
This is a note on Markov Chain.

## Stochastic Process

A **stochastic process** is a collection of random variables $X_t$ indexed by time $t$.

## Markov Property

A stochastic process $\{X_t\}$ has the **Markov property** if for any $n \geq 0$, any $t_0 < t_1 < \cdots < t_n$, and any $B \in \mathcal{B}$, we have

$$
P(X_{t_{n+1}} \in B | X_{t_0}, X_{t_1}, \ldots, X_{t_n}) = P(X_{t_{n+1}} \in B | X_{t_n}).
$$

## Markov Chain

A **Markov chain** is a stochastic process with the Markov property.

### Transition Probability

The **transition probability** at time $n$ of a Markov chain is defined as

$$
P_{ij} = P(X_{n+1} = j | X_n = i).
$$

### Transition Probability Matrix

The **transition probability matrix** at time $n$ of a Markov chain is a matrix $P = (P_{ij})$ where $P_{ij}$ is the transition probability from state $i$ to state $j$.

### n-Step Transition Probability

The **n-step transition probability** from time $m$ to $n+m$ of a Markov chain is defined as

$$
P_{ij}^{(n)} = P(X_{n+m} = j | X_m = i).
$$

### Time-Homogeneous Markov Chain

A Markov chain is **time-homogeneous** if the transition probabilities are independent of time.

> In this note, we focus on time-homogeneous Markov chains.

## Chapman-Kolmogorov Equation

The **Chapman-Kolmogorov equation** states that for a time-homogeneous Markov chain, the $(n+m)\text{-step}$ transition probability is given by

$$
P_{ij}^{(n+m)} = \sum_{k} P_{ik}^{(n)} P_{kj}^{(m)}.
$$

## Hitting Probability

The **hitting probability** of a state $j$ starting from state $i$ is defined as

$$
f_{ij} = P(\text{state } j \text{ is hit starting from state } i)
$$

By intuition, the hitting probability could be calculated by

$$
f_{ij} = \sum_{n=1}^{\infty} P(\text{state } j \text{ is first hit at time } n | X_0 = i).
$$

Thus, we have the following definition.

### First-Visit(Hitting) Probability

The **first-visit probability** of a state $j$ starting from state $i$ is defined as

$$
f_{ij}^{(n)} = P(\text{state } j \text{ is first hit at time } n | X_0 = i).
$$

#### Convergence of First-Visit Probability

The first-visit probability will always converge to $0$ as $n$ goes to infinity.

Proof:

If this is not the case, then the sum of hitting probabilities will be greater than $1$, which is a contradiction.

### Hitting Time for a Given Path

The **hitting time** of a state $j$ to state $i$ for a given path $i_0, i_1, \ldots$ is defined as

$$
T_{j}^i = \min\{n \geq 1: X_n = i | X_0 = j\}
$$

Sometimes we use $T^i$ if it is clear that the starting state is $j$.

### Mean Hitting Time (Mean First-Visit Time)

The **average hitting time** of a state $j$ to state $i$ is defined as

$$
\beta_{j}^i = E[T_{j}^i | X_0 = j].
$$

## Communication

We say that the state $i$ communicates with state $j$ if

$$
P_{ij}^{(n)} > 0 \text{ for some } n \geq 0
$$

and is denoted by $i \rightarrow j$.

### Inter-Communication

Two states $i$ and $j$ are said to **inter-communicate** if $i \rightarrow j$ and $j \rightarrow i$.

### Irreducible Class of States

A set of states $C$ is said to be an **irreducible class of states** if any state in $C$ communicates with any other state in $C$.

### Closed Class of States

A set of states $C$ is said to be a **closed class of states** if any state in $C$ communicates only with other states in $C$.

### Absorbing State

A state $i$ is said to be an **absorbing state** if itself forms a closed class of states.

## Recurrence

### Recurrent and Transient State

A state $i$ is said to be **recurrent** if the hitting probability

$$
f_{ii} = 1.
$$

A state $i$ is said to be **transient** if it is not recurrent.

### Recurrent and Transient Class

A class of states $C$ is said to be **recurrent** if any state in $C$ is recurrent.

A class of states $C$ is said to be **transient** if any state in $C$ is transient.

### Two Important Theorems

1. A state $i$ is recurrent if and only if

$$
\sum_{n=1}^{\infty} P_{ii}^{(n)} = \infty.
$$

Proof:

We define the following generating function:

$$
f(s) = \sum_{n=1}^{\infty} f_{ii}^{(n)} s^n
$$

Then, the probability of hitting state $i$ at time $n$ for the k-th time is the coefficient of $s^n$ in the expansion of $f(s)^k$.

Thus, we the transition probability:

$$
P_{ii}^{(n)} = \sum_{k=1}^{n} P(\text{hitting }i\text{ at time }n\text{ for the }k\text{-th time})
$$

which should equal to the coefficient of $s^n$ in the expansion of $\sum_{k=1}^{n}f^k(s)$.

As the coefficient of $s^i$ for $i<k$ in $f(s)^k$ is 0, we have

$$
\begin{align}
    P_{ii}^{(n)} &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n} f^k(s) \\
    &= \text{coefficient of }s^n\text{ in }f^{n+1}(s) + \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n} f^k(s) \\
    &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{n+1} f^k(s) \text{(by repeating similar steps)} \\
    &= \text{coefficient of }s^n\text{ in }\sum_{k=1}^{\infty} f^k(s)
\end{align}
$$

Thus, we have

$$
\sum_{n=1}^{\infty} P_{ii}^{(n)} = \sum_{n=1}^{\infty} f^k(1)
$$

If $i$ is recurrent, then $f(1) = 1$ the previous sum diverges.

If $i$ is transient, then $f(1) < 1$ and the previous sum is a sum of a geometric series, which converges.

2. If $i$ is transient, then for all $j$

$$
P_{ji}^{(n)} \rightarrow 0 \text{ as } n \rightarrow \infty.
$$

Proof:

If the condition does not satisfy,
we assume that $a=\lim_{n\rightarrow\infty}P_{ji}^{(n)}$,
and as $i$ is transient,
let $\alpha = f_{ii}<1$.
We choose $\epsilon$, such that $\frac{a(1-\alpha)}{3+\alpha} > \epsilon >0$.
Then there exist $N$ such that for all $n\ge N$, $a-\epsilon<P_{ji}^{(n)}<a+\epsilon$,
and $f_{ji}^{(n)} < \epsilon$ and $\sum_{k=1}^{n}f_{ii}^{(n)} > \alpha-\epsilon$

For $n>2N+1$, we have

$$
\begin{align}
    P_{ji}^{(n)} &= f_{ji}^{(n)} + \sum_{k=1}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\
    &\le \epsilon+\sum_{k=1}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\
    &\le \epsilon+\epsilon+\sum_{k=N}^{n-1}P_{ji}^{k}f_{ii}^{n-k} \\
    &\le 2\epsilon+\sum_{k=N}^{n-1}(a+\epsilon)f_{ii}^{n-k} \\
    &= 2\epsilon+(a+\epsilon)\sum_{k=N}^{n-1}f_{ii}^{n-k} \\
    &\le 2\epsilon + (a+\epsilon)\alpha \\
    &< a-\epsilon
\end{align}
$$

which contradicts the assumption.

### Finite State Space Must have Recurrent State

If the state space is finite, then there must be at least one recurrent state.

Proof:

If all state are transient,
then by the second theorem in [previous section](#two-important-theorems),
we have $P_{ji}^{(n)} \rightarrow 0$ as $n \rightarrow \infty$ for all $i$ and $j$,
thus $\sum_{i\in S} P_{ji}^{(n)}$ tends to 0 as $n$ goes to infinity.

However, $\sum_{i\in S} P_{ji}^{(n)}$ is constantly 1 as it is the sum of the transition probability of state $j$ to all other states,
which is a contradiction.

### Intercommunication and Recurrence

If two states intercommunicate, then they are either both recurrent or both transient.

Proof:

Given two states $i$ and $j$ that intercommunicate.

Suppose $i$ is recurrent,
then

$$
\sum_{n=1}^{\infty} P_{ii}^{(n)} = \infty
$$

As $i$ and $j$ intercommunicate,
there either exists $n$ such that $P_{ij}^{(n)} > 0$,
and $m$ such that $P_{ji}^{(m)} > 0$,

Then we have

$$
P_{jj}^{(n+k+m)} \ge P_{ij}^{(n)}P_{jj}^{(k)}P_{ji}^{(m)}
$$

Thus,

$$
\begin{align}
    \sum_{\alpha=1}^{\infty} P_{jj}^{(\alpha)} &\ge 
    \sum_{\alpha=n+m+1}^{\infty} P_{jj}^{(\alpha)} \\
    &\ge \sum_{\alpha=1}^{\infty} P_{ij}^{(n)}P_{jj}^{(\alpha)}P_{ji}^{(m)} \\
    &= P_{ij}^{(n)}P_{ji}^{(m)}\sum_{\alpha=1}^{\infty} P_{jj}^{(\alpha)}
\end{align}
$$

which is infinite as the sum of $P_{jj}^{(\alpha)}$ is infinite.

Then, $j$ is recurrent.

### Number of Visits to a State for a Given Path

The number of visits to a state $j$ for a given path $i_0, i_1, \ldots$ is defined as

$$
N_j = \sum_{n=1}^{\infty} I(X_n = j).
$$

#### Number of Visits at Infinity

$$
\begin{align}
    P(N_j = \infty) = 1 &\text{ if } j \text{ is recurrent} \\
    P(N_j = \infty) = 0 &\text{ if } j \text{ is transient}
\end{align}
$$

### Decomposition of State Space

The state space can be decomposed into a set of Irreducible Closed Recurrent classes
and a set of Transient classes.

Proof:

Given state space $I$,
as the InterCommunicate relation is an equivalence relation,
we have a partition of $I$ into Inter Communicate classes.
Let the classes be $I_{\alpha}, \alpha \in A$.

As all states in an InnerCommunicate class have the same Recurrence,
let

$$
\begin{equation}
B = \{\alpha \in A | I_{\alpha} \text{ is Transient}\}
\end{equation}
$$

Let $C = A - B$.
Then, for all $\alpha \in C$, $I_{\alpha}$ is Recurrent.

We then prove that $I_{\alpha}$ is Closed for all $\alpha \in C$.,
which finished the proof of this theorem.

If $I_{\alpha}$ is not closed,
then there exists $i \in I_{\alpha}$ and $j \notin I_{\alpha}$,
and $n \in \mathbb{N}$
such that $P_{ij}^{(n)} > 0$.

Also, as $i \in I_{\alpha}$, $i$ is Recurrent, and thus

$$
\begin{equation}
f_{ii} = 1
\end{equation}
$$

Also, in this case, $j$ can not Communicate with $i$,
as if it does, $j$ will be in $I_{\alpha}$,
which means

$$
\begin{equation}
f_{ji} = 0
\end{equation}
$$

Then we have

$$
\begin{align*}
1 &= f_{ii} \\
&= \sum_{k\in I} P_{ik}^{(n)} f_{ki} \\
&= P_{ij}^{(n)}*f_{ji} + \sum_{k\in I, k\neq j} P_{ik}^{(n)} f_{ki} \\
&= \sum_{k\in I, k\neq j} P_{ik}^{(n)} f_{ki} \\
&\le \sum_{k\in I, k\neq j} P_{ik}^{(n)} \\
&= 1- P_{ij}^{(n)} \\
&< 1
\end{align*}
$$

This is a contradiction. Thus, $I_{\alpha}$ is closed.

### Mean Recurrence Time

The **mean recurrence time** of a state $i$ is defined as

$$
\mu_i = E[T_i^i | X_0 = i] = \sum_{n=1}^{\infty} n f_{ii}^{(n)}.
$$

#### Mean Recurrence Time for a Transient State

The mean recurrence time for a transient state is infinite.

#### Positive and Null Recurrent State

A state $i$ is said to be **positive recurrent** if the mean recurrence time:

$$
\mu_i < \infty.
$$

A state $i$ is said to be **null recurrent** if the mean recurrence time:

$$
\mu_i = \infty.
$$

#### Identification of Positive and Null Recurrent State

A null recurrent state has similar properties to a transient state:

A recurrent state is null recurrent if and only if

$$
\lim_{n\rightarrow \infty}P_{ii}^{(n)} = 0
$$

#### State in the Same Irreducible Class have the Same Recurrence (Positive or Null)

If two states are in the same irreducible class, then they are either both positive recurrent or both null recurrent.

#### Finite State Space Must have Positive Recurrent State

If the state space is finite, then there must be at least one positive recurrent state.

## Periodicity

### Period of a State

The **period** of a state $i$ is defined as

$$
d_i = \gcd\{n \geq 1: P_{ii}^{(n)} > 0\}
$$

If $d_{i}$ is $1$, then the state is said to be **aperiodic**.

where $\gcd$ is the greatest common divisor.

### Period of a Class

All states in the same irreducible class have the same period.

## Stationary Distribution

### Stationary Distribution

A distribution $\pi = (\pi_1, \pi_2, \ldots)$ is said to be a **stationary distribution** of a Markov chain if

$$
\pi = \pi P
$$

where $P$ is the transition probability matrix.

### Existence and Uniqueness of Stationary Distribution

If a Markov chain is irreducible and positive recurrent, then it has a unique stationary distribution.

And the stationary distribution is given by

$$
\pi_i = \frac{1}{\mu_i}
$$

where $\mu_i$ is the mean recurrence time of state $i$.

### Ergodic Markov Chain

A Markov chain is said to be **ergodic** if it is irreducible, positive recurrent and aperiodic.

#### Limiting Distribution

Given an ergodic Markov chain, and any vector $\pi^{(0)}$,

The limit:

$$
\lim_{n\rightarrow \infty} \pi^{(0)} P^n = \pi
$$

where $\pi$ is the stationary distribution.

## Discrete Renewal Process

Define a sequence of independent distributed random variables $\{X_n\}$ with positive integer state space.

Define a stochastic process $\{T_n\}$ as

$$
T_n = T_{n-1} + X_n
$$

where $T_0 = 0$.

Then, $\{T_n\}$ is a **discrete renewal process**.

### Other Ways to Define Discrete Renewal Process

This first definition is defined by the **inter-arrival time**.
We can directly define the process by the **renewal time $T_n$**, which are not independent.

We can also define the process by assign each time $n$ a **renewal probability** $u_n$,
which is the probability that the process is renewed at time $n$.

Or we can define the process by the **number of renewal** $N_n$,
which is the number of renewals in the time interval $[0, n]$.

### Mean of Number of Renewals

If we are given the renewal probability $u_n$,
then the mean of the number of renewals in the time interval $[0, n]$ is

$$
E[N_n] = \sum_{k=1}^{n} u_k.
$$

### Renewal Process and Markov Chain

If we are given a Markov chain with state space $S = \{1, 2, \ldots\}$,
and we set $Y_0 = 1$,
and we said the process is renewed whenever it hits state $1$,
then the process is a discrete renewal process.

And the renewal probability is given by

$$
u_n = P(\text{the process is renewed at time } n) = P(Y_n = 1|Y_0=1) = P_{11}^{(n)}
$$

#### Inter-renewal Distribution of Discrete Renewal Process Defined by Markov Chain

The inter-renewal distribution of the discrete renewal process defined by a Markov chain is given by

$$
f_k = P(\text{the precess first reach }1\text{ at } k | Y_0 = 1) = f_{11}^{(k)}
$$

which is the first renewal probability.

#### The Limit Renewal Theorem of Ergodic Markov Chain

Given an ergodic Markov chain with state space $S = \{1, 2, \ldots\}$,
and we set $Y_0 = 1$,
and we said the process is renewed whenever it hits state $1$,
and we set the number of renewals in the time interval $[0, n]$ as $N_n$,
then

$$
\lim_{n\rightarrow \infty} \frac{E[N_n]}{n} = \frac{1}{E[T_1]}
$$

where $E[T_1]$ is the mean recurrence time of state $1$.

## Branching Process

### Offspring Distribution

An **offspring distribution** is a probability distribution of the number of offspring of each individual.

### Discrete Branching Process

A **discrete branching process** with offspring distribution $Z$ is a stochastic process $\{X_n\}$ defined as

$$
X_n = \sum_{i=1}^{X_{n-1}} Z
$$

And $X_n$ is called the size of the nth generation.

### The Probability Generating Function of Discrete Branching Process

The **probability generating function** of a discrete branching process is defined as

$$
F(s) = E[s^{Z}]
$$

where $Z$ is the offspring distribution.

### Mean Generation Size

If we are starting from $X_{0} = 1$,
then the mean generation size of the nth generation is given by

$$
E[X_n] = E[Z]^n
$$

where $E[Z]$ is the mean of the offspring distribution.

### Total Progeny

The total progeny of the branching process is defined as
the total number of individuals that ever exist in the process.

$$
T = \sum_{n=0}^{\infty} X_n
$$

#### Probability Generating Function of Total Progeny

Let $G(s)$ be the probability generating function of the total progeny $T$,
then

$$
G(s) = sF(G(s))
$$

### Extinction

We say that the branching process is **extinct** if $X_n = 0$ for some $n$.

#### Probability of Extinction

The probability of extinction is given by the ever visit probability of state $0$ starting from state $1$.

$$
\gamma = f_{10} = P(\text{extinction}) = \sum_{n=1}^{\infty} f_{10}^{(n)}
$$

##### Calculation of Probability of Extinction

The probability of extinction is the least non-negative solution of the equation

$$
s = F(s)
$$

## Random Walk

### Simple Random Walk

A **simple random walk** is a stochastic process $\{X_n\}$ defined as

$$
X_n = X_{n-1} + Y_n
$$

where $Y_n$ is a sequence of independent and identically distributed random variables.

In this note, we only focusing on $Y$ to be a Bernoulli random variable with

$$
P(Y = 1) = p, P(Y = -1) = 1 - p
$$

#### Symmetric Random Walk

If $p = 0.5$, then the random walk is said to be **symmetric**.

#### Number of Possible Path

The number of possible paths of a simple random walk from $(0,i)$ to $(n,j)$ is denoted as $N_n(i,j)$.

#### Crossing Condition

Let the number $N_{n}^{m}(i,j)$ denote the number of possible paths of a simple random walk from $(0,i)$ to $(n,j)$ that is at $(k,m)$ for some $k$.

Markov Chain Note

Quantum Mechanics Note

Probability Note

Classical Mechanics Notes


## The Seven Snow White Princesses and the Envious Dwarf

Once upon a time, in a distant kingdom, there was a towering castle. In this castle lived a cruel and hypocritical giant dwarf named Grindolf. Though he was short in stature, his heart was filled with jealousy and ambition.

In his youth, Grindolf was once one of the most respected wizards in the kingdom, beloved for his exceptional magic skills. However, as time passed, he became intoxicated with his own power, yearning to rule the entire kingdom and become its sole sovereign.

To achieve his goals, Grindolf used various cunning tactics to eliminate his rivals and ultimately usurped the kingdom's rule. He oppressed the people using his magical powers, instilling fear and tyranny, while outwardly portraying himself as a benevolent and just ruler.

On the edge of the kingdom, there was a mysterious forest where seven beautiful and kind-hearted Snow White princesses lived. They were the guardians of the forest, frolicking joyfully every day, nurturing the flowers, grass, and trees under their care.

One day, while the seven Snow White princesses were playing in the forest, they unexpectedly stumbled upon a dwarf sleeping amidst a bed of flowers. Though his appearance was ordinary, he emitted an aura of kindness and purity. The princesses brought him back to their cottage and tenderly cared for him until he slowly awakened.

Upon awakening, the dwarf was surprised to find himself in unfamiliar surroundings, surrounded by seven beautiful Snow White princesses. They greeted him warmly, introducing themselves by name, and the dwarf politely reciprocated the introduction. Soon, the dwarf and the Snow White princesses became good friends, deciding to spend their time together.

However, when Grindolf learned of the dwarf's existence, his jealousy and hatred flared up. Disguising himself as an old woman, he tempted the dwarf with a poisoned apple. The moment the dwarf bit into the apple, he fell into a deep sleep.

Returning home, the Snow White princesses discovered the dwarf's sleeping figure and were filled with sorrow. They placed him on the bed, surrounding him with many candles, and kept vigil by his side. After many days of guarding him, one day, a brave girl arrived at the cottage, determined to save him.

The girl kissed the dwarf's lips, breaking Grindolf's curse, and the dwarf awakened from his slumber. He lived happily ever after with the girl, and the seven Snow White princesses were relieved to see their friend find true love.

Grindolf, exposed for his evil deeds, was banished to a solitary island to reflect on his actions. Since then, the kingdom no longer suffered from his brutal methods, and the seven Snow White princesses continued to guard their beloved forest, bringing peace and happiness to all.

## 七个白雪公主与嫉妒的大矮人

很久很久以前，在一个遥远的王国里，有一座高耸入云的城堡。在这个城堡中，住着一个残暴且伪善的大矮人，他的名字叫做格林多夫。他曾经是王国中最受人尊敬的魔法师之一，但他的心中渐渐充满了嫉妒和野心。

年轻时，格林多夫曾因为他出色的魔法技艺而受到王国人民的喜爱和尊重。然而，随着时间的推移，他开始渐渐沉湎于自己的力量之中，他渴望统治整个王国，成为唯一的统治者。

为了达到他的目的，格林多夫用各种诡计暗中排除了他的竞争对手，并最终篡夺了王国的统治权。他利用他的魔法力量压迫人民，使他们生活在恐惧和压迫之中，而外表上，他却装作一个善良公正的统治者。

在王国的边缘，有一片神秘的森林，那里生活着七个美丽善良的白雪公主。她们是这片森林的守护者，每天都在森林中快乐地奔跑，让花草树木在她们的照料下茁壮成长。

一天，七个白雪公主在森林里玩耍时，意外地发现了一个沉睡在鲜花丛中的小矮人。他的容颜虽然普通，但身上散发出一股善良与纯洁的气息。白雪公主们将他带回了她们的小屋，小心照料他，直到他缓缓醒来。

当小矮人醒来时，他惊讶地发现自己置身于陌生的地方，周围是七个美丽的白雪公主。她们友好地向他打招呼，告诉他她们的名字，而小矮人也礼貌地向她们自我介绍。很快，小矮人和白雪公主们成为了好朋友，并决定一起度过这段时光。

然而，格林多夫听说了小矮人的存在，他心中的嫉妒与恨意燃起。于是，他化身为一个老妇人，用一个毒苹果诱惑了小矮人。当小矮人咬下苹果的一刹那，他便陷入了深深的沉睡。

白雪公主们回到家中，发现了小矮人沉睡的身影，她们悲伤万分。她们将小矮人放在床上，周围点燃了许多蜡烛，守候在他的身边。经过多日的守护，终于有一天，一个勇敢的女孩来到了小屋，她听说了小矮人的故事，决心要拯救他。

女孩吻了小矮人的唇，这吻打破了格林多夫的诅咒，小矮人从沉睡中苏醒了过来。他和女孩幸福地生活在一起，而七个白雪公主们也为他们的朋友找到了真爱而感到欣慰。

格林多夫，在他的恶行被揭露后，被放逐到了一个孤岛上，反思他的所作所为。从那以后，王国里再也没有人遭受到他那残暴的手段，而七个白雪公主们也继续守护着她们心爱的森林，带来美好与和平。

> This story is totally fictional and created by AI. Any resemblance to actual events or locales or persons, living or dead, is entirely coincidental.

![The Seven Snow White Princesses and the Envious Dwarf](/static/img/2024-05-10-TheSevenSnowWhitePrincessesAndTheEnviousDwarf/1.png)

> This proudly generated by [stable diffusion](/tools/stable_diffusion) hosted on my personal blog.


The Seven Snow White Princesses and the Envious Dwarf


## Introduction

For long, MacOS users have been left out of the gaming world.
But with the advent of
[Wine](https://en.wikipedia.org/wiki/Wine_(software)),
it's now possible to run Windows games on MacOS.
In this tutorial, I'll show you how to run Witcher 3 on MacOS using
[Whisky](https://github.com/Whisky-App/Whisky)
\- a modern Wine wrapper built for MacOS.

All software used in this tutorial is free and open-source
except for Steam and Witcher 3, which are proprietary.

## Prerequisites

### Steam Account

You'll need to have a Steam account and have already purchased Witcher 3.

### System Requirements

You'will have to have Rosetta 2 installed on your system.

From [Apple's Official Documentation](https://support.apple.com/en-gb/102527):

> Open any app that needs Rosetta. If the app opens, Rosetta is already installed and working.
> If Rosetta is not installed, you’ll be automatically asked to install it.

I am using MacOS Sonoma 14.2 on a MacBook Pro (M1, 2021).
I do not ensure that this method will work on other systems.

### Download Necessary Files

#### Download Whisky

Download the latest version of Whisky from the official
[GitHub Releases Page](https://github.com/Whisky-App/Whisky/releases).
Just use the latest version available.

Download the `.zip` file or `.dmg` file as I highlighted in the image below.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/1.png)

#### Download Steam Installer for Windows

Download the Steam installer for **Windows** from the official
[Steam Website](https://store.steampowered.com/about/).

**Important**: What you need is the Windows installer, not the MacOS installer.
The file should be named something like `SteamSetup.exe`.

To download it, you'll need to visit the [Official Steam Website](https://store.steampowered.com/about/) and click on the **Windows Icon** below the "INSTALL STEAM" button as I highlighted in the image below.

![Steam](/static/img/2024-03-07-Run-Witcher3-On-MacOS/2.png)

## Installation

### Install Whisky

Open the `.zip` or `.dmg` file you downloaded on [Download Whisky step](#download-whisky) and drag the Whisky app to your Applications folder. Or just leave it anywhere on your computer if you prefer.

### Install Necessary Dependencies for Whisky

Open Whisky app, and it will ask you to install GPTK. Click on "Next" and wait for the installation to finish.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/3.png)

### Create a New Bottle

Create a new bottle by clicking on the "+" button on the right top corner of the Whisky app.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/4.png)

Just put arbitrary bottle name and leave all other settings as default and click "Create". You will need to wait some time for the bottle to be created.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/5.png)

### Install Steam

On the right bottom corner of the Whisky app, click on the "Run..." button.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/6.png)

Select the Steam installer you downloaded on [Download Steam Installer for Windows step](#download-steam-installer-for-windows). And then click "Open".

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/7.png)

You will see the Steam installer running. Just click "Next" and "Install" as you would do on Windows.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/8.png)

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/9.png)

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/10.png)

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/11.png)

Just wait a bit, and your Steam will be installed and opened.

![Whisky](/static/img/2024-03-07-Run-Witcher3-On-MacOS/12.png)

### Install Witcher 3

Now, you can install Witcher 3 as you would do on Windows. Just search for Witcher 3 in your library and click on "Install".

## Final Thoughts

This method also allows you to run other Windows games on MacOS.
However, not all games are supported, and some games may not run as smoothly as they would on Windows.

For some common issues and solutions, you can check the
[Whisky's Official Documentation](https://github.com/Whisky-App/Whisky/wiki/Game-Support).


Running Witcher 3 on MacOS


HLS (HTTP Live Streaming) is a widely used streaming protocol that breaks video
files into small chunks and serves them over HTTP.
It's a popular choice for streaming video content on the web,
and it's used by many platforms.

In this tutorial, I'll show you how to download encrypted HLS content.

## Prerequisites

You'll need to have `ffmpeg` installed on your system.

## Find the M3U8 file

The first step is to find the M3U8 file.
This file contains the URLs of the video chunks and the encryption key.

You can find the M3U8 file by inspecting the network requests
in your browser's developer tools when loading the page.

## Download the M3U8 file

Once you have the M3U8 file, you can download it using `ffmpeg`:

```bash
ffmpeg -i "https://example.com/video.m3u8" -c copy -bsf:a aac_adtstoasc output.mp4
```


Download encrypted HLS content with ffmpeg


If you visited my blog around three months ago,
you might recall the vibrant theme featuring fancy animations, colors,
and a rotating torus knot implemented with three.js.
While this theme garnered appreciation from friends and myself,
I've made a deliberate decision to move away from it.
The reasons are straightforward.

Managing and updating my blog became increasingly time-consuming
and complex as I added more features to the intricate design.
The simplicity I initially enjoyed was overshadowed
by the demands of maintaining such a detailed theme.

Crucially, my aesthetic preferences have
evolved towards a desire for simplicity.
I now advocate for a minimalist approach—no need
for elaborate colors or animations.
I've realized the importance of focusing on
substance rather than the outward appearance.

I've spent considerable time on superficial aspects,
not just in my blog but in various facets of my life.
Now, my goal is to streamline the visual aspects and dedicate
more time to exploring profound and intellectually stimulating subjects,
such as algebraic topology.

Algebraic topology, with its intricate concepts, fascinates me.
I've observed individuals sharing visually striking
higher-dimensional objects online,
seemingly not mentioned the rich mathematical content behind the images.
I appreciate the beauty, but there's a deeper layer that goes unnoticed.

Sharing such images without delving into the underlying concepts on platforms
where few understand algebraic topology can come across as mere showmanship.
I aim to shift the focus from superficial displays
to a genuine appreciation for the intricate
and fascinating world of serious and interesting subjects.


Transitioning Themes: Embracing Simplicity and Serious Exploration

General Topology Notes


Recently, I have upgraded my Postgresql database for my freshrss instance from 14 to 15. Here are the steps I followed to do it.

First, we need to local the binary of the old version of Postgresql.
In most of the cases, it can be found by

```bash
which psql
```

In my case, it is `/usr/local/bin/psql`.

Then, we need to find the data directory of the old version of Postgresql.
In my case, it is at `/var/lib/postgres/14`.
And the `postgresql.conf` file is at `/var/lib/postgres/14/data/postgresql.conf`.

Then, we need to install the new version of Postgresql.
I use `nix` to manage my packages,
so I just change the version of Postgresql in my `configuration.nix` file and run `nixos-rebuild switch`.

After that, we need to find the binary of the new version of Postgresql.
Which can be found by

```bash
which psql
```

Then, we need to find the data directory of the new version of Postgresql.
In my case, it is at `/var/lib/postgres/15`.
And the `postgresql.conf` file is which you should copy and modify from the old version of Postgresql.

We need to first stop the old version of Postgresql.

```bash
systemctl stop postgresql
```

And `su` to the `postgres` user.

```bash
sudo su - postgres
```

Then, we need to run the following command to first initialize the new version of Postgresql.

```bash
initdb -D /var/lib/postgres/15/
```

Then, we need 


Then, we need to run the following command to migrate the data from the old version of Postgresql to the new version of Postgresql.
With `--check` option, it will check if the migration is possible.

```bash
/usr/lib/postgresql/15/bin/pg_upgrade \
  --old-datadir=$OLD_DATADIR \
  --new-datadir=$NEW_DATADIR \
  --old-bindir=$OLD_BINDOR \
  --new-bindir=$NEW_BINDIR \
  --old-options '-c config_file=$OLD_CONFIG_FILE' \
  --new-options '-c config_file=$NEW_CONFIG_FILE' \
  --check
```

If the check is successful, we can run the following command to do the migration.

```bash
/usr/lib/postgresql/15/bin/pg_upgrade \
  --old-datadir=$OLD_DATADIR \
  --new-datadir=$NEW_DATADIR \
  --old-bindir=$OLD_BINDOR \
  --new-bindir=$NEW_BINDIR \
  --old-options '-c config_file=$OLD_CONFIG_FILE' \
  --new-options '-c config_file=$NEW_CONFIG_FILE'
```

After that, we need to check if the migration is successful.
If it is successful, we can then remove the old version of Postgresql.


Upgrade Postgresql From 14 to 15


XTX Markets is launching a new $10mn challenge fund, the Artificial Intelligence Mathematical Olympiad Prize (AI-MO Prize). The fund intends to spur the development of AI models that can reason mathematically, leading to the creation of a publicly-shared AI model capable of winning a gold medal in the International Mathematical Olympiad (IMO).

The grand prize of $5mn will be awarded to the first publicly-shared AI model to enter an AI-MO approved competition and perform at a standard equivalent to a gold medal in the IMO.

There will also be a series of progress prizes, totalling up to $5mn, for publicly-shared AI models that achieve key milestones towards the grand prize.

> The AI-MO Prize [..] will help compare different AI problem solving strategies at a technical level, in a manner that will be accessible and appealing to the broader public.
>> Terence Tao, UCLA

> I am sure that many people will be following the AI-MO Prize with great excitement, to see when, in the future, AI will match the world's brightest young minds.
>> Gregor Dolinar, IMO President

XTX Markets is launching a new $10mn challenge fund, the Artificial Intelligence Mathematical Olympiad Prize (AI-MO Prize). The fund intends to spur the development of AI models that can reason mathematically, leading to the creation of a publicly-shared AI model capable of winning a gold medal in the International Mathematical Olympiad (IMO).

$10mn AI Mathematical Olympiad Prize Launches


一种展示摘掉眼镜后真实视力的方法是，使用相机对准眼镜并拍摄照片，然后在iPhone上长按“AF锁定”按钮。这样一来，当眼镜被移开后，你通过相机屏幕看到的画面将反映佩戴者的实际视力状况。这个方法不仅能够帮助视力正常的人理解近视者的视觉体验，还可能对店铺或城市规划产生积极影响。其原理基于相机的自动对焦（AF）功能，通过在对准眼镜并长按以锁定自动对焦时，相机会根据眼镜的光学特性调整焦距。因此，当眼镜被移开时，相机的焦点仍然保持在根据眼镜调整的位置，从而通过相机屏幕模拟出佩戴者的视力状况。

![AF Lock](/static/img/2023-11-29-让别人看到你裸眼视力的方法/optimize.gif)

一种展示摘掉眼镜后真实视力的方法是，使用相机对准眼镜并拍摄照片，然后在iPhone上长按“AF锁定”按钮。这样一来，当眼镜被移开后，你通过相机屏幕看到的画面将反映佩戴者的实际视力状况。这个方法不仅能够帮助视力正常的人理解近视者的视觉体验，还可能对店铺或城市规划产生积极影响。其原理基于相机的自动对焦（AF）功能，通过在对准眼镜并长按以锁定自动对焦时，相机会根据眼镜的光学特性调整焦距。因此，当眼镜被移开时，相机的焦点仍然保持在根据眼镜调整的位置，从而通过相机屏幕模拟出佩戴者的视力状况。

让别人看到你裸眼视力的方法


## Introduction

In the expansive and complex realm of the internet, a foundational principle governs communication - IP addresses. Short for Internet Protocol, these addresses serve as digital fingerprints for devices interconnected within the vast web. This article seeks to simplify the intricacies surrounding IP addresses, illuminating key aspects that act as a guide for those navigating the digital domain.

## Understanding IP Addresses:

At its essence, an IP address is a unique numerical label assigned to each device in a computer network that communicates through the Internet Protocol. Analogous to a home address in the physical world, it functions as a digital identifier, ensuring that data packets navigate the intricate web to reach their intended destinations.

## Types of IP Addresses:

In the expansive realm of IP addresses, familiarity with locating your public IP address becomes crucial for navigating the digital landscape. Two predominant versions, IPv4 (Internet Protocol version 4) and IPv6 (Internet Protocol version 6), dominate this scene. While IPv4 utilizes a 32-bit address scheme, limiting unique addresses, IPv6, its modernized successor, employs a 128-bit format, vastly expanding the pool of available addresses to meet the demands of a growing number of internet-connected devices.

## Components of IP Addresses:

Analyzing the structure of IP addresses reveals a numeric composition separated by periods. In the case of IPv4, the format is xxx.xxx.xxx.xxx, with each "xxx" ranging from 0 to 255. Conversely, IPv6 addresses, represented in hexadecimal notation, such as 2001:0db8:85a3:0000:0000:8a2e:0370:7334, offer a more extensive and flexible range.

## Static vs. Dynamic IP Addresses:

The dynamism of IP addresses extends to their classification as static or dynamic. A static IP address remains constant and is manually assigned, ideal for devices like servers requiring a consistent point of access. In contrast, dynamic IP addresses are automatically assigned by a network's DHCP server, offering flexibility for devices with intermittent network connections.

## Significance of IP Addresses in Networking:

At the heart of computer networks lies the omnipresent IP address, a vital element ensuring the smooth flow of data. Whether it's a computer, smartphone, or any networked device, each relies on its unique IP address to establish structured communication, forming the backbone of connectivity across the internet.

## Subnetting for Efficient IP Address Management:

To optimize IP address management, subnetting divides large networks into smaller, more manageable segments. Beyond efficient IP address utilization, subnetting enhances security and performance by strategically allocating addresses, reducing congestion, and improving data transmission efficiency.

## Public and Private IP Addresses:

IP addresses serve dual roles as public and private identifiers. Public IP addresses, visible on the internet, represent a network to the external world. Private IP addresses operate within local networks, enabling internal device communication without exposure to the broader internet.

## NAT and IP Address Conservation:

With the proliferation of devices, Network Address Translation (NAT) conserves IP addresses by allowing multiple devices within a local network to share a single public IP address. This solution extends the usability of IPv4 addresses, mitigating the threat of address exhaustion.

## IP Address Geolocation:

A fascinating aspect of IP addresses is geolocation, providing the general location of a device based on its IP address. Despite not being pinpoint accurate, this feature finds applications in enhancing online security, delivering localized content, and services.

## Emerging Trends in IP Addressing:

In the ever-evolving technological landscape, IP addressing evolves alongside progress. The Internet of Things (IoT) and the surge in smart devices increase the demand for IP addresses. IPv6, with its expansive address space, accommodates this surge, ensuring digital infrastructure evolves with technological advancements.

## IPv6 Adoption:

Embracing IPv6 becomes crucial in addressing the limitations posed by IPv4. The 32-bit address scheme of IPv4 faces challenges as connected devices increase. IPv6, with its 128-bit format, accommodates this proliferation, enhancing the efficiency and resilience of internet infrastructure.

## IP Address Management Tools:

As networks become intricate ecosystems, IP address management tools streamline allocation and monitoring processes. These tools play a critical role in network administration, ensuring smooth operations amid the increasing complexity of modern infrastructures, contributing to order, security, and optimal performance.


In the expansive and complex realm of the internet, a foundational principle governs communication - IP addresses. Short for Internet Protocol, these addresses serve as digital fingerprints for devices interconnected within the vast web. This article seeks to simplify the intricacies surrounding IP addresses, illuminating key aspects that act as a guide for those navigating the digital domain.

Understanding IP Address - Some Basics



After more than thirty years of searching,
with the assistance of supercomputers,
mathematicians have discovered the ninth Dedekind number.
Dedekind numbers were defined by German mathematician Richard Dedekind in 1897
and are based on Boolean functions (values of false or true).
These functions take $n$ Boolean variables as input and generate
another Boolean variable as output.
The first six Dedekind numbers are quite simple, with $D(1)$ being 2,
followed by $3$, $6$, $20$, $168$, and $7581$,
with subsequent numbers growing increasingly larger.
In 1991, one of the most powerful supercomputers at the time,
the Cray-2, took 200 hours to calculate $D(8)$.
Thirty-two years later,
mathematicians at the University of Paderborn in Germany
used the Noctua 2 supercomputer and field-programmable gate arrays
(FPGA) to calculate $D(9)$ as
$286,386,577,668,298,411,128,469,151,667,598,498,812,366$
— a 42-digit number.
Discovering $D(10)$ will undoubtedly take a considerable amount of time as well.



After more than 30 years of search and computational effort, mathematicians recently revealed the intricate 42-digit structure of the ninth Dedekind number, showcasing the complexity of mathematical exploration and technological progress.

Discovery of the Ninth Dedekind Number: A Decades-Long Search and Computational Breakthrough


## Intro

This is my personal note for Galois Theory.

This  is a work in progress.
When I have time, I will add more content to it.

## Concepts of Ring

<div class="math_definition_env">
<span id="definition:ring" className="math_definition">Ring</span>
is a set equipped with two binary operations, usually referred to as
addition $+$ and multiplication $·$,
that satisfy the following conditions:

1. The set is an abelian group under the operation $+$
   with an identity element usually denoted as 0.

2. The operation $·$ is associative,
   meaning that for all $a, b, c$ in the set, $(a·b)·c$ equals $a·(b·c)$.

3. The operation $·$ is distributive over the operation $+$,
   meaning that for all $a, b, c$ in the set, $a·(b+c)$ equals
   $a·b + a·c$ and $(a+b)·c$ equals $a·c + b·c$.

4. The operation $·$ has an identity element, usually denoted as $1$.

5. The operation $·$ is closed, meaning that for all $a, b$ in the set,
$a·b$ is also in the set.

6. A ring is called a commutative ring if the operation $·$ is commutative,
   meaning that for all $a, b$ in the set, $a·b$ equals $b·a$.
</div>

<div class="math_definition_env">
<span id="definition:field" class="math_definition"> Field </span>
is a ring in which every non-zero element is invertible.
</div>

<div class="math_definition_env">
<span id="definition:subring" class="math_definition"> Subring </span>
is a subset of a ring
that is a ring with the restriction of the ring operations.
</div>

<div class="math_definition_env">
<span id="definition:subfield" class="math_definition"> Subfield </span>
is a subset of a field
that is a field with the restriction of the field operations.
</div>

<div class="math_definition_env">
<span id="definition:ring-homomorphism" class="math_definition"> Ring Homomorphism </span>
is a function between two rings
that preserves the operations of the rings.

A ring homomorphism is called an
<span id="definition:ring-homomorphism:isomorphism" class="math_definition"> isomorphism </span>
if it is bijective.
</div>

<div class="math_definition_env">
<span id="definition:ring-homomorphism" class="math_definition"> Ideal </span>
is a subset of a ring that is closed under addition, negation,
and multiplication by any element in the ring.
</div>

<div class="math_definition_env">
<span id="definition:ring-homomorphism" class="math_definition"> Principal Ideal </span>
is an ideal that is generated by a single element.
</div>

<div class="math_definition_env">
<span id="definition:integral-domain" class="math_definition"> Integral Domain </span>
is a commutative ring with identity
in which the product of any two non-zero elements is non-zero.
</div>

<div class="math_definition_env">
<span id="definition:quotient-ring" class="math_definition"> Quotient Ring </span>
is a ring constructed from a ring $R$ and an ideal $I$ of $R$.
Where the elements of the quotient ring are the cosets of $I$ in $R$.
</div>

## Definitions

<ul>
    <li><a href="#definition:ring">Ring</a></li>
    <li><a href="#definition:ring">Commutative Ring</a></li>
    <li><a href="#definition:ring-homomorphism">Ring Homomorphism</a></li>
    <li><a href="#definition:field">Field</a></li>
    <li><a href="#definition:subring">Subring</a></li>
    <li><a href="#definition:subfield">Subfield</a></li>
    <li><a href="#definition:ideal">Ideal</a></li>
    <li><a href="#definition:ring-homomorphism:isomorphism">Ring Isomorphism</a></li>
    <li><a href="#definition:principal-ideal">Principal Ideal</a></li>
    <li><a href="#definition:integral-domain">Integral Domain</a></li>
    <li><a href="#definition:quotient-ring">Quotient Ring</a></li>
</ul>


Galois Theory Notes


《自然》撤回了罗彻斯特大学助理教授 Ranga P. Dias 作为通讯作者的室温超导论文。
这是他被撤回的第三篇论文。
《自然》在撤回声明中称，因所发表的论文没有准确反映所研究材料的来源、
所进行的实验测量等原因，应多位署名作者的要求，撤回论文
Dias 本人尚未回复是否同意此次撤稿。
涉事论文的标题是《氮掺杂氢化镥的近环境超导性证据》
（Evidence of near-ambient superconductivity in a N-doped lutetium hydride）。 
论文称，他们研发出一种含镥、氢、氮的材料，在约 20.6 摄氏度的室温和 10 千巴（约 1 万倍大气压）
的压力下表现出超导特性。该论文于 3 月 8 日发表在《自然》上。
随后国际上多个研究团队宣称，未能复现论文报告的结果。

《自然》撤回室温超导论文


This is some generalization of a question.

Link to the pdf file:
[Addition Of Algebraic Element](/static/math/AdditionOfAlgebraicElement/main.pdf)

Link to the source code:
[Addition Of Algebraic Element](/static/math/AdditionOfAlgebraicElement/main.tex)

::: question
**Question 1**. *Proof that
$\sqrt{1} + \sqrt{2} + \sqrt{3} \dots + \sqrt{n}$ for $n > 2$ is
irrational.*
:::

However, if we consider the linear combination of $\sqrt{1}$,
$\sqrt{2}$, $\sqrt{3}$ $\dots$ $\sqrt{n}$ over $\mathbb{Q}$, then it is
likely that we will get a similar result as we kick out some special
cases.

::: question
**Question 2**. *Given $n$ positive integers, $a_1 < a_2 < \dots < a_n$,
such that $\sqrt{\frac{a_{j}}{a_{i}}} \notin \mathbb{Q}$ for all $i < j$
and $\sqrt{a_{i}} \notin \mathbb{Q}$ for all $i$. Also given
$\lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{Q}$. Prove that
$\sum_{i=1}^{n} \lambda_{i} \sqrt{a_{i}} \in \mathbb{Q}$ if and only if
$\lambda_{1} = \lambda_{2} = \dots = \lambda_{n} = 0$.*
:::

::: proof
*Proof.* Suppose this stand for $n=k$, then it is suffice to prove
$n=k+1$.

Suppose there exist
$\lambda_{1}, \lambda_{2} \dots \lambda_{k+1} \in \mathbb{Q}$ such that
$$\sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} = q \in \mathbb{Q}$$

If $\lambda_{j} = 0$, then
$-\lambda_{j}\sqrt{a_{j}} + \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} = q \in \mathbb{Q}$,
and we are done according to the assumption.

If $\lambda_{j} \neq 0$ for all $j$, then we have
$$\sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}} = q - \lambda_{k+1}\sqrt{a_{k+1}}$$

Clearly, $q - \lambda_{k+1}\sqrt{a_{k+1}}$ is a root of a irreducible
quadratic polynomial $g \in \mathbb{Q}[x]$. And the polynomial must have
another root which is $q + \lambda_{k+1}a_{k+1}$.

Also, the polynomial
$$f = \prod (x - ( \pm \lambda_{1}\sqrt{a_{1}} + \pm \lambda_{2}\sqrt{a_{2}} + \dots + \pm \lambda_{n}\sqrt{a_{n}} ))$$
is also in $\mathbb{Q}[x]$ by induction.

Also,
$$
\begin{aligned}
            f( q - \lambda_{k+1}\sqrt{a_{k+1}} ) &= f( \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} ) \\
            &= 0
        
\end{aligned}
$$

As $g$ is irreducible over $\mathbb{Q}$, $g$ must divides $f$. Thus
$$f( q + \lambda_{k+1}\sqrt{a_{k+1}} ) = 0$$

Thus, there exits $\alpha_{1}, \alpha_{2} \dots \alpha_{n}$ which is
either $1$ or $-1$ such that
$$\sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} = q + \lambda_{k+1}\sqrt{a_{k+1}}$$

Thus
$$
\begin{cases}
                \sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} = q + \lambda_{k+1}\sqrt{a_{k+1}} \\
                \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}} = q - \lambda_{k+1}\sqrt{a_{k+1}}
\end{cases}
$$

$$
\begin{aligned}
            \sum_{i=1}^{k} \alpha_{i} \lambda_{i}\sqrt{a_{i}} + \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{i}}
            &= q + \lambda_{k+1}\sqrt{a_{k+1} }+ q - \lambda_{k+1}\sqrt{a_{k+1}} \\
            \sum_{i=1}^{k} ( \alpha_{i} + 1 ) \lambda_{i}\sqrt{a_{i}} &= 2q \in \mathbb{Q} \\
        
\end{aligned}
$$

By the induction assumption, we have $(\alpha_{i} + 1)\lambda_{i} = 0$
for all $i \le k$. As $\lambda_{i} \neq 0$ for all $i$, we have
$\alpha_{i} = -1$ for all $i \le k$.

Thus,
$$
\begin{aligned}
            \sum_{i=1}^{k} ( \alpha_{i} + 1 ) \lambda_{i}\sqrt{a_{i} } &= 0
        
\end{aligned}
$$
which implies $q = 0$.

Thus,
$$
\begin{aligned}
            \sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}} &= 0 \\
            \sqrt{a_{k+1}} (\sum_{i=1}^{k+1} \lambda_{i}\sqrt{a_{i}}) &= 0 \\
            \sum_{i=1}^{k} \lambda_{i}\sqrt{a_{k+1}a_{i}} &= - \lambda_{k+1} a_{k+1} \in \mathbb{Q}\\
        
\end{aligned}
$$

The only thing left to proof is that $a_{k+1}a_{i}, i \le k$ satisfy the
assumption in the question, and then the induction assumption will
apply, which proves that $\lambda_{i} = 0, i \le k$.

Given any $i < j$, we have
$$
\begin{aligned}
            \sqrt{\frac{a_{j}a_{k+1}}{a_{i}a_{k+1}}} &= \sqrt{\frac{a_{j}}{a_{i}}} \notin \mathbb{Q} \\
            \sqrt{a_{j}a_{k+1}} &= a_{k+1} \sqrt{\frac{a_{j}}{a_{k+1}}} \notin \mathbb{Q}
        
\end{aligned}
$$
◻
:::

As, we does not use any special property of $\mathbb{Q}$ other then the
fact that it is a field. So, the proof should be valid for any field
$\mathbb{F}$ that characteristic equals 0, if we rephrase the question
in term of $\mathbb{F}$.

::: question
**Question 3**. *Given $n$ distinct element
$a_{1}, a_{2}, \dots, a_{n} \in \mathbb{F}$, let $b_{i}$ be a root of
$x^{2} - a_{i}$ for all $i$. And $b_{i}b_{j}^{-1} \notin \mathbb{F}$ for
all $i \neq j$, and $b_{i} \notin \mathbb{F}$ for all $i$. Then given
$\lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}$,
$\sum_{i=0}^{n} \lambda_{i}b_{i} \in \mathbb{F}$ if and only if
$\lambda_{i} = 0$ for all $i$.*
:::

If we try to shift $b_{k}$ to $b_{k} + c_{k}$ such that
$c_{k} \in \mathbb{F}$, then the result clearly still stand. So, the
previous question can be generalised a bit.

::: question
**Question 4**. *Given $n$ distinct irreducible quadratic polynomial
$f_{1}, f_{2} \dots f_{n}$ in $\mathbb{F}[x]$.*

*And given any $i \neq j$, $f_{j}$ does not have root in
$\mathbb{F}[x]/f_{i}$.*

*And $b_{i}$ be all roots of $f_{i}$ for all $i$. Then given
$\lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}$,
$\sum_{i=0}^{n} \lambda_{i}b_{i} \in \mathbb{F}$ if and only if
$\lambda_{i} = 0$ for all $i$.*
:::

If we are not satisfied with the quadratic polynomials, we can try to
generalise the question to any polynomial. However, we may need to have
a more strict condition on the roots.

::: question
**Question 5**. *Given $n$ distinct irreducible polynomial
$f_{1}, f_{2} \dots f_{n}$ in $\mathbb{F}[x]$ with orders greater or
equal than 2.*

*Given any $i \neq j$. $f_{j}$ does not have root in
$\mathbb{F}[x]/f_{i}$.*

*And $b_{i}$ be all roots of $f_{i}$ for all $i$. Then given
$\lambda_{1}, \lambda_{2} \dots \lambda_{n} \in \mathbb{F}$,
$\sum_{i=0}^{n} \lambda_{i}b_{i,1} \in \mathbb{F}$ if and only if
$\lambda_{i} = 0$ for all $i$.*
:::

To prove the above question, we need another definition and some
relating lemma.

::: definition
**Definition 1**. *Given a polynomial $f(x) \in \mathbb{F}[x]$,
$$f(x) = x^{n} + a_{1}x^{n-1} + a_{2}x^{n-2} + \dots + a_{n-1}x + a_{n}$$*

*Define the derivative $f'(x)$ of $f(x)$ as
$$f'(x) = n x^{n-1} + (n-1)a_{1}x^{n-2} + (n-2)a_{2}x^{n-3} + \dots + 2a_{n-2}x + a_{n-1}$$*
:::

Through some simple calculation, we can prove the following lemma.

::: lemma
**Lemma 1**. *Given polynomials $f(x), g(x) \in \mathbb{F}[x]$,
$$(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$$*
:::

::: lemma
**Lemma 2**. *Given a irreducible polynomial $f(x) \in \mathbb{F}[x]$,
and a field $\mathbb{K}$, such that $\mathbb{F} \subseteq \mathbb{K}$
and $f(x)$ can be factorized into product of linear factors in
$\mathbb{K}$.*

*Chose a root $\alpha$ of $f(x)$ in $\mathbb{K}$. Then $(x-\alpha)^2$
divides $f(x)$ in $\mathbb{K}$, if and only if $f'(x) = 0$.*
:::

As $f'(x)$ is still a polynomial in $\mathbb{F}[x]$, and we can choose
arbitrary $\mathbb{K}$. This lemma implies that irreducible polynomial
$f(x)$ have distinct roots if and only if $f'(x) \neq 0$.

::: proof
*Proof.* If $(x-\alpha)^2$ divides $f(x)$ in $\mathbb{K}$.

Let, $$f(x) = (x-\alpha)^{2}g(x)$$

Then, $$f'(x) = 2(x-\alpha)g(x) + (x-\alpha)^{2}g'(x)$$

Then $f'(\alpha) = 0$. As $f(x)$ is irreducible, and $f(x)$ and $f'(x)$
have common root. So, $f(x)$ divides $f'(x)$.

As, the degree of $f'(x)$ is less than $f(x)$, then $f'(x) = 0$.

Conversely, if $f'(x) = 0$.

Let, $$f(x) = (x-\alpha)g(x)$$

Then, $$f'(x) = g(x) + (x-\alpha)g'(x)$$

Then,
$$
\begin{aligned}
            0 & =  f'(\alpha) \\
            & =  g(\alpha) + (\alpha-\alpha)g'(\alpha) \\
            & =  g(\alpha)
        
\end{aligned}
$$

So, $\alpha$ is a root of $g(x)$.

Thus, $(x-\alpha)^2$ divides $f(x)$ in $\mathbb{K}$. ◻
:::

::: lemma
**Lemma 3**. *Given distinct irreducible polynomials
$f(x), g(x) \in \mathbb{F}[x]$, where the characteristic of $\mathbb{F}$
equals $0$.*

*Chose any field $\mathbb{K}$, such that
$\mathbb{F} \subseteq \mathbb{K}$, and $f(x)$ and $g(x)$ can be
factorized into product of linear factors in $\mathbb{K}$.*

*Let $a_{1}, a_{2} \dots a_{n}$ be all roots of $f(x)$ in $\mathbb{K}$,
and $b_{1}, b_{2} \dots b_{m}$ be all roots of $g(x)$ in $\mathbb{K}$.*

*Then, $$h(x) = \prod_{i=1}^{n}\prod_{j=1}^{m}(x-(a_{i} + b_{j}))$$ have
coefficients in $\mathbb{F}$,*
:::

Addition Of Algebraic Element


## Intro

Some note for NixOS configurations.

## Useful Commands

### Rebuild the System

Need `sudo`.

```bash
# test the configuration
sudo nixos-rebuild test --flake "/path/to/config#tag" --show-trace

# switch to the configuration
sudo nixos-rebuild switch --flake "/path/to/config#tag" --show-trace
```

## Useful Websites

[NixOS Full Options List](https://nixos.org/manual/nixos/stable/options.html#opt-environment.etc)

[NixOS Package List](https://search.nixos.org)

[GitHub User Public Key](https://github.com/Alex222222222222.keys)

## Put File Under /etc

Check [ ](https://nixos.org/manual/nixos/stable/options.html#opt-environment.etc)

```nix
{
    example-configuration-file =
    { 
        source = "./file.conf.example";
        mode = "0440";
    };
    "default/useradd".text = "GROUP=100 ...";
}
```

## Reference Store Path of a Nix Package

```nix
''
${pkgs.hello}/bin/hello
''
```

## Use Agenix to Manage Secrets

Check [`agenix GitHub`](https://github.com/ryantm/agenix#age-module-reference).

### Install Agenix Via Flakes

```nix
{
  inputs.agenix.url = "github:ryantm/agenix";
  # optional, not necessary for the module
  #inputs.agenix.inputs.nixpkgs.follows = "nixpkgs";
  # optionally choose not to download darwin deps (saves some resources on Linux)
  #inputs.agenix.inputs.darwin.follows = "";

  outputs = { self, nixpkgs, agenix }: {
    # change `yourhostname` to your actual hostname
    nixosConfigurations.yourhostname = nixpkgs.lib.nixosSystem {
      # change to your system:
      system = "x86_64-linux";
      modules = [
        ./configuration.nix
        agenix.nixosModules.default
      ];
    };
  };
}
```

### Get Secrets

`agenix` use `ssh keys` as keys.

System host `ssh key` is at `/etc/ssh/`.

User's `ssh key` is at `~/.ssh`.

### Mkdir For Agenix

```bash
mkdir -p ./secrets
```

### Add Secrets Keys

Keys should be put at `./secrets/secrets.nix`.

```nix
# ./secrets/secrets.nix
let
  user1 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIL0idNvgGiucWgup/mP78zyC23uFjYq0evcWdjGQUaBH";
  user2 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAILI6jSq53F/3hEmSs+oq9L4TwOo1PrDMAgcA1uo1CCV/";
  users = [ user1 user2 ];

  system1 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIPJDyIr/FSz1cJdcoW69R+NrWzwGK/+3gJpqD1t8L2zE";
  system2 = "ssh-ed25519 AAAAC3NzaC1lZDI1NTE5AAAAIKzxQgondgEYcLpcPdJLrTdNgZ2gznOHCAxMdaceTUT1";
  systems = [ system1 system2 ];
in
{
  "secret1.age".publicKeys = [ user1 system1 ];
  "secret2.age".publicKeys = users ++ systems;
}
```

This file declare that the `secret1.age` can be decrypted by `user1` and `system1`.
`secret2.age` can be decrypted by all users in `users` and `systems`.

### Add Secrets

```bash
# You need to be in the same directory as secrets.nix
cd ./secrets

# This will open your editor ($EDITOR) to create the secret
agenix -e secret1.age
```

### Add Secrets to NixOS Configurations

```nix
{
  # this will allow you to use the secrets in your NixOS configuration as a string
  age.secrets.secret1.file = ../secrets/secret1.age;
}
```

Or,

```nix
{
  # this will allow you to access the at the path /etc/secret1 with the correct permissions
  age.secrets.secret1 = {
    file = ../secrets/secret1
    path = "/etc/secret1";;
    mode = "770";
    owner = "nginx";
    group = "nginx";
  };
}
```

### Use Secrets

#### Use Path of Secrets

```nix
{
  users.users.user1 = {
    isNormalUser = true;
    passwordFile = config.age.secrets.secret1.path;
  };
}
```

#### Replace Inplace Strings With Secrets

Considering that there still might be some modules which doesn't support reading secrets from a file, you could provide a placeholder string instead of a clear-text password and replace this placeholder with the secret provided by Agenix.

In the following example, the Dex module creates the config file /run/dex/config.yaml containing the placeholder string @dex-user-password@. The activation script will read the Agenix secret from config.age.secret.dex-user-password.path and replace the placeholder string with the actual secret.

```nix
system.activationScripts."dex-user-secret" = ''
  secret=$(cat "${config.age.secrets.dex-user-password.path}")
  configFile=/run/dex/config.yaml
  ${pkgs.gnused}/bin/sed -i "s#@dex-user-password@#$secret#" "$configFile"
'';
```

## Mount WebDav Using AutoFS

I found this two links really helpful.

[https://askubuntu.com/questions/821123/autofs-and-webdav-how-to-make-them-work-together](https://askubuntu.com/questions/821123/autofs-and-webdav-how-to-make-them-work-together)

[https://www.reddit.com/r/NixOS/comments/b5p6f7/how_do_i_use_davfs2/](https://www.reddit.com/r/NixOS/comments/b5p6f7/how_do_i_use_davfs2/)

First you need to install [`agenix`](#use-agenix-to-manage-secrets) as a way to manage your `WebDav` secrets.

Then you need to add the following to your `configuration.nix`.
This will mount your `WebDav` to `/mnt/{{ mount-dir }}`.

```nix
{
  age.secrets = {
    webdav-secrets = {
      file = ./secrets/{{ your secret file }};
      owner = "root";
      group = "root";
      mode = "600";
      # this is the path where the secret will be mounted
      path = "/etc/davfs2/secrets";
    };
  };

  # put the configuration for davfs2 in the /etc/davfs2 directory
  environment.etc."davfs2/auto.mount" = {
    # rw means read-write, ro means read-only
    # you need to add backslash `\` before ':' and '#' in your url
    text = ''
      {{ mount-dir }} -fstype=davfs,conf=/etc/davfs2/conf,rw {{ your webdav url }}
    '';
    mode = "0440";
  };
  environment.etc."davfs2/conf" = {
    text = ''
      secrets /etc/davfs2/secrets
    '';
    mode = "0440";
  };

  # enable the service
  services.davfs2.enable = true;
  services.autofs = {
    enable = true;
    autoMaster = "/mnt/ /etc/davfs2/auto.mount";
  };
}
```

## Clean the System & Nix Store

```nix
# clean journalctl
services.cron = {
  enable = true;
  systemCronJobs = [
    "0 0 * * * journalctl --vacuum-time=7d 1>/dev/null"
  ];
};

# garbange collection check https://nixos.wiki/wiki/Nix_Cookbook#Reclaim_space_on_Nix_install.3F
nix.gc.automatic = true;
nix.settings.auto-optimise-store = true;
```

## Enable the Mosh Server

```nix
# Enable mosh, the ssh alternative when client has bad connection
# Opens UDP ports 60000 ... 61000
programs.mosh.enable = true;
networking.firewall.allowedTCPPortRanges = [
  {
    from = 60000;
    to = 61000;
  }
];
networking.firewall.allowedUDPPortRanges = [
  {
    from = 60000;
    to = 61000;
  }
];
```

## Enable Fail2Ban

```nix
services.fail2ban = {
  enable = true;
  maxretry = 5;
  ignoreIP = [
    "127.0.0.0/8" 
    "10.0.0.0/8" 
    "172.16.0.0/12" 
    "192.168.0.0/16"
    "8.8.8.8"
  ];
};
```

NixOS Configurations | NixOS 不完全手记


## GreenFiled The City In Minecraft

Greenfield was founded by `THEJESTR` back in August 2011. The map started with a single gas station, built by him, that still resides in its original form and location in `Hunterspoint`. In the last 11 years, he has led over 20 different staff members and 400+ builders to make the map what it is today.

Greenfield is a fictional city designed to resemble the West Coast of America, laboriously inspired by Los Angeles. Built to a one-to-one scale, every single building in Greenfield has substantial detail with carefully detailed elaborate surroundings. With a substantially sizable airport to lavish homes and businesses, Greenfield is one of the largest, most complex, and detailed original cities in Minecraft. As of July 2022, the in-progress map is now about 6500x8800 blocks, or about 57 square kilometers.

Download Resource Pack:
[Link1](https://www.greenfieldmc.net/downloads/)
[Link2](https://ipfs.io/ipfs/bafybeigswzgajk2cin3r66pcrjq2i6yt4gnufaf7egxzq33b3hqoa6nmci/)
[Link3](https://01c16de193.acl.multichain.storage/ipfs/QmVoJT55q3ituxSMyu6ZMdJ6cMFsizwtmkpqTxNWyn8974?filename=Greenfield.Texture.Pack.1.17.zip)


Download Map:
[Link1](https://www.greenfieldmc.net/downloads/)
[Link2](https://ipfs.io/ipfs/bafybeigvxjcnz5ey2mphjpaepdf6vnlgu3sathwobgzh2lhlm5q2zhs4wq/)
[Link3](https://ipfs.io/ipfs/bafybeidvl5nvhqkspcqsb4hotxdpfzsluiohy6n4qljlyrizra3cvbwlte/)
[Link5](https://01c16de193.acl.multichain.storage/ipfs/QmRwB777gUymVb3td3UJLoExW5uiHZSeXAdZQ2MqNdyvxN?filename=Greenfield.zip)
[Link6](https://01c16de193.acl.multichain.storage/ipfs/QmVkh2Fs7K77QsuuUAPEAAdp9C4q8wQRntCUstYVE5P8A4?filename=Greenfield_v0.5.4.zip)

[GreenField](https://www.greenfieldmc.net/about/)
![GreenField Image](/static/img/2023-06-05-greenfield-city-in-minecraft/1.png)
![GreenField Image](/static/img/2023-06-05-greenfield-city-in-minecraft/2.png)

GreenField The City In Minecraft


## Introduction

Uninstalling a software on a system has long being a headache problem.
If you install a software using a package manager like `brew`, or `apt`.
It is easy. You simply do

```shell
# brew
brew remove [package-name]
# use auto remove to remove the dependency
brew autoremove

# apt
apt purge [package-name]
# use auto remove to remove the dependency
apt autoremove
```

If you install the app using app store.
Just uninstall it in the app store.

However, if you install the app by just copy the app into `/Applications`.
Then, simply remove the app itself is clearly not enough.
For instance, steam store all the game you download in the '~/Library' folder.
If you just remove the app itself, there will be tons of files left.

And there is a app that help us do that.
The [AppCleaner](https://freemacsoft.net/appcleaner/).

## Download

Just go to the webpage [AppCleaner](https://freemacsoft.net/appcleaner/),
and select the latest version.

## Uninstall App

Open the app,
click the list icon in the right up conner,
type in the app you want to uninstall,
for instance `Steam`.
Select the App you want to uninstall,
then AppCleaner will search for all the files of the App,
you could unselect the files you do not want to delete.
Click remove, then, you are all set.

![Steam Sample](/static/img/2023-04-26-Uninstall-App-On-MacOS/1.png)


How to uninstall a app on MacOS


## Introduction

GPG Suite is an open-source software package for macOS
that provides encryption and decryption of emails, files,
and directories using the OpenPGP encryption standard.
OpenPGP is a widely used standard for secure email communication
that provides end-to-end encryption,
which means that only the sender and recipient of an encrypted message
can read its contents.

GPG Suite includes several tools,
including GPGMail for encrypting and signing emails,
GPG Keychain for managing OpenPGP keys,
and GPG Services for encrypting and decrypting files and directories.
It also includes GPGPreferences,
which provides a user interface for configuring GPG Suite settings.

GPG Suite is popular among users who require secure communication and privacy protection,
such as journalists, activists, and government officials.
It is available for free download and use,
although users can make a donation to support the ongoing development of the software.

![Screenshot of GPG Suite](/static/img/2023-04-17-GPGMail-Mac/1.png)

## Installation

Check [Free-GPGMail](https://github.com/Free-GPGMail/Free-GPGMail) project.

1. Download the latest version of Free-GPGMail.

    Using Homebrew:

    ```bash
    brew install --cask free-gpgmail
    ```

2. Restart Mail.app, go to `Preferences -> General -> Manage Plugins....`

    If the "Manage Plugins" button is not present in the lower left corner, open a terminal and run:

    ``` bash
    sudo defaults write “/Library/Preferences/com.apple.mail” EnableBundles 1
    defaults write com.apple.mail EnableBundles -bool true
    ```

    Then restart Mail.app again.

    - Make sure that `GPGMailLoader_*.mailbundle`, if present, is disabled.
    - Enable the `Free-GPGMail_<version>.mailbundle`.
    - Apply and Restart Mail.

3. In Mail.app, check `Preferences -> Free-GPGMail`. If it says that you are in Trial Mode or Decryption Only Mode, hit Activate. It will perform a dummy activation routine.

## Manage Keys

Also check [GPGTools](https://gpgtools.tenderapp.com/kb/how-to/first-steps-where-do-i-start-where-do-i-begin-setup-gpgtools-create-a-new-key-your-first-encrypted-email).

How to set up GPGSuite and GPGMail on Mac


> This article is originally done by [basicappleguy](https://basicappleguy.com/basicappleblog/bigstarrysur), and I am reposting it here for my own reference.

![Apple’s Big Sur meets Van Gogh’s Starry Night.](/static/img/2023-04-09-Big-Starry-Sur/1.png)

This image was generated using Midjourney with keyword “Big Sur” and “Starry Night".

## DOWNLOADS

![Downloads](/static/img/2023-04-09-Big-Starry-Sur/2.png)

[Mac](/static/img/2023-04-09-Big-Starry-Sur/StarrySur_Mac.heic)
|
[iPhone](/static/img/2023-04-09-Big-Starry-Sur/StarrySur_iPhone.png)
|
[iPad](/static/img/2023-04-09-Big-Starry-Sur/StarrySur_iPad.png)

Big Starry Sur


## Intro

This is a personal collection of math and computer science textbooks.
Each book contains:
- Name
- ISBN-10 ISBN-13
- Language
- Author
- Date of Publication
- Area
- Review / Description
- Preview Image
- Solution Manual (if available)

## Elements of Algebraic Topology

Name: Elements of Algebraic Topology

ISBN-10: 0201627280

ISBN-13: 978-0201627282

Language: English

Author: [James Raymond Munkres](https://en.wikipedia.org/wiki/James_Munkres)

Date of Publication: 1993

Area: Math, Algebraic Topology

Review / Description: Not finished reading yet.

![Elements of Algebraic Topology](/static/img/2023-03-05-List-of-Textbooks/1.jpg)

## Functional Analysis

Name: Functional Analysis

ISBN-10: 0070619883

ISBN-13: 978-0070619883

Language: English

Author: [Walter Rudin](https://en.wikipedia.org/wiki/Walter_Rudin)

Date of Publication: 2007

Area: Math, Functional Analysis

Review / Description: Not finished reading yet.

![Functional Analysis](/static/img/2023-03-05-List-of-Textbooks/2.jpg)

## Abstract Algebra

Name: Abstract Algebra

ISBN-10: 0471368792

ISBN-13: 978-0471368793

Language: English

Author: [Israel Nathan Herstein](https://en.wikipedia.org/wiki/Israel_Nathan_Herstein)

Date of Publication: 1996

Area: Math, Abstract Algebra

Review / Description: Not finished reading yet.

![Abstract Algebra](/static/img/2023-03-05-List-of-Textbooks/3.png)

## Principles of Mathematical Analysis / Baby Rudin

Name: Principles of Mathematical Analysis

ISBN-10: 007054235X

ISBN-13: 978-0070542358

Language: English

Author: [Walter Rudin](https://en.wikipedia.org/wiki/Walter_Rudin)

Date of Publication: 1976

Area: Math, Mathematical Analysis

Review / Description: Good book. Rudin's style.

Preview Image:

![Principles of Mathematical Analysis](/static/img/2023-03-05-List-of-Textbooks/4.png)

Solution Manual: [Check](https://minds.wisconsin.edu/handle/1793/67009)


## Real and Complex Analysis

Name: Real and Complex Analysis

ISBN-10: 0070542341

ISBN-13: 978-0070542341

Language: English

Author: [Walter Rudin](https://en.wikipedia.org/wiki/Walter_Rudin)

Date of Publication: 1976

Area: Math, Mathematical Analysis

Review / Description: Not finished reading yet.

![Real and Complex Analysis](/static/img/2023-03-05-List-of-Textbooks/5.png)

## 微积分学教程

Name: 微积分学教程

第一卷：

ISBN-10: 704018303X

ISBN-13: 9787040183030

第二卷：

ISBN-10: 7040183048

ISBN-13: 9787040183047

第三卷：

ISBN-10: 7040183056

ISBN-13: 9787040183054

Language: Chinese

Author: [Grigorii Fichtenholz](https://en.wikipedia.org/wiki/Grigorii_Fichtenholz)

Date of Publication: 2008

Area: Math, Mathematical Analysis

Review / Description: 

微积分学教程 is a Chinese translation of the original Russian textbook "Курс дифференциального и интегрального исчисления" (Course of Differential and Integral Calculus) by Grigorii Fichtenholz. The book is a comprehensive introduction to calculus and covers topics such as limits, derivatives, integrals, applications of calculus, and series.

The book is well-regarded by many mathematicians and educators, and is often used as a textbook for calculus courses in universities in Russia and China. The book is known for its clear and concise explanations, rigorous proofs, and wide range of examples and exercises.

One potential drawback of the book for some readers is that it is quite dense and may require a strong mathematical background to fully understand. Additionally, some readers may prefer a more modern approach to teaching calculus that incorporates more visual aids or real-world applications.

Overall, 微积分学教程 by Grigorii Fichtenholz is a highly respected textbook on calculus, and would likely be a valuable resource for students and educators looking to gain a deep understanding of the subject.

![微积分学教程](/static/img/2023-03-05-List-of-Textbooks/6.jpg)

## Algebra

Name: Algebra

ISBN-10: 0132413779
ISBN-13: 9780132413770

Language: English

Author: [Michael Artin](https://en.wikipedia.org/wiki/Michael_Artin)

Date of Publication: 2010

Area: Math, Abstract Algebra, Linear Algebra

Review / Description: 

Algebra is a comprehensive textbook on algebraic structures, including groups, rings, and fields. The book covers topics such as group actions, Galois theory, and homological algebra. The text is known for its clear and concise explanations, as well as its emphasis on geometric interpretations of algebraic concepts.

The book is well-regarded by many mathematicians and educators, and is often used as a textbook for advanced undergraduate or graduate-level courses in algebra. It is particularly popular among students and researchers interested in algebraic geometry and number theory.

One potential drawback of the book for some readers is that it may be too advanced for those without a strong mathematical background. Additionally, some readers may prefer a more traditional approach to algebra, rather than the geometric viewpoint presented in the book.

Overall, Algebra by Michael Artin is a highly respected textbook on algebra, and would likely be a valuable resource for students and educators looking to gain a deep understanding of the subject.

![Algebra](/static/img/2023-03-05-List-of-Textbooks/7.jpg)

Personal Collection of Textbooks


## Pyenv

### Install the pyenv
```bash
brew install pyenv-virtualenv
```

Add the following line to .zshrc or .zprofile to run pyenv
```bash
# python env auto activation
eval "$(pyenv init -)"
eval "$(pyenv virtualenv-init -)"
if which pyenv-virtualenv-init > /dev/null; then eval "$(pyenv virtualenv-init -)"; fi
```

To run pyenv in the current session, run
```bash
source ~/.zshrc
```

Install python version
List all available python version
```bash
pyenv install —list 
```

Install a specific version
```bash
pyenv install $VERSION 
```

List all installed versions
```bash
P
```

### Change versions
Set the global python version for the system
```bash
pyenv global $VERSION
```

Set local version
```bash
pyenv local $VERSION 
```

Create a specific virtualenv for repo
```bash
pyenv virtualenv $PYTHON_VERSION $NAME
```

Use virtualenv for repo
```bash
pyenv local $NAME 
```



How to Install and Use Pyenv


A personal collection of beautiful hugo themes.

- [Kembang](https://themes.gohugo.io/themes/kembang/) by [Aziz](https://github.com/httpsecure)
- [Hugo Ficurinia](https://themes.gohugo.io/themes/hugo-ficurinia/) by [Gabriele Musco](https://gabmus.org/)
- [PaperMod](https://themes.gohugo.io/themes/hugo-papermod/) by [Aditya Telange](https://github.com/adityatelange/)
- [Vno](https://themes.gohugo.io/themes/vno-hugo/) by [xsling](https://github.com/xslingcn)
- [Hugo.386](https://themes.gohugo.io/themes/hugo.386/) by [JM Fergeau](https://jmf-portfolio.netlify.com/)
- [Stack](https://themes.gohugo.io/themes/hugo-theme-stack/) by [Jimmy Cai](https://jimmycai.com/)
- [Compose](https://themes.gohugo.io/themes/compose/) by [Weru](https://themes.gohugo.io/themes/compose/)
- [TailBliss](https://themes.gohugo.io/themes/tailbliss/) by [Jeremy Nusser](https://nusserstudios.com/)
- [Clarity](https://themes.gohugo.io/themes/hugo-clarity/) by [Chip Zoller](https://neonmirrors.net/), [Dan Weru](https://github.com/onweru), [Ivan Boothe](https://rootwork.org/)
- [Clean White](https://themes.gohugo.io/themes/hugo-theme-cleanwhite/) by [Huabing Zhao](https://github.com/zhaohuabing)
- [Jane](https://themes.gohugo.io/themes/hugo-theme-jane/) by [xianmin](https://xianmin.org/)
- [PaperCSS](https://themes.gohugo.io/themes/papercss-hugo-theme/) by [Zachary Betz](https://themes.gohugo.io/themes/papercss-hugo-theme/)
- [terminal](https://themes.gohugo.io/themes/hugo-theme-terminal/) by [panr](https://radoslawkoziel.pl/)


Beautiful Hugo Theme


### Apple电池健康

![Apple电池健康](/static/img/2022-5-25-Apple电池健康/2022-05-25-Apple-Battery-Health.jpeg)

#### 🔋 关于电池健康度

[原文章见telegram](https://t.me/AppleNuts/1446)

通常一部全新的 iPhone 电池健康度会维持相当长一段时间的 100%. 不管你如何祸害你的设备。

但是，一旦 iPhone 电池的最大容量百分比不再是 100% 时，这个百分比会迅速下降，不管你如何呵护它。

这是由于 iOS 系统的电池算法问题导致的。按照算法，多数设备首次使用时，电池容量是 > 100% 的，只是以 100% 显示。其实电池是在持续而相对匀速老化的。直到低于算法计算的 100%，用户才会看到数值变化。

个人建议最好的办法是通过iMazing / 日志报告等方式查看并估算 iPhone 实际循环充电次数，来获得一个大致印象。

当 iPhone 电池容量低于86% 时，多数用户应该会明显感觉到，在不借助外部电源的情况下，设备续航已经很难维持一天使用需求。但只有低于 80% 时，官方才会承认电池需要维修。

iPad 同样拥有电池健康度计算功能，且与 iOS 的计算机制完全一致。只是 Apple 没有让它在 iPadOS 系统设置中明确显示。大概因为 iPad 电池容量更大，完整充电周期更长，因此损耗更慢。

而 Mac 电池只是会笼统显示电池状态和维修建议。

#### 💻 你的 Mac 电池健康吗？

[原文章见telegram](https://t.me/AppleNuts/1450)

在之前的推文提及过，Mac 显示电池健康度的方式非常笼统，似乎不是很希望让用户了解到真正的电池健康度。 https://t.me/AppleNuts/1446

普通用户虽然可以借助第三方工具去查询电池健康度，如 Coconut Battery，但是还要下载第三方软件，而且软件的部分功能需要收费。

所以，顺手写了一个捷径用来计算 Mac 电池的健康度，运行捷径后可以瞬间得到 循环充电次数和健康度。

* 基于捷径 + zsh 开发，信息准确，执行速度极快
* 同时兼容 Intel / M1 系列 Mac
* 在没有电池的 Mac 设备无法运行（如：Mac mini）

[Shortcut](https://www.icloud.com/shortcuts/314cf272715b4f6387554d57b221132c)

Apple电池健康


### 方舟像素字体 / Ark Pixel Font 

![方舟像素字体](/static/img/2022-5-25-方舟像素字体/2022-05-25-itch-io-banner.png)

开源的泛中日韩像素字体。支持 10、12 和 16 像素。目标是为像素风格的游戏开发提供一套可用于正文的开箱即用的字体解决方案。

这个项目不仅提供了全部的字形设计源文件，也提供了构建字体所需要的完整程序

该字体目前还处于早期开发阶段，所有尺寸仍然缺少大量汉字。有一个临时的可用于生产的替代方案，可查看[缝合怪像素字体项目](https://github.com/TakWolf/fusion-pixel-font)。

目前 12 像素为主要开发目标。10、16 像素为实验性质

[项目地址](https://github.com/TakWolf/ark-pixel-font)



方舟像素字体 / Ark Pixel Font


费米实验室的最新研究结果表明 W 玻色子的质量严重偏离标准模型的理论预言。W玻色子传递弱相互作用（物理学中的基本力之一）。因为粒子物理标准模型（SM）对W玻色子的质量有严格的限制，所以测量它的质量可以检验标准模型。费米实验室（CDF）的对撞机探测器现在报告了从 Tevatron 粒子加速器的数据中提取的W玻色子质量的精确测量值（参见 Campagnari 和 Mulders 的方法）。令人惊讶的是，研究人员发现W玻色子的质量明显高于标准模型的预测，相差7个标准差。

论文：[https://www.science.org/doi/10.1126/science.abk1781](https://www.science.org/doi/10.1126/science.abk1781)

High-precision measurement of the W boson mass with the CDF II detector


NASA 邀请民众把自己的名字加入 Artemis 宇宙飞船

登记的名字将存在闪存/U盘里，届时由 Artemis 宇宙飞船带着名字环绕月球

谁都可以登记，登记的人，都可以得到 NASA 颁发的“登机牌”

[登记](https://www.nasa.gov/send-your-name-with-artemis/)


![登机牌](/static/img/2022-3-22-Artemis-Fly-your-name-around-the-Moon/2022-03-22-NASA.jpg)




Artemis! Fly your name around the Moon!


### The first Asahi Linux Alpha Release is here!

[Asahi Linux](https://asahilinux.org/2022/03/asahi-linux-alpha-release/)

![Asahi Linux](/static/img/2022-3-21-The-First-Linux-System-Support-Mac-M1/2022-03-21-Asahi-Linux.jpg)

首个支持苹果 M1 / Pro / Max Mac 的 Linux 发行版发布，面向所有用户开放下载
https://asahilinux.org/2022/03/asahi-linux-alpha-release/

#### What works

All M1, M1 Pro, and M1 Max devices are supported except for the Mac Studio.

- Wi-Fi
- USB2 (Thunderbolt ports)
- USB3 (Mac Mini Type A ports)
- Screen (no GPU)
- NVMe
- Lid switch
- Power button
- Built-in display (framebuffer only)
- Built-in keyboard/touchpad
- Display backlight on/off
- Battery information / charge control
- RTC
- Ethernet (desktops)
- SD card reader (M1 Pro/Max)
- CPU frequency switching

M1 machines only (no Pro/Max):
- Headphones jack (might be flaky)

Mac Mini only:
- HDMI output

Not yet, but coming soon:

- USB3
- Speakers
- Display controller (backlight brightness control, V-Sync, proper DPMS)

#### What doesn’t

Everything else, but notably:
- DisplayPort
- Thunderbolt
- HDMI on the MacBooks
- Bluetooth
- GPU acceleration
- Video codec acceleration
- Neural Engine
- CPU deep idle
- Sleep mode
- Camera
- Touch Bar
Note: on the 13" MacBook Pro, you can use Fn + the number row keys (1-9, 0, and the next two) as F1..F12 in lieu of the Touch Bar.

#### Known bugs

If Wi-Fi doesn't work, try toggling it off and on in the network management menu
If the headphones jack doesn't work or only one channel works, try rebooting. There's a flakiness issue.



The First Linux System Support Mac M1


[RSS2Email](https://github.com/rss2email/rss2email)

send your RSS Feed To your Mail Box

Send RSS Feed Through Email


功能齐全的开发者工具，包括html encode decode, url encode decode, json beautifier, json to yaml converter, hash tools, check sum tools。很实用

[DevToys](https://github.com/ObuchiYuki/DevToysMac/)

![DevToys](/static/img/2022-2-15-DevToys-Swift-Army-Knife-DevTools-For-Mac/2022-02-15_1.png)

![DevToys](/static/img/2022-2-15-DevToys-Swift-Army-Knife-DevTools-For-Mac/2022-02-15_2.png)

![DevToys](/static/img/2022-2-15-DevToys-Swift-Army-Knife-DevTools-For-Mac/2022-02-15_3.png)

7 Days	0	Site 7 Days	0
30 Days	0	Site 30 Days	0
Total	0	Site Total	0