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Differential Operator Rings form the magma category RngDiffOp.
The notional power structures exist as parents of differential operator rings.
Type(R) : RngDiffOp -> RngDiffOp
The category, or type, of the differential operator ring R.
The power structure of the differential operator ring R.
As outlined in the introduction, a differential operator ring R is of the
form F[D], for a differential ring F.
The ring F is called the base ring or coefficient ring of R.
CoefficientRing(R) : RngDiffOp -> Rng
The base ring, or coefficient ring, of the differential operator ring R.
The constant ring of the differential ring operator R.
By construction the variable D of a differential operator ring
F[D] is related to the derivation δF.
That is why δF is also considered to be the derivation of R.
The derivation of the differential operator ring R.
The differential belonging to the derivation of the differential
operator ring R.
The derivation must have been constructed in such a way that
it is defined by a differential.
> F<z> := RationalDifferentialField(Rationals());
> R<D> := DifferentialOperatorRing(F);
> BaseRing(R) eq F;
true
> Derivation(R);
Mapping from: RngDiff: F to RngDiff: F given by a rule [no inverse]
> Differential(R);
(1) d(z)
Returns true if and only if the differential operator rings R and
F are the same.
Returns true if and only if the differential operator rings R and
F are identical.
Returns true if and only if the given argument is
a differential operator ring.
Returns true iff R is defined over a ring F with
derivation weakly of the form (F.1).d/d(F.1).
Returns true iff the base ring of R is an algebraic differential
field or a differential series ring F such that the derivation of R
acts as a (weak) zero derivation on F.1.
> F<z> := RationalDifferentialField(Rationals());
> R<D> := DifferentialOperatorRing(F);
> IsDifferentialOperatorRing(F);
false
> IsDifferentialOperatorRing(R);
true
> Derivation(R)(z);
1
> HasProjectiveDerivation(R);
false
> HasProjectiveDerivation(ChangeDerivation(R,z));
true
> HasZeroDerivation(R);
false
> S<t> := DifferentialLaurentSeriesRing(Rationals());
> V<W> := DifferentialOperatorRing(S);
> IsDifferentialOperatorRing(V);
true
> Derivation(V)(t);
t
> HasProjectiveDerivation(V);
true
> HasZeroDerivation(V);
false
> P<Q>, mp := ChangeDerivation(V,3/t);
> IsDifferentialOperatorRing(P);
true
> HasProjectiveDerivation(P);
false
> X<y> := BaseRing(P);
> Q*y;
y*Q + 3
The relative precision of the derivation of an operator ring over a
Laurent series ring.
This example illustrates the relative precision of derivations
of differential operatorrings.
> S<t>:=DifferentialLaurentSeriesRing(Rationals());
> RS<DS> := DifferentialOperatorRing(S);
> RelativePrecisionOfDerivation(RS);
Infinity
> RV<DV> := ChangeDerivation(RS, t^2+O(t^8));
> relprec := RelativePrecisionOfDerivation(RV);
> relprec;
6
> RelativePrecisionOfDerivation(BaseRing(RV)) eq relprec;
true
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