo(Integer)

LinearlyExplicitOver(R)

AbelianMonoidRing(R, IndexedExponents(Var))

CommutativeStar

AbelianMonoid

MagmaWithUnit

RightModule(%)

AbelianProductCategory(R)

Module(%)

CoercibleTo(OutputForm)

SemiRng

LinearlyExplicitOver(Integer)

Monoid

NonAssociativeAlgebra(R)

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(Integer)

LeftModule(Fraction(Integer))

IndexedDirectProductCategory(R, IndexedExponents(Var))

SetCategory

CoercibleFrom(Fraction(Integer))

NonAssociativeRng

CoercibleFrom(R)

BiModule(Fraction(Integer), Fraction(Integer))

RetractableTo(Fraction(Integer))

RetractableTo(R)

AbelianGroup

AbelianSemiGroup