Mathematics

I recently got interested in singular perturbation theory , and to get help with the algebra, I turned to SymPy. I had tried to use Mathematica in graduate school in the early 90s, but the experience had been sufficiently frustrating that I had steered clear of computer algebra systems since.

SymPy presents itself as a “friendly”, less intimidating alternative, with a more familiar and conventional language and operating model.

Formal perturbation theory, and in particular singular perturbation theory, are standard topics in applied mathematics, but seem to be largely unknown outside that specific community. Which is a shame, because they are both useful, and intellectually fascinating.

Perturbation expansions, in particular for higher orders, have a reputation for being “cumbersome”, as the say: the algebra quickly becomes both tedious and error-prone. Which is true, but it turns out that the SymPy computer-algebra system can be put to good use in this context.

The functional equation of the logarithm is well-known:

\[ \log(xy) = \log(x) + \log(y) \]

But why is this so?

When I discovered the first edition of Carl D. Meyer’s book on “Matrix Analysis and Applied Linear Algebra”, I considered it the best book on Linear Algebra I was aware of — and surely the best available book for an application-minded reader. In short: the book I really wished had been available when I was a graduate student and encountering this material for the first time.

So it was with great excitement and curiosity when I saw that a new, second edition was available. How could a near-perfect book be made even better?

The proof that $\sqrt{2}$ is irrational is part of the standard high-school curriculum. The same cannot be said for the proof that $e$, the base of the natural logarithm, is irrational as well. Yet the proof is short, simple, elegant.

Every once in a while, I need to evaluate the normal distribution function $\Phi(x)$:

$$ \Phi(x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x \! e^{-\frac{1}{2}t^2} \, dt $$

Unfortunately, it is not always available in the standard math libraries, and hence I have to implement a “good-enough” version myself. Here are some options.

Let $G$ be a finite group with $n$ elements, and $H$ a subgroup of $G$ with $m$ elements. Then $m$ is a divisor of $n$: for a finite group, the order of any subgroup divides the order of the group.

In my previous post on one-dimensional heat flow, I encountered sums of the form:

$$ \sum_{n=0}^\infty \frac{\pm 1}{2n+1} \exp \left( - (2n+1)^2 x \right) $$

A plot (involving the first 1000 terms) is shown below, and looks reasonable enough. Is this curve, which forms the limit of the series, a known function?

Imagine a rod that is initially at temperature $T_1$ and then brought into an environment with a lower temperature $T_0 < T_1$. How quickly does the body cool down? When will it have reached the environment’s temperature? What is the temperature profile throughout the rod, as a function of time?

This is essentially a worked homework set: a complete, step-by-step solution of the diffusion (or heat) equation in one dimension.

The (notorious) Dirac delta-function is usually introduced as a function with the following properties: \[ \delta(x) = \begin{cases} 0 & \qquad x \neq 0 \\ \text{“$\infty$”} & \qquad x = 0 \end{cases} \]