Month: January 2010

Calendars, Connections, and Cats

Posted on by John

Logo from Broadway musical Cats

James Burke had a television series Connections in which he would create a connection between two very different things. For example, in one episode he starts with the discovery of the touchstone for testing precious metals and tells a winding tale of how the touchstone led centuries later to the development of nuclear weapons.

I had a Connections-like moment when a calendar led to some physics, which then lead to Andrew Lloyd Webber’s musical Cats.

A few days ago I stumbled on Ron Doerfler’s graphical computing calendar and commented on the calendar here. When I discovered Ron Doerfler’s blog, I bookmarked his article on Oliver Heaviside to read later. (Heaviside was a pioneer in what was later called distribution theory, a way of justifying such mathematical mischief as differentiating non-differentiable functions.) As I was reading the article on Heaviside, I came to this line:

At one time the ionosphere was called the Heaviside layer …

Immediately the lyrics “Up, up, up to the Heaviside layer …” started going through my head. These words come from the song “The Journey to the Heaviside Layer” from Cats. I had never thought about “Heaviside” in that song as being related to Mr. Heaviside. I’ve never seen the lyrics in print, so I thought the words were “heavy side” and didn’t stop to think what they meant.

Andrew Lloyd Webber based Cats on Old Possum’s Book of Practical Cats by T. S. Eliot. The song “The Journey to the Heaviside Layer” in particular is based on the poem Old Deuteronomy from Eliot’s book. Webber used the Heaviside layer as a symbol for heaven, based on an allusion in one of T. S. Eliot’s letters. The symbolism is obvious in the musical, but I hadn’t thought about “Heaviside layer” as meaning “the heavens” (i.e. the upper atmosphere) as well as heaven in the theological sense.

How the central limit theorem began

Posted on by John

The Central Limit Theorem says that if you average enough independent copies of a random variable, the result has a nearly normal (Gaussian) distribution. Of course that’s a very rough statement of the theorem. What are the precise requirements of the theorem? That question took two centuries to resolve. You can see the final answer here.

The first version of the Central Limit Theorem appeared in 1733, but necessary and sufficient conditions weren’t known until 1935. I won’t recap the entire history here. I just want to comment briefly on how the Central Limit Theorem began and how different the historical order of events was from the typical order of presentation.

A typical probability course might proceed as follows.

  1. Define the normal distribution.
  2. State and prove a special case of the Central Limit Theorem.
  3. Present the normal approximation to the binomial as a corollary.

This is the opposite of the historical order of events.

Abraham de Moivre discovered he could approximate binomial distribution probabilities using the integral of exp(-x2) and proved an early version of the Central Limit Theorem in 1733. At the time, there was no name given to his integral. Only later did anyone think of exp(-x2) as the density of a probability distribution. De Moivre certainly didn’t use the term “Gaussian” since Gauss was born 44 years after de Moivre’s initial discovery. De Moivre also didn’t call his result the “Central Limit Theorem.” George Pólya gave the theorem that name in 1920 as it was approaching its final form.

For more details, see The Life and Times of the Central Limit Theorem.

Related links:

Regular expressions in Mathematica

Posted on by John

Regular expressions are fairly portable. There are two main flavors of regular expressions—POSIX and Perl—and more languages these days use the Perl flavor. There are some minor differences in what it means to be “like Perl” but for the most part languages that say they follow Perl’s lead specify regular expressions the same way. The differences lie in how you use regular expressions: how you form matches, how you replace strings, etc.

Mathematica uses Perl’s regular expression flavor. But how do you use regular expressions in Mathematica? I’ll give a few tips here and give more details in the notes Regular expressions in Mathematica.

First of all, unlike Perl, Mathematica specifies regular expressions with ordinary strings. This means that metacharacters have to be doubly escaped. For example, to represent the regular expression d{4} you must use the string "\d{4}".

The function StringCases returns a list of all matches of a regular expression in a string. If you simply want to know whether there was a match, you can use the function StringFreeQ. However, note the you probably want the opposite of the return value from StringFreeQ because it returns whether a string does not contain a match.

By default, the function StringReplace replaces all matches of a regular expression with a given replacement pattern. You can limit the number of replacements it makes by specifying an addition argument.

Related links:

2010 calendar of lost mathematical art

Posted on by John

Rod Carvalho wrote a post this morning announcing a beautiful 2010 calendar created by Ron Doerfler. Doerfler’s blog is entitled Dead Reckonings: Lost Art in the Mathematical Sciences. The calendar is an example of such lost art. It is illustrated with nomograms, ingenious ways of computing with graphs before electronic calculators were common. The illustrations are pleasant to look at even if you have no idea what they mean.

Image via Ron Doerfler.

Related posts

Spherical trig is a lost art. Why care about spherical trig?

The Gudermannian function gd(x) is another interesting relic of an early time. It is closely related to the Mercator projection and shows how to relate ordinary and hyperbolic trig functions without using complex numbers.

The image above shows solutions to the equation u + v + w = uvw. Here’s a post explaining the significance of that equation.