LinearOrdinaryDifferentialOperatorCategory AΒΆ
lodo.spad line 1 [edit on github]
A: Ring
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)
- ^: (%, NonNegativeInteger) -> %
from MagmaWithUnit
- ^: (%, PositiveInteger) -> %
from Magma
- adjoint: % -> %
adjoint(a)
returns the adjoint operator of a.
- annihilate?: (%, %) -> Boolean
from Rng
- antiCommutator: (%, %) -> %
- apply: (%, A, A) -> A
- associates?: (%, %) -> Boolean if A has EntireRing
from EntireRing
- associator: (%, %, %) -> %
from NonAssociativeRng
- binomThmExpt: (%, %, NonNegativeInteger) -> % if % has CommutativeRing
- characteristic: () -> NonNegativeInteger
from NonAssociativeRing
- charthRoot: % -> Union(%, failed) if A has 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 A has CommutativeRing and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has IntegralDomain
from Algebra %
- coerce: % -> OutputForm
from CoercibleTo OutputForm
- coerce: A -> %
from Algebra A
- coerce: Fraction Integer -> % if A has RetractableTo Fraction Integer or A has Algebra Fraction Integer
from CoercibleFrom Fraction Integer
- coerce: Integer -> %
from NonAssociativeRing
- 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)
- D: () -> %
D()
provides the operator corresponding to a derivation in the ringA
.
- 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 operatorc
of minimal order such that the nullspace ofc
is generated by all the sums of a solution ofa
by a solution ofb
.
- exquo: (%, %) -> Union(%, failed) if A has EntireRing
from EntireRing
- exquo: (%, A) -> Union(%, failed) if A has EntireRing
- fmecg: (%, NonNegativeInteger, A, %) -> %
- ground: % -> A
- latex: % -> String
from SetCategory
- leadingCoefficient: % -> A
from IndexedProductCategory(A, NonNegativeInteger)
- leadingMonomial: % -> %
from IndexedProductCategory(A, NonNegativeInteger)
- leadingTerm: % -> Record(k: NonNegativeInteger, c: A)
from IndexedProductCategory(A, NonNegativeInteger)
- leftDivide: (%, %) -> Record(quotient: %, remainder: %) if A has Field
- leftExactQuotient: (%, %) -> Union(%, failed) if A has Field
- leftExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) 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
from FreeModuleCategory(A, NonNegativeInteger)
- listOfTerms: % -> List Record(k: NonNegativeInteger, c: A)
- mainVariable: % -> Union(SingletonAsOrderedSet, failed)
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
- map: (A -> A, %) -> %
from IndexedProductCategory(A, NonNegativeInteger)
- mapExponents: (NonNegativeInteger -> NonNegativeInteger, %) -> %
- monicLeftDivide: (%, %) -> Record(quotient: %, remainder: %) if A has IntegralDomain
- monicRightDivide: (%, %) -> Record(quotient: %, remainder: %) if A has IntegralDomain
- 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)
- one?: % -> Boolean
from MagmaWithUnit
- opposite?: (%, %) -> Boolean
from AbelianMonoid
- plenaryPower: (%, PositiveInteger) -> % if A has Algebra Fraction Integer or A has CommutativeRing
from NonAssociativeAlgebra A
- pomopo!: (%, A, NonNegativeInteger, %) -> %
- primitiveMonomials: % -> List %
from MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
- 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: % -> %
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
- rightDivide: (%, %) -> Record(quotient: %, remainder: %) if A has Field
- rightExactQuotient: (%, %) -> Union(%, failed) if A has Field
- rightExtendedGcd: (%, %) -> Record(coef1: %, coef2: %, generator: %) 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
from FreeModuleCategory(A, NonNegativeInteger)
- symmetricPower: (%, NonNegativeInteger) -> % if A has Field
symmetricPower(a, n)
computes an operatorc
of minimal order such that the nullspace ofc
is generated by all the products ofn
solutions ofa
.
- symmetricProduct: (%, %) -> % if A has Field
symmetricProduct(a, b)
computes an operatorc
of minimal order such that the nullspace ofc
is generated by all the products of a solution ofa
by a solution ofb
.
- symmetricSquare: % -> % if A has Field
symmetricSquare(a)
computessymmetricProduct(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
AbelianMonoidRing(A, NonNegativeInteger)
Algebra % if A has CommutativeRing and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has IntegralDomain
Algebra A if A has CommutativeRing
Algebra Fraction Integer if A has Algebra Fraction Integer
BiModule(%, %)
BiModule(A, A)
BiModule(Fraction Integer, Fraction Integer) if A has Algebra Fraction Integer
canonicalUnitNormal if A has canonicalUnitNormal
CharacteristicNonZero if A has CharacteristicNonZero
CharacteristicZero if A has CharacteristicZero
CoercibleFrom Fraction Integer if A has RetractableTo Fraction Integer
CoercibleFrom Integer if A has RetractableTo Integer
CommutativeRing if % has VariablesCommuteWithCoefficients and A has IntegralDomain or A has CommutativeRing and % has VariablesCommuteWithCoefficients
CommutativeStar if A has CommutativeRing and % has VariablesCommuteWithCoefficients 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)
IndexedDirectProductCategory(A, NonNegativeInteger)
IndexedProductCategory(A, NonNegativeInteger)
IntegralDomain if A has IntegralDomain and % has VariablesCommuteWithCoefficients
LeftModule Fraction Integer if A has Algebra Fraction Integer
LinearlyExplicitOver Integer if A has LinearlyExplicitOver Integer
MaybeSkewPolynomialCategory(A, NonNegativeInteger, SingletonAsOrderedSet)
Module % if A has CommutativeRing and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has IntegralDomain
Module A if A has CommutativeRing
Module Fraction Integer if A has Algebra Fraction Integer
NonAssociativeAlgebra % if A has CommutativeRing and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has IntegralDomain
NonAssociativeAlgebra A if A has CommutativeRing
NonAssociativeAlgebra Fraction Integer if A has Algebra Fraction Integer
noZeroDivisors if A has EntireRing
RetractableTo Fraction Integer if A has RetractableTo Fraction Integer
RetractableTo Integer if A has RetractableTo Integer
RightModule Fraction Integer if A has Algebra Fraction Integer
RightModule Integer if A has LinearlyExplicitOver Integer
TwoSidedRecip if A has CommutativeRing and % has VariablesCommuteWithCoefficients or % has VariablesCommuteWithCoefficients and A has IntegralDomain