LinearOrdinaryDifferentialOperator(A, diff)

lodo.spad line 152 [edit on github]

LinearOrdinaryDifferentialOperator defines a ring of differential operators with coefficients in a ring A with a given derivation. Multiplication of operators corresponds to functional composition: (L1 * L2).(f) = L1 L2 f

* : (%, %) -> %
from Magma
* : (%, A) -> %
from RightModule(A)
* : (%, Fraction(Integer)) -> % if A has Algebra(Fraction(Integer))
from RightModule(Fraction(Integer))
* : (%, Integer) -> % if A has LinearlyExplicitOver(Integer)
from RightModule(Integer)
* : (A, %) -> %
from LeftModule(A)
* : (Fraction(Integer), %) -> % if A has Algebra(Fraction(Integer))
from LeftModule(Fraction(Integer))
* : (Integer, %) -> %
from AbelianGroup
* : (NonNegativeInteger, %) -> %
from AbelianMonoid
* : (PositiveInteger, %) -> %
from AbelianSemiGroup
+ : (%, %) -> %
from AbelianSemiGroup
- : % -> %
from AbelianGroup
- : (%, %) -> %
from AbelianGroup
/ : (%, A) -> % if A has Field
from AbelianMonoidRing(A, NonNegativeInteger)
0 : () -> %
from AbelianMonoid
1 : () -> %
from MagmaWithUnit
= : (%, %) -> Boolean
from BasicType
D : () -> %
from LinearOrdinaryDifferentialOperatorCategory(A)
^ : (%, NonNegativeInteger) -> %
from MagmaWithUnit
^ : (%, PositiveInteger) -> %
from Magma
adjoint : % -> %
from LinearOrdinaryDifferentialOperatorCategory(A)
annihilate? : (%, %) -> Boolean
from Rng
antiCommutator : (%, %) -> %
from NonAssociativeSemiRng
apply : (%, A, A) -> A
from UnivariateSkewPolynomialCategory(A)
associates? : (%, %) -> Boolean if A has EntireRing
from EntireRing
associator : (%, %, %) -> %
from NonAssociativeRng
binomThmExpt : (%, %, NonNegativeInteger) -> % if % has CommutativeRing
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
characteristic : () -> NonNegativeInteger
from NonAssociativeRing
charthRoot : % -> Union(%, "failed") if A has CharacteristicNonZero
from CharacteristicNonZero
coefficient : (%, List(SingletonAsOrderedSet), List(NonNegativeInteger)) -> %
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
coefficient : (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
coefficient : (%, NonNegativeInteger) -> A
from AbelianMonoidRing(A, NonNegativeInteger)
coefficients : % -> List(A)
from FreeModuleCategory(A, NonNegativeInteger)
coerce : % -> % if % has VariablesCommuteWithCoefficients and A has IntegralDomain or % has VariablesCommuteWithCoefficients and A has CommutativeRing
from Algebra(%)
coerce : A -> %
from CoercibleFrom(A)
coerce : Fraction(Integer) -> % if A has Algebra(Fraction(Integer)) or A has RetractableTo(Fraction(Integer))
from CoercibleFrom(Fraction(Integer))
coerce : Integer -> %
from CoercibleFrom(Integer)
coerce : % -> OutputForm
from CoercibleTo(OutputForm)
commutator : (%, %) -> %
from NonAssociativeRng
construct : List(Record(k : NonNegativeInteger, c : A)) -> %
from IndexedProductCategory(A, NonNegativeInteger)
constructOrdered : List(Record(k : NonNegativeInteger, c : A)) -> %
from IndexedProductCategory(A, NonNegativeInteger)
content : % -> A if A has GcdDomain
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
degree : (%, List(SingletonAsOrderedSet)) -> List(NonNegativeInteger)
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
degree : % -> NonNegativeInteger
from AbelianMonoidRing(A, NonNegativeInteger)
degree : (%, SingletonAsOrderedSet) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
directSum : (%, %) -> % if A has Field
from LinearOrdinaryDifferentialOperatorCategory(A)
elt : (%, A) -> A
from Eltable(A, A)
exquo : (%, %) -> Union(%, "failed") if A has EntireRing
from EntireRing
exquo : (%, A) -> Union(%, "failed") if A has EntireRing
from UnivariateSkewPolynomialCategory(A)
fmecg : (%, NonNegativeInteger, A, %) -> %
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
ground : % -> A
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
ground? : % -> Boolean
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
latex : % -> String
from SetCategory
leadingCoefficient : % -> A
from IndexedProductCategory(A, NonNegativeInteger)
leadingMonomial : % -> %
from IndexedProductCategory(A, NonNegativeInteger)
leadingSupport : % -> NonNegativeInteger
from IndexedProductCategory(A, NonNegativeInteger)
leadingTerm : % -> Record(k : NonNegativeInteger, c : A)
from IndexedProductCategory(A, NonNegativeInteger)
leftDivide : (%, %) -> Record(quotient : %, remainder : %) if A has Field
from UnivariateSkewPolynomialCategory(A)
leftExactQuotient : (%, %) -> Union(%, "failed") if A has Field
from UnivariateSkewPolynomialCategory(A)
leftExtendedGcd : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if A has Field
from UnivariateSkewPolynomialCategory(A)
leftGcd : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
leftLcm : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
leftPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
leftPower : (%, PositiveInteger) -> %
from Magma
leftQuotient : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
leftRecip : % -> Union(%, "failed")
from MagmaWithUnit
leftRemainder : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
linearExtend : (Mapping(A, NonNegativeInteger), %) -> A if A has CommutativeRing
from FreeModuleCategory(A, NonNegativeInteger)
listOfTerms : % -> List(Record(k : NonNegativeInteger, c : A))
from IndexedDirectProductCategory(A, NonNegativeInteger)
mainVariable : % -> Union(SingletonAsOrderedSet, "failed")
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
map : (Mapping(A, A), %) -> %
from IndexedProductCategory(A, NonNegativeInteger)
mapExponents : (Mapping(NonNegativeInteger, NonNegativeInteger), %) -> %
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
minimumDegree : % -> NonNegativeInteger
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
monicLeftDivide : (%, %) -> Record(quotient : %, remainder : %) if A has IntegralDomain
from UnivariateSkewPolynomialCategory(A)
monicRightDivide : (%, %) -> Record(quotient : %, remainder : %) if A has IntegralDomain
from UnivariateSkewPolynomialCategory(A)
monomial : (%, List(SingletonAsOrderedSet), List(NonNegativeInteger)) -> %
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
monomial : (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
monomial : (A, NonNegativeInteger) -> %
from IndexedProductCategory(A, NonNegativeInteger)
monomial? : % -> Boolean
from IndexedProductCategory(A, NonNegativeInteger)
monomials : % -> List(%)
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
numberOfMonomials : % -> NonNegativeInteger
from IndexedDirectProductCategory(A, NonNegativeInteger)
one? : % -> Boolean
from MagmaWithUnit
opposite? : (%, %) -> Boolean
from AbelianMonoid
plenaryPower : (%, PositiveInteger) -> % if A has Algebra(Fraction(Integer)) or A has CommutativeRing
from NonAssociativeAlgebra(%)
pomopo! : (%, A, NonNegativeInteger, %) -> %
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
primitiveMonomials : % -> List(%)
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
primitivePart : % -> % if A has GcdDomain
from FiniteAbelianMonoidRing(A, NonNegativeInteger)
recip : % -> Union(%, "failed")
from MagmaWithUnit
reducedSystem : Matrix(%) -> Matrix(A)
from LinearlyExplicitOver(A)
reducedSystem : Matrix(%) -> Matrix(Integer) if A has LinearlyExplicitOver(Integer)
from LinearlyExplicitOver(Integer)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(A), vec : Vector(A))
from LinearlyExplicitOver(A)
reducedSystem : (Matrix(%), Vector(%)) -> Record(mat : Matrix(Integer), vec : Vector(Integer)) if A has LinearlyExplicitOver(Integer)
from LinearlyExplicitOver(Integer)
reductum : % -> %
from IndexedProductCategory(A, NonNegativeInteger)
retract : % -> A
from RetractableTo(A)
retract : % -> Fraction(Integer) if A has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retract : % -> Integer if A has RetractableTo(Integer)
from RetractableTo(Integer)
retractIfCan : % -> Union(A, "failed")
from RetractableTo(A)
retractIfCan : % -> Union(Fraction(Integer), "failed") if A has RetractableTo(Fraction(Integer))
from RetractableTo(Fraction(Integer))
retractIfCan : % -> Union(Integer, "failed") if A has RetractableTo(Integer)
from RetractableTo(Integer)
rightDivide : (%, %) -> Record(quotient : %, remainder : %) if A has Field
from UnivariateSkewPolynomialCategory(A)
rightExactQuotient : (%, %) -> Union(%, "failed") if A has Field
from UnivariateSkewPolynomialCategory(A)
rightExtendedGcd : (%, %) -> Record(coef1 : %, coef2 : %, generator : %) if A has Field
from UnivariateSkewPolynomialCategory(A)
rightGcd : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
rightLcm : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
rightPower : (%, NonNegativeInteger) -> %
from MagmaWithUnit
rightPower : (%, PositiveInteger) -> %
from Magma
rightQuotient : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
rightRecip : % -> Union(%, "failed")
from MagmaWithUnit
rightRemainder : (%, %) -> % if A has Field
from UnivariateSkewPolynomialCategory(A)
right_ext_ext_GCD : (%, %) -> Record(generator : %, coef1 : %, coef2 : %, coefu : %, coefv : %) if A has Field
from UnivariateSkewPolynomialCategory(A)
sample : () -> %
from AbelianMonoid
smaller? : (%, %) -> Boolean if A has Comparable
from Comparable
subtractIfCan : (%, %) -> Union(%, "failed")
from CancellationAbelianMonoid
support : % -> List(NonNegativeInteger)
from FreeModuleCategory(A, NonNegativeInteger)
symmetricPower : (%, NonNegativeInteger) -> % if A has Field
from LinearOrdinaryDifferentialOperatorCategory(A)
symmetricProduct : (%, %) -> % if A has Field
from LinearOrdinaryDifferentialOperatorCategory(A)
symmetricSquare : % -> % if A has Field
from LinearOrdinaryDifferentialOperatorCategory(A)
totalDegree : % -> NonNegativeInteger
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
totalDegree : (%, List(SingletonAsOrderedSet)) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
totalDegreeSorted : (%, List(SingletonAsOrderedSet)) -> NonNegativeInteger
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
unit? : % -> Boolean if A has EntireRing
from EntireRing
unitCanonical : % -> % if A has EntireRing
from EntireRing
unitNormal : % -> Record(unit : %, canonical : %, associate : %) if A has EntireRing
from EntireRing
variables : % -> List(SingletonAsOrderedSet)
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
zero? : % -> Boolean
from AbelianMonoid
~= : (%, %) -> Boolean
from BasicType

CharacteristicNonZero

Module(Fraction(Integer))

NonAssociativeSemiRing

BiModule(%, %)

canonicalUnitNormal

Rng

UnivariateSkewPolynomialCategory(A)

CoercibleFrom(Integer)

TwoSidedRecip

CancellationAbelianMonoid

LeftModule(A)

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

IndexedDirectProductCategory(A, NonNegativeInteger)

CoercibleTo(OutputForm)

RightModule(Integer)

noZeroDivisors

Magma

SemiGroup

BiModule(A, A)

LeftModule(%)

AbelianProductCategory(A)

NonAssociativeRing

CharacteristicZero

FullyRetractableTo(A)

Algebra(%)

RetractableTo(A)

CommutativeRing

CoercibleFrom(Fraction(Integer))

RightModule(Fraction(Integer))

NonAssociativeAlgebra(A)

NonAssociativeSemiRng

RetractableTo(Integer)

LinearOrdinaryDifferentialOperatorCategory(A)

CommutativeStar

FreeModuleCategory(A, NonNegativeInteger)

AbelianMonoid

MagmaWithUnit

Comparable

LinearlyExplicitOver(A)

RightModule(%)

IndexedProductCategory(A, NonNegativeInteger)

FiniteAbelianMonoidRing(A, NonNegativeInteger)

Module(%)

CoercibleFrom(A)

AbelianMonoidRing(A, NonNegativeInteger)

SemiRng

LinearlyExplicitOver(Integer)

Monoid

Eltable(A, A)

NonAssociativeAlgebra(%)

Algebra(Fraction(Integer))

BasicType

Ring

RightModule(A)

LeftModule(Fraction(Integer))

AbelianSemiGroup

IntegralDomain

SetCategory

Algebra(A)

NonAssociativeRng

MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)

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

RetractableTo(Fraction(Integer))

FullyLinearlyExplicitOver(A)

AbelianGroup

Module(A)