On 08/21/2006 07:18 PM, Martin Rubey wrote:
Ralf Hemmecke <[EMAIL PROTECTED]> writes:
Well, but how can you tell this to Axiom? It should be impossible to construct
the domain UnivariateTaylorSeries(Expression Integer,x,0).
I guess the Axiom designers thought that returning that domain for taylor x
would be reasonable. I must say, I question that.
UTS(Fraction Integer, x, 0) would have been sufficient and you wouldn't have
the trouble.
No, I disagree. As long as we don't have domains UnivariateExpression and
MultivariateExpression that take variables as parameters, things like UTS(EXPR
INT, x, 0) are necessary.
Otherwise you cannot create series like
(67) -> series(sin(y+x), x=0)
(67)
sin(y) 2 cos(y) 3 sin(y) 4 cos(y) 5
sin(y) + cos(y)x - ------ x - ------ x + ------ x + ------ x
2 6 24 120
+
sin(y) 6 cos(y) 7 sin(y) 8 cos(y) 9 sin(y) 10 11
- ------ x - ------ x + ------ x + ------ x - ------- x + O(x )
720 5040 40320 362880 3628800
Type: UnivariatePuiseuxSeries(Expression Integer,x,0)
Looking at this thing I would say that if you take
R = Q[s,c] -- polynomial ring in two variables over rationals
I = (s^2+c^2-1)R -- ideal in R
A = R/I -- factor structure
S = A[[x]] -- formal power series
then S would be a perfect candidate for the result type of the above
expression. And there is no "Expression Integer".
While constructing the result of "series", Axiom should try hard to get
a reasonable (in some sense minimal) type for the result.
Note that you get "PositiveInteger if you just type 1 on the axiom
prompt. Would you be happy, if Axiom returned [EMAIL PROTECTED](Integer)?
I agree that taking EXPR INT as the default domain is a bad idea though.
I think, we are on the same side anyway. ;-)
I think there is one important point for MMA, Maple and MuPAD to make: as long
as the mathematics of a particular topic is unclear, good expression domains
are useful and sometimes maybe even necessary.
Good point and I agree wholeheatedly. But Axiom allows after
understanding the structure of things to encode even that structure
(into the type system). The latter is THE feature of Axiom and should
not be forgotten due to laziness.
By the way, just today I was led to use the domain SUP SUP INT. (In code, of
course)
What is so special about that? It is just Z[x][y], only that the names
of x an y are not given. If you, however, think of a univariate
polynomial ring over a ring R as the collection of functions N -> R with
finite support, you don't even see the x or y.
Ralf
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