Misc Math

Mathematical art By PeterMcB on Mathematics Justin Mullins, a British artist, takes photographs of mathematical equations and diagrams. He currently has an exhibition of his work at Lauderdale House, in Highgate, London, UK. Comments Keep New: Posted on: Fri, Feb 3 2006 3:14 AM Email This Clip/Blog This

Dantas on All of Physics By Walt on Uncategorized If you want to learn all of modern physics, Christine Dantas is here to help. Comments Keep New: Posted on: Thu, Feb 9 2006 11:51 PM Email This Clip/Blog This

Two-Envelope Paradox By Walt on Uncategorized Have any of you ever heard of the two-envelope paradox? It’s a paradox so important that Wikipedia manages to have two articles on it: Two envelopes problem and Envelopes paradox. The only thing that puzzles me about it is that I’m having trouble seeing how it’s a paradox — unlike, say, the Monty Hall problem, the naive answer is the correct one. Comments Keep New: Posted on: Mon, Apr 3 2006 10:01 PM Email This Clip/Blog This

Behavioral Economics By Walt on Economics I spotted a survey article, Behavioral Economics: Past, Present, and Future, which gives a guide to this fairly-new field of economics. The subject was born from a mathematical failure. Economists had given precise axioms as to how people would take into account time and uncertainty when making decisions. The axioms allowed precise predictions that (unlike most economics) could be tested in small-scale experiments with a few test subjects. The result was almost-total failure: nearly every prediction turned out to be wrong. Instead of this being the last word on the subject, this has inspired large amounts of research into finding empirical regularities in the discrepancies between the predictions and the experimental results, and formulating a new theory that is both precise and correct. It’s interesting because the original failure could have led to a turn away from mathematical modeling altogether, but it instead has led to research in improved mathematical modeling. Comments Keep New: Posted on: Tue, Apr 11 2006 10:03 PM Email This Clip/Blog This

Easwaran on Conditional Probability By Walt on Mathematics Frequent commenter Kenny Easwaran (who also has a weblog, Antimeta, devoted to philosophy of math) has written several interesting essays on the interpretation of conditional probability: What Conditional Probability Must (Almost) Be Two Analyses of Conditional Probability in Terms of Unconditional The question is practically and philosophically interesting in the case that the event you are conditioning on occurs with probability zero. Comments Keep New: Posted on: Wed, May 17 2006 9:19 PM Email This Clip/Blog This

TeX Finally Made Functional By Walt on Computer science This is more of a computer programming post than a math post, but since we all use TeX, I thought it might be of general interest. I came across this paper by Heckman and Wilhelm, which describes how to implement TeX’s algorithm for typesetting math formulas in a functional language (namely, ML). We see the output of this algorithm every time we look at a math paper, but until now it lacked a precise formal description other than what was embedded Knuth’s Pascal code. Since Knuth wrote TeX back when computers were much slower and compilers were much dumber, the code contains many hand optimizations that make the logic hard to follow. (The algorithm itself is too complicated to lend itself to a purely verbal description.) By rewriting it in a functional language, the authors are able to turn the algorithm into a well-defined mathematical function. Comments Keep New: Posted on: Thu, Nov 22 2007 9:08 PM Email This Clip/Blog This

Bases in Banach Spaces By Walt on Mathematics For a succinct introduction to bases in Banach spaces, Christopher Heil has provided A Basis Theory Primer. (The first 20 pages are a quick review of Banach space theory. The rest is devoted to bases in Banach spaces.) Comments Keep New: Posted on: Sun, Jan 6 2008 11:09 PM Email This Clip/Blog This

Statistics Not Sadistic By Walt on Mathematics Not only is John Armstrong a failed crackpot, he is wrong about statistics. Statistics is, from the mathematical point of view, a perfectly interesting subject; this fact is carefully concealed from us by statisticians. For example, most mathematicians know the central limit theorem, which says that the sum of large numbers of independent, identically distributed (iid) random variables tend to be normally distributed. This even has an elegant proof in terms of Fourier analysis, where addition of random variables because multiplication of Fourier transforms. What mathematicians don’t know is that almost every other statistic ever defined also satisfies the central limit theorem. The median of a large number of iid random variables? Normally distributed. The mode of a large number of iid random variables (where the underlying distribution has a single mode)? Normally distributed. The cosine of the seventeenth percentile? Normally distributed. The simplest explanation for this cavalcade of normality involves the Gâteaux derivative in functional analysis. Comments Keep New: Posted on: Wed, Feb 6 2008 11:00 PM Email This Clip/Blog This