Mathematics

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: \[ \delta(x) = \begin{cases} 0 & \qquad x \neq 0 \\ \text{“$\infty$”} & \qquad x = 0 \end{cases} \]

The “Newsvendor Problem” is a classic problem in inventory and supply-chain management: how much product to carry in stock in the face of uncertain demand?

The problem is obviously of interest in its own right, but it is also an archetypical problem, meaning that variations of it arise frequently and in different contexts. It is therefore valuable to know “how to think about” this kind of problem; in particular, since in its simplest form, it has a closed-form, analytic solution.

The Altitude Theorem or Geometric Mean Theorem is a result from high-school geometry. In a right triangle, the altitude $h$ on the hypotenuse divides the hypotenuse into two segments, $p$ and $q$. The theorem now states that $h^2 = pq$ or, equivalently, $h = \sqrt{pq}$: the altitude equals the geometric mean of the segments of the hypotenuse.

The content of this theorem is a bit surprising, because the altitude and the hypotenuse segments seem geometrically somewhat “unrelated”: it’s not clear how one could (geometrically) be transformed into the other. And although the theorem can be proven in many different ways, many of the proofs are at least partially algebraic, and therefore do not provide an intuitive, geometric sense why it is true. But it turns out that a very elegant, strictly geometric proof of this proposition can be constructed.