I've been looking at some of the trac tickets. Two of them caught my eye, and I'm curious about them:
http://trac.sagemath.org/sage_trac/ticket/2515 http://trac.sagemath.org/sage_trac/ticket/1915 Both of them deal with Infinity. In the first one, someone suggests doing away with ExtendedRationalField and ExtendedIntegerRing, because Infinity should be handled by the code in sage.rings/infinity.py. In the second one, it is pointed out that evaluating 1/Infinity yields "Zero" in the "infinity ring", whereas maybe it should yield the element 0 in some "numeric domain". The documentation in infinity.py says that in the infinity ring > The rules for arithmetic are that the unsigned infinity ring does not > canonically > coerce to any other ring, and all other rings canonically coerce to > the unsigned infinity ring, sending all elements to the single element > ``a number less than infinity'' of the unsigned infinity ring. I have two issues with this: first, things don't quite behave as expected: sage: a=GF(2)(1) sage: a/Infinity # expect this to give Zero, or maybe "A number less than infinity" Traceback (click to the left for traceback) ... TypeError: unsupported operand parent(s) for '/': 'Finite Field of size 2' and 'The Infinity Ring' However, this does work: sage: UnsignedInfinityRing(a) A number less than infinity Second, I disagree with the mathematics described in the documentation (and I agree more or less with the first aspect above of Sage's behavior): I don't think there is any good notion of infinity associated to, say, a finite field. I object to elements of a finite field being coerced to something *less* than infinity, since there is no intrinsic ordering on them. I also object to elements of, say, a finite field of order 4, or a polynomial ring, being coerced to something called "a number". Thus I'm happy with a/Infinity being undefined if a is an element of GF(2). I don't know anything about coercion, so I don't know if this makes sense, but it seems to me that any ring R could have a method R.has_infinity() (or R.is_extended() or something like that), which is True if there is a good notion of infinity associated to that ring, False otherwise. Then if R.has_infinity() is True and if r is an element of R, evaluating r / Infinity would yield the zero element of R. Other arithmetic, like r * Infinity, would coerce to Infinity in the unsigned infinity ring. You could also have R.has_signed_infinity(). For example, if R is the complex numbers, then R.has_infinity() would be True but R.has_signed_infinity() would be false, while if R is the integers, then R.has_signed_infinity() would be True. Having this in place would let you replace some of the coercion code for the InfinityRing_class, say converting elif isinstance(x, (sage.rings.integer.Integer, sage.rings.rational.Rational, sage.rings.real_double.RealDoubleElement, sage.rings.real_mpfr.RealNumber)) or isinstance(x, (int,long,float)) into something like elif x.parent().has_signed_infinity() or isinstance(x, (int,long,float)): Among my many questions about this: 1. Does this make any sense? 1a. If so, is it feasible to implement? 2. Why does Sage behave the way it does with a=GF(2)(1); a/Infinity I guess the issue is that Infinity is an element of InfinityRing, not UnsignedInfinityRing, and there is no coercion taking place. Doing 'a/ UnsignedInfinityRing(Infinity)' returns 'A number less than infinity', though. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
