I have worked around with Mul and Add, so I know pretty well how they
work, but I haven't worked with Symbol as much. Ondrej and others who
wrote the core will know much better. Also, is there a specific class
that you are considering if it should be a subclass of Symbol?
Aaron Meurer
On Aug 1, 2009, at 12:53 PM, Marco wrote:
>
> 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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