Monthly Archives: November 2011

Not so Nobel today

When I was in high school, a good friend of mine wrote a haiku that ended up being published in the Atlanta Journal Constitution. It goes something like this

Turkeys flee in fright
Have a happy Thanksgiving
Bye-bye Indians

I remember it to this day. And the AJC thought it fit to publish – which is pleasant.

I also wanted to note that Thanksgiving is not only a welcome vacation and a promise of delicious food. It’s also the time for the Ig Nobel awards. Such notable (real) studies include No evidence of contagious yawning in the red-footed tortoise!

The math award this year was not as exciting as the others, in my opinion. It went to a series of people who predicted the world would end… and were wrong (it would seem). This is supposed to teach people to be careful about making mathematical assumptions and calculations.

In fact, I find it particularly uninspired.  Nonetheless, it is very entertaining, and I highly recommend it.

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Points under Parabola

In my last post, I mentioned I would post my article proper on WordPress. Someone then told me about latex2wp, a python script that will translate a tex file into something postable on WordPress. So I did it, and it works pretty well! Other than changing references (removing them) and a few stylistic things here and there, and any \begin{align} type environments, it works perfectly.

So here it is:


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Finding the Number of Lattice Points Under a Quadratic

I always keep an eye on the Polymath Projects, ever since I became interested in Polymath 4 (link to Polymath4 wiki). While I worked on Polymath4 as an REU student under Dr. Croot, I fell upon a method to ‘quickly’ count the number of lattice points under a quadratic (with no linear term and rational coefficient). Unfortunately, it didn’t lead to direct improvement, so I didn’t post it on the wiki.

But I did a short write-up of the method, and it’s here: Points under Parabola.

At some point, I’ll try to write it up on this blog proper.

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Two short problems

A brief post today:

I was talking about an algebraic topology problem from Hatcher’s book (available freely on his website) with two of my colleagues. In short, we were finding the fundamental group of some terrible space, and we thought that there might be a really slick almost entirely algebraic way to do a problem. We had a group $latex G$ and the exact sequence $latex 0 \to \mathbb{Z} \to G \to \mathbb{Z} \to 0$, in short, and we wondered what we could say about $latex G$. Before I go on, I mention that we had been working on things all day, and we were a bit worn. So the calibre of our techniques had gone down.

In particular, we could initially think of only two examples of such a $latex G$, and we could show one of them didn’t work. Of the five of us there, two of us thought that there might be a whole family of nonabelian groups that we were missing, but we couldn’t think of any. And if none of us could think of any, could there be any? At the time, we decided no, more or less. So $latex G \approx Z \times Z$, which is what we wanted in the sense that we sort of knew this was the correct answer. As is often the case, it is very easy to rationalize poor work if the answer that results is the correct one.

We later made our work much better (in fact, we can now show that our group in question is abelian, or calculate is in a more geometric way). But this question remained – what counterexamples are there? There are infinitely many nonabelian groups satisfying that exact sequence! But I’ll leave this question for a bit –

Find a nonabelian group (or family of groups) that satisfy $latex 0 \to \mathbb{Z} \to G \to \mathbb{Z} \to 0$

The second quick problem of this post. It’s found in Ahlfors – Find a closed form for $latex \displaystyle \sum_{n \in \mathbb{Z}} \frac{1}{z^3 – n^3}$.

When I first had to do this, I failed miserable. I had all these divergent series about, and it didn’t go so well. I try to factor $latex z^3 – n^3 = (z – n)(z – \omega n)(z – \omega^2 n)$, use partial fractions, and go. And… it’s not so fruitful. You get three terms, each of which diverge (if taken independently from each other) for a given $latex z $. And you can do really possibly-witty things, like find functions that have the same poles and try to match the poles, and such. But the divergence makes things hard to deal with. But if you do $latex -(n^3 – z^3) = -[(n-z)(n-\omega z)(n – \omega^2 z)]$, everything works out very nicely. That’s the thing with complex numbers – the ‘natural factorization’ may not always be unique.

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