I think we are thinking of these indetermanents wrongly--they are
dummy variables and have should really have no context outside of the
function definition. Another problem is that, if we say f = sin(a)+cos
(x), then f(1,2) is ambiguous. Here's what I think should happen:
f = sin(x) should be illegal (though I'm not sure how to throw an
informative error here). Instead one should have to type f(x) = sin
(x). This would be pre-parsed to something like
with inject_indetermanents('x'):
f = sin(x)
Here inject_indetermanents would be assign x to an indetermanent that
knows both its name (for printing) and its position (in the tuple if
f is called with multiple arguments, and x is recursively called down
the tree). It would be a morphism of the category of sets and always
act as the identity function on __call__. It could, say, know how to
differentiate itself (return 1 if the variable with respect to
integration matches self.name, 0 (or dx/dother) otherwise. It would
support addition, exponentiation, etc. via a generic "sum" class that
takes two functions and returns a function. Coercible into R[x], etc.
(I'm not sure how contexts work, perhaps the all the indetermanents
in the block would have to be wrapped... this is still doable via
regular expressions...)
Any comments?
- Robert
On Dec 4, 2006, at 11:26 AM, William Stein wrote:
> On Mon, 04 Dec 2006 06:38:16 -0800, Joel B. Mohler
> <[EMAIL PROTECTED]> wrote:
>> I'm not sure I entirely agree with this. I see that it could be done
>> this
>> way, but it seems quite mathematically strange to me. The fraction
>> field of
>> a ring is a well-known mathematical object, but the idea of putting a
>> generator for a polynomial ring in an exponent is entirely
>> undefined in
>> any
>> abstract sense (unless there's some mathematics I'm missing).
>
> It defines the formal function x |--> x^x, which is defined on all
> numbers x such that x^x is defined. It is at least as meaningful as
> the following:
>
> def f(x):
> return x^x
>
> William
>
>
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