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RE: [Axiom-developer] about Expression Integer
From: |
Bill Page |
Subject: |
RE: [Axiom-developer] about Expression Integer |
Date: |
Tue, 21 Feb 2006 18:41:37 -0500 |
On February 21, 2006 4:53 PM Ralf Hemmecke wrote:
> ...
> > (4) -> differentiate(ex1,x)$S
>
> > This function seems to come from 'DifferentialExtension'
>
> No. I rather think it comes from 'PartialDifferentialRing(Symbol)'.
> Look at Hyperdoc. Browse SUP and view Operations->Origins (Filter
> "differentiate").
I think Hyperdoc is confused.
>From http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PolySpad
SparseUnivariatePolynomial(R:Ring): UnivariatePolynomialCategory(R) with ...
>From http://wiki.axiom-developer.org/axiom--test--1/src/algebra/PolycatSpad
UnivariatePolynomialCategory(R:Ring): Category ==
Join(PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet),
Eltable(R, R), Eltable(%, %), DifferentialRing,
DifferentialExtension R) with ...
>From http://wiki.axiom-developer.org/axiom--test--1/src/algebra/CatdefSpad
DifferentialExtension(R:Ring): Category == Ring with ...
if R has PartialDifferentialRing(Symbol) then
PartialDifferentialRing(Symbol)
add ...
if R has PartialDifferentialRing Symbol then
differentiate(x:%, v:Symbol):% ==
differentiate(x, differentiate(#1, v)$R)
------------
>
> > So I am confused as to why
> > 'differentiate(univariate(p,v))' does not seem to yield this
> > same result. Can you help?
>
I think what we need really is:
differentiate(univariate(p),v)
but I am having a disagreement with SPAD about how to write this.
See:
http://wiki.axiom-developer.org/SandBoxPolynomial
> Not really. I think there is a bug here, since x is sometimes
> treated as the indeterminate of SUP and sometimes not.
This makes sense to me:
'x' is an indeterminante (kernel?) of 'SUP(R:Ring)' if 'x' is
an indeterminante of R,
i.e. by extension in the same sense in which differentiation
is extended to SUP, see ')sh SUP':
differentiate : (%,Symbol) -> % if R has PDRING SYMBOL
But the list of indeterminantes of 'SUP' can be larger than the
list of variables (fixed at 1 and denoted '?' for 'SUP').
Maybe that is a bit confusing compared to common usage. :(
>
> > It seems to me that the interrelationship between all of these
> > polynomial categories, domains and packages is remarkably
> > convoluted.
>
> I don't really think so. The problematic thing is to take
> "Expression Integer" as an argument in the construction of SUP.
>
> Could you explain in which sense SUP(Expression Integer) should be
> semantically different from Expression(Integer)?
SUP(Expression Integer) has an extra (but unspecified) variable:
the "polynomial variable".
>
> Isn't an element of SUP(Expression Integer) also an expression?
No, only the members of 'SUP(EXPR INT)' of degree 0. For example:
monomial(x,1)$SUP EXPR INT
cannot be converted to 'EXPR INT'.
> It is obvious that Axiom can construct SUP(Expression Integer),
> but does this make sense?
Yes, I think so.
>
> I do not fully understand the "Expression" constructor, but
> I somehow consider it as a domain that contains every expression.
No. The 'Rep' of 'Expression x' is 'FRAC MPOLY x'. It cannot
possibly contain every expression.
> Therefore, I think constructing higher domains that are built on
> Expression should not yield a bigger class than Expression itself.
>
Unfortunately 'Expression' is not a "GeneralExpression" or even
Aldor's 'ExpressionTree'. But I agree that really Axiom needs a
'GeneralExpression' of this kind.
> What would be the relation between SUP(Expression Integer) and
> Expression(Integer) on a semantic level? Equality, subset, superset?
>
There is a sense in which 'EXPR INT' is a SubDomain of 'SUP EXPR INT'
in so much as it is always possible to coerce a member of 'EXPR INT'
to 'SUP EXPR INT' but the reverse conversion is only possible for
members of 'SUP EXPR INT' of degree 0.
Regards,
Bill Page.
- Re: [Axiom-developer] about Expression Integer, (continued)
- Re: [Axiom-developer] about Expression Integer, Martin Rubey, 2006/02/19
- DMP([x], DMP([x], Integer)) (was Re: [Axiom-developer] about Expression Integer), Ralf Hemmecke, 2006/02/20
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/20
- Re: [Axiom-developer] about Expression Integer, Francois Maltey, 2006/02/20
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/20
- Re: [Axiom-developer] about Expression Integer, Ralf Hemmecke, 2006/02/21
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/21
- Re: [Axiom-developer] about Expression Integer, Ralf Hemmecke, 2006/02/21
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/21
- Re: [Axiom-developer] about Expression Integer, Ralf Hemmecke, 2006/02/21
- RE: [Axiom-developer] about Expression Integer,
Bill Page <=
- Re: [Axiom-developer] about Expression Integer, Martin Rubey, 2006/02/22
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/23
- Re: [Axiom-developer] about Expression Integer, Martin Rubey, 2006/02/22
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/22
- Re: [Axiom-developer] about Expression Integer, Francois Maltey, 2006/02/22
- Re: [Axiom-developer] about Expression Integer, Martin Rubey, 2006/02/23
- RE: [Axiom-developer] about Expression Integer, Bill Page, 2006/02/23
- Re: [Axiom-developer] about Expression Integer, Ralf Hemmecke, 2006/02/24
- Re: [Axiom-developer] about Expression Integer, William Sit, 2006/02/24
- Re: [Axiom-developer] about Expression Integer, Ralf Hemmecke, 2006/02/20