Recursive Quickie

A recursive quickie to get back into the swing of things. We’re going to define two sequences of numbers, X_n and Y_n in the following way. Define a number Z, such that

Now, notice that when you multiply two numbers of the form , you get another number of that same form. This is due to the ring structure they have, but don’t pay attention to that right now : )

So then, consider Z raised to the nth power. It will also be of that form, so we’re going to define X_n to be the non-root 2 coefficient on Zⁿ, and Y_n to be the root 2 coefficient. That is,

Now, the question is twofold. Consider the ratio, X_n/Y_n as n goes to infinity. Does this converge, and if so, to what? More technically, what restrictions must we place on the value of X₀ and Y₀ to guarantee convergence?

So Busy: Movie Grab Bag

Finals! Finals finals finals. This fox is right in the middle of finals week (though it’s more like finals 10-days). My math studies final is in less than 24 hours, a 48 hour take home exam from Tuesday at 3 to Thursday at 3. It will likely be one of the most grueling tests I’ve ever taken, marking the end of probably the most grueling math course I will take for a good long while. I like to think of it like sky diving. I’ve jumped, I’m 20 feet from the ground, and this test is me looking back to see whether or not my parachute is open. I don’t know if this is a good thing or a bad thing, but I’m excited. Rar!

But it’s leaving me precious little time for maths. Which is a shame. But soon, summer!

Until then, movies.

• Iron Man – This movie looks like a lot of fun. I haven’t seen it, but I’m very very excited. Not least because Robert Downy Jr. is so damn awesome. Go see it. Go go go.
• Law and Order: Criminal Intent – Certainly not a movie, but netflix just got the 4th and 5th seasons on instant viewing, and I’ve been working through them. A lot of people really don’t like Criminal Intent, and I guess I can see that – some of the episodes feel like they’ve been done before. But Vincent D’Onofrio is a lot of fun to watch. Indeed, he’s the only reason I’d recommend Sherlock, to see him as the Professor.
• Cube – Seven strangers wake up inside a maze of cubic rooms. They can’t remember who they are, but they all have certain qualities. A cop, an engineer, an escape artist, doctor, a mathematician, etc. The movie is them trying to work out how to get out of the maze, and avoid being killed by the horrific booby traps contained in some rooms. It’s a pretty good psychological horror movie, I would recommend it if you like that sort of thing. Especially memorable, I say this as a mathematician, is the scene where the mathematician declares the answer to everything …. Permutations! … I guess you have to see it to understand.
• The Orphanage – Terrifying movie. Terrrrrifying. Also very Spanish. What could be more terrifying then the ghosts of children? Imagine The Ring meets Pan’s Labyrinth. The ending absolutely chilled me. This movie is something special.
• Braindead – Peter Jackson brings us one of the greatest zombie movies of all time. The bite of a Sumatran rat monkey spreads the zombie-infection in post-war New Zealand, instigating one of the bloodiest movies in history. Some scenes in this, especially the one with the lawn mower (just think – zombie movie? lawn mower?), you could imagine Peter Jackson standing off to one side with a hose, just spraying everyone down with fake blood. I like to think that if Peter Jackson were to remake this with a $100 million budget, he would make it exactly the same as before, except spend the left over $97 million on more fake blood and intestines.

Anway. Back to studying. More math soon enough.