LinearOrdinaryDifferentialOperatorCategory AΒΆ

lodo.spad line 1 [edit on github]

LinearOrdinaryDifferentialOperatorCategory is the category 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

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

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)

=: (%, %) -> Boolean

from BasicType

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

adjoint: % -> %

adjoint(a) returns the adjoint operator of 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: (%, NonNegativeInteger) -> A

from AbelianMonoidRing(A, NonNegativeInteger)

coefficient: (%, SingletonAsOrderedSet, NonNegativeInteger) -> %

from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)

coefficients: % -> List A

from FreeModuleCategory(A, NonNegativeInteger)

coerce: % -> % if % has VariablesCommuteWithCoefficients and A has CommutativeRing or % has VariablesCommuteWithCoefficients and A has IntegralDomain

from Algebra %

coerce: % -> OutputForm

from CoercibleTo OutputForm

coerce: A -> %

from CoercibleFrom A

coerce: Fraction Integer -> % if A has RetractableTo Fraction Integer or A has Algebra Fraction Integer

from CoercibleFrom Fraction Integer

coerce: Integer -> %

from CoercibleFrom Integer

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)

D: () -> %

D() provides the operator corresponding to a derivation in the ring A.

degree: % -> NonNegativeInteger

from AbelianMonoidRing(A, NonNegativeInteger)

degree: (%, List SingletonAsOrderedSet) -> List NonNegativeInteger

from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)

degree: (%, SingletonAsOrderedSet) -> NonNegativeInteger

from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)

directSum: (%, %) -> % if A has Field

directSum(a, b) computes an operator c of minimal order such that the nullspace of c is generated by all the sums of a solution of a by a solution of b.

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?: % -> Boolean

from FiniteAbelianMonoidRing(A, NonNegativeInteger)

ground: % -> A

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: (NonNegativeInteger -> A, %) -> 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: (A -> A, %) -> %

from IndexedProductCategory(A, NonNegativeInteger)

mapExponents: (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?: % -> Boolean

from IndexedProductCategory(A, NonNegativeInteger)

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)

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 CommutativeRing or A has Algebra Fraction Integer

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 %, 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

reducedSystem: Matrix % -> Matrix A

from LinearlyExplicitOver A

reducedSystem: Matrix % -> Matrix 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

right_ext_ext_GCD: (%, %) -> Record(generator: %, coef1: %, coef2: %, coefu: %, coefv: %) if A has Field

from UnivariateSkewPolynomialCategory A

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

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

symmetricPower(a, n) computes an operator c of minimal order such that the nullspace of c is generated by all the products of n solutions of a.

symmetricProduct: (%, %) -> % if A has Field

symmetricProduct(a, b) computes an operator c of minimal order such that the nullspace of c is generated by all the products of a solution of a by a solution of b.

symmetricSquare: % -> % if A has Field

symmetricSquare(a) computes symmetricProduct(a, a) using a more efficient method.

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

AbelianGroup

AbelianMonoid

AbelianMonoidRing(A, NonNegativeInteger)

AbelianProductCategory A

AbelianSemiGroup

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

Algebra A if A has CommutativeRing

Algebra Fraction Integer if A has Algebra Fraction Integer

BasicType

BiModule(%, %)

BiModule(A, A)

BiModule(Fraction Integer, Fraction Integer) if A has Algebra Fraction Integer

CancellationAbelianMonoid

canonicalUnitNormal if A has canonicalUnitNormal

CharacteristicNonZero if A has CharacteristicNonZero

CharacteristicZero if A has CharacteristicZero

CoercibleFrom A

CoercibleFrom Fraction Integer if A has RetractableTo Fraction Integer

CoercibleFrom Integer if A has RetractableTo Integer

CoercibleTo OutputForm

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

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

Comparable if A has Comparable

Eltable(A, A)

EntireRing if A has EntireRing

FiniteAbelianMonoidRing(A, NonNegativeInteger)

FreeModuleCategory(A, NonNegativeInteger)

FullyLinearlyExplicitOver A

FullyRetractableTo A

IndexedDirectProductCategory(A, NonNegativeInteger)

IndexedProductCategory(A, NonNegativeInteger)

IntegralDomain if % has VariablesCommuteWithCoefficients and A has IntegralDomain

LeftModule %

LeftModule A

LeftModule Fraction Integer if A has Algebra Fraction Integer

LinearlyExplicitOver A

LinearlyExplicitOver Integer if A has LinearlyExplicitOver Integer

Magma

MagmaWithUnit

MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)

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

Module A if A has CommutativeRing

Module Fraction Integer if A has Algebra Fraction Integer

Monoid

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

NonAssociativeAlgebra A if A has CommutativeRing

NonAssociativeAlgebra Fraction Integer if A has Algebra Fraction Integer

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors if A has EntireRing

RetractableTo A

RetractableTo Fraction Integer if A has RetractableTo Fraction Integer

RetractableTo Integer if A has RetractableTo Integer

RightModule %

RightModule A

RightModule Fraction Integer if A has Algebra Fraction Integer

RightModule Integer if A has LinearlyExplicitOver Integer

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

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

unitsKnown

UnivariateSkewPolynomialCategory A