## Archive of posts in 2023

### The duality between two plane trajectories related by a conformal map

The conformal map $\fc wz$ transforms the trajectory with energy $-B$ in potential $\fc Uz\ceq A\v{\d w/\d z}^2$ into the trajectory with energy $-A$ in potential $\fc Vw\ceq B\v{\d z/\d w}^2$. I will prove this beautiful result and show some implications of it.

- Categories: physics
- Tags: complex, classical mechanics, canonical transformation, kepler problem, mathematical physics, vector analysis, long paper

### Embed the latest Mastodon post in my website

I show the latest Mastodon post in my website hosted on GitHub Pages, and it does not require JavaScript on the client side. How do I do that?

- Categories: update
- Tags: web, jekyll, selfhosting

### Rotational symmetry of plane lattices as a simple example of algebraic number theory

For a plane lattice, there is only a finite number of different rotational symmetries that are compatible with the discrete translational symmetry. For example, the 5-fold rotational symmetry is not one of them. Why is that? It turns out that whether an $m$-fold symmetry is compatible with translational symmetry is the same as whether $\varphi(m)\le2$.

- Categories: math
- Tags: complex, condensed matter physics, algebraic number theory, lattice, mathematical physics

### I restructured my blog

If you have not noticed, the equations on my blog are now rendered server-side. This change makes it possible for those who turn off JavaScript on their browsers to see the equations. I also updated the theme from Minima v2 to Minima v3.

### An 🎃 easy 👻 but 🍬

*spooky*💀 Ruby challenge“It’s the weekend and you’ve just completed a seance with friends. After communing with the dead, you realize a mysterious message was left behind.” What is the decoded message? Use your Ruby skills to find out!

- Categories: programming
- Tags: ruby, code golf, fooling around

### You can replace $\mathrm i$ with $-\mathrm i$ in the Schrödinger equation?

When someone asks you why it is $-\mathrm i$ here instead of $\mathrm i$ or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.

- Categories: physics
- Tags: quantum mechanics, letter, complex

### How to self-host Overleaf (ShareLaTeX)

It has been a pain setting up Overleaf on my own server. I have finally figured it out and I am sharing my notes here.

- Categories: guide
- Tags: linux, selfhosting, tex

### My old computers and my experience of daily driving Linux and self-hosting

I have been using Linux and self-hosting for over a year now. Overall, I would like to say that it is rather satisfactory and that I will continue to use Linux as my daily driver in future. Thinking of that, I felt a little bit nostalgic about my old computers. So I decided to write an article about my old computers and my experience of daily driving Linux now.

- Categories: misc
- Tags: linux, selfhosting

### I created Sunniesnow, a rhythm game similar to Lyrica

In order to build a better player community for the rhythm game Lyrica, I created a new rhythm game called Sunniesnow. Learn more about Sunniesnow by joining our Discord server.

- Categories: update
- Tags: update, rhythm game, web

### The distribution when indistinguishable balls are put into boxes

Suppose there are $n$ distinguishable boxes and $k$ indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains $m$ balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit $n,k\to\infty$ with $k/n$ fixed, the distribution tends to be a geometric distribution.

- Categories: math
- Tags: probability, combinatorics, ruby