Small Perturbation

Happy leap-day to anyone reading this! How often does a leap-year occur? It is not once every 4 years anymore. It is more like 97 times every 400 years! Adding a day once every 4 years would be correct if the year was 365.25 days long. This is what the Julian calendar did. But then it was discovered that a year is more like 365.2425 days which led to the Gregorian calendar. I like the story of how this was implemented. Centuries of not accounting for this difference caused a 10 day discrepancy between the night sky of 4 October 1582 and the night sky of 4 October in the year that the Julian calendar started. Therefore Pope Gregory XII pronounced that the next day would be 15 October 1582 making the month have 21 days.

The system now is to add an extra day to years that are divisible by 400 and to years that are divisible by 4 but not 100. This means that while the year 2000 was a leap-year, the year 2100 will not be. My prediction is that at least a few printed calendars will be incorrect in the year 2100.

Anyway, what I actually want to talk about is a series of Pronunciation Manual videos on YouTube. The channel parodies instructional language videos by telling you the worst possible way to pronounce something. The meme started in April 2011, I think as a way to take advantage of video monetization. It is stupid simple to make a PronunciationManual video. Each one is 8 seconds long and just consists of a printed word with a voice over.

What's a fun thing to do when you learn a less than intuitive concept? Searching the web to find another person's opinion of it! I recently learned about Grassman numbers and my search turned up a blog post by a professor named Luboš Motl who makes some pretty debatable claims. After reading the post, I found out that he is actually quite famous. So yes, he probably knows much more than me about the subject, but I must still object to how complacent he is with using an object and not defining it.

First of all, happy holidays to anyone reading this! You didn't think I was going to let Christmas / Grav-mass go by without a post did you? Well I absolutely would have if I didn't have this post ready in time. Spreading the spirit of the season can be done in a small number of words - and all of the posts I write have to be long. No exceptions!

Anyway, like many people who have a piano at home, I sometimes hear a great piece of orchestral music in a movie and try to play an approximation to it on the piano. Usually what I try to play is full of mistakes and I lose interest after half an hour. However, my approximations to two Star Wars songs have evolved into fairly well defined pieces that I can play from start to finish. Want to guess which ones?

Both pieces are ending themes so they have to merge into this music that plays during the black and blue credits of every Star Wars movie. That should narrow it down significantly. Anyway, in the rest of this post you can find audio files and sheet music. Learning to play this was satisfying enough. But then I realized that this was a perfect opportunity to learn the music TeX packages so it's a win-win situation.

There are plenty of cases where a proof written down by a physicist is worse than a proof written down by a mathematician, but this is a particularly bad one. In one of my courses, we got to derive the Dirac matrices, which are instrumental in describing spin 1/2 particles. These four matrices are written as with an index. One definition of them says that they should satisfy the anti-commutation relations of the Clifford algebra:

where is the Minkowski metric from special relativity.

How big do our matrices have to be in order to satisfy this? They obviously cannot be 1x1 matrices because these are just numbers that commute. It turns out that they have to be at least 4x4 but all published sources I have seen fail at explaining why. I will go through the physics proof that is often given and then set the record straight by writing a real proof. If it appears nowhere else, let it appear here!

Normally when I see an article about numerology, astrology or homoeopathy, I don't give it the time of day. But this one is interesting because it sounds like the author actually made an honest effort to read up on the science related to the fine structure constant and just got it horribly wrong.

The article is The Mystery of 137 and it lives on a site dedicated to the new age philosopher Ken Wilber. Who would've guessed that a site like that would actually have a correct equation that comes up all the time in quantum electrodynamics?

Awhile ago, my friend showed me Pimp My Gun. This site has a Flash driven app that lets you assemble the weapon of your dreams. It is basically a drawing program that has a library of hundreds of firearm components. There is so much room for customization. I tried it out and came up with the following guns:

So we sent a gun salesman to your mother's house, so you can nag mum to buy a magnum!

I never turn down a chance to be a smart-ass. One of the best things higher mathematics can teach you is how to go back and correct almost everyone who claimed to be teaching you math. It's almost impossible to a cover a decent amount of material in a math course without sacrificing correctness. This is true in grade school when you learn tons of stuff that isn't real math and it is true in grad school when writing one proof that is perfectly rigorous takes two weeks. Here are some common questions that need to be rephrased before they make any sense. The links point to where I found the questions but they could've come from anywhere. If they look like they were taken straight out of your high school calculus textbook, they probably were.

Stupid

Smart

As some of you may know, the programming languages I use the most are C and Python. One reason for this is popularity - I want to learn something that will help me edit the programs I use. I also think it's good to know at least one compiled language and one interpreted language. Interpreted languages or "scripting languages" are more convenient in most respects but they take longer to run. I already knew Python would be slower than C but I wanted to see how much slower.

To make the above plot, I used C and Python codes to diagonalize an n by n matrix and kept track of their execution times. Once you get past the small matrices, the trend that begins to emerge is that Python is ~30 times slower than C.

As part of a summer internship, I got to put together several electronic components, and for the first time, use something more permanent than a breadboard. I found my first circuit very frustrating because my solder connections kept coming loose and I was told to make it as small as possible. But seriously... am I so used to learning about algebraic varieties and Feynman diagrams that I have become allergic to learning a real transferable skill?

The need for my first circuit arose because the photodiode that we used to measure the power in various lasers had a proportionality constant that was too small. For every Watt of power, the diode was calibrated to put only across two pins. We wanted to amplify this to a larger value. The component typically used for these applications that you can buy off the shelf is the operational amplifier.