From Big to Bigger

I would like to start this post off by saying congratulations to all the 2008 graduates; high school, college, graduate school, medical school, etc. it doesn't matter. The road has been long and hard (some harder than others) but you have finally made it. Looking back on the last four or five (or more in some cases) years, wonderful memories rush to the forefront of you mind at incomprehensible speeds. Times spent with friends doing "stupid" things, things you regret ever happening but because they did you are a better person. (We all have things we regret it's just a matter of how we deal with it, so I don't want to hear any "I don't have any regrets about my life" bull s**t) So of you will be continuing your education or looking to work in academia, while others will be starting "life". Which ever category you fall into I wish you the best of luck.


A few days ago (May 22nd) I attended my siblings' high school convocation, well actually many of my friends were there but it would take too much time to list them, so I'll just list a few: Ann, Clara, Roland, and Jeremy. Now that the event is over I am glad that I decided to attend. Ever time someone I was close to won an award it made me proud since I would be able to say "I know them and we are great friends" in addition they deserved the award for all of the hard work they put forward both in and out of school. I can only imagine how their parents must have felt.

Today was my sibling's graduation. It's not until you have to sit through one of those as a nonparticipant that you see how fast they move. For some reason I remember my graduation taking forever, but then again I was anxious for it to all finally be over. The speeches given by the saluditorian and valedictorian were both quite good. The saluditorian is a good friend of mine and a funny guy so naturally there were some (well really many) hidden comments in his speech. The only part I regret is that my parents decided to show up only five minutes before graduation started so parking was hell and it was near impossible to find seating. We ended up sitting above the tunnel where all the graduates entered and left, so as my siblings were leaving they were able to see us. Also there was no way to get any good pictures of what was happening on stage because we were so far away. Oh well, being there is what really counts, and I managed to get a few good pictures.

Sum of Two Squares

I know that I haven't had a post about math or anything math related for a while, so I think it is time to change this.

Well the problem discussed here really isn't that difficult but it is interesting enough to mention. "Given an integer determine if can be written as the sum of two squares." For example if , however, if instead we had it can be shown that it is not possible to have where .

Sure you could try using "brute force" to solve this but that is boring and definitely not worthy of its own post. The necessary key insight is a theorem by Fermat. The theorem states that a prime number can be expressed as the sum of two squares if and only if . The proof of one direction is not that difficult but a proof of the other direction (that if then is the sum of two squares) requires a little more work. I don't feel like posting them here but feel free to look on Wikipedia, where you will find numerous proofs. The proof by Euler is quite straight forward but is quite long and split into many sections (your call on whether this is a good thing or not). My personal favorite is the first proof by Dedekind using Gaussian Integers.

Now another important fact to know is the following. If two integers, and , can each be written as the sum of two squares, then their product, can be written as the sum of two squares. The proof is just a basic exercise in equation manipulation, if you have not proven this fact for yourself I suggest you try to in your spare time.

Using these two facts the problem basically boils down to "factor ." Actually our task is even easier than this, we only need to find the prime factors, , of such that . Thus if factors as and for some , we have that and is odd, then can not be expressed as the sum of two squares.

Pretty simple, now writing a program to actually do all of these things is a little harder but not by that much.

Deja Vu

Today just like any other day had its ups and downs. A few days ago I decided to try Wubi, which is a way to try Ubuntu before you decide on whether or not you actually want to install it on your computer. Well I was not able to get my wifi working, a serious issue for any computer user in these times. I spent basically all day trying to get this right but it seems that my efforts were all in vain. Thus I decided to uninstall Wubi, however, even after this when I started my computer there would always be an option to start up Ubuntu (which of course didn't work with out Wubi). So I reverted my laptop back to the factory shipped condition. The next few hours were spent removing all the unwanted programs and reinstalling those which I had saved before the restore.

I wasn't at the computer during the entire restore process. Went out with Ann to a restaurant, but am not able to recall it's name. the food was pretty good and not over prices. For some reason I let her talk me into having the sweet potato side. Honestly it wasn't all that bad but not as good as the sweet potatoes that my mom makes when I come home. Looking back the only negative about our lunch outing was that almost as soon as we got to the restaurant it started to rain, so we were wet while we waited to be seated.

After lunch we went back to my place and had some fun with the internet and learning about phone numbers. Just in case you were wondering it is not possible to have pi and "e" as phone numbers. They both violate some rule about the placement of the number "1". Well now I will have to find a new mathematical constant to have as my future phone number. If you know of one that is available feel free to leave it in a comment.

Today I downloaded the new Lupe Fiasco album, The Cool. I haven't really had the chance to listen to all of the tracks but some stand out more than others. As of now my favorite is Go Go Gadget Flow, and the one I skip most often would have to be Superstar, due mainly to the fact that I hear it everywhere I go. Below is a video of another one of the better songs called "Little Weapon", listening to the lyrics makes you think how sad and it is even worse because it is true.

Sales Tax and Apples

A few days ago I was standing in line at a Subway looking at the prices of the menu items; $4.99, $5.89, $0.99, etc. This just got me thinking, why don't they just give you the price after tax? Just imagine being an immigrant and going to the store and buying two products both listed at $5.99. You go to the register and expect to pay $11.98, but you have to pay $12.82 after 7% sales tax. The first thing that you think is "What is going on here? Why do I have to give you more money than the listed price?" The truth of the matter is that most other countries the price that is shown is the one after tax, so you can imagine how this would cause confusion. Honestly, how difficult would it be for stores to just calculate the tax and add it on to the already listed price? Not very I hope.

I know that no matter how much I, or anyone else for that matter, rants about this, this stupid policy is not likely to change.

Just browsing around the internet I came upon an all too familiar forum discussion topic: "Why are Macs better than PCs". Of course as expected I saw the typical "viruses" claim. What made this thread interesting however, was that there were other claims that were made, some of them irrational but at least they tried to be original. After the virus statements, I would have to say that the next most popular claim was that Macs had better graphics. Now I don't know how true this is since I have never sat the two down side by side and compared so I have no real opinion on this. Some other listed reasons included, easier to use, being faster, and just look better. Now Macs looking better is obviously not a reason that they are better than PCs so I will just disregard this one. Whoever made the being faster claim obviously knows nothing about computers. It's not the computer, but the processor, RAM, and amount of programs running that determine speed. Finally, as far as being easier to use this is really just a matter of getting acclimated. Every time I have to use a Mac I have the hardest time navigating, not because I don't know how to find what I am looking for, but because the keyboard shortcuts are different. This and the way you interact with open windows is a little different when it comes to maximizing/minimizing and other such task.

Amidst all this PC bashing there was the occasional "voice of reason". One of the reasons for the "lack of Mac viruses" is obviously the small portion of market power they have but also because of how the OS and software are closely integrated. One of my favorite statements in the whole thread went something like this, "I don't have anything against Macs, however, what I do have an issue with is the air of superiority that seems to surround almost all Mac users. They seem to think that just because they own a Mac that they are so much better than all PC users. Really, from the looks of things it seems that the extra money you pay for a Mac is not for a better computer but rather for this false sense of superiority." I would have to agree with this statement. Almost all the Mac users I know seem to always try to mention how much better their computer is than mine. Even though it is true it's not because I am using a PC but because I have less RAM, a slower processor (which is about to change), and the amount of programs I run simultaneously. When I was looking for laptops I originally though about getting an iBook or Powerbook. However, as it turns out they were selling for about $2,000 and if I wanted to get the same thing on a PC I would only have to pay about $600-700. So it's no surprise I went with my HP Pavilion dv6700.

Small Intermission

Finals ended last week and I am glad that I will never have to see some of those classes again. Looking back on the semester, I realize that French was not all that bad. Honestly I am going to miss the subject, but not the having class every morning. Another class I'm not going to miss all that much is Abstract Algebra. Now I know that I might regret this but for some reason the topic doesn't appeal to me quite as much as number theory, analysis, or other branches of mathematics. The funny thing is that algebra and number theory are very closely related. Just about every theorem we learned in algebra could be stated in terms that wold make it applicable to a number theory class. Overall the semester wasn't that bad, surprisingly advanced calculus turned out to be ,what I considere, my easiest class. I still have to wait another week or so to find out my final grades but I am pretty confident that I made As in all of my classes.

When I returned home I found my laptop waiting for me. In case you are wondering I mentioned the laptop in one of my last two post. Setting everything up was pretty straight forward, no real surprises. The only real issues were getting used to the new feel of vista and removing all of the unnecessary preinstalled software. Really I could group these into the same problem since it took me a while to find the "add/remove programs" function in vista, which happens to be called "programs and features". After finding this I spent the next hour or so finding unwanted material and then waiting for Windows to uninstall them. Overall Windows Vista isn't all that bad, I would rather be using XP but after spending some time with it you realize that in all actuality it isn't as bad as some reports make it out to be.

Here are a few screen shots of my new set up.

In use:



Desktop:















As you probably noticed this is not the usual Windows set up. I have "replaced" the task bar and system try with an application called ObjectDock, which imitates the thing that is used my Macs. On the side is Google Desktop's side bar, personally I prefer it to the one that comes with Vista.

Finally, my birthday was a little while ago. I didn't really do all that much, just had lunch with two of my best friends and then spent time with the family. Even though it doesn't sound like much it turned out pretty well.