Warning You need to know some basic group theory terminology to appreciate(I hope you do) the following content. Burnside’s lemma Here is the statement of the Burnside’s lemma: Let $G$ be a group that actions on a set $X$. Denote $X^{g}$ the set of fixed points of $g$ i.e. $\{x \in X | g \cdot x = x\}$, then the number of orbits of the action is equal to $\dfrac{1}{|G|}\sum\limits_{g \in G} |X^g|$....
Problem Let $I_{m} = \int^{2\pi}_{0} \cos(x)\cos(2x)\dots\cos(mx) dx$. For $m$ in $1, 2, \dots, 10$, for which $m$ is $I_m \neq 0$ ? Solution $I_m$ is non-zero if and only if $m \equiv 0, 3 \pmod{4}$. Main idea The official(seemingly? It is in the putnam problem book and a few solutions I found online do the same.) solution is to substitute $\cos x = \frac{e^{ix} + e^{-ix}}{2}$ followed by grouping the terms into $\cos x \cos (2x) \dots \cos (mx) = e^{\text{something}}$, and analyze the something....
Info Updates on 20250408: Fixed a bug in parsing memory operand. Improved error message displaying. I happen to have to learn (just a bit of) RISC-V assembly, so I made a simple risc-v instructions explainer so that I don’t have to do multiple google searches whenever I have troubles in reading some risc-v instructions. It can fully run in the browser, feel free to try it out at https://katsuragicsl.github.io/riscv-explainer/.
Info Updates: After giving a second thought on the topic and reorganizing the materials, I had a sharing session with my teammates and decided to update this article accordingly. Updates include more suitable examples and graphics. Info Updates 2: It is revised again and published as a preprint. Now you can see it at https://arxiv.org/abs/2503.17813 or https://katsuragicsl.github.io/papers/connectedness/ An empty business lingo or a good quantification? We hope to, and probably need to, quantify the severity of security bugs....
After the huge attention gained by deepseek R1, I have seen a good amount of people immediately went to looking for censorships (which are expected to exist), laughed at it, and looked down on it. They totally missed the points and these behaviors made them look like slaves of their very own defense mechanism1. Deepseek did make some techonological breakthroughs. Indeed the censorship is a defect of such a great product....