# Pelican Signals

Pelican的插件系统是使用blinkersignal实现的。Pelican所有可以用的signals可以在signals.py找到。本文的目的是记录这些signals是在Pelican运行中什么时候发出的。

(1) Pelican有一个叫做Pelican的类，含有程序的主体框架。当Pelican的一个实例pelican初始化完成之后（基本设置，加载插件），发出第一个signal。

signals.initialized.send(pelican)


# Lambda Calculus

This post is my note for What is seminar on Lambda Calculus.

Lambda calculus was created by Alonzo Church in the 1930s, and was used by him to solve Entscheidungsproblem in 1936, which is related to Hilbert's tenth problem. In the same year, Alan Turing independently solved Entscheidungsproblem using his invention Turing machine. Shortly after, Turing realized that these two models are actually equivalent as models of computation.

In this note, I will first give the formal definition of lambda expressions. Then with the help of Python, I am going to show how to do Boolean algebra and basic arithmetic using lambda calculus, which to some extend gives an illustration that Turing machine and lambda calculus are equivalent.

## Definition

Lambda calculus consists of lambda expressions and operations on them. There are three basic elements in Lambda expression:

1. variables: x, y, z, ...
2. symbols in abstraction: λ and .
3. parentheses for association: ()

# Cassini Ovals

This post is my note for What is seminar on Cassini Ovals.

### Definition

An ellipse is a geometric object formed by locus of points which have fixed sum of distances to two fixes foci.

In the above figure, we can express the definition of ellipses in one simple equation:

$$|PF_1|+|PF_2|=c,$$

for some $c>0$. What if we change the addition in the above equation to multiplication? This comes the definition of Cassini ovals.

This post is my note for What is…? seminar on Rademacher function.

For a real number $x$ in $[0, 1]$, let $(a_1(x), a_2(x), \cdots, a_n(x), \cdots)$ be its binary representation. That is to say, $$$x = \sum_{n=1}^\infty \frac{a_n(x)}{2^n}. \label{20170623-1}$$$

For some $x$, there might be two possible binary representation. Take $\frac{1}{2}$ for example, it can be represented as $(1, 0, 0, \cdots)$ or $(0, 1, 1, \cdots)$. In this situation, we always prefer the former representation, in which all terms become $0$ eventually.

Definition 1. Let $n \ge 1$ be an integer. The $n$-th Rademacher function $r_n: [0, 1] \to \{-1,1\}$ is defined to be $r_n(x) = 1 - 2a_n(x).$

# Lüroth expansion

This post is my note for What is…? seminar on Lüroth expansion.

Definition 1. A Lüroth expansion of a real number $x \in (0, 1]$ is a (possibly) infinite sequence $(a_1, a_2, \cdots, a_n, \cdots)$ with $a_n \in \N$ and $a_n \ge 2$ for all $n \ge 1$ such that $$$x = \frac{1}{a_1} + \frac{1}{a_1(a_1-1)a_2} + \cdots + \frac{1}{a_1(a_1-1)\cdots a_{n-1}(a_{n-1}-1)a_n} + \cdots$$$

By abuse of notation, we will something write the right hand side of Definition 1 as $(a_1, a_2, \cdots, a_n, \cdots)$.

Given a system of representation of real numbers in $(0, 1]$, we need to determine whether these representations are 1 to 1 corresponding the numbers in $(0, 1]$. Aslo we should study the (Lüroth) shift map: $$$T: (a_1, a_2, \cdots, a_n, \cdots) \mapsto (a_2, \cdots, a_n, \cdots). \label{20170620-2}$$$

# 编写Hexo插件让主页显示特定的文章

(I) 让主页只显示特定文章（或者说过滤掉特定文章）。

(II) 建立一个页面只显示特定文章。

# Cabaret

Cabaret是一部音乐剧，一部充满黄段子的音乐剧。Cabaret一开幕我就明白朋友路上说的少儿不宜是什么意思了。虽然我并不能完全明白剧中的对话（估计错过了好多黄段子 😛 ），但是剧中的大致剧情还是明白的。这里我就不剧透了，有兴趣的朋友也很容易在网上找到简介。一开始根据剧中反映的性泛滥以及对金钱的崇拜，我还以为是说战后经济大低谷时期的事情。后来剧中出现了纳粹红袖章我才意识到自己搞错了。原来剧中的人们正活着1928～1930期间，正是纳粹主义开始猖獗的时候。

# Idempotented algebra

This post is a reading note to Bernstein’s Le “centre” de Bernstein, on the part of idempotented algebras (section 1.1 to 1.7, ibid.). Let $R$ be a ring, an element $e$ is called an idempotent if $e^2=e$. Let $k$ be a field.

Definition 1. A $k$-idempotented algebra $(H, I)$ is a $k$-algebra $H$ together with the set idempotents $I$ satisfying

(*) For any finite family $(x_i)$, there exists an idempotent $e \in I$ such that $ex_ie = x_i$ for all $i$.

The $k$-structures of $H$ is not involved in the discussion of this post, so we will ignore it. Let $(H, I)$ be an idempotented algebra for the rest of the post.

Lemma 2. Let $e, f \in I$. The following are equivalent.

(1) $eHe \subset fHf$
(2) $e \in fHf$
(3) $e = fef$