Ars Mathematica

The Moody Muon

Walt — Sun, 14 Mar 2010 22:33:23 +0000

Tomasso Dorigo has a new, non-technical, post explaining muon decay. Tomasso characterized his description as “although not perfectly correct in the physics, was actually not devoid of some didactic power.”

Weyl Tube Formula

Walt — Sat, 13 Mar 2010 22:09:06 +0000

Simon Willerton has written an introduction to the Weyl tube formula.

More details can be found in this review of Alan Gray’s book on the subject.

Applications of Fractional Derivatives

Walt — Fri, 12 Mar 2010 21:16:36 +0000

In ordinary calculus, you define “integral derivatives” — the first derivative, second derivative, etcetera. If you think of differentiation as an operator D that takes functions to functions, then the higher-order derivatives are just Dⁱ for natural numbers i. As far back as Liouville, mathematicians have defined fractional derivatives, extensions of this definition to real numbers, i. There is more than one possible definition, Wikipedia page gives the usual definition, which is in terms of the Laplace transform.

I’ve never known if fractional derivatives were good for anything, or were just a historical curiosity. (They are a special case of singular integral operators, which are useful in PDEs.) This very brief paper discusses an application of fractional derivatives to models of particles in a liquid. This sounds like it should be related to Brownian motion, and it is, but the processes that arise are related to more general Lévy processes than just Brownian motion.

Magistra’s Mathematical Ghost

Walt — Fri, 12 Mar 2010 06:52:43 +0000

I ran across this interesting post by a historian who was an undergraduate mathematics major. She found her old linear algebra notes, and was surprised to find how little of it she still understood:

It’s not just that I can’t answer this question now, it’s that I can barely comprehend even what it means. The terminology bounces through my brain, stirring vague imprecise echoes… It’s disconcerting enough to come across the ghost of your own self. It would be even more disconcerting to know for sure that part of your intellect is now forever closed off to you.

Class Notes in Probability, Statistics, and Operations Research

Walt — Thu, 11 Mar 2010 06:31:27 +0000

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.

Hilbert Space Methods for Differential Equations

Walt — Wed, 10 Mar 2010 01:22:17 +0000

In addition to individual articles, the Electronic Journal of Differential Equations also publishes monographs in the area of differential equations. The first one was a book that I think I remember seeing on library bookshelves: Showalter’s Hilbert Space Methods for Differential Equations.

Someone’s Math Teacher Weeps

Walt — Mon, 08 Mar 2010 12:39:20 +0000

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.

Algebraic Number Theory at Leiden

Walt — Mon, 08 Mar 2010 00:33:09 +0000

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.

Stevenhagen’s notes on class field theory look particularly interesting. They start with particular examples, and explain what the theory means in those particular examples.

Stevenhagen on Number Rings

Walt — Sat, 06 Mar 2010 21:23:07 +0000

These notes by Stevenhagen provide an elementary introduction to rings of algebraic numbers.

The Geometry Center

Walt — Fri, 05 Mar 2010 21:23:25 +0000

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.