# LinearOrdinaryDifferentialOperator2(A, M)ΒΆ

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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (%, A) -> %

from RightModule A

*: (%, Fraction Integer) -> % if A has Algebra Fraction Integer
*: (A, %) -> %

from LeftModule A

*: (Fraction Integer, %) -> % if A has Algebra Fraction Integer
*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

/: (%, A) -> % if A has Field
=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

adjoint: % -> %
annihilate?: (%, %) -> Boolean

from Rng

antiCommutator: (%, %) -> %
apply: (%, A, A) -> A
associates?: (%, %) -> Boolean if A has EntireRing

from EntireRing

associator: (%, %, %) -> %
binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
characteristic: () -> NonNegativeInteger
charthRoot: % -> Union(%, failed) if A has CharacteristicNonZero
coefficient: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
coefficient: (%, NonNegativeInteger) -> A
coefficient: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
coefficients: % -> List A
coerce: % -> % if A has IntegralDomain and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has CommutativeRing

from Algebra %

coerce: % -> OutputForm
coerce: A -> %

from Algebra A

coerce: Fraction Integer -> % if A has RetractableTo Fraction Integer or A has Algebra Fraction Integer
coerce: Integer -> %
commutator: (%, %) -> %
construct: List Record(k: NonNegativeInteger, c: A) -> %
constructOrdered: List Record(k: NonNegativeInteger, c: A) -> %
content: % -> A if A has GcdDomain
D: () -> %
degree: % -> NonNegativeInteger
degree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger
degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger
directSum: (%, %) -> % if A has Field
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
fmecg: (%, NonNegativeInteger, A, %) -> %
ground?: % -> Boolean
ground: % -> A
hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

latex: % -> String

from SetCategory

leadingCoefficient: % -> A
leadingMonomial: % -> %
leadingSupport: % -> NonNegativeInteger
leadingTerm: % -> Record(k: NonNegativeInteger, c: A)
leftDivide: (%, %) -> Record(quotient: %, remainder: %) if A has Field
leftExactQuotient: (%, %) -> Union(%, failed) if A has Field
leftExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if A has Field
leftGcd: (%, %) -> % if A has Field
leftLcm: (%, %) -> % if A has Field
leftPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower: (%, PositiveInteger) -> %

from Magma

leftQuotient: (%, %) -> % if A has Field
leftRecip: % -> Union(%, failed)

from MagmaWithUnit

leftRemainder: (%, %) -> % if A has Field
linearExtend: (NonNegativeInteger -> A, %) -> A if A has CommutativeRing
listOfTerms: % -> List Record(k: NonNegativeInteger, c: A)
mainVariable: % -> Union(SingletonAsOrderedSet, failed)
map: (A -> A, %) -> %
mapExponents: (NonNegativeInteger -> NonNegativeInteger, %) -> %
minimumDegree: % -> NonNegativeInteger
monicLeftDivide: (%, %) -> Record(quotient: %, remainder: %) if A has IntegralDomain
monicRightDivide: (%, %) -> Record(quotient: %, remainder: %) if A has IntegralDomain
monomial?: % -> Boolean
monomial: (%, List SingletonAsOrderedSet, List NonNegativeInteger) -> %
monomial: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %
monomial: (A, NonNegativeInteger) -> %
monomials: % -> List %
numberOfMonomials: % -> NonNegativeInteger
one?: % -> Boolean

from MagmaWithUnit

opposite?: (%, %) -> Boolean

from AbelianMonoid

pomopo!: (%, A, NonNegativeInteger, %) -> %
primitiveMonomials: % -> List %
primitivePart: % -> % if A has GcdDomain
recip: % -> Union(%, failed)

from MagmaWithUnit

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
reducedSystem: Matrix % -> Matrix A

from LinearlyExplicitOver A

reducedSystem: Matrix % -> Matrix Integer if A has LinearlyExplicitOver Integer
reductum: % -> %
retract: % -> A

from RetractableTo A

retract: % -> Fraction Integer if A has RetractableTo Fraction Integer
retract: % -> Integer if A has RetractableTo Integer
retractIfCan: % -> Union(A, failed)

from RetractableTo A

retractIfCan: % -> Union(Fraction Integer, failed) if A has RetractableTo Fraction Integer
retractIfCan: % -> Union(Integer, failed) if A has RetractableTo Integer
right_ext_ext_GCD: (%, %) -> Record(generator: %, coef1: %, coef2: %, coefu: %, coefv: %) if A has Field
rightDivide: (%, %) -> Record(quotient: %, remainder: %) if A has Field
rightExactQuotient: (%, %) -> Union(%, failed) if A has Field
rightExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) if A has Field
rightGcd: (%, %) -> % if A has Field
rightLcm: (%, %) -> % if A has Field
rightPower: (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower: (%, PositiveInteger) -> %

from Magma

rightQuotient: (%, %) -> % if A has Field
rightRecip: % -> Union(%, failed)

from MagmaWithUnit

rightRemainder: (%, %) -> % if A has Field
sample: %

from AbelianMonoid

smaller?: (%, %) -> Boolean if A has Comparable

from Comparable

subtractIfCan: (%, %) -> Union(%, failed)
support: % -> List NonNegativeInteger
symmetricPower: (%, NonNegativeInteger) -> % if A has Field
symmetricProduct: (%, %) -> % if A has Field
symmetricSquare: % -> % if A has Field
totalDegree: % -> NonNegativeInteger
totalDegree: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
totalDegreeSorted: (%, List SingletonAsOrderedSet) -> NonNegativeInteger
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
zero?: % -> Boolean

from AbelianMonoid

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra % if A has IntegralDomain and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has CommutativeRing

Algebra A if A has CommutativeRing

BasicType

BiModule(%, %)

BiModule(A, A)

CancellationAbelianMonoid

CommutativeRing if % has VariablesCommuteWithCoefficients and A has CommutativeRing or A has IntegralDomain and % has VariablesCommuteWithCoefficients

CommutativeStar if A has IntegralDomain and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has CommutativeRing

Comparable if A has Comparable

Eltable(A, A)

Eltable(M, M)

EntireRing if A has EntireRing

IntegralDomain if A has IntegralDomain and % has VariablesCommuteWithCoefficients

Magma

MagmaWithUnit

Module % if A has IntegralDomain and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has CommutativeRing

Module A if A has CommutativeRing

Monoid

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if A has EntireRing

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

TwoSidedRecip if A has IntegralDomain and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has CommutativeRing

unitsKnown