On Mar 18, 2011, at 11:33 PM, Ronan Lamy wrote: > 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.
What is the adherence of f(A)? By the way, I think the axiom of choice is needed to choose a filter on R to use for a set of equivalence classes for the hyperreals (something like that). > > 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+, …) Yes, I think this is similar to the hyperreals. It seems to me like your 0+ is like an infinitesimal. Aaron Meurer > > > >> 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.
