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.

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 be the set of limit points of a set Prove that 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 and a point So clearly is not a limit point of thus there is a neighborhood of which contains only finitely many points of Now if this neighborhood contained any point then there is a neighborhood of such that it is a subset of the neighborhood of But then since is a limit point of every one of it's neighborhoods contains infinitely many points of which contradicts the facts that the neighborhood of only has a finite number of points from . Therefore this neighborhood of is a subset of 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 , then , so just choose and you see where the problem arises.

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 is an odd prime, and are relatively prime to then

If is a polynomial then

Isn't it just beautiful. :D

Change of Interest

Recently I have been somewhat distracted. Not distracted in the normal sense of the word, it's just that I have been adsorbed in different things. Until recently I used to be all about Analysis and Putnam preparation. However, now I am quite determined to become better at coding. Currently I am only working in C++ since it was the first language I learned and decided that I should become somewhat efficient with it before learning others. If things go as planned I should start learning Java by the beginning of my junior year. So far I understand the basics; loops, cases/switches, variable types, and functions, mainly I need to learn more about what the various libraries have to offer and how to best use them. A few days ago I was introduced to the algorithm library, and needless to say this saved me a lot of coding. Now instead of having to write a function to sort the elements of a list/vector/array I can just use the sort function and then move on the more important things. Also looked through the vector library and found some useful functions there, but have not needed to use them as often as the ones in the algorithm library.
To get practice I have been doing problems at SPOJ, my only problem with the site is the insane number of test cases they require your code to run, usually around 20000 or more. So even if your code runs perfectly it might not pass because of the time limit constraint. When this happens I just move on to the next problem and might one day come back and attempt to find a solution with better time complexity (faster algorithm in short). Even though not much "high level" mathematics is involved it is a great place to get practice with problem solving since there is usually an obvious (usually brute force) solution but you try to find one that is more efficient. Kind of like finding the elegant solution to a problem. Another site with good problems is Project Euler, even though their problems are geared towards the mathematically enticed programmer.
Don't get me wrong I am still preparing for the Putnam; currently reading Problem Solving Through Problems (notice how it is no longer under my books I wish to own section). The book is better than all they hype. Sure there are no solutions but if you actually read the sections then there is no real need for one, even though there will be the occasional problem that is just beyond your immediate grasp. Honestly I wish I had purchased this book sooner.
Here is what I will be taking during the spring '08 semester:

Advanced Calculus 2 (MAA 4212) - MWF
Abstract Algebra (MAS 4301) - MWF
Beginning French 2 (FRE 1131) - MTWRF
Managerial Economics (ECP 3703) - Online

Apart from classes I plan to continue with my coding en devours, as well as read a textbook called "Mathematical Economics and Finance". Still debating with the prospect of taking more differential equations classes, so I might go and review my notes from Elementary Differential Equations and actually read the book or buy an actual differential equations book geared for the "mathematically gifted".

Putnam - 2

Well as promised here is my next Putnam problem post. Luckily I had the dilemma of choosing between two questions. Both questions came from the 1988 exam, A2 and A6. In the end I decided to post problem A6 with a solution and post problem A2 and with proof for another day.

Putnam - 1

Well I have been spending between half an hour and one hour every day solving Putnam problems. Actually I should say attempting to solve Putnam problems, but it is the same concept I suppose.

Here is the most recent problem I have been able to solve on my own.

Overall I would say that it took me a little over an hour to come up with this solution. I have looked at other solutions and have yet to find one like mine which kind of makes me feel proud. It should be noted that in the following proof I leave out showing that the sum does in fact converge absolutely, something I would have done if under actually testing conditions.


Well the next time I manage to actually solve a problem that I think is worth posting I shall do so. Hopefully that won't be too far away.

Faces Without Names

It happens to everyone at least once in their life. They are at an event with a friend and then they are suddenly introduced to a large group of new people. You honestly know that there is no way in hell you will remember everyone's names but you pretend to do so. Then after five minutes or so you realize that the only names you remember are those of the last person to be introduced and the member of the group you found most attractive. At the end of the evening when everyone is heading home you think to yourself, "Thank God, there is a chance that I will never see them again , and thus not embarrass myself by not remembering their names."
Well just when you thought the worst was over you run into at least one of the group members in public, and you are not with your friend. Damn they spotted you. Your mind races trying to remember their name, but with no success you resort to saying something along the lines of "Hey how are you?" and skillfully conduct the conversation in a way that lets you avoid using their name.

Well things like this happened to me a few weeks ago. While I was walking from my dorm to flag football practice I ran into someone who knew me. Now since I didn't recognize his face I was almost certain that we had never actually formally met. Well as it turns out we were both members of MAO in high school and he recognized me from there. Needless to say I had no idea who he was because he was not on stage often if at all. After talking to him I got to thinking about just how many people know me because of MAO. I'm fine with it if I am actually friends with the person now and talk to them on a semi-regular basis but it just scares me a little if I don't really know the person.

Classes are going alright so far. French is a lot of work, I am really beginning to wish that I had just taken either Spanish or Portuguese. Learning vocabulary is just taking up way too much time. It is becoming difficult to find time to work on programming or relearning Linear Algebra and Elementary Differential Equations. At least I still have the weekends. Putnam practices start tomorrow [a little late in my opinion but not much I can do about it, so I started my studying a while ago] and it should be interesting to see who will be there. Hopefully it will be more structured that it was last year where the professor really didn't have a plan for what material would be covered at what time. I would really like if my Number Theory teacher ran the practices since he is very enthusiastic and seems to communicate better with students. Sadly however, he is very busy so it is going to be extremely unlikely that he will be running the practices this year. Maybe next year will be different...or so I hope.

As things are now I am still uncertain about what I really want to do over the summer. If I have the choice to choose between attending and REU or and Actuarial Internship (and I hope I have to make this choice later in the year) I am not sure which I would choose. Really I suppose it comes down to which company/companies offer me internships and which REU(s) I get accepted for participation. In terms of internships I only have a list of two companies I would really like to work for, one is a consulting company and the other I like because of its location (yes there are other reasons for liking this company). In terms of REUs I would like to do one in any of the following fields (in order of preference); financial mathematics, stochastic processes and probability, differential equations, and number theory.