cases.tex

-a \over b
ax^2 + bx + c = 0
_1^2 He_30^40
{\det\matrix[ 1 , 0 , 0 ;0, 1, 0;0, 0, 1] \over x} + 2
\hat[x] + x
\norm{x}
\sqrt{2}
\sum_{i=0}^\infty {1\over x}
{a\choose b + 1} + 1 \over x
\int_0^1 x^2 dx
N_2 + 2H_2 -->^^\text{ reaction } 2NH_3
{_82^196 Pb} + {_{-1}^0 e} --> {_81^196 Tl}
(1\over2)^2
\lim_{x --> \infty} {1\over x} = 0
\matrix(1&0&0\\0&x^2&0;   ;0&0&1)
a &= b + c; \nonumber &+ d; &+ e \nonumber; &= f + g
\argmin__{x \in \reals}
a &= b+1\\&= c^2
\mathbb{NZQARCHO}
\cancel{x-x} = 0
\class{red}{X}
\text{We have that $1$ and $2$ are numbers} + 1
y = \cases{0 & \text{if $x < 0$} \\ 1 & \text{otherwise}}
\left)x\middle(y\right(
\begin{pmatrix}1&0\\0&1\end{pmatrix}
\series_{i=0}^{4} x_i^i
A_{ 1 , 2 \, 3 , 4 } + B^{7 \over 2}
% # empty string
(5\;5)

1 + 1 = 2 % this is simple \\ 2+2
\inner{x, y}
\prescript_1^2 He_30^40
{1 \bover 2}
\underset{n\in\integers}{\argmax} 2-x^2
P \overset{?}{=} NP
x = "x"
\frac{1}2
\binom a b
\root[3]{2}
\displaystyle{\frac 1 2}
x + \pad{x} + x = 3x
\underbrace{x}__\text[this is it]

1 + 2 + 3 = 6
\int_0^1_000_000 \d x = [x]_0^1_000_000
\begin{pmatrix} a & b \\ c & d \end{pmatrix}
He^III
\alpha + \beta = \gamma
\argmin__{n \in \naturals}
\hat{x_2}
_1^2 X_3^4
\class{red}{X + Y}
\sum_{i=1}^3 {x^i \over 3} = \series_{i=1}^{3} {x_i \over 3}
x = \begin{cases} 0 & \text{if $x <    $.} \\ 1 & \text{otherwise}\end{cases}
x + 2 &= 20\\x &= 18
x ==> 2
x = (1, 2, ..., 3)
\tau \approx 6,28

HI_t = \frac{t}{T}
RUL_t=\frac{(1-HI_t)\cdot t}{HI_t}
||\mathbf{x}||_\infty=\max\left(|x_1|, |x_2|, ..., |x_n| \right)
\underset{\mu_t,\sigma^2_t}{\mathrm{argmax}} p_{\mu_t,\sigma^2_t}(HI_t | \mathbf{x}_t)
\underset{\mu_t,\sigma^2_t}{\mathrm{argmax}} p_{\mu_t,\sigma^2_t}(HI_t | \mathbf{x}_t) p_t(\mathbf{x}_t)
\theta = \underset{\theta}{\mathrm{argmin}} l(\mu_t, \sigma^2_t, HI_t)
l_{MLE}(\mu_t, \sigma^2_t, HI_t) = -\frac{1}{N} \sum_{t\in B} \log p_\theta (HI_t | \mathbf{x}_t)
l_{MLE}(\mu_t, \sigma^2_t, HI_t) &= -\frac{1}{N} \sum_{t\in B} \log p_\theta (HI_t | \mathbf{x}_t) \\ &= -\frac{1}{N} \sum_{t\in B} \log \left(\frac{1}{\sigma_t\sqrt{2\pi}}\exp \left(-\frac{1}{2\sigma_t^2}(HI_t-\mu_t)^2 \right) \right) \\ &= -\frac{1}{N} \sum_{t\in B} \log \left(\frac{1}{\sigma_t\sqrt{2\pi}}\right) - \frac{1}{2\sigma_t^2}(HI_t-\mu_t)^2 \\ &= \frac{1}{N} \sum_{t\in B} \log(\sigma_t\sqrt{2\pi}) + \frac{1}{2\sigma_t^2}(HI_t-\mu_t)^2\\ &= \frac{1}{N} \sum_{t\in B} \log(\sigma_t) + \log(\sqrt{2\pi}) + \frac{1}{2\sigma_t^2}(HI_t-\mu_t)^2\\ &= \frac{1}{N} \sum_{t\in B} \frac{1}{2}\log(\sigma_t^2) + \frac{1}{2}\log(2\pi) + \frac{1}{2\sigma_t^2}(HI_t-\mu_t)^2\\ &= \frac{1}{2N} \sum_{t\in B} \log(\sigma_t^2) + \log(2\pi) + \frac{1}{\sigma_t^2}(HI_t-\mu_t)^2
l_{MLE}(\mu_t, a_t, HI_t) = \frac{1}{2N} \sum_{t\in B} a_t + \log(2\pi) + \frac{(HI_t-\mu_t)^2}{\exp(a_t)}
l_{MSE}(z_t, HI_t) = \frac{1}{N}\sum_{t\in B}(HI_t - z_t)^2
l(\mu_t, a_t, z_t, HI_t) &= \beta\cdot l_{MLE} + l_{MSE} \\ &= \frac{\beta}{2N} \sum_{t\in B}\left( a_t + \frac{(HI_t-\mu_t)^2}{\exp(a_t)}\right) + \frac{1}{N}\sum_{t\in B}\left(HI_t - z_t\right)^2 \\ &= \frac{1}{N} \sum_{t\in B}\frac{\beta a_t}{2} + \frac{\beta(HI_t-\mu_t)^2}{2\exp(a_t)} + (HI_t - z_t)^2

\sin
\ddot{f}(x)
\bar{\log}(x+y)
-2+2 = 0

\text{\% percent}
n! = \prod_{k=1}^n k

x =^^? 2 +_4 \mod 3

\binom{x}{y} == {x \choose y}

OH^- +H_3O^+ <-> 2H_2O

\[a\(b\)c\]