Popping the Hood
Lots of things appear to be magic: Computers, the Banach-Tarski Paradox, cars, the phenomenal success of companies like Facebook, airplanes, the Internet, the Central Limit Theorem, and the fact that...
View ArticleFinding Isomorphisms Between Finite Groups
One of the most interesting problems I came across as I was building my Abstract Algebra package was that of finding an isomorphism between two finite groups G and H, represented by their Cayley...
View ArticleHow My Chess Engine Works
I have always had a lot of respect for chess, despite the fact that I'm not very good at it myself. As I learned more about the game, I also heard about the successes of computer chess AIs, in...
View ArticleCSS Gotchas
I'm still learning html and css. As I debug simple websites I write, (including this one), I've encountered a lot of behavior that seemed very counterintuitive to me. I thought I'd share some of this...
View ArticleIncompetence as Encouragement
A few weeks ago, I was looking for a good python package that could convert user-written strings into numbers. I thought for sure a ton of people would have done this, but I was actually unable to find...
View ArticleBeing the First
I was at the climbing gym a few weeks ago with a friend of mine. We were just messing around, making up with bouldering problems for ourselves to do. One of the problems my friend came up with was a...
View ArticleDon't Trust Asymptotics: Part I
Suppose I give you a sequence of real numbers $x_n$, and tell you that $\lim_n x_n = \infty$. What can you tell me about $x_{100}$? How about $x_{1,000,000}$?
View ArticleGoldbach's Conjecture and Coding Length
Goldbach's conjecture is that every even integer greater or equal to four can be written as the sum of two prime numbers. (Try it: $4 = 2 + 2$, $6 = 3 + 3$, $8 = 3 + 5$, $10 = 3 + 7$, $12 = 5 + 7$...)...
View ArticleThe Hottest Person in the Group
Suppose you think you're pretty good-looking. In fact, you think you're at about the 90th percentile--you're hotter than 90% of people and not as hot as the other 10%. If I put you with another nine...
View ArticleMemory Locality and Python Objects
I've been obsessed with sorting over the last few weeks as I write a python C-extension implementing a lazily-sorted list. Among the many algorithms that this lazily-sorted list implements is...
View ArticleVisualizing the James-Stein Estimator
In the words of one of my professors, "Stein's Paradox may very well be the most significant result in Mathematical Statistics since World War II." The problem is this: You observe $X_1, \ldots, X_n...
View ArticleMarkov Chain Implications Graph
I've been studying for quals for the last several weeks. Today, I was reviewing basic Markov Chain theory, and decided to understand it by drawing a graph of various statements about Markov Chains....
View ArticleMartingale Implications Graph
Here's another directed graph of statement implications that I used to study for quals. This one is about convergence of stochastic processes with martingales. Like my Markov Chain Implications Graph,...
View ArticleThe Zero Times Infinity Problem
There are two ways to keep yourself safe while rock climbing: The first option is to protect yourself carefully with ropes and gear, so that if you fall you won't fall too far or hard. The second...
View ArticleThe Ten Best Ideas in Statistics
I've been studying Statistics for six years now, seriously for the last four years, and as my main focus for the last three. Now that I've finished the core PhD curriculum at Stanford, I've spent some...
View ArticleThe LaTex Numbers
Let's define the LaTex numbers to be the set of all real numbers that can be unambiguously expressed with the LaTex type system. This set of numbers has a few fun properties, not least of which, as...
View ArticleHow I Got 2x Speedup with One Line of Code
If you had asked me whether or not it was possible to get a 2x speedup for my LazySorted project by adding a single line of code, I would have told you "No way, substantial speedups can really only...
View ArticleVisualizing K-Means Clustering
Suppose you plotted the screen width and height of all the devices accessing this website. You'd probably find that the points form three clumps: one clump with small dimensions, (smartphones), one...
View ArticleHow to Solve Problems
I spend a lot of my time solving problems: I solve well-defined math problems in grad school, open-ended problems in statistical consulting, physical puzzles when I'm rock climbing, architectural...
View ArticleA Statistical Analysis of Climbing
Recently, 28 of us on the Stanford Climbing Team completed a short survey on our climbing abilities. Although the survey was intended to assess our interest in different clinics, the answers to the...
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