Hi Aaron,
Can you explain the thinking behind the Symbol class? I looked at the
doc string but it's not clear how general this kind of object is.
Thanks
On Jul 31, 11:16 pm, "Aaron S. Meurer" <asmeu...@gmail.com> wrote:
> On Jul 31, 2009, at 8:42 PM, Marco wrote:
>
>
>
> > Hi Aaron,
>
> > The Set class is important to have as a master class for all kinds of
> > other classes which should exist, such as Group, Field, Vector space,
> > Algebra, etc. As it is in Category theory.
>
> > A Python set may indeed be a good place to start, but as you suggest,
> > it would only be a finite set of objects. There should be a more
> > general Set class, so that for example we would have a set called the
> > Rationals, which itself is a class, and this set would have properties
> > such that 5/6 in Rationals would evaluate as True, even though 5/6 was
> > not specifically instantiated beforehand.
>
> This should be easy. We have a Rational class that is what every
> rational number is an instance of. For example, 5/6 as a rational
> number in SymPy would be represented as Rational(5, 6). It would be
> easy to create a RationalSet class that checks to see if something is
> an instance of Rational or if it has Rational assumptions on it in
> RationalSet.__contains__(). So for example,
> >>> Rational(5, 6) in RationalSet
> and
> >>> a = Symbol('a', rational=True)
> >>> a in RationalSet
>
> would both return True. This should work very nicely with our new
> assumptions system that we are getting.
>
>
>
> > In terms of what I have in mind for calculations, there are so many!
>
> > For example, out of a manifold one creates many vector bundles and
> > vector spaces, such as the tangent bundle and the vector space of
> > differential forms (these are possibly the most famous but there are
> > many others) There are many calculations which can be done with
> > forms, taking interior and exterior products, derivatives, etc. Most
> > of the time we don't do these calculations in coordinates - they are
> > coordinate-free (this is where their main power comes from). Even
> > when we work on specific manifolds like spheres or tori, we still do
> > not do our calculations in coordinates except in the most necessary
> > cases, and in this case we choose coordinates in special ways.
>
> > As an example, you could calculate the following fact: if X is a
> > vector field and a is a 1-form, and p is a differential form (of mixed
> > degree) then there is a universal identity
>
> > (X + a)(X + a)p = a(X) p,
>
> I am not familiar with the theory of manifolds, but I think I can see
> that there really are things to be calculated here.
>
>
>
>
>
> > where the left hand side is interior product by X and exterior product
> > by a, done twice, and the right hand side is just multiplication by
> > a scalar a(X).
>
> > This kind of calculation is usually done by hand, without coordinates,
> > just using the intrinsic behaviour of the quantities X, a, p.
>
> > Another issue is that we often combine vector spaces - we take direct
> > sums, tensor products, exterior products of vector bundles and of
> > vector spaces... To extend all our operations to these spaces should
> > be easy if they inherit from the right master classes... usually in
> > the maple or mathematica packages i've used, the operations are not
> > defined in this object-oriented way and so you have to teach the
> > computer all your operations again, even though it should be obvious
> > what they mean, because of inheritance.
>
> I think having a master Set class would even help for some of the
> other thing that SymPy does. For example, I would assume that Set
> would be able to easily represent that it is the union of two other
> sets, either by having that build in to the class or by having a Union
> subclass. We could then make Interval a subclass of set and it would
> be easy to represent something like [1, 2]U[3, 4]. We could even have
> infinite union support and be able to easily represent something like
> R - Z as ...U(-1, 0)U(0, 1)U(1, 2)U.... It would fit in nicely if,
> for example, an object has the assumptions real=True and
> integer=False, if and only if it is a member of that set.
>
> Great! Looking forward.
>
> Aaron Meurer
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