README.md

이 파일은 유튜브 진우영 채널 (@cwychin)의 LaTeX 입문: 실습 중심의 강의 영상을 기반으로 하고 있습니다.

latex_example.tex

\documentclass{amsart}

\usepackage{amssymb}
\usepackage{hyperref}

\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\rank}{\mathrm{rank}}
\DeclareMathOperator{\Var}{\mathrm{Var}}
\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[section]{Definition}
\newtheorem{example}[theorem]{Example}

\theoremstyle{remark}
\newtheorem{remark}[section]{Remark}
\newtheorem*{claim}{Claim}

\numberwithin{equation}{section}

\title{Let's Get Started}
\author{Haizn Z. Scîe}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
    Warning: the content of this article will make no sense!
\end{abstract}

\section{American Roulette}

In American roulette, the house has an edge of 5.26\%\footnote
{It is higher than that of French roulette.} in average,
and thus \emph{gambler's ruin} applies.

\section{\LaTeX{} Skills}

\subsection{Names}

I can typeset the names of mathematicians such as Arzel\`a, \v Cech, Erd\H os,
G\"odel, and l'H\^opital.

\subsection{\TeX}

In the TeXbook, Knuth says ``the displaced `E' is a reminder that \TeX{}
is about typesetting, and it destinguished \TeX{} from other system names."

\subsection{Three kinds of Dashes}

There are three different kinds of dashes: one in X-ray, one in page 17--22,
and a punctuation dash---like this.

\section{Math Exercises 1}

\subsection{Roots and superscripts}

The two roots of $x^2 = 2$ are $\pm \sqrt 2$.
If $\sqrt{2} = p / q$ for some integers $p$ and $q$,
then we have $p^2 = 2q^2$, which is impossible.
A similar argument holds for $\sqrt[n]{2}$ with $n \geq 3$.

\subsection{Pythagorean theorem}

The Pythagorean theorem says $\alpha^2 + \beta^2 = \gamma^2$.
It follows that $\gamma = \sqrt{\alpha^2 + \beta^2}$.

\subsection{Fibonacci numbers}

Fibonacci numbers satisfy $$F_n = F_{n-1} + F_{n-2}$$
for $n = 1,2,\dots$.

\subsection{Inner products}

If $\vec x = \left( x_1, \dots, x_n \right)$ and $\vec y = \left( y_1, \dots, y_n \right)$,
then $$\vec x \cdot \vec y = x_1y_1 + \cdots + x_ny_n.$$

\subsection{Repeating decimals}

If $0.\overline 9 < 1$, then $0 < 1 - 0.\overline 9 \leq \epsilon$ should hold for
every $\epsilon > 0$.

\subsection{Approximating sine}

We have $\sin x \approx x$ for small $x$.

\subsection{Closures}

If $A_1, \dots, A_n \subset X$, then
$\overline{A_1 \cup \cdots \cup A_n} = \overline{A_1} \cup \cdots \cup \overline{A_n}$

\subsection{Inverse function}
If $f \circ g$ and $g \circ f$ are both identities,
then $g$ is denoted by $f^{-1}$.

\subsection{Set builder notation}

If $R = \left\{ x \mid x \not \in x \right\}$,
then $R \in R$ and $R \not \in R$ are both true.

\section{Math Exercises 2}

\subsection{Quadratic formula}

If $a$, $b$, and $c$ are complex numbers where $a \neq 0$,
the roots of $ax^2 + bx + c = 0$ are given by
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

\subsection{Basel sum}

In 1734, Euler proved that
$$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$$

\subsection{Euler's infinite product}

Euler's infinite product formula says
$$\frac{\sin x}{x} = \prod_{n=1}^\infty \left( 1 - \frac{x^2}{n^2\pi^2} \right).$$

\subsection{Definition of derivative}

If $y = f(x)$, we have
$$\frac{dy}{dx} = f^\prime(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$

\subsection{Stokes' theorem}

Let $S$ be an oriented surface, and $F$ be a $C^1$ vector field on $S$.
Then,
$$\iint_S \left( \nabla \times F\right) \cdot \,dS = \int_{\partial S} F \cdot \,dS.$$

\subsection{Integration by parts} \label{ssec:int}

The variance of the standard normal random variable can be computed by the
following integration by parts:
$$\int_{-\infty}^\infty x^2 \cdot \frac{e^{-x^2/2}}{\sqrt 2 \pi} = \left[ \frac{-xe^{-x^2/2}}{\sqrt 2 \pi} \right]^\infty_{x=-\infty} + \int_{-\infty}^\infty \frac{e^{-x^2/2}}{\sqrt 2 \pi} \,dx.$$

\section{Math Exercises 3}

\subsection{Command declarations}

If $n \in \Z$, then $n \in \N$.
$\inner{a}{b} = \vec a \cdot \vec b$.
$n = \rank A^TA$ and $\sigma^2 = \Var X$.

\subsection{A random set}

$$\left\{ \frac 1 p + \frac 1 q : p \text{ and } q \text{ are primes}\right\}$$

\subsection{L\'evy equivalence theorem}

If $X_1, X_2, \dots$ are independent, then
$\sum_{n=1}^\infty X_n$ converges $\text{a.s.}$
iff $\sum_{n=1}^\infty X_n$ converges in distribution.

\subsection{Completing the computation}

We continue the computation given in \ref{ssec:int}:
$$\begin{aligned}
    \int_{-\infty}^{\infty} x^2 \cdot \frac{e^{-x^2/2}}{\sqrt 2 \pi} \,dx
    &= \left[ \frac{e^{-x^2/2}}{\sqrt 2 \pi} \right]_{x=-\infty}^\infty + \int_{-\infty}^{\infty} \frac{e^{-x^2/2}}{\sqrt 2 \pi} \,dx \\
    &= (0 - 0) + 1 \\
    &= 1.
\end{aligned}$$

\subsection{A long inequality}

$$\begin{aligned}
    |\mathbf E [f(Z)] - \mathbf E [f(S)] - \mathbf E [f^{\prime\prime}(S)] \mathbf E [Y^2] / 2 | \\
    \leq \frac{\epsilon}{2} \mathbf E[Y^2] + M\mathbf E[Y^2; |Y| > \delta].
\end{aligned}$$

\subsection{Definition of cross product}

The cross product of $\mathbf x = (x_1, x_2, x_3)$ and $\mathbf y = (y_1, y_2, y_3)$
is given by
$$\mathbf x \times \mathbf y = \begin{vmatrix}
    \mathbf i & \mathbf j & \mathbf k \\
    x_1 & x_2 & x_3 \\
    y_1 & y_2 & y_3
\end{vmatrix}.$$

\subsection{Large matrices}

$$\begin{pmatrix}
    0 & x_{12} & x_{13} & \cdots & x_{1n} \\
    x_{21} & 0 & x_{23} & \cdots & x_{2n} \\
    x_{31} & x_{32} & 0 & \cdots & x_{3n} \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    x_{n1} & x_{n2} & x_{n3} & \cdots & 0
\end{pmatrix}$$

\section{Exercise 1}

\theorem[Baum--Katz]{ \label{th:42}
    If either \begin{itemize}
        \item $t \leq 2$ and $r > t/2$; or
        \item $0 < t < 2$ and $r \geq 2$,
    \end{itemize}
    then
    \begin{equation} \label{eq:41}
        \mathbf E X = 0 \text{ (in case $t \geq 1$ and $r/t \leq 1$)} \quad \text{ and } \quad \mathbf E |X|^t < \infty
    \end{equation}
    is equivalent to
    \begin{equation} \label{eq:42}
        \sum_{n=1}^\infty n^{r-2} \mathbf P\left( |S_n| > n^{r/t}\epsilon \right) < \infty \text{ for all } \epsilon > 0.
    \end{equation}

    \begin{proof}
        It is trivial that \eqref{eq:41} implies \eqref{eq:42}.
        We leave the other direction as an exercise.
    \end{proof}
}

\theorem[continuity]{
    For any function $f: \R \to \R$, the following are equivalent:
    \begin{itemize}
        \item $f$ is continuous;
        \item $f^{-1}(U)$ is open for any open $U \subset \R$;
        \item $f^{-1}(C)$ is closed for any closed $C \subset \R$.
    \end{itemize}

    \corollary{ \label{cor:43}
        The composition of any two continuous functions
        from $\R$ to itself is continuous.
    }

    \remark{
        The previous corollary holds for any continuous maps between
        arbitrary topological spaces.
    }

    \begin{proof}[Proof of Corollary \ref{cor:43}]
        Let $f,g : \R \to \R$ be continuous maps.
        For any open $U \subset \R$,
        the set $f^{-1}(U)$ is open by Theorem \ref{th:42},
        and so is $g^{-1}(f^{-1}(U))$ by the same theorem.
        Since $g^{-1}(f^{-1}(U)) = (f \circ g)^{-1}(U)$,
        the continuity of $f \circ g$ follows.
    \end{proof}
}

\end{document}