Welcome to Gaia! ::

In defense of the lack of mathematical rigor in QFT:
"Just be impressed that we waved our hands and learned how to fly"

Ah I love Gnuplot. It adds a little flare to my otherwise bland terminal output.

I deleted a very important makefile today. ********!!! stressed scream burning_eyes wahmbulance cheese_whine

Today I got permanently kicked out of a class (Automata, Computability, and Complexity) in which I was not enrolled.

What would happen if I went back?

I corrected the teacher a few too many times. Fortunately, he never asked me for my name, so he didn't realize I wasn't supposed to be there.

It is sad. I was kind of looking forward to taking the final.

Today I realized, once again, the absurdity of higher mathematics when one of the applied math undergrads saw a bunch of us working on algebra and exclaimed "Why are those limits going backwards?"

Limits are defined as follows:
Consider a partially ordered set with what is called the Moore-Smith property: Given a and b in the set, there is a g in the set such that a ≤ g and b ≤ g.
A right mapping family is a set of objects O(a) and maps such that if a ≤ b, m(a, b) sends O(a) to O(b), where the maps preserve whatever structure o(a) has.
A left mapping family is a set of objects P(a) and maps such that if a ≤ b, n(b, a) sends P(b) to P(a).

In calculus, for a function the indexing set would be the reals and O(a) would be the function evaluated at a. The maps just take the value of the function at one point to the value of the function at another point.

Given a right mapping family, the right limit is the unique object O such that given an object S and a family of maps f(a):O(a) → S such that f(b)◦m(a,b) = f(a), there is a unique map from O to S. This is written as O = lim_{→} O(a)
Given a left mapping family, the left limit is the unique object P such that given an object T and a family of maps g(a):T → P(a) such that n(b,a)◦g(b) = g(a), there is a unique map from T to P. This is written as P = lim_{←} P(a)

As you can see, the left limit does appear to be going backwards. While one could think of it as simply going down the indexing set instead of up it, there are also the maps outside the mapping family that need to be considered.

Examples: Let the indexing set be the positive integers with Artin ordering (a ≤ b if a|b), let O(a) be Z for all a, and for a ≤ b, let the map m(a,b) be multiplication by b/a. We then get that the right limit of O(a) is Q. I'd explain how this works but there's a lot of machinery I'd need to set up first.
A more useful example is given a left mapping family of finite Abelian groups, the dual (specifically, the Fourier transform) of the left limit of this family is the right limit of the dual of each of the groups, and similarly the dual of the right limit is the left limit of the dual. Furthermore, this occurs in a manner that preserves the standard group topology that you apply to each object, so that a continuous map dualizes to a continuous map.
So we get that you can apply Fourier inversion to groups that are limits of families of finite groups. Specifically, this gives us the Fourier transform on Q/Z, which extends to the Fourier transform on R/Z, which you'll recognize as the Fourier transform for periodic functions.

Long story short, one use of limits in both directions is to generalize the Fourier transform from the two standard examples, R and R/Z, to a more general class of groups.
There are other uses of limits, but that's the one that showed up on the homework I was doing yesterday.

smile I think I have some sense of the first part but I'm trying to understand the rest. Is this the same concept as right hand and left hand limits first introduced in calc but more formally? Are m and n mapping operators and the right mapper is on the right side?

What subject is this a part of? Abstract algebra or topology or something else?