Like many people, while a graduate student Roger L. Goodwin took a wide variety of courses in applied probability, statistics, and operations research. Unlike the others, though, he compiled lecture notes from all of his classes into one big book, available for download.
I was reading some political blogs this morning, when I came across this quote:
Look, there’s an endless list of topics I don’t understand at all. I went through an entire semester of pre-Calculus in high school and was never able to understand what a function is. I still don’t. It’s a complicated subject and I was a lazy student.
I don’t know what to say to that.
Qiaochu Yuan recommended Stevenhagen’s mathematical writings in general, so I did some additional searching. I found this page of lecture notes for algebraic number theory courses at Leiden University.
These notes by Stevenhagen provide an elementary introduction to rings of algebraic numbers.
The Geometry Center at the University of Minnesota was a pioneer in putting mathematics on the web. The Center specialized in visualization of advanced geometric topics.
The Center itself was closed in 1998, but their website is still available. The site is quite old (the pages that note that Netscape 2.0 is required are particularly poignant reminders), and many parts of it no longer work, but much of the content is still there.
The Economist has an article on the question of the hardest language to learn. They suggest that a language called Tuyuca is the answer. What makes Tuyuca unusual is that verbs carry an ending that indicates whether the statement is thought to be true or known to be true with certainty. Imagine a language with one tense for conjectures, and another for theorems.
The idea of a homomorphism extends neatly to general signatures. A function between two objects with the same signature is a homomorphism if it preserves all function and relation symbols. So φ is a homomorphism if for each n-ary function symbol f
φ( f(x1, …, xn) ) = f( φ(x1, …, φ(xn) )
and each n-ary relation symbol R
R(x1, …, xn) implies R(φ(x1, …, φ(xn))
This coincides with the usual definition of homomorphism for groups and rings. For partially-ordered sets, homomorphisms correspond to order-preserving maps.
Previous post here.
This post at Geomblog is a nice survey of the different approaches to computational topology, which includes both computational approaches to topology and applications of topology to particular areas within computation.