# Mathematical Diversions

A collection of mathematical miscellany that diverts me from my main research interests. Some of the posts are interesting (to me) mathematical problems with their solutions. Others are mostly recreational things that distracted me at one time or another. The intended audience for posts in this category are my sole self, and so I make no great effort to polish the writing. Do send me an email with questions if you find anything here interesting or confusing.

### Joris' Theorem

A proof is given of Joris' theorem, which states that if two coprime powers of a function are both smooth, then so is the original …

### When are curves tangent to hypersurfaces?

Given a curve and a hypersurface, both in Euclidean space, how can we tell if the curve is tangent to the hypersurface?

### Torsion, curvature, and parallel transport

An expository note on the notions of curvature and torsion of a general linear connection on the tangent bundle.

### Simulating Closed Cosmic Strings

One of my current research interests is in the geometry of constant-mean-curvature time-like submanifolds of Minkowski space. A special …

### Shooting particles with Python

Numpy simulation of how classical and quantum particles interact with potential barriers.

### Purely kinetic initial data for Schwarzschild

Can initial data for Einstein's equation be specified in such a way that all the 'energy' are kinetic and not potential?

### Repeating Decimals

When does the decimal expansion of a number end with one digit repeated?

### Where Does The π Go

It is fairly well known that experts in partial differential equations and experts in harmonic analysis prefer to define the Fourier …

### Symmetry and Uniformization

A bit of trivia (can't think of any use of it now) Theorem   Let $(M,g)$ be a two-dimensional compact Riemannian manifold with …

### A Little Hilbert Space Problem

First let us consider the following question on a finite dimensional vector space. Let $(V, \langle\cdot\rangle)$ be a $k$-dimensional …