## Laplace's Theorem

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.

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.

Analytic number theory is the application of methods from analysis to the study of integers, in particular primes. This may seem paradoxical: at the heart of analysis lie notions of continuity and differentiability — and what could be more discrete and discontinuous than the set of primes?

Everyone knows Euler’s famous identity, linking the imaginary unit to trigonometric functions:

$$ \mathrm{e}^{\mathrm{i} \pi} = -1 $$

But another remarkable identity is also due to Euler, this one linking the set of prime numbers to an analytical function:

Linear Algebra is one of the foundational topics for all applied mathematics. But compared to Analysis, it initially often feels stranger and less familiar. Although technically not hard, the level of abstraction is higher, making it hard to see what all the formalism is supposed to achieve.