The terms covariant and contravariant come up in many contexts. An earlier post discussed how the terms are used in programming and category theory. The meaning in programming is an instance of the general use in category theory. Vector fields…

The terms covariant and contravariant come up in many contexts. An earlier post discussed how the terms are used in programming and category theory. The meaning in programming is an instance of the general use in category theory. Vector fields…

I started a discussion on Twitter this evening about consulting company names. Here are some of the names. Turing Machine Computing: If we can’t do it, it can’t be done. Heisenberg Consulting: You can have speed or quality, but not…

This morning I ran across Pulp-o-mizer and decided my series of posts on mechanical vibrations could use a little sensational promotion. The posts are Part I: Introduction and free, undamped vibrations. Part II: Free, damped vibrations (under-damping, critical damping, over-damping)…

This is the final post in a four-part series on vibrating systems. The first three parts were free, undamped vibrations free, damped vibrations driven, undamped vibrations and now we consider driven, damped vibrations. We are looking at the equation m…

Pick a number x between 0 and 1. Then repeatedly replace x with 4x(1-x). For almost all starting values of x, the result exhibits chaos. Two people could play this game with starting values very close together, and eventually their…

Rachel Kroll wrote a blog post last Friday entitled I mortgaged my future with a Mac. The part I found most interesting is near the end of the post. Instead of staying with my wonky-but-free ways of doing things, I…

Posted in Computing

Here’s an amusing sentence I ran across this morning: The code was simplified in 2003 and is harder to understand. Maybe the author is being sarcastic, but I doubt it. I believe he means something like this: In 2003, the…

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Toward the end of his life, Beethoven added metronome markings to the scores of his symphonies to indicate exactly how fast they should be performed. The tempos indicated in the scores are consistently faster than how the symphonies are usually…

This is the third in a series of four blog posts on mechanical vibrations. The first two posts were Part I: Introduction and free undamped vibrations Part II: Free undamped vibrations

From the preface to Physics for Mathematicians: In addition to presenting the advanced physics, which mathematicians find so easy, I also want to explore the workings of elementary physics, and mysterious maneuvers — which physicists seem to find so natural…

Can regular expressions parse HTML? There are several answers to that question, both theoretical and practical. First, let’s look at theoretical answers. When programmers first learn about regular expressions, they often try to use them on HTML. Then someone wise…

This is the second post in a series on vibrations determine by the equation m u” + γ u’ + k u = F cos ωt The first post in the series looked at the simplest case, γ = 0…

Stéfan van der Walt wrote some Python code to animate the system described in yesterday’s post on mechanical vibrations. Stéfan posted his code on github. It currently illustrates undamped free vibrations, but could be modified to work with damped or…

I always thought that Fourier was the first to come up with the idea of expressing general functions as infinite sums of sines and cosines. Apparently this isn’t true. The idea that various functions can be described in terms of…

I’ve just started reading Invisible in the Storm: The Role of Mathematics in Understanding Weather. The subtitle may be a little misleading. There is a fair amount of math in the book, but the ratio of history to math is…