Traffine I/O

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2022-12-16

LaTeXの数式記述

LaTeXとは

LaTeXとは、数式などの複雑な構造を含む文書を美しく出力することができるマークアップ言語であり、数学や物理学などの科学分野でよく用いられます。記号$$で文字列を挟むことによりLaTeX文中内で数式を表すことができます。

LaTeXの数式記述一覧

数式 LaTeX
\equiv $\equiv$
\approx $\approx$
\fallingdotseq $\fallingdotseq$
\risingdotseq $\risingdotseq$
\sim $\simeq
\geq $\geq
\geqq $\geqq
\leq $\leq
\leqq $\leqq
\gg $\gg
\ll $\ll
\N $\N
\Z $\Z
\R $\R
\exist $\exist
\forall $\forall
\times $\times
\ast $\ast
\div $\div
\pm $\pm
\mp $\mp
\oplus $\oplus
\ominus $\ominus
\otimes $\otimes
\oslash $\oslash
\circ $\circ
\ltimes $\ltimes
\rtimes $\rtimes
\in $\in
\ni $\ni
\notin $\notin
\subset $\subset
\supset $\supset
\subseteq $\subseteq
\supseteq $\supseteq
\nsubseteq $\nsubseteq
\nsupseteq $\nsupseteq
\cap $\cap
\cup $\cup
\emptyset $\emptyset
\varnothing $\varnothing
\parallel $\parallel
x^2 $x^2$
x^{10} $x^{10}$
x^{y+1} $x^{y+1}$
x_i $x_i$
x_i^2 $x_i^2$
_n C _r $_n C _r$
\mathrm{e}^x $\mathrm{e}^x$
\pi $\pi$
\alpha $\alpha$
\beta $\beta$
\gamma $\gamma$
\mu $\mu$
\nu $\nu$
\theta $\theta$
\eta $\eta$
\delta $\delta$
\zeta $\zeta$
\ell $\ell$
\epsilon $\epsilon$
\sigma $\sigma$
\lambda $\lambda$
\tau $\tau$
\omega $\omega$
\phi $\phi$
\chi $\chi$
\nabla $\nabla$
\psi $\psi$
\kappa $\kappa$
\xi $\xi$
\varepsilon $\varepsilon$
\vartheta $\vartheta$
\varpi $\varpi$
\varsigma $\varsigma$
\varphi $\varphi$
\Gamma $\Gamma$
\Delta $\Delta$
\Omega $\Omega$
\to $\to$
\rightarrow $\rightarrow$
\Rightarrow $\Rightarrow$
\leftarrow $\leftarrow$
\Leftarrow $\Leftarrow$
\leftrightarrow $\leftrightarrow$
\Leftrightarrow $\Leftrightarrow$
\models $\models$
\cdots $\cdots$
\sin(x) $\sin(x)$
\cos(x) $\cos(x)$
\tan(x) $\tan(x)$
\neq $\neq$
x_1, x_2, \cdots, x_n $x_1, x_2, \cdots, x_n$
\dot{x} $\dot{x}$
\ddot{x} $\ddot{x}$
\vec{x} $\vec{x}$
\hat{x} $\hat{x}$
\bar{x} $\bar{x}$
\tilde{x} $\tilde{x}$
\overrightarrow{x} $\overrightarrow{x}$
\overleftarrow{x} $\overleftarrow{x}$
\infty $\infty$
\int $\int$
\lim $\lim$
\lim_{n\to \infty} a_n $\lim_{n\to \infty} a_n$
\mathrm{d} x $\mathrm{d} x$
F(x)=\int f(x) \mathrm{d} x $F(x) = \int f(x) \mathrm{d} x$
\sqrt{x} $\sqrt{x}$
\sqrt[n]{a} $\sqrt[n]{a}$
\int_{-\infty}^{\infty} \mathrm{e}^{-x^2} \mathrm{d} x $\int_{-\infty}^{\infty} \mathrm{e}^{-x^2} \mathrm{d} x$
\sum $\sum$
\sum_i^N x_i $\sum_i^N x_i$
\prod $\prod$
\prod_i x_i $\prod_i x_i$
\frac{1}{2} $\frac{1}{2}$
\sum_{k=0}^\infty \frac{h^k f^{k}(x)}{k!} $\sum_{k=0}^\infty \frac{h^k f^{k}(x)}{k!}$
a \mathrm{a} $a \mathrm{a}$
\left( \right) $\left \right)$
\left( \frac{x}{y} \right) $\left( \frac{x}{y} \right)$
\left[ \frac{x}{y} \right] $\left[ \frac{x}{y} \right]$
\left( \frac{x}{y} \right. $\left( \frac{x}{y} \right.$
\partial $\partial$
\frac{\partial f}{\partial x} $\frac{\partial f}{\partial x}$
\\ $\\$
\quad $\quad$
\underset{k}{\textrm{argmax}} $\underset{k}{\textrm{argmax}}$
\textrm{precision} × \textrm{recall} $\textrm{precision} × \textrm{recall}$

微分

\left. \frac{dy}{dx} \right|_{x=1}
$$
\left. \frac{dy}{dx} \right|_{x=1}
$$

複数行の式

\begin{aligned} (x+y)^2 &= (x+y) (x+y) \\ &= x(x+y) + y (x+y) \\ &= x^2 + xy + yx + y^2 \\ &= x^2 + 2xy+y^2 \end{aligned}
$$
\begin{aligned}
  (x+y)^2 &= (x+y) (x+y) \\
  &= x(x+y) + y (x+y) \\
  &= x^2 + xy + yx + y^2 \\
  &= x^2 + 2xy+y^2
\end{aligned}
$$

行列

\begin{pmatrix} a & b\\ c & d \end{pmatrix}
$$
\begin{pmatrix}
  a & b\\
  c & d
\end{pmatrix}
$$
\begin{bmatrix} a & b\\ c & d \end{bmatrix}
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\begin{Vmatrix} a & b \\ c & d \end{Vmatrix}
$$
\begin{Vmatrix}
  a & b \\
  c & d
\end{Vmatrix}
$$
\left( \begin{array}{c} (x^{1})^T\\ \vdots\\ (x^{N})^T\\ \end{array} \right)
$$
\left(
  \begin{array}{c}
    (x^{1})^T\\
    \vdots\\
    (x^{N})^T\\
  \end{array}
\right)
$$
A = \left( \begin{array}{ccccc} 1 & 1 & 4 & 5 & 1 \\ 1 & 0 & 5 & 0 & 4 \\ 4 & 5 & 1 & 4 & 0 \end{array} \right)
$$
A = \left(
  \begin{array}{ccccc}
    1 & 1 & 4 & 5 & 1 \\
    1 & 0 & 5 & 0 & 4 \\
    4 & 5 & 1 & 4 & 0
  \end{array}
\right)
$$
\begin{pmatrix} a_{11} & \cdots & a_{1i} & \cdots & a_{1n}\\ \vdots & \ddots & & & \vdots \\ a_{i1} & & a_{ii} & & a_{in} \\ \vdots & & & \ddots & \vdots \\ a_{n1} & \cdots & a_{ni} & \cdots & a_{nn} \end{pmatrix} \times \begin{vmatrix} a_{11} & \cdots & a_{1i} & \cdots & a_{1n}\\ \vdots & \ddots & & & \vdots \\ a_{i1} & & a_{ii} & & a_{in} \\ \vdots & & & \ddots & \vdots \\ a_{n1} & \cdots & a_{ni} & \cdots & a_{nn} \end{vmatrix}
$$
\begin{pmatrix}
  a_{11} & \cdots & a_{1i} & \cdots & a_{1n}\\
  \vdots & \ddots &        &        & \vdots \\
  a_{i1} &        & a_{ii} &        & a_{in} \\
  \vdots &        &        & \ddots & \vdots \\
  a_{n1} & \cdots & a_{ni} & \cdots & a_{nn}
\end{pmatrix}

\times

\begin{vmatrix}
  a_{11} & \cdots & a_{1i} & \cdots & a_{1n}\\
  \vdots & \ddots &        &        & \vdots \\
  a_{i1} &        & a_{ii} &        & a_{in} \\
  \vdots &        &        & \ddots & \vdots \\
  a_{n1} & \cdots & a_{ni} & \cdots & a_{nn}
\end{vmatrix}
$$

条件分岐

f(x) = \left\{ \begin{array}{ll} 1 & (x = 1) \\ 0 & (otherwise) \end{array} \right.
$$
f(x) = \left\{
  \begin{array}{ll}
    1 & (x = 1) \\
    0 & (otherwise)
  \end{array}
\right.
$$

式番号

\tag{1.0.2} x_i = \sigma (W_i + \beta)
$$
\tag{1.0.2}
x_i = \sigma (W_i + \beta)
$$

線形計画問題

\begin{equation*} \begin{aligned} & \text{minimize} & {\bf c}^T{\bf x} \\ & \text{s.t.} & A{\bf x} \le {\bf b} \\ & & {\bf x} \ge {\bf 0} \\ \end{aligned} \end{equation*}
$$
\begin{equation*}
  \begin{aligned}
    & \text{minimize}
      & {\bf c}^T{\bf x} \\
    & \text{s.t.}
      & A{\bf x} \le {\bf b} \\
      & & {\bf x} \ge {\bf 0} \\
  \end{aligned}
\end{equation*}
$$

シンプレックス法

\begin{array}{rrrrrrr} x_3 & = & 1 & + & x_1 & - & x_2 \\ x_4 & = & 3 & - & x_1 & & \\ x_5 & = & 2 & & & - & x_2 \\ \hline z & = & & & x_1 & + & x_2 \end{array}
$$
\begin{array}{rrrrrrr}
  x_3 & = & 1 & + & x_1 & - & x_2 \\
  x_4 & = & 3 & - & x_1 &   &     \\
  x_5 & = & 2 &   &     & - & x_2 \\ \hline
  z   & = &   &   & x_1 & + & x_2
\end{array}
$$

Ryusei Kakujo

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