# Commutators

In quantum mechanics we assume the following:

1. Each observable is associated with an Hermitian operator of a Hilbert space H. Its eigenvalues must be real and the eigenstates are orthogonal to each other, thus form a set of basis of H.
2. Upon observation, one of the eigenvalues will be the quantity and the wave function will collapse onto one of the corresponding eigenstates.

Here I don’t want to discuss the deep insights, which I have no idea of. That’s why I take Copenhagen interpretation, so shut up and compute!

# Practical Theology

Those who know me in person may already know that I’ve become religious. The foundation of my belief comes from Christianity, Calvinist branch of Protestantism, to be specific. It was a spectacular psychological journey in which I did much philosophical and sci-fi pondering, and my basic understanding of life, universe and everything has altered a lot. Of course the faith didn’t come out from nowhere. I’ve repeatedly experienced certain kinds of revelation (facts that hard to reasoning via atheists’ perspective), as the lyric goes: “how precious did that grace appear, the hour I first believed …”

# Shannon–Hartley theorem

1. 其带宽为 $W$.
2. 其噪声为功率谱密度为 $N_0$ 的高斯白噪声
3. 信号的平均功率为 $P$.

# The Noisy-Channel Coding Theorem

## Definitions & Notations

• 用 $X, Y, Z$ 来表示随机变量，$\mathcal{A}_{X}$ 表示变量的取值集合，$X^N$ 则表示将 $N$ 个独立的 $X$ 组成的整体作为一个随机变量
• 信息熵： $H(X) = -\sum\limits_{p_i}\log{p_i}$
• 条件信息熵： $H(X|Y) = \sum\limits_{y \in \mathcal{A}_Y} P(y) H(X|Y=y) = - \sum\limits_{xy \in \mathcal{A}_X\mathcal{A}_Y} P(x, y) \log{P(x|y)}$，满足:

$H(X, Y) = H(X) + H(Y|X) = H(Y) + H(X|Y)$

• 互信息： $I(X; Y) \equiv H(X) - H(X|Y) = H(Y) - H(Y|X)$，度量了两个随机变量相互蕴含了多少关于对方的信息
• 信道容量： $C \equiv \underset{\mathcal{P}_X}{\mathrm{max}}\ I(X;Y)$，表示输出变量 $Y$ 最多能够蕴含多少关于输入 $X$ 的信息

# Style mod for GoodReader with vimperator

## The Pain Point

Like I’ve mentioned in other posts, most of my reading is done with digital materials. And the application that I use most often for that purpose is named GoodReader(IOS platform). There certainly are better alternatives, nevertheless, I’m too lazy(poor) to alter.

The old fashioned UI design of the app seems a little bit complicated and confusing, however the front-end page for file transfer via a WLAN is too simple to be satisfying.

• no design not all
• progressing info is too inconspicuous

# Hammersley Clifford Theorem

## Preface

PRML 中 8.3.2 小节简单描述了 Markov Random Fields 的分解特性，其中最核心的部分就是 Hammersley Clifford Theorem, 然而它并没有证明这个定理，只是在末尾的时候提到了这个结论，导致我在阅读中间部分的时候一头雾水。好在我 google 到了一个优雅的证明，顺便翻译在此。

# Add TOC indexes to PDF

## PDF books without Indexes are like Planets with Individuals

I prefer e-books to paper books, actually I may have read only 1-2 paper books in recent 3 years, while e-books of a much larger amount. The pros of e-books are really obvious:

1. portability
2. context can be copied
3. easy to manage
4. easy to navigate

I don’t want to argue about the superiorities here. I just wanna to convey that the 4th feature is a crucial one, and it is mainly achieved via indexes, bookmarks, reference links, etc. There’s no doubt that among those approaches, indexes play the most significant role. So I say that e-books without indexes are like planets with individuals, i.e. huge gaps among separated pieces of text.