Lagrange’s four-square theorem is a beautiful result in number theory. However, to the best of my knowledge, I haven’t encountered it often in theoretical computer science. Today’s entry discusses two interesting stories related to this theorem.
Gee Law’s Blog
姚栋想要知道如何在 Windows 中使用命令解压缩 ZIP 文件夹。他表示如果要用 .NET Framework 提供的 API，则需要 4.5，但环境要求 4.0；如果用 PowerShell Expand-Archive 命令，则需要 5.1，但环境只允许 2.0；想用 cmd，但不知道怎么做。然而应该时刻记住：Windows 自带的 GUI 实际上经常是 COM 的图形版，在 COM 中寻找几乎总是可以找到你想要的图形操作的命令版本。
Starting from the classic puzzles of finding patterns in a sequence, this entry explores the framework to define persuasive patterns. The framework is found to be quite self-contained in the sense that it is ‘asymptotically invariant’ to the choice of ‘language of expression’. The only short-coming of this framework is that it works only for computable sequences, yet the process of pattern discovery is incomputable.
In this entry, I discuss a pathological construction of general terms of a class of integer-valued sequences with the widely accepted concept of ‘elementary functions’. The idea here is to simply ‘concatenate’ the integers in a real number, then extract the appropriate digits for each term. It turns out that this definition characterises the elementariness very well (note that the definition is also self-referencing). The extended inspection leaves a problem open: Are all integer-valued sequences elementary? Update: The question is solved with an affirmative answer.
In this entry, I discuss a kind of widely posed exam problems for elementary calculus learners. It concerns a technique of application of (differential) mean value theorems. Speficially, a class of such problems can be solved dogmatically if the equation to prove resembles a ‘factorisable’ linear differential equation.
I allowed myself some CHANEL products to ‘reward’ myself for the acceptance of my first cryptography paper. I also consider spending money at its best value an interesting game. A commonly-seen discount aux Galeries Lafayette Xidan (Beijing) leads to an obviously NP-complete problem. Haha!
An algorithmic imagination of how Sina Weibo could implement Frequented Visitees, a feature that gives users the serveral accounts which they visit the most often.