## Thursday, December 13, 2007

### Four to Nine Years

Well the semester is finally over. I am not 100% sure but I believe I have A's (to see the list of classes look at this post).

Enough about that though since really there isn't too much I can say that is interesting enough for me to type. A few days ago I attended an information session for my school's MSF (Masters of Science in Finance) program. Even though I am an undergraduate student I am able to apply for their Combined Degree program, and would be able to obtain the MSF degree in 2010, when I would normally graduate. The application process doesn't look too difficult or complicated; all you have to do is get at least a 650 on the GMAT, a B+ or better in FIN3403, and submit a statement of purpose. Now I don't know how hard the GMAT is but after reading what kind of skills it "test" it sounds just like the SAT, so I should be fine if I buy a study guide or 2. Even though I haven't taken FIN 3403 the speaker said that it would be alright as long as I was taking it the semester I submitted my application. However, after reading the section on quantitative analytics from a book they handed out, I was not sure whether it would be worth my time to get the MSF or wait and get a MMF (Masters of Science in Mathematical Finance), an equivalent degree in Financial Engineering, or a PhD in Financial Mathematics/Applied Mathematics/Operations Research. The reason is that after looking at the course offerings and descriptions for the MSF program it sounded like they were more interested in the financial side and less on the applied math side. Just to make sure though I am planning on making an appointment to speak with the program head and voice my concerns.

So far I have a list of schools I am considering for graduate school. Here they are broken down based on the degree I would be pursuing.
MMF/Financial Engineering: Chicago, Stanford
PhD in Pure Mathematics: Chicago, Stanford
PhD in Applied Mathematics: Cal Tech., Carnegie Mellon, Chicago, Duke, Georgia Tech., Stanford
PhD in Financial Mathematics/Operations Research: Carnegie Mellon, Cornell
I am sure that this list will change some as I get closer to graduation and talk to more faculty members about what programs would best meet my needs.

## Monday, December 10, 2007

### To Clarify A Question on Limits

Today I went to my advanced calculus professor's office hours. I didn't really have any questions but I knew that other people would be there so maybe one of them would ask a question I had failed to think of myself. Sadly this did not happen, however, one person did ask a question for which I was able to think of two different solutions.

Let $E^'$ be the set of limit points of a set $E.$ Prove that $E^'$ is closed. The proof the professor gave assumed knowledge of convergent sequences. This is actually the simplest proof I have seen, so it will now be the one documented in this post.

My proof: Consider the set $F=(E^')^c$ and a point $p\in F.$ So clearly $p$ is not a limit point of $E,$ thus there is a neighborhood of $p$ which contains only finitely many points of $E.$ Now if this neighborhood contained any point $q\in E^'$ then there is a neighborhood of $q$ such that it is a subset of the neighborhood of $p.$ But then since $q$ is a limit point of $E$ every one of it's neighborhoods contains infinitely many points of $E,$ which contradicts the facts that the neighborhood of $p$ only has a finite number of points from $E$. Therefore this neighborhood of $p$ is a subset of $(E^')^c,$ thus making it an open set. At this point we are done.

Now that wasn't too bad. I think this is the proof that Rudin was looking for since this problem was asked before the introduction of sequences. Just a side note, if you replace "finitely many" with "no" in the above proof then you will run into some issues. Just consider $E=\left\{\frac 1n : n\in \mathbb{N}\right\}$, then $E^'=\{0\}$, so just choose $p=\frac 12$ and you see where the problem arises.

## Sunday, December 09, 2007

### Short Thanks For LaTex Help

Well I have finally found a solution to my "Blogger needs LaTex compatibility" problem. For those of you who have a similar problem just follow this link and read the directions. Honestly it takes less thank 5 minuets to set up.

If $p$ is an odd prime, and $a,b$ are relatively prime to $p$ then $(a+b)^p\equiv a^p+b^p\equiv a+b\bmod p.$

If $F(x)$ is a polynomial then $a-b|F(a)-F(b)$

Isn't it just beautiful. :D