A Party to the World » proofs

Desert Island

Dan — Mon, 17 Nov 2008 14:16:56 +0000

One of those relatively common “getting to know you” games is “Desert Island.” You’re given a class of objects (like, CDs), and a number indicating how many you can bring. In short, if you had to listen to 5 albums for the rest of your life, or read five books, or play the same five board games, which would you choose?

Today’s xkcd was on the subject of being stranded, isolated, with essentially large amounts of time on your hands. I was so happy reading it, because in one panel he describes that during the character’s infinite solitude, he “rederived modern math in the sand… and then some.” This is what I had always imagined when I’ve played Desert Island. No distractions – no women, enough food to support me, no video games, no internet. Just a (let’s say noble) quest to further my own knowledge. Discover. Explore. I think of it like being in an enormous library of blank books except for a few basic texts, which you know by heart. Eternity to fill them.

Courtesy of xkcd.com

Sum of Combinatoric Terms

Dan — Fri, 03 Oct 2008 17:42:14 +0000

Matt Matteson had a homework problem dealing with (n+a)^b, and finding a bound for it. Expanding this and evaluating it at n=a piqued my curiosity about the sum of each of the binomial expansion terms. That is to say, the sum of bC0 + bC1 + … + bCb. Well, I explored it a little bit, and did a quick-and-dirty writeup of my findings:

Sum of Combinatoric Terms

Heapsort

Dan — Wed, 25 Jun 2008 00:05:43 +0000

I was reading about Algorithm Geeks, a Google Group dedicated to algorithm-related questions apparently. I looked at one of the topics, Unique Elements in an Array, wondering what the chatter looked like.

To remove duplicate elements of an array, the best way I know is to sort your array, and then do a linear traversal, comparing each to its successor, and when they match, deleting the one of them. Quicksort is generally the algorithm-of-choice in large part because of its simplicity, but someone on this thread suggested heapsort – an algorithm of which I had never heard. I looked it over, and was drawn to the article on heaps. On that page, I was looking at an analysis of the time complexity for building the heap, and I got curious about the summation they present (h/2^h). So, I took some time and proved its convergence:

Convergence of Infinite Sum