LinearOrdinaryDifferentialOperator2(A, M)

lodo.spad line 191 [edit on github]

• A : DifferentialRing
• M : LeftModule(A) with

  differentiate : % -> %

LinearOrdinaryDifferentialOperator2 defines a ring of differential operators with coefficients in a differential ring A and acting on an A-module M. 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)

elt : (%, M) -> M

from Eltable(M, M)

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)

Eltable(M, M)

CoercibleFrom(Integer)

TwoSidedRecip

CancellationAbelianMonoid

LeftModule(A)

SemiRing

EntireRing

NonAssociativeAlgebra(Fraction(Integer))

unitsKnown

IndexedDirectProductCategory(A, NonNegativeInteger)

CoercibleTo(OutputForm)

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)

RightModule(Integer)

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)