Today in complex analysis…
I completed the proof that I’ve been working on for about two weeks— if a sequence 
is bounded, and 
is such that 
, then the series 
converges. The proposition itself didn’t interest me too much, it was the tools necessary to prove it that I wanted: the triangle inequality, the existence of supremums and infimums, and the nested interval theorem. After having taken 2 other upper level courses with Dr. Johnson, those (except the triangle inequality) are topics that we spend a lot of time on— we managed to free up a significant amount of time this semester for new material by proving all that stuff early on.
Dr. Johnson made two important observations that added further tools to the kit: first, in the process, we had shown that any infinite bounded set has a limit point, and second, as a consequence, Cauchy sequences converge. At least, Cauchy sequences converge if in the concerned number system it is true that every set bounded above has a least upper bound.
He also defined the Stieljes integral today, and gave a nice introductory/motivatory talk on it. Here’s my paraphrasal of his definition:
Let
be such that
is within the intial set of
. Then
is said to be
integrable on
and a subdivision
of
, for all subdivisions
of
![\left| M - \sum_{t=1}^m f(r_{2t}) \left[ g(r_{2t+1}) - g(r_{2t-1}) \right] \right| < c.](/cz/latexrender/pictures/d4c760f0ea3bf56baeb4b52dbe8a3b1b.png "\left| M - \sum_{t=1}^m f(r_{2t}) \left[ g(r_{2t+1}) - g(r_{2t-1}) \right] \right| < c.")
I think that gets the gist of it. One sweet thing about this integral is that it makes some of the vague mathematical statements made in the sciences more precise; his example was 
. If we’re considering a rod whose volume 
and density 
vary along its length, how would we do that integration except via the Stieljes integral? I almost wished I hadn’t completed all my required mechanics classes, so I could bust out with that one on a homework assignment :).
Actually, since we’re dealing with a physical phenomena, I suspect all the concerned functions would be differentiable, and the Stieljes integral could be reduced to the form 
. I suspect this mainly because as that difference between the 
decreases, the difference term would approach 
. We’ll see; supposedly he’s going to add derivatives to the toolbox sometime soon.
Moore’s method is such a neat way to learn math.
Possibly relevant posts:
- One direction in Khintchine’s inequality for Rademacher sums (9/26/2008)
- Problems with Reimann integration (8/26/2003)
- Regions are Connected (6/16/2005)
Moore’s Method?
Comment by Ronald — 2/23/2005 @ 2:56 pm
Moore’s method is like a more streamlined version of the Socratic method: the prof provides the definitions and propositions to be proved, then an individual attempts to prove one of them to the point that the class is satisfied. The only function of the prof is to provide definitions, occasional helpful comments, and catch errors that everyone in the class misses. It’s the best way to learn how to do proofs
It’s called Moore’s method because it was developed by the topologist R.L. Moore at the University of Texas; it’s also called the Texas style. This article has some details on how one guy implemented it in his classes, and the results.
Comment by Alex — 2/24/2005 @ 4:08 pm