Test formula

https://www.latex-tutorial.com/tutorials/beginners/latex-amsmath/

https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

$\frac{5ab-4x^2-\frac{1}{2}\sqrt{x}}{-\frac{2}{3}y+\frac{5}{6}} = \frac{x^2 - \frac{1}{3}x - 3\frac{ab}{s}}{3x - \frac{4}{5}y+2}$

$K_D = \frac{[A_{total}]-[AB]}{[AB]}\cdot [B]$
$\frac{[AB]}{[A_{total}]} = \frac{1}{1+\frac{K_D}{[B]}}$


$1 - (1-p)^N \sim Np$
$E_{out} = E_{in} + E_{reg}$ $\lambda$ $g$  $g(\mathbf{w^*} \vert_{\lambda^*})$
${\rm sign}(\mathbf{w_n}^\intercal \mathbf{x}_2) = -y_2$
$f(\mathbf{x}_1) = f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0)(\mathbf{x}_1-\mathbf{x}_0)$
$f(\mathbf{x}_1) - f(\mathbf{x}_0) = - \eta \nabla f(\mathbf{x}_0)\nabla f(\mathbf{x}_0)$
$\nabla f(\mathbf{x}) = \sum_{i=1}^N \; \nabla f(\mathbf{x}_i)$
$N \to \infty$
$y = w_0 + w_1x_1 + w_2x_2 + \cdots + w_dx_d + w_{11}x_1^2 + w_{12}x_1x_2 + \cdots + w_{(d-1)d}x_{d-1}x_d + w_{dd}x_d^2$
$\sum_{i=1,d} \; \vert w_i \vert \le 1.0$
$E_{out}=E_{in} \pm \sigma$
$\max \sigma = \frac{1}{2\sqrt{N}}$
$\mathbf{w} = [w_0, w_1, w_2, \ldots, w_d]^\intercal$
$\mathbf{w} = [ w_1, w_2, \ldots, w_d]^\intercal$
$y = \{+1, -1\}$
$s = \frac{1}{2}at^2$f(x) -
$s = c_0 + c_1t+ c_2t^2+ c_3t^3$
$\mathbb{E}_\mathbf{x} \Vert f(\mathbf{x}) - g^D(\mathbf{x}) \Vert^2 = \int {\Vert f(\mathbf{x}) - g^D(\mathbf{x}) \Vert^2 p(\mathbf{x})d\mathbf{x}}$
$E_{out} = E_{in} + \lambda {\Vert \tilde{\mathbf{w}}\Vert}^2$
$Z^\intercal Z^\prime=$
$p(\mathbf{w}, \mathcal{D}) = p(\mathbf{w}| \mathcal{D})p(\mathcal{D}) = p(\mathcal{D}| \mathbf{w})p(\mathbf{w})$
$w_1x_1 + w_2x_2 + \cdots + w_dx_d \ge b$
${\rm sign}(w_0 \cdot 1 + w_1x_1 + w_2x_2 + \cdots + w_dx_d)$
${\rm sign}(\mathbf{w}^\intercal \mathbf{x})$
$\mathbf{w}^\intercal\mathbf{x}_1=y_1$
$\mathbf{w}^\intercal\mathbf{x}_2=y_2$
$\cdots$
$\mathbf{w}^\intercal\mathbf{x}_N=y_N$
$E = \sum_{i=1}^N \; (\mathbf{w}^\intercal \mathbf{x}_i + b - y_i)^2 + \lambda \mathbf{w}^\intercal \mathbf{w}$
$\frac{\partial{E}}{\partial{b}} = 0 \to b = \frac{1}{N} \sum(y_i - \mathbf{w}^\intercal \mathbf{x}_i) = \frac{1}{N} \sum y_i$
$\frac{\partial{E}}{\partial{\mathbf{w}}} = 0 \to (X^\intercal X + \lambda I)\mathbf{w} = y - b$
$$\begin{equation} \phi(\mathbf{x}) = \exp(-\gamma (x_1^2+x_2^2)) \cdot \left[ 1, \frac{\sqrt{2 \gamma}}{\sqrt{1!}}x_1, \frac{\sqrt{2 \gamma}}{\sqrt{1!}}x_2, \frac{{(\sqrt{2 \gamma}x_1)}^2}{\sqrt{2!}}, 2 \gamma x_1x_2, \frac{{(\sqrt{2 \gamma}x_2)}^2}{\sqrt{2!}}, o(x_1^2 + x_1x_2 + x_2^2) \right] \end{equation}$$ $$\begin{equation} \phi(\mathbf{x}^\prime) = \exp(-\gamma ({x_1^\prime}^2+{x_2^\prime}^2)) \cdot \left[ 1, \frac{\sqrt{2 \gamma}}{\sqrt{1!}}x_1^\prime, \frac{\sqrt{2 \gamma}}{\sqrt{1!}}x_2^\prime, \frac{{(\sqrt{2 \gamma}x_1^\prime)}^2}{\sqrt{2!}}, 2 \gamma x_1^\prime x_2^\prime, \frac{{(\sqrt{2 \gamma}x_2^\prime)}^2}{\sqrt{2!}}, o({x_1^\prime}^2 + x_1^\prime x_2^\prime + {x_2^\prime}^2) \right] \end{equation}$$ $$\begin{equation} \phi(x) = \exp(-\gamma x^2) \cdot \left[ 1, \frac{(\sqrt{2 \gamma}x)}{\sqrt{1!}}, \frac{{(\sqrt{2 \gamma}x)}^2}{\sqrt{2!}}, \frac{{(\sqrt{2 \gamma}x)}^3}{\sqrt{3!}}, \ldots \right] \end{equation}$$
$$\begin{equation} \phi(x^\prime) = \exp(-\gamma {x^\prime}^2) \cdot \left[ 1, \frac{(\sqrt{2 \gamma}x^\prime)}{\sqrt{1!}}, \frac{{(\sqrt{2 \gamma}x^\prime)}^2}{\sqrt{2!}}, \frac{{(\sqrt{2 \gamma}x^\prime)}^3}{\sqrt{3!}}, \ldots \right] \end{equation}$$

$$\frac{[AB]}{[AB]+[A]} = \frac{1}{1+\frac{[A]}{[AB]}} = \frac{1}{1+\frac{K_D}{[B]}} \approx \frac{1}{1+\frac{K_D}{[B_{total}]}}$$

$y={\rm sign}(\mathbf{w}\cdot \mathbf{z}  + b)$
$s_1 = \mathbf{w}_1\mathbf{x}$
$s_i = \mathbf{w}_i\mathbf{x}$
$s_K = \mathbf{w}_K\mathbf{x}$
$f(x) \; x \; x_1 \; x_2 \; x_j \; x_m \; h \; z \; w_1 \; w_2 \; z_1 \; z_2 \; z_i \; z_{m^\prime}  w_{i1} \; w_{i2} \; w_{ij} \; w_{im}$
$s_1 = \mathbf{w}_1\mathbf{z}$
$s_i = \mathbf{w}_i\mathbf{z}$
$s_K = \mathbf{w}_K\mathbf{z}$
$f_1 = \frac{e^{s_1}}{1+e^{s_1}}$
$f_i = \frac{e^{s_i}}{1+e^{s_i}}$
$f_K = \frac{e^{s_K}}{1+e^{s_K}}$
$f_1 = \frac{e^{s_1}}{\sum_j e^{s_j}}$
$f_i = \frac{e^{s_i}}{\sum_j e^{s_j}}$
$f_K = \frac{e^{s_K}}{\sum_j e^{s_j}}$
$s_{ji} = \mathbf{w}_j\mathbf{x}_i$
$f_{ji} = \frac{e^{s_{ji}}}{1+ e^{s_{ji}}}$
$J(\theta_1, \theta_2, \theta_3, \ldots, \theta_i, \ldots, \theta_n)$
$J^\prime(\theta_1, \theta_2, \theta_3, \ldots, \theta_i+\Delta\theta_i, \ldots, \theta_n)$
$\frac{\partial{J}}{\partial{\theta_i}} = \frac{J^\prime - J}{\Delta\theta_i}$
$\cos(x)\sin(y)+\frac{y}{x}$
$\mathbf{h}_{t-1}=\begin{bmatrix} x_{t-3} \\ x_{t-2} \\ x_{t-1} \end{bmatrix}$
$\begin{bmatrix} h_{\Sigma,t-1} \\ h_{\nabla,t-1} \\ h_{mem,t-1} \end{bmatrix}$