Le vendredi 18 mars 2011 à 22:40 -0600, Aaron S. Meurer a écrit : > On Mar 18, 2011, at 10:24 PM, Ronan Lamy wrote: > > > Le vendredi 18 mars 2011 à 16:29 +0530, Hector a écrit : > > > > > >> Yes, exactly. > >> It seems more consistent that way. Has this issue already been > >> discussed? If not, do we need to report this? > >> Since I am looking forward to apply for GSoC, and would love to work > >> with SymPy, this would be good start for me. > >> > >> If core developers/contributors agree, I would like to work on this > >> regardless GSoC. Because with thing implemented, we can never go for > >> limit of multi-variable functions. > > > > I thought about this before, but I didn't reach the point where I could > > suggest a design. > > > > I think there needs to be an object describing the "destination" of a > > limit (i.e. "0", "0+", "0-", "+oo", ...), so the syntax for limit would > > be limit(f(x), x, <something that means "0+">) instead of limit(f(x), x, > > 0, dir="+"). These destination objects would also be used by series() > > and the like and would be passed around in the internal code. > > > > The proper mathematical notion corresponding to this is a filter[*]. It > > applies also in the multivariate case and allows to describe limits when > > |(x,y)| -> 0, or when (x,y) -> (0,0) along a particular ray or some more > > complicated curve, etc. > > > > [*]: See http://fr.wikipedia.org/wiki/Filtre_%28math%C3%A9matiques%29 > > (in French) - I couldn't find a good description in English. The > > en.Wikipedia article in particular is so full of category-theoretic > > abstract nonsense as to be useless. > > Isn't a filter some kind of set of subsets that satisfies some > properties? One of my professors introduced these to me when > describing the hyperreal system. But I don't understand how that lets > you define a "limit path" like you want.
> And I really don't understand how that could help with an > implementation. I remember that when using filters to describe the > hyperreal system, you have to use the axiom of choice to get what you > want, i.e., it is only useful as a theoretical tool. > Yes, it's a set F of subsets with the properties "Union(A, B) is in F for any A, B in F" and "if A is in F and B contains A, then B is in F", but you don't need the axiom of choice to define or use them. Basically, the limit of f along a filter F is y0 iff y0 is in the adherence of f(A) for every A in F (and is the only such point). For instance, the filter that defines the limit of a function in 0+ is the set of parts of R that contain an interval ]0, r[, with r>0. But the actual definition of filters isn't terribly important for us - besides the fact that it exists and is consistent. What matters is that it captures well the notion of limit and that it's possible to do some arithmetic with them (1/0+ = +oo, 0- * 0- = 0+, ...) > Aaron Meurer > -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to sympy@googlegroups.com. To unsubscribe from this group, send email to sympy+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
