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physikerwelt/500longestMathFormulae.json

Created November 7, 2015 16:57
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500longestMathFormulae.json
[{
"math_input": \begin{align}
\boldsymbol\tau &= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i))] \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times -(\boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega))] \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{0 - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\}] \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{[\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i)] - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\}]\;\ldots\;\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) = 0 \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{[\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)] - \boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i)\}]\;\ldots\text{ addition associativity} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\} - \boldsymbol\omega\times\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i)]\;\ldots\text{ cross-product distributivity over addition} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\} - (\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i)(\boldsymbol\omega\times\boldsymbol\omega)]\;\ldots\text{ cross-product scalar multiplication} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\} - (\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i)(0)]\;\ldots\text{ self cross-product} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{\boldsymbol\omega(\boldsymbol\Delta\mathbf{r}_i\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega)\}] \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\omega\times \{\boldsymbol\Delta\mathbf{r}_i \times (\boldsymbol\omega \times \boldsymbol\Delta\mathbf{r}_i)\}]\;\ldots\text{ vector triple product} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times -(\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\alpha) + \boldsymbol\omega\times \{\boldsymbol\Delta\mathbf{r}_i \times -(\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\omega)\}]\;\ldots\text{ cross-product anticommutativity} \\
&= -\sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\alpha) + \boldsymbol\omega\times \{\boldsymbol\Delta\mathbf{r}_i \times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\omega)\}]\;\ldots\text{ cross-product scalar multiplication} \\
&= -\sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\alpha)] + -\sum_{i=1}^n m_i [\boldsymbol\omega\times \{\boldsymbol\Delta\mathbf{r}_i \times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\omega)\}]\;\ldots\text{ summation distributivity} \\
\boldsymbol\tau &= -\sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\alpha)] + \boldsymbol\omega\times -\sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i \times (\boldsymbol\Delta\mathbf{r}_i \times \boldsymbol\omega)]\;\ldots\;\boldsymbol\omega\text{ is not characteristic of particle P}_i
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\phi & = \cfrac{\partial \phi}{\partial r}~\mathbf{e}_r + \cfrac{1}{r}~\cfrac{\partial \phi}{\partial \theta}~\mathbf{e}_\theta + \cfrac{\partial \phi}{\partial z}~\mathbf{e}_z\ \\
\boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial v_r}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left(\cfrac{\partial v_r}{\partial \theta} - v_\theta\right)~\mathbf{e}_r\otimes\mathbf{e}_\theta + \cfrac{\partial v_r}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z \\
& + \cfrac{\partial v_\theta}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} + v_r \right)~\mathbf{e}_\theta\otimes\mathbf{e}_\theta + \cfrac{\partial v_\theta}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z \\
& + \cfrac{\partial v_z}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\cfrac{\partial v_z}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_\theta + \cfrac{\partial v_z}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\\
\boldsymbol{\nabla}\boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rr}}{\partial \theta} - (S_{\theta r}+S_{r\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{rr}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\
& + \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{r\theta}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{r\theta}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\
& + \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rz}}{\partial \theta} -S_{\theta z}\right]~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{rz}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\
& + \frac{\partial S_{\theta r}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta r}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\
& + \frac{\partial S_{\theta\theta}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta\theta}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\
& + \frac{\partial S_{\theta z}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta z}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\
& + \frac{\partial S_{zr}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\
& + \frac{\partial S_{z\theta}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{z\theta}}{\partial \theta} + S_{zr}\right]~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\
& + \frac{\partial S_{zz}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}~\frac{\partial S_{zz}}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_z
\end{align}
},
{
"math_input":
\begin{align}
I_1&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+}} \\
&\times
\ln\left(\frac{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2
\Theta_+}(\Delta^{(p)}_1+\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2}{-(\Delta^{(p)}_2)
^2-4p_+^2p_-^2\sin^2\Theta_+
-\sqrt{(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2 \Theta_+}(\Delta^{(p)}_1-\Delta^{(p)}_2)+\Delta^{(p)}_1\Delta^{(p)}_2
}\right) \\
&\times\left[-1-\frac{c\Delta^{(p)}_2}{p_-(E_+-cp_+\cos\Theta_+)}+\frac{p_+^2c^2\sin^2\Theta_+}
{(E_+-cp_+\cos\Theta_+)^2}-\frac{2\hbar^2\omega^2p_-\Delta^{(p)}_2}{c(E_+-cp_+\cos
\Theta_+)((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)}\right], \\
I_2&=\frac{2\pi Ac}{p_-(E_+-cp_+\cos\Theta_+)}\ln\left(
\frac{E_-+p_-c}{E_--p_-c}\right), \\
I_3&=\frac{2\pi A}{\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+
}} \\
&\times\ln\Bigg(\Big((E_-+p_-c)(4p_+^2p_-^2\sin^2\Theta_+(E_--p_-c)+(\Delta^{(p)}_1+\Delta^{(p)}_2)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\
&-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)\Big((E_--p_-c)
(4p_+^2p_-^2\sin^2\Theta_+(-E_--p_-c) \\
&+(\Delta^{(p)}_1-\Delta^{(p)}_2)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)-\sqrt{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}))\Big)^{-1}\Bigg) \\
&\times\left[\frac{c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{p_-(E_+-cp_+\cos\Theta_+)}\right.\\
&+\Big[((\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+)(E_-^3+E_-p_-c)+p_-c(2
((\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)E_-p_-c \\
&+\Delta^{(p)}_1\Delta^{(p)}_2(3E_-^2+p_-^2c^2))\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1} \\
&+\Big[-8p_+^2p_-^2m^2c^4\sin^2\Theta_+(E_+^2+E_-^2)-2\hbar^2\omega^2p_+^2\sin^2\Theta_+p_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c) \\
&+2\hbar^2\omega^2p_- m^2c^3(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)\Big]
\Big[(E_+-cp_+\cos\Theta_+)((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)\Big]^{-1} \\
&+\left.\frac{4E_+^2p_-^2(2(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2-4m^2c^4p_+^2p_-^2\sin^2\Theta_+)(\Delta^{(p)}_1E_-+\Delta^{(p)}_2p_-c)}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}\right], \\
I_4&=\frac{4\pi Ap_-c(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)}{(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+}+\frac{16\pi E_+^2p_-^2
A(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2}{((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)^2}, \\
I_5&=\frac{4\pi A}{(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)
((\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+)} \\
&\times\left[\frac{\hbar^2\omega^2p_-^2}{E_+cp_+\cos\Theta_+}
\Big[E_-[2(\Delta^{(p)}_2)^2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+8p_+^2p_-^2\sin^2\Theta_+((\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2)]
\right.\\
&+p_-c[2\Delta^{(p)}_1\Delta^{(p)}_2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)+16\Delta^{(p)}_1\Delta^{(p)}_2p_+^2p_-^2\sin^2\Theta_+]\Big]\Big[(\Delta^{(p)}_2)^2+4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\
&+ \frac{2\hbar^2\omega^2 p_{+}^2 \sin^2\Theta_+(2\Delta^{(p)}_1\Delta^{(p)}_2
p_-c+2(\Delta^{(p)}_2)^2E_-+8p_+^2p_-^2\sin^2\Theta_+ E_-)}{E_+-cp_+\cos\Theta_+}\\
&-\Big[2E_+^2p_-^2\{2((\Delta^{(p)}_2)^2-(\Delta^{(p)}_1)^2)(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2
+8p_+^2p_-^2\sin^2\Theta_+[((\Delta^{(p)}_1)^2+(\Delta^{(p)}_2)^2)(E_-^2+p_-^2c^2)\\
&+4\Delta^{(p)}_1\Delta^{(p)}_2E_-p_-c]\}\Big]\Big[(\Delta^{(p)}_2E_-+\Delta^{(p)}_1p_-c)^2+4m^2c^4p_+^2p_-^2\sin^2\Theta_+\Big]^{-1}\\
&-\left.\frac{8p_+^2p_-^2\sin^2\Theta_+(E_+^2+E_-^2)(\Delta^{(p)}_2p_-c +\Delta^{(p)}_1
E_-)}{E_+-cp_+\cos\Theta_+}\right], \\
I_6&=-\frac{16\pi E_-^2p_+^2\sin^2\Theta_+ A}{(E_+-cp_+\cos\Theta_+)^2
(-(\Delta^{(p)}_2)^2+(\Delta^{(p)}_1)^2-4p_+^2p_-^2\sin^2\Theta_+)}
\end{align}
},
{
"math_input": \begin{align}
j(i) &= j \left( \tfrac{1 + i}{2} \right) = 1 \\
j\left(\sqrt{2}i\right) &= \big(\tfrac{5}{3}\big)^3 \\
j(2i) &= \big(\tfrac{11}{2}\big)^3 \\
j\left(2\sqrt{2}i\right) &= \tfrac{125}{216} \left(19 + 13\sqrt{2} \right)^3\\
j(4i) &= \tfrac{1}{64} \left(724 + 513\sqrt{2} \right)^3\\
j\left( \tfrac{1 + 2i}{2} \right) &= \tfrac{1}{64} \left(724 - 513\sqrt{2} \right)^3\\
j\left( \tfrac{1 + 2\sqrt{2}i}{3} \right) &= \tfrac{125}{216} \left(19 - 13\sqrt{2} \right)^3\\
j(3i) &= \tfrac{1}{27} \left(2 + \sqrt{3}\right)^2 \left(21 + 20\sqrt{3}\right )^3 \\
j\left(2\sqrt{3}i\right) &= \tfrac{125}{16} \left(30 + 17\sqrt{3}\right)^3\\
j\left( \tfrac{1 + 7\sqrt{3}i}{2} \right) &= -\tfrac{64000}{7} \left(651 + 142\sqrt{21} \right)^3\\
j\left(\tfrac{1 + 3\sqrt{11}i}{10} \right) &= \tfrac{64}{27} \left(23 - 4\sqrt{33}\right)^2 \left(-77 + 15\sqrt{33} \right)^3\\
j\left(\sqrt{21}i\right) &= \tfrac{1}{32} \left(5 + 3\sqrt{3}\right)^2 \left(3 + \sqrt{7} \right)^2 \left( 65 + 34\sqrt{3} + 26\sqrt{7} + 15\sqrt{21}\right)^3\\
j\left( \tfrac{\sqrt{30}i}{1} \right) &= \tfrac{1}{16} \left(10 + 7\sqrt{2} + 4\sqrt{5} + 3\sqrt{10} \right)^4 \left( 55 + 30\sqrt{2} + 12\sqrt{5} + 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{2} \right) &= \tfrac{1}{16} \left(10 + 7\sqrt{2} - 4\sqrt{5} - 3\sqrt{10} \right)^4 \left( 55 + 30\sqrt{2} - 12\sqrt{5} - 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{5} \right) &= \tfrac{1}{16} \left(10 - 7\sqrt{2} + 4\sqrt{5} - 3\sqrt{10} \right)^4 \left( 55 - 30\sqrt{2} + 12\sqrt{5} - 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{10} \right) &= \tfrac{1}{16} \left(10 - 7\sqrt{2} - 4\sqrt{5} + 3\sqrt{10} \right)^4 \left( 55 - 30\sqrt{2} - 12\sqrt{5} + 10\sqrt{10} \right)^3\\
j\left(\tfrac{1+\sqrt{31}i}{2}\right)&=\left(1-\left(1+\tfrac{\sqrt{19}}{2}\left(\left(\tfrac{13-\sqrt{93}}{13+\sqrt{93}}\right)^{1/2}\left(\tfrac{\sqrt{31}+\sqrt{27}}{\sqrt{31}-\sqrt{27}}\right)^{1/3}+\left(\tfrac{13+\sqrt{93}}{13-\sqrt{93}}\right)^{1/2}\left(\tfrac{\sqrt{31}-\sqrt{27}}{\sqrt{31}+\sqrt{27}}\right)^{1/3}\right)\right)^2\right)^3\\
j(\sqrt{70}i) &= \left(1 + \tfrac{9}{4}\left(303 + 220\sqrt{2} + 139\sqrt{5} + 96\sqrt{10}\right)^2 \right)^3\\
j(\sqrt{94}i) &= \left(1 + \tfrac{9}{64}\left(2454 + 1736\sqrt{2} + \left(546 + 384\sqrt{2}\right)\sqrt{9 + 8\sqrt{2}} + \left(527 + 373\sqrt{2} + \left(117 + 83\sqrt{2}\right)\sqrt{9 + 8\sqrt{2}}\right)\sqrt{3 + 4\sqrt{2} + 3\sqrt{9 + 8\sqrt{2}}}\right)^2\right)^3\\
j(7i) &= \left( 1 + \tfrac{9}{4}\sqrt{21+8\sqrt{7}} \left(30 + 11\sqrt{7} + \left (6+\sqrt{7} \right )\sqrt{21+8\sqrt{7}}\right)^2 \right)^3\\
j(8i) &= \left( 1 + \tfrac{9}{4} \sqrt[4]{2} \left (1 + \sqrt{2} \right) \left(123 + 104\sqrt[4]{2} + 88\sqrt{2} + 73\sqrt[4]{8}\right)^2 \right)^3\\
j(10i) &= \left(1 + \tfrac{9}{8}\left(2402 + 1607\sqrt[4]{5} + 1074\sqrt[4]{25} + 719\sqrt[4]{125}\right)^2 \right)^3\\
j \left( \frac{5 \, i}{2} \right) &= \left(1 + \tfrac{9}{8}\left(2402 - 1607\sqrt[4]{5} + 1074\sqrt[4]{25} - 719\sqrt[4]{125}\right)^2 \right)^3\\
j(2\sqrt{58}i) &= \left(1+\tfrac{9}{256}\left(1+\sqrt{2}\right)^5\left(5+\sqrt{29}\right)^5\left(793+907\sqrt{2}+237\sqrt{29}+103\sqrt{58}\right)^2\right)^3\\
j\left( \tfrac{1 + \sqrt{1435}i}{2} \right) &= \left( 1 - 9 \left ( 9892538 + 4424079\sqrt{5} + 1544955\sqrt{41} + 690925\sqrt{205} \right )^2 \right)^3\\
j\left( \tfrac{1 + \sqrt{1555}i}{2} \right) &= \left( 1 - 9 \left ( 22297077 + 9971556\sqrt{5} + \left ( 3571365 + 1597163\sqrt{5} \right ) \sqrt{\tfrac{31 + 21\sqrt{5}}{2}} \right)^2 \right)^3\\
\end{align}
},
{
"math_input":
\begin{align}
I_1&=\frac{2\pi A}{\sqrt{\Delta_2^2+4p_i^2p_f^2\sin^2\Theta_i}}
\ln\left(
\frac{\Delta_2^2+4p_i^2p_f^2\sin^2\Theta_i-\sqrt{\Delta_2^2+4p_i^2p_f^2\sin^2
\Theta_i}(\Delta_1+\Delta_2)+\Delta_1\Delta_2}{-\Delta_2^2-4p_i^2p_f^2\sin^2\Theta_i
-\sqrt{\Delta_2^2+4p_i^2p_f^2\sin^2 \Theta_i}(\Delta_1-\Delta_2)+\Delta_1\Delta_2
}\right) \\
&\times\left[1+\frac{c\Delta_2}{p_f(E_i-cp_i\cos\Theta_i)}-\frac{p_i^2c^2\sin^2\Theta_i}
{(E_i-cp_i\cos\Theta_i)^2}-\frac{2\hbar^2\omega^2p_f\Delta_2}{c(E_i-cp_i\cos
\Theta_i)(\Delta_2^2+4p_i^2p_f^2\sin^2\Theta_i)}\right],\\
I_2&=-\frac{2\pi Ac}{p_f(E_i-cp_i\cos\Theta_i)}\ln\left(
\frac{E_f+p_fc}{E_f-p_fc}\right), \\
I_3&=\frac{2\pi A}{\sqrt{(\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i
}} \\
&\times\ln\Bigg(\Big((E_f+p_fc)(4p_i^2p_f^2\sin^2\Theta_i(E_f-p_fc)+(\Delta_1+\Delta_2)
((\Delta_2E_f+\Delta_1p_fc) \\
&-\sqrt{(\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i}))\Big)\Big((E_f-p_fc)
(4p_i^2p_f^2\sin^2\Theta_i(-E_f-p_fc) \\
&+(\Delta_1-\Delta_2)
((\Delta_2E_f+\Delta_1p_fc)-\sqrt{(\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i}))\Big)^{-1}
\Bigg) \\
&\times\left[-\frac{(\Delta_2^2+4p_i^2p_f^2\sin^2\Theta_i)(E_f^3+E_fp_f^2c^2)+p_fc(2
(\Delta_1^2-4p_i^2p_f^2\sin^2\Theta_i)E_fp_fc+\Delta_1\Delta_2(3E_f^2+p_f^2c^2))}{(\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i}\right.\\
&-\frac{c(\Delta_2E_f+\Delta_1p_fc)}{p_f(E_i-cp_i\cos\Theta_i)} \\
&-\frac{4E_i^2p_f^2(2(\Delta_2E_f+\Delta_1p_fc)^2-4m^2c^4p_i^2p_f^2\sin^2\Theta_i)(\Delta_1E_f+\Delta_2p_fc)}{((\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i)^2} \\
&+\left.\frac{8p_i^2p_f^2m^2c^4\sin^2\Theta_i(E_i^2+E_f^2)-2\hbar^2\omega^2p_i^2\sin^2\Theta_ip_fc(\Delta_2E_f+\Delta_1p_fc)+
2\hbar^2\omega^2p_f m^2c^3(\Delta_2E_f+\Delta_1p_fc)}
{(E_i-cp_i\cos\Theta_i)((\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i)}\right], \\
I_4&=-\frac{4\pi Ap_fc(\Delta_2E_f+\Delta_1p_fc)}{(\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i}
-\frac{16\pi E_i^2p_f^2
A(\Delta_2E_f+\Delta_1p_fc)^2}{((\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i)^2}, \\
I_5&=\frac{4\pi A}{(-\Delta_2^2+\Delta_1^2-4p_i^2p_f^2\sin^2\Theta_i)
((\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i)}\\
&\times\left[\frac{\hbar^2\omega^2p_f^2}{E_i-cp_i\cos\Theta_i}\right.\\
&\times\frac{E_f[2\Delta_2^2(\Delta_2^2-\Delta_1^2)+8p_i^2p_f^2\sin^2\Theta_i(\Delta_2^2+\Delta_1^2)]
+p_fc[2\Delta_1\Delta_2(\Delta_2^2-\Delta_1^2)+16\Delta_1\Delta_2p_i^2p_f^2\sin^2\Theta_i]}{\Delta_2^2+4p_i^2p_f^2\sin^2\Theta_i}\\
&+ \frac{2\hbar^2\omega^2 p_i^2\sin^2\Theta_i(2\Delta_1\Delta_2
p_fc+2\Delta_2^2E_f+8p_i^2p_f^2\sin^2\Theta_i E_f)}{E_i-cp_i\cos\Theta_i}\\
&+\frac{2E_i^2p_f^2\{2(\Delta_2^2-\Delta_1^2)(\Delta_2E_f+\Delta_1p_fc)^2
+8p_i^2p_f^2\sin^2\Theta_i[(\Delta_1^2+\Delta_2^2)(E_f^2+p_f^2c^2)
+4\Delta_1\Delta_2E_fp_fc]\}}{((\Delta_2E_f+\Delta_1p_fc)^2+4m^2c^4p_i^2p_f^2\sin^2\Theta_i)}\\
&+\left.\frac{8p_i^2p_f^2\sin^2\Theta_i(E_i^2+E_f^2)(\Delta_2p_fc +\Delta_1
E_f)}{E_i-cp_i\cos\Theta_i}\right],\\
I_6&=\frac{16\pi E_f^2p_i^2\sin^2\Theta_i A}{(E_i-cp_i\cos\Theta_i)^2
(-\Delta_2^2+\Delta_1^2-4p_i^2p_f^2\sin^2\Theta_i)},
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla} \boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rr}}{\partial \theta} - (S_{\theta r}+S_{r\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{rr}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{r\theta}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{r\theta}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{rz}}{\partial \theta} -S_{\theta z}\right]~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{rz}}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta r}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta r}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta\theta}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta\theta}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{\theta z}}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{\theta z}}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{zr}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{zr}}{\partial \theta} - S_{z\theta}\right]~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_\theta +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{z\theta}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{z\theta}}{\partial \theta} + S_{zr}\right]~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_\theta +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \frac{\partial S_{zz}}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}~\frac{\partial S_{zz}}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z\otimes\mathbf{e}_z
\end{align}
},
{
"math_input": \begin{align}
j(i) &= j \left( \tfrac{1 + i}{2} \right) = 1 \\
j\left(\sqrt{2}i\right) &= \big(\tfrac{5}{3}\big)^3 \\
j(2i) &= \big(\tfrac{11}{2}\big)^3 \\
j\left(2\sqrt{2}i\right) &= \tfrac{125}{216} \left(19 + 13\sqrt{2} \right)^3\\
j(4i) &= \tfrac{1}{64} \left(724 + 513\sqrt{2} \right)^3\\
j\left( \tfrac{1 + 2i}{2} \right) &= \tfrac{1}{64} \left(724 - 513\sqrt{2} \right)^3\\
j\left( \tfrac{1 + 2\sqrt{2}i}{3} \right) &= \tfrac{125}{216} \left(19 - 13\sqrt{2} \right)^3\\
j(3i) &= \tfrac{1}{27} \left(2 + \sqrt{3}\right)^2 \left(21 + 20\sqrt{3}\right )^3 \\
j\left(2\sqrt{3}i\right) &= \tfrac{125}{16} \left(30 + 17\sqrt{3}\right)^3\\
j\left( \tfrac{1 + 7\sqrt{3}i}{2} \right) &= -\tfrac{64000}{7} \left(651 + 142\sqrt{21} \right)^3\\
j\left(\tfrac{1 + 3\sqrt{11}i}{10} \right) &= \tfrac{64}{27} \left(23 - 4\sqrt{33}\right)^2 \left(-77 + 15\sqrt{33} \right)^3\\
j\left(\sqrt{21}i\right) &= \tfrac{1}{32} \left(5 + 3\sqrt{3}\right)^2 \left(3 + \sqrt{7} \right)^2 \left( 65 + 34\sqrt{3} + 26\sqrt{7} + 15\sqrt{21}\right)^3\\
j\left( \tfrac{\sqrt{30}i}{1} \right) &= \tfrac{1}{16} \left(10 + 7\sqrt{2} + 4\sqrt{5} + 3\sqrt{10} \right)^4 \left( 55 + 30\sqrt{2} + 12\sqrt{5} + 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{2} \right) &= \tfrac{1}{16} \left(10 + 7\sqrt{2} - 4\sqrt{5} - 3\sqrt{10} \right)^4 \left( 55 + 30\sqrt{2} - 12\sqrt{5} - 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{5} \right) &= \tfrac{1}{16} \left(10 - 7\sqrt{2} + 4\sqrt{5} - 3\sqrt{10} \right)^4 \left( 55 - 30\sqrt{2} + 12\sqrt{5} - 10\sqrt{10} \right)^3\\
j\left( \tfrac{\sqrt{30}i}{10} \right) &= \tfrac{1}{16} \left(10 - 7\sqrt{2} - 4\sqrt{5} + 3\sqrt{10} \right)^4 \left( 55 - 30\sqrt{2} - 12\sqrt{5} + 10\sqrt{10} \right)^3\\
j\left(\tfrac{1+\sqrt{31}i}{2}\right)&=\left(1-\left(1+\tfrac{\sqrt{19}}{2}\left(\left(\tfrac{13-\sqrt{93}}{13+\sqrt{93}}\right)^{1/2}\left(\tfrac{\sqrt{31}+\sqrt{27}}{\sqrt{31}-\sqrt{27}}\right)^{1/3}+\left(\tfrac{13+\sqrt{93}}{13-\sqrt{93}}\right)^{1/2}\left(\tfrac{\sqrt{31}-\sqrt{27}}{\sqrt{31}+\sqrt{27}}\right)^{1/3}\right)\right)^2\right)^3\\
j(\sqrt{70}i) &= \left(1 + \tfrac{9}{4}\left(303 + 220\sqrt{2} + 139\sqrt{5} + 96\sqrt{10}\right)^2 \right)^3\\
j(7i) &= \left( 1 + \tfrac{9}{4}\sqrt{21+8\sqrt{7}} \left(30 + 11\sqrt{7} + \left (6+\sqrt{7} \right )\sqrt{21+8\sqrt{7}}\right)^2 \right)^3\\
j(8i) &= \left( 1 + \tfrac{9}{4} \sqrt[4]{2} \left (1 + \sqrt{2} \right) \left(123 + 104\sqrt[4]{2} + 88\sqrt{2} + 73\sqrt{2}\sqrt[4]{2}\right)^2 \right)^3\\
j\left( \tfrac{1 + \sqrt{1435}i}{2} \right) &= \left( 1 - 9 \left ( 9892538 + 4424079\sqrt{5} + 1544955\sqrt{41} + 690925\sqrt{205} \right )^2 \right)^3\\
j\left( \tfrac{1 + \sqrt{1555}i}{2} \right) &= \left( 1 - 9 \left ( 22297077 + 9971556\sqrt{5} + \left ( 3571365 + 1597163\sqrt{5} \right ) \sqrt{\tfrac{31 + 21\sqrt{5}}{2}} \right)^2 \right)^3
\end{align}
},
{
"math_input": \begin{align}
r:\ &\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + \frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_r}{\partial \phi} +
\frac{u_{\theta}}{r} \frac{\partial u_r}{\partial \theta} - \frac{u_{\phi}^2 + u_{\theta}^2}{r}\right) =
-\frac{\partial p}{\partial r} + \rho g_r + \\
&\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_r}{\partial r}\right) +
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_r}{\partial \phi^2} +
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_r}{\partial \theta}\right) - 2\frac{u_r +
\frac{\partial u_{\theta}}{\partial \theta} + u_{\theta} \cot(\theta)}{r^2} - \frac{2}{r^2 \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi}
\right] \\
\phi:\ &\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} +
\frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\phi}}{\partial \phi} + \frac{u_{\theta}}{r} \frac{\partial u_{\phi}}{\partial \theta} +
\frac{u_r u_{\phi} + u_{\phi} u_{\theta} \cot(\theta)}{r}\right) =
-\frac{1}{r \sin(\theta)} \frac{\partial p}{\partial \phi} + \rho g_{\phi} + \\
&\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\phi}}{\partial r}\right) +
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\phi}}{\partial \phi^2} +
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\phi}}{\partial \theta}\right) +
\frac{2 \sin(\theta) \frac{\partial u_r}{\partial \phi} + 2 \cos(\theta) \frac{\partial u_{\theta}}{\partial \phi} -
u_{\phi}}{r^2 \sin(\theta)^2}
\right] \\
\theta:\ &\rho \left(\frac{\partial u_{\theta}}{\partial t} + u_r \frac{\partial u_{\theta}}{\partial r} +
\frac{u_{\phi}}{r \sin(\theta)} \frac{\partial u_{\theta}}{\partial \phi} +
\frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta} + \frac{u_r u_{\theta} - u_{\phi}^2 \cot(\theta)}{r}\right) =
-\frac{1}{r} \frac{\partial p}{\partial \theta} + \rho g_{\theta} + \\
&\mu \left[\frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial u_{\theta}}{\partial r}\right) +
\frac{1}{r^2 \sin(\theta)^2} \frac{\partial^2 u_{\theta}}{\partial \phi^2} +
\frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta}\left(\sin(\theta) \frac{\partial u_{\theta}}{\partial \theta}\right) +
\frac{2}{r^2} \frac{\partial u_r}{\partial \theta} - \frac{u_{\theta} +
2 \cos(\theta) \frac{\partial u_{\phi}}{\partial \phi}}{r^2 \sin(\theta)^2}
\right].
\end{align}
},
{
"math_input": {}_{{}_{\frac{\sqrt[7]{23328}}{6}\left\{\left[-\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}}{12}+ {\rm{i}} \left(-\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right]
\sqrt[7]{-2+\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+
\left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+{\rm{i}}\left(\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right]
\sqrt[7]{-2+\sqrt[3]{28+84\sqrt3{\rm{i}}}+\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+
\left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+{\rm{i}}\left(\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{-52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right]
\sqrt[7]{-2+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+
\left[ -\frac{1}{6}+\frac{\sqrt[3]{28+84\sqrt{3}{\rm{i}}}+\sqrt[3]{28-84\sqrt{3}{\rm{i}}}}{12}+ {\rm{i}} \left(-\frac{\sqrt7}{6}+\frac{-1-\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7+12\sqrt{21}{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{24}\sqrt[3]{52\sqrt7-12\sqrt{21}{\rm{i}}}\right)\right]
\sqrt[7]{-2+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+
\sqrt[7]{-2+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(-14+\sqrt[3]{2548+588\sqrt3{\rm{i}}}+\sqrt[3]{2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}+
\sqrt[7]{-2+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{28+84\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{28-84\sqrt3{\rm{i}}}+\frac{\sqrt7}{7}\left(14+\frac{-1+\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548+588\sqrt3{\rm{i}}}+\frac{-1-\sqrt3{\rm{i}}}{2}\sqrt[3]{-2548-588\sqrt3{\rm{i}}}\right){\rm{i}}}\right\}}}\,
},
{
"math_input":
\begin{matrix}
z_0 = -1 & f[z_0] = 2 & & & & & & & & \\
& & \frac{f'(z_0)}{1} = -8 & & & & & & & \\
z_1 = -1 & f[z_1] = 2 & & \frac{f''(z_1)}{2} = 28 & & & & & & \\
& & \frac{f'(z_1)}{1} = -8 & & f[z_3,z_2,z_1,z_0] = -21 & & & & & \\
z_2 = -1 & f[z_2] = 2 & & f[z_3,z_2,z_1] = 7 & & 15 & & & & \\
& & f[z_3,z_2] = -1 & & f[z_4,z_3,z_2,z_1] = -6 & & -10 & & & \\
z_3 = 0 & f[z_3] = 1 & & f[z_4,z_3,z_2] = 1 & & 5 & & 4 & & \\
& & \frac{f'(z_3)}{1} = 0 & & f[z_5,z_4,z_3,z_2] = -1 & & -2 & & -1 & \\
z_4 = 0 & f[z_4] = 1 & & \frac{f''(z_4)}{2} = 0 & & 1 & & 2 & & 1 \\
& & \frac{f'(z_4)}{1} = 0 & & f[z_6,z_5,z_4,z_3] = 1 & & 2 & & 1 & \\
z_5 = 0 & f[z_5] = 1 & & f[z_6,z_5,z_4] = 1 & & 5 & & 4 & & \\
& & f[z_6,z_5] = 1 & & f[z_7,z_6,z_5,z_4] = 6 & & 10 & & & \\
z_6 = 1 & f[z_6] = 2 & & f[z_7,z_6,z_5] = 7 & & 15 & & & & \\
& & \frac{f'(z_7)}{1} = 8 & & f[z_8,z_7,z_6,z_5] = 21 & & & & & \\
z_7 = 1 & f[z_7] = 2 & & \frac{f''(z_7)}{2} = 28 & & & & & & \\
& & \frac{f'(z_8)}{1} = 8 & & & & & & & \\
z_8 = 1 & f[z_8] = 2 & & & & & & & & \\
\end{matrix}
},
{
"math_input":
\begin{array}{lll}
& L_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 2 & . & . & . & . & . \\
. & . & 3 & . & . & . & . \\
. & . & . & 4 & . & . & . \\
. & . & . & . & 5 & . & . \\
. & . & . & . & . & 6 & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & . & . & . & . & . & . \\
1 & 1 & . & . & . & . & . \\
1 & 2 & 1 & . & . & . & . \\
1 & 3 & 3 & 1 & . & . & . \\
1 & 4 & 6 & 4 & 1 & . & . \\
1 & 5 & 10 & 10 & 5 & 1 & . \\
1 & 6 & 15 & 20 & 15 & 6 & 1
\end{smallmatrix}
\right ]
;\quad
\\
\\
& U_7=\exp
\left (
\left [
\begin{smallmatrix}
. & 1 & . & . & . & . & . \\
. & . & 2 & . & . & . & . \\
. & . & . & 3 & . & . & . \\
. & . & . & . & 4 & . & . \\
. & . & . & . & . & 5 & . \\
. & . & . & . & . & . & 6 \\
. & . & . & . & . & . & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
. & 1 & 2 & 3 & 4 & 5 & 6 \\
. & . & 1 & 3 & 6 & 10 & 15 \\
. & . & . & 1 & 4 & 10 & 20 \\
. & . & . & . & 1 & 5 & 15 \\
. & . & . & . & . & 1 & 6 \\
. & . & . & . & . & . & 1
\end{smallmatrix}
\right ]
;
\\
\\
\therefore & S_7
=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 2 & . & . & . & . & . \\
. & . & 3 & . & . & . & . \\
. & . & . & 4 & . & . & . \\
. & . & . & . & 5 & . & . \\
. & . & . & . & . & 6 & .
\end{smallmatrix}
\right ]
\right )
\exp
\left (
\left [
\begin{smallmatrix}
. & 1 & . & . & . & . & . \\
. & . & 2 & . & . & . & . \\
. & . & . & 3 & . & . & . \\
. & . & . & . & 4 & . & . \\
. & . & . & . & . & 5 & . \\
. & . & . & . & . & . & 6 \\
. & . & . & . & . & . & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
1 & 3 & 6 & 10 & 15 & 21 & 28 \\
1 & 4 & 10 & 20 & 35 & 56 & 84 \\
1 & 5 & 15 & 35 & 70 & 126 & 210 \\
1 & 6 & 21 & 56 & 126 & 252 & 462 \\
1 & 7 & 28 & 84 & 210 & 462 & 924
\end{smallmatrix}
\right ].
\end{array}
},
{
"math_input":
\begin{align}
\ln q_\mu^*(\mu) &= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ \sum_{n=1}^N (x_n-\mu)^2 + \lambda_0(\mu-\mu_0)^2 \} + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ \sum_{n=1}^N (x_n^2-2x_n\mu + \mu^2) + \lambda_0(\mu^2-2\mu_0\mu + \mu_0^2) \} + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ (\sum_{n=1}^N x_n^2)-2(\sum_{n=1}^N x_n)\mu + \sum_{n=1}^N \mu^2) + \lambda_0\mu^2-2\lambda_0\mu_0\mu + \lambda_0\mu_0^2) \} + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ (\lambda_0+N)\mu^2 -2(\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n)\mu + (\textstyle\sum_{n=1}^N x_n^2) + \lambda_0\mu_0^2 \} + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ (\lambda_0+N)\mu^2 -2(\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n)\mu \} + C_4 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ (\lambda_0+N)\mu^2 -2\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}(\lambda_0+N) \mu \right\} + C_4 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ (\lambda_0+N)\left(\mu^2 -2\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N} \mu\right) \right\} + C_4 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ (\lambda_0+N)\left(\mu^2 -2\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N} \mu + \left(\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}\right)^2 - \left(\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}\right)^2\right) \right\} + C_4 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ (\lambda_0+N)\left(\mu^2 -2\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N} \mu + \left(\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}\right)^2 \right) \right\} + C_5 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ (\lambda_0+N)\left(\mu-\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}\right)^2 \right\} + C_5 \\
&= - \frac{1}{2} \left\{ (\lambda_0+N)\operatorname{E}_{\tau}[\tau] \left(\mu-\frac{\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n}{\lambda_0+N}\right)^2 \right\} + C_5 \\
\end{align}
},
{
"math_input": \left(
\begin{array}{llll}
\left\{\Gamma _{tt}^t,\Gamma _{tr}^t,\Gamma _{t\theta }^t,\Gamma _{t\phi }^t\right\} & \left\{\Gamma _{rt}^t,\Gamma _{rr}^t,\Gamma
_{r\theta }^t,\Gamma _{r\phi }^t\right\} & \left\{\Gamma _{\theta t}^t,\Gamma _{\theta r}^t,\Gamma _{\theta \theta }^t,\Gamma _{\theta
\phi }^t\right\} & \left\{\Gamma _{\phi t}^t,\Gamma _{\phi r}^t,\Gamma _{\phi \theta }^t,\Gamma _{\phi \phi }^t\right\} \\
\left\{\Gamma _{tt}^r,\Gamma _{tr}^r,\Gamma _{t\theta }^r,\Gamma _{t\phi }^r\right\} & \left\{\Gamma _{rt}^r,\Gamma _{rr}^r,\Gamma
_{r\theta }^r,\Gamma _{r\phi }^r\right\} & \left\{\Gamma _{\theta t}^r,\Gamma _{\theta r}^r,\Gamma _{\theta \theta }^r,\Gamma _{\theta
\phi }^r\right\} & \left\{\Gamma _{\phi t}^r,\Gamma _{\phi r}^r,\Gamma _{\phi \theta }^r,\Gamma _{\phi \phi }^r\right\} \\
\left\{\Gamma _{tt}^{\theta },\Gamma _{tr}^{\theta },\Gamma _{t\theta }^{\theta },\Gamma _{t\phi }^{\theta }\right\} & \left\{\Gamma
_{rt}^{\theta },\Gamma _{rr}^{\theta },\Gamma _{r\theta }^{\theta },\Gamma _{r\phi }^{\theta }\right\} & \left\{\Gamma _{\theta t}^{\theta
},\Gamma _{\theta r}^{\theta },\Gamma _{\theta \theta }^{\theta },\Gamma _{\theta \phi }^{\theta }\right\} & \left\{\Gamma _{\phi
t}^{\theta },\Gamma _{\phi r}^{\theta },\Gamma _{\phi \theta }^{\theta },\Gamma _{\phi \phi }^{\theta }\right\} \\
\left\{\Gamma _{tt}^{\phi },\Gamma _{tr}^{\phi },\Gamma _{t\theta }^{\phi },\Gamma _{t\phi }^{\phi }\right\} & \left\{\Gamma _{rt}^{\phi
},\Gamma _{rr}^{\phi },\Gamma _{r\theta }^{\phi },\Gamma _{r\phi }^{\phi }\right\} & \left\{\Gamma _{\theta t}^{\phi },\Gamma _{\theta
r}^{\phi },\Gamma _{\theta \theta }^{\phi },\Gamma _{\theta \phi }^{\phi }\right\} & \left\{\Gamma _{\phi t}^{\phi },\Gamma _{\phi
r}^{\phi },\Gamma _{\phi \theta }^{\phi },\Gamma _{\phi \phi }^{\phi }\right\}
\end{array}
\right)
},
{
"math_input":
\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\
\begin{array}{|l|} \hline \angle' \\ \alpha \\ \alpha\;\angle' \\ \hline\beta \\ \beta\;\angle' \\ \gamma \\ \hline\gamma\;\angle' \\ \delta \\ \delta\;\angle' \\ \hline\varepsilon \\ \varepsilon\;\angle' \\ \stigma \\ \hline\stigma\;\angle' \\ \zeta \\ \zeta\;\angle' \\ \hline \end{array} & \begin{array}{|r|r|r|} \hline\circ & \lambda\alpha & \kappa\varepsilon \\ \alpha & \beta & \nu \\ \alpha & \lambda\delta & \iota\varepsilon \\ \hline \beta & \varepsilon & \mu \\ \beta & \lambda\zeta & \delta \\ \gamma & \eta & \kappa\eta \\ \hline \gamma & \lambda\vartheta & \nu\beta \\ \delta & \iota\alpha & \iota\stigma \\ \delta & \mu\beta & \mu \\ \hline \varepsilon & \iota\delta & \delta \\ \varepsilon & \mu\varepsilon & \kappa\zeta \\ \stigma & \iota\stigma & \mu\vartheta \\ \hline \stigma & \mu\eta & \iota\alpha \\ \zeta & \iota\vartheta & \lambda\gamma \\ \zeta & \nu & \nu\delta \\ \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \nu \\ \hline \circ & \alpha & \beta & \nu \\ \circ & \alpha & \beta & \mu\eta \\ \circ & \alpha & \beta & \mu\eta \\ \hline\circ & \alpha & \beta & \mu\eta \\ \circ & \alpha & \beta & \mu\zeta \\ \circ & \alpha & \beta & \mu\zeta \\ \hline \circ & \alpha & \beta & \mu\stigma \\ \circ & \alpha & \beta & \mu\varepsilon \\ \circ & \alpha & \beta & \mu\delta \\ \hline \circ & \alpha & \beta & \mu\gamma \\ \circ & \alpha & \beta & \mu\beta \\ \circ & \alpha & \beta & \mu\alpha \\ \hline \end{array}
\end{array}
},
{
"math_input":
\begin{align}
\mathbf{w}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{w} & =
\left(\sum_{i=1}^l\alpha_i\phi^{\text{T}}(\mathbf{x}_i)\right)\left(\sum_{j=1,2}\sum_{n =1}^{l_j}(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})^{\text{T}}\right)
\left(\sum_{k=1}^l\alpha_k\phi(\mathbf{x}_k)\right)\\
& = \sum_{j=1,2}\sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\alpha_i\phi^{\text{T}}(\mathbf{x}_i)(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})^{\text{T}}
\alpha_k\phi(\mathbf{x}_k) \\
& = \sum_{j=1,2}\sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l \left(\alpha_ik(\mathbf{x}_i,\mathbf{x}_n^j)-\frac{1}{l_j}\sum_{p=1}^{l_j}\alpha_ik(\mathbf{x}_i,\mathbf{x}_p^j)\right)
\left(\alpha_kk(\mathbf{x}_k,\mathbf{x}_n^j)-\frac{1}{l_j}\sum_{q=1}^{l_j}\alpha_kk(\mathbf{x}_k,\mathbf{x}_q^j)\right) \\
& = \sum_{j=1,2}\left( \sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\Bigg( \alpha_i\alpha_kk(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_n^j)\right.\\
& \left.{} - \frac{2\alpha_i\alpha_k}{l_j}\sum_{p=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_p^j)
\left. + \frac{\alpha_i\alpha_k}{l_j^2}\sum_{p=1}^{l_j}\sum_{q=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_p^j)k(\mathbf{x}_k,\mathbf{x}_q^j) \right)\right) \\
& = \sum_{j=1,2}\left( \sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\left( \alpha_i\alpha_kk(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_n^j)
- \frac{\alpha_i\alpha_k}{l_j}\sum_{p=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_p^j) \right)\right) \\
& = \sum_{j=1,2} \mathbf{\alpha}^{\text{T}} \mathbf{K}_j\mathbf{K}_j^{\text{T}}\mathbf{\alpha} - \mathbf{\alpha}^{\text{T}} \mathbf{K}_j\mathbf{1}_{l_j}\mathbf{K}_j^{\text{T}}\mathbf{\alpha} \\
& = \mathbf{\alpha}^{\text{T}}\mathbf{N}\mathbf{\alpha}.
\end{align}
},
{
"math_input":
\begin{align}
\cfrac{\partial W}{\partial I_1} & =
\cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_1} +
\cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_1} +
\cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_1} \\
& = I_3^{-1/3}~\cfrac{\partial W}{\partial \bar{I}_1}
= J^{-2/3}~\cfrac{\partial W}{\partial \bar{I}_1} \\
\cfrac{\partial W}{\partial I_2} & =
\cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_2} +
\cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_2} +
\cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_2} \\
& = I_3^{-2/3}~\cfrac{\partial W}{\partial \bar{I}_2}
= J^{-4/3}~\cfrac{\partial W}{\partial \bar{I}_2} \\
\cfrac{\partial W}{\partial I_3} & =
\cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_3} +
\cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_3} +
\cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_3} \\
& = - \cfrac{1}{3}~I_3^{-4/3}~I_1~\cfrac{\partial W}{\partial \bar{I}_1}
- \cfrac{2}{3}~I_3^{-5/3}~I_2~\cfrac{\partial W}{\partial \bar{I}_2}
+ \cfrac{1}{2}~I_3^{-1/2}~\cfrac{\partial W}{\partial J} \\
& = - \cfrac{1}{3}~J^{-8/3}~J^{2/3}~\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}
- \cfrac{2}{3}~J^{-10/3}~J^{4/3}~\bar{I}_2~\cfrac{\partial W}{\partial \bar{I}_2}
+ \cfrac{1}{2}~J^{-1}~\cfrac{\partial W}{\partial J} \\
& = -\cfrac{1}{3}~J^{-2}~\left(\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}+
2~\bar{I}_2~\cfrac{\partial W}{\partial \bar{I}_2}\right) +
\cfrac{1}{2}~J^{-1}~\cfrac{\partial W}{\partial J}
\end{align}
},
{
"math_input":
\begin{align}
\begin{bmatrix}
\dot{\mathbf{e}}_1\\
\dot{\mathbf{e}}_2\\
\vdots\\
\dot{\mathbf{e}}_i\\
\vdots\\
\dot{\mathbf{e}}_{n-1}\\
\dot{\mathbf{e}}_n
\end{bmatrix}
&=
\mathord{\overbrace{
\begin{bmatrix}
\dot{h}_1(\mathbf{x})\\
\dot{h}_2(\mathbf{x})\\
\vdots\\
\dot{h}_i(\mathbf{x})\\
\vdots\\
\dot{h}_{n-1}(\mathbf{x})\\
\dot{h}_n(\mathbf{x})
\end{bmatrix}
}^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{x})}}
-
\mathord{\overbrace{
M(\hat{\mathbf{x}}) \, \operatorname{sgn}( V(t) - H(\hat{\mathbf{x}}(t)) )
}^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(\mathbf{\hat{x}})}}
=
\begin{bmatrix}
h_2(\mathbf{x})\\
h_3(\mathbf{x})\\
\vdots\\
h_{i+1}(\mathbf{x})\\
\vdots\\
h_n(\mathbf{x})\\
L_f^n h(\mathbf{x})
\end{bmatrix}
-
\begin{bmatrix}
m_1 \operatorname{sgn}( v_1(t) - h_1(\hat{\mathbf{x}}(t)) )\\
m_2 \operatorname{sgn}( v_2(t) - h_2(\hat{\mathbf{x}}(t)) )\\
\vdots\\
m_i \operatorname{sgn}( v_i(t) - h_i(\hat{\mathbf{x}}(t)) )\\
\vdots\\
m_{n-1} \operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\hat{\mathbf{x}}(t)) )\\
m_n \operatorname{sgn}( v_n(t) - h_n(\hat{\mathbf{x}}(t)) )
\end{bmatrix}\\
&=
\begin{bmatrix}
h_2(\mathbf{x}) - m_1(\hat{\mathbf{x}}) \operatorname{sgn}( \mathord{\overbrace{ \mathord{\overbrace{v_1(t)}^{v_1(t) = y(t) = h_1(\mathbf{x})}} - h_1(\hat{\mathbf{x}}(t)) }^{\mathbf{e}_1}} )\\
h_3(\mathbf{x}) - m_2(\hat{\mathbf{x}}) \operatorname{sgn}( v_2(t) - h_2(\hat{\mathbf{x}}(t)) )\\
\vdots\\
h_{i+1}(\mathbf{x}) - m_i(\hat{\mathbf{x}}) \operatorname{sgn}( v_i(t) - h_i(\hat{\mathbf{x}}(t)) )\\
\vdots\\
h_n(\mathbf{x}) - m_{n-1}(\hat{\mathbf{x}}) \operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\hat{\mathbf{x}}(t)) )\\
L_f^n h(\mathbf{x}) - m_n(\hat{\mathbf{x}}) \operatorname{sgn}( v_n(t) - h_n(\hat{\mathbf{x}}(t)) )
\end{bmatrix}.
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\sigma} & =
\cfrac{2}{\sqrt{I_3}}\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + 2\sqrt{I_3}~\cfrac{\partial\hat{W}}{\partial I_3}~\boldsymbol{\mathit{1}} \\
& = \cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{B} -
\cfrac{1}{J^{4/3}}~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\
& \qquad \qquad + \left[\cfrac{\partial\bar{W}}{\partial J} - \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\
& = \cfrac{2}{J}\left[\left(\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\bar{\boldsymbol{B}} -
\cfrac{\partial\bar{W}}{\partial \bar{I}_2}~\bar{\boldsymbol{B}} \cdot\bar{\boldsymbol{B}} \right] + \left[\cfrac{\partial\bar{W}}{\partial J} - \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\
& = \cfrac{\lambda_1}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{\lambda_2}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \cfrac{\lambda_3}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) &= -[\boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i)] \\
&= -[(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)(\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i(\boldsymbol\omega\cdot(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i))]\;\ldots\text{ vector triple product} \\
&= -[(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)(\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i(\boldsymbol\Delta\mathbf{r}_i\cdot(\boldsymbol\omega\times\boldsymbol\omega))]\;\ldots\text{ scalar triple product} \\
&= -[(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)(\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i) - \boldsymbol\Delta\mathbf{r}_i(\boldsymbol\Delta\mathbf{r}_i\cdot(0))]\;\ldots\text{ self cross-product} \\
&= -[(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)(\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i)] \\
&= -[\boldsymbol\omega\times(\boldsymbol\Delta\mathbf{r}_i (\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i))]\;\ldots\text{ cross-product scalar multiplication} \\
&= \boldsymbol\omega\times -(\boldsymbol\Delta\mathbf{r}_i (\boldsymbol\omega\cdot\boldsymbol\Delta\mathbf{r}_i))\;\ldots\text{ cross-product scalar multiplication} \\
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) &= \boldsymbol\omega\times -(\boldsymbol\Delta\mathbf{r}_i (\boldsymbol\Delta\mathbf{r}_i \cdot \boldsymbol\omega))\;\ldots\text{ dot-product commutativity} \\
\end{align}
},
{
"math_input": \begin{align}
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) + \boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i) + (\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times(\boldsymbol\Delta\mathbf{r}_i\times\boldsymbol\omega) &= 0 \\
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) + \boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i) + (\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times -(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i) &= 0\;\ldots\text{ cross-product anticommutativity} \\
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) + \boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i) + -[(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)] &= 0\;\ldots\text{ cross-product scalar multiplication} \\
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) + \boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i) + -[0] &= 0\;\ldots\text{ self cross-product} \\
\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)) + \boldsymbol\omega\times((\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i)\times\boldsymbol\Delta\mathbf{r}_i) &= 0
\end{align}
},
{
"math_input":
\begin{align}
& \varepsilon\,
\left\{
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right\}
\\
& + \varepsilon^2\,
\left\{
\frac{\partial^2 \Phi_2}{\partial t^2}
+ g\, \frac{\partial \Phi_2}{\partial z}
+ \eta_1\, \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right)
+ \frac{\partial}{\partial t} \left( |\mathbf{u}_1|^2 \right)
\right\}
\\
& + \varepsilon^3\,
\left\{
\frac{\partial^2 \Phi_3}{\partial t^2}
+ g\, \frac{\partial \Phi_3}{\partial z}
+ \eta_1\, \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi_2}{\partial t^2}
+ g\, \frac{\partial \Phi_2}{\partial z}
\right)
\right.
\\ & \qquad \quad \left.
+ \eta_2\, \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right)
+ 2\, \frac{\partial}{\partial t} \left( \mathbf{u}_1 \cdot \mathbf{u}_2 \right)
\right.
\\ & \qquad \quad \left.
+ \tfrac12\, \eta_1^2\,
\frac{\partial^2}{\partial z^2}
\left(
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right)
+ \eta_1\, \frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}_1|^2 \right)
+ \tfrac12\, \mathbf{u}_1 \cdot \boldsymbol{\nabla} \left( |\mathbf{u}_1|^2 \right)
\right\}
\\ &
+ \mathcal{O}\left( \varepsilon^4 \right)
= 0,
\qquad \text{at } z=0.
\end{align}
},
{
"math_input": \begin{align}
r:\ &\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} +
\frac{u_{\phi}}{r} \frac{\partial u_r}{\partial \phi} + u_z \frac{\partial u_r}{\partial z} - \frac{u_{\phi}^2}{r}\right) = {}\\
&-\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) +
\frac{1}{r^2}\frac{\partial^2 u_r}{\partial \phi^2} + \frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2} -
\frac{2}{r^2}\frac{\partial u_\phi}{\partial \phi} \right] + \rho g_r \\
\phi:\ &\rho \left(\frac{\partial u_{\phi}}{\partial t} + u_r \frac{\partial u_{\phi}}{\partial r} +
\frac{u_{\phi}}{r} \frac{\partial u_{\phi}}{\partial \phi} + u_z \frac{\partial u_{\phi}}{\partial z} + \frac{u_r u_{\phi}}{r}\right) = {}\\
&-\frac{1}{r}\frac{\partial p}{\partial \phi} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_{\phi}}{\partial r}\right) +
\frac{1}{r^2}\frac{\partial^2 u_{\phi}}{\partial \phi^2} + \frac{\partial^2 u_{\phi}}{\partial z^2} + \frac{2}{r^2}\frac{\partial u_r}{\partial \phi}-\frac{u_{\phi}}{r^2}\right] + \rho g_{\phi} \\
z:\ &\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + \frac{u_{\phi}}{r} \frac{\partial u_z}{\partial \phi} +
u_z \frac{\partial u_z}{\partial z}\right) = {}\\
&-\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) +
\frac{1}{r^2}\frac{\partial^2 u_z}{\partial \phi^2} + \frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z.
\end{align}
},
{
"math_input":
\begin{align}
_{(x)}\Gamma^m_{ij} & = G^{mk}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k} \,_{(X)}\Gamma_{\alpha\beta\gamma} + G^{mk}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\frac{\partial x^k}{\partial X^\rho}~g^{\nu\rho}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~\frac{\partial x^k}{\partial X^\rho}~g^{\nu\rho}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\delta^\gamma_\rho~g^{\nu\rho}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~\delta^\beta_\rho~g^{\nu\rho}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~g^{\nu\gamma}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma_{\alpha\beta\gamma} +
\frac{\partial x^m}{\partial X^\nu}~g^{\nu\beta}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~g_{\alpha\beta} \\
& = \frac{\partial x^m}{\partial X^\nu}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\frac{\partial x^m}{\partial X^\nu}~\delta^{\nu}_{\alpha}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}
\end{align}
},
{
"math_input":
\begin{align}
\ln q_\mu^*(\mu) &= \operatorname{E}_{\tau}\left[\ln p(\mathbf{X}\mid \mu,\tau) + \ln p(\mu\mid \tau) + \ln p(\tau)\right] + C \\
&= \operatorname{E}_{\tau}\left[\ln p(\mathbf{X}\mid \mu,\tau)\right] + \operatorname{E}_{\tau}\left[\ln p(\mu\mid \tau)\right] + \operatorname{E}_{\tau}\left[\ln p(\tau)\right] + C \\
&= \operatorname{E}_{\tau}\left[\ln \prod_{n=1}^N \mathcal{N}(x_n\mid \mu,\tau^{-1})\right] + \operatorname{E}_{\tau}\left[\ln \mathcal{N}(\mu\mid \mu_0, (\lambda_0 \tau)^{-1})\right] + C_2 \\
&= \operatorname{E}_{\tau}\left[\ln \prod_{n=1}^N \sqrt{\frac{\tau}{2\pi}} e^{-\frac{(x_n-\mu)^2\tau}{2}}\right] + \operatorname{E}_{\tau}\left[\ln \sqrt{\frac{\lambda_0 \tau}{2\pi}} e^{-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}}\right] + C_2 \\
&= \operatorname{E}_{\tau}\left[\sum_{n=1}^N \left(\frac{1}{2}(\ln\tau - \ln 2\pi) - \frac{(x_n-\mu)^2\tau}{2})\right)\right] + \operatorname{E}_{\tau}\left[\frac{1}{2}(\ln \lambda_0 + \ln \tau - \ln 2\pi) - \frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}\right] + C_2 \\
&= \operatorname{E}_{\tau}\left[\sum_{n=1}^N -\frac{(x_n-\mu)^2\tau}{2}\right] + \operatorname{E}_{\tau}\left[-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}\right] + \operatorname{E}_{\tau}\left[\sum_{n=1}^N \frac{1}{2}(\ln\tau - \ln 2\pi)\right] + \operatorname{E}_{\tau}\left[\frac{1}{2}(\ln \lambda_0 + \ln \tau - \ln 2\pi)\right] + C_2 \\
&= \operatorname{E}_{\tau}\left[\sum_{n=1}^N -\frac{(x_n-\mu)^2\tau}{2}\right] + \operatorname{E}_{\tau}\left[-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}\right] + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \left\{ \sum_{n=1}^N (x_n-\mu)^2 + \lambda_0(\mu-\mu_0)^2 \right\} + C_3
\end{align}
},
{
"math_input": \begin{align}
&-\frac{4}{3}i \left [ \oint_{C_1} \frac{\frac{z}{(z+\sqrt{3}i)(z-\sqrt{3}i)\left(z+\frac{i}{\sqrt{3}} \right)}}{z-\frac{i}{\sqrt{3}}}\,dz +\oint_{C_2} \frac{\frac{z}{(z+\sqrt{3}i)(z-\sqrt{3}i)\left(z-\frac{i}{\sqrt{3}}\right)}}{z+\frac{i}{\sqrt{3}}} \right ] \\
&= -\frac{4}{3}i \left[ 2\pi i \left(\frac{z}{(z+\sqrt{3}i)(z-\sqrt{3}i)(z+\frac{i}{\sqrt{3}})}\right)\Bigg|_{z=\frac{i}{\sqrt{3}}} + 2\pi i \left(\frac{z}{(z+\sqrt{3}i)(z-\sqrt{3}i)(z-\frac{i}{\sqrt{3}})} \right)\Bigg|_{z=-\frac{i}{\sqrt{3}}}\right] \\
&= \frac{8\pi}{3} \left[\frac{\frac{i}{\sqrt{3}}}{(\frac{i}{\sqrt{3}}+\sqrt{3}i)(\frac{i}{\sqrt{3}}-\sqrt{3}i)(\frac{i}{\sqrt{3}}+\frac{i}{\sqrt{3}})} + \frac{-\frac{i}{\sqrt{3}}}{(-\frac{i}{\sqrt{3}}+\sqrt{3}i)(-\frac{i}{\sqrt{3}}-\sqrt{3}i)(-\frac{i}{\sqrt{3}}-\frac{i}{\sqrt{3}})} \right] \\
&= \frac{8\pi}{3} \left[\frac{\frac{i}{\sqrt{3}}}{(\frac{4}{\sqrt{3}}i)(-\frac{2}{i\sqrt{3}})(\frac{2}{\sqrt{3}i})}+\frac{-\frac{i}{\sqrt{3}}}{(\frac{2}{\sqrt{3}}i)(-\frac{4}{\sqrt{3}}i)(-\frac{2}{\sqrt{3}}i)}\right] \\
&= \frac{8\pi}{3}\left[\frac{\frac{i}{\sqrt{3}}}{i(\frac{4}{\sqrt{3}})(\frac{2}{\sqrt{3}})(\frac{2}{\sqrt{3}})}+\frac{-\frac{i}{\sqrt{3}}}{-i(\frac{2}{\sqrt{3}})(\frac{4}{\sqrt{3}})(\frac{2}{\sqrt{3}})}\right] \\
&= \frac{8\pi}{3}\left[\frac{\frac{1}{\sqrt{3}}}{(\frac{4}{\sqrt{3}})(\frac{2}{\sqrt{3}})(\frac{2}{\sqrt{3}})}+\frac{\frac{1}{\sqrt{3}}}{(\frac{2}{\sqrt{3}})(\frac{4}{\sqrt{3}})(\frac{2}{\sqrt{3}})}\right] \\
&= \frac{8\pi}{3}\left[\frac{\frac{1}{\sqrt{3}}}{\frac{16}{3\sqrt{3}}}+\frac{\frac{1}{\sqrt{3}}}{\frac{16}{3\sqrt{3}}} \right] \\
&= \frac{8\pi}{3}\left[\frac{3}{16} + \frac{3}{16} \right] = \pi.
\end{align}
},
{
"math_input": \begin{array}{lll}
f(a\mathbf{i} + b\mathbf{j} + c\mathbf{k})
&=& \frac{1 + \mathbf{i} + \mathbf{j} + \mathbf{k}}{2} (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) \frac{1 - \mathbf{i} - \mathbf{j} - \mathbf{k}}{2} \\
&=& \frac{1}{4} ( (a\mathbf{i} + b\mathbf{j} + c\mathbf{k}) +(- a + b\mathbf{k} - c\mathbf{j}) + (-a\mathbf{k} - b +c\mathbf{i}) + (a\mathbf{j} - b\mathbf{i} - c))\\
&& (1 - \mathbf{i} - \mathbf{j} - \mathbf{k})\\
&=& \frac{1}{4} ( (-a - b - c) + (a - b+ c) \mathbf{i} + (a + b - c) \mathbf{j} + (-a + b + c) \mathbf{k})\\
&& (1 - \mathbf{i} - \mathbf{j} - \mathbf{k})\\
&=& \frac{1}{4} ( ( (-a - b - c) + (a - b + c) \mathbf{i} + (a + b - c) \mathbf{j} + (-a + b + c) \mathbf{k})\\
&&+ ( (a + b + c) \mathbf{i} + (a - b + c) + (a + b - c) \mathbf{k} + (a - b - c) \mathbf{j})\\
&&+ ( (a + b + c) \mathbf{j} + (-a + b - c) \mathbf{k} + (a + b - c) + (-a + b + c) \mathbf{i})\\
&&+ ( (a + b + c) \mathbf{k} + (a - b + c) \mathbf{j} + (-a - b + c) \mathbf{i} + (-a + b + c))\\
&=& \frac{1}{4} ( ( (-a - b - c) + (a - b + c) + (a + b - c) + (-a + b + c) )\\
&&+ ( (a - b + c) + (a + b + c) + (-a + b + c) + (-a - b + c) ) \mathbf{i}\\
&&+ ( (a + b - c) + (a - b - c) + (a + b + c) + (a - b + c) ) \mathbf{j}\\
&&+ ( (-a + b + c) + (a + b - c) + (-a + b - c) + (a + b + c) ) \mathbf{k})\\
&=& \frac{1}{4} (0 + 4c \mathbf{i} + 4a \mathbf{j} + 4b \mathbf{k})\\
&=&c\mathbf{i} + a\mathbf{j} + b\mathbf{k}
\end{array}
},
{
"math_input": \begin{align}
d_1 &= \frac{1}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_1}{p_s}\right) + \left(r + \frac{a_1^{2}}{2}\right)t_2\right] \\
d_2 &= \frac{1}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_1}{p_s}\right) + \left(r - \frac{a_1^{2}}{2}\right)t_2\right] \\
d_3 &= \frac{1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_1}{g_1}\right) + \left(r + \frac{a_2^{2}}{2}\right)t_2\right] \\
d_4 &= \frac{1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_1}{g_1}\right) + \left(r - \frac{a_2^{2}}{2}\right)t_2\right] \\
d_5 &= \frac{a_1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1}\right) + \left(r + \frac{a_2^{2}}{2a_1^{2}}\right)t_2\right] \\
d_6 &= \frac{a_1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1}\right) + \left(r - \frac{a_2^{2}}{2a_1^{2}}\right)t_2\right] \\
d_7 &= \frac{a_2}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1}\right) + \left(r + \frac{a_1^{2}}{2a_2^{2}}\right)t_2\right] \\
d_8 &= \frac{a_2}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1}\right) + \left(r - \frac{a_1^{2}}{a_2^{2}2}\right)t_2\right] \\
d_9 &= \frac{a_1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1+p_s}\right) + \left(r + \frac{a_2^{2}}{2a_1^{2}}\right)t_2\right] \\
d_{10} &= \frac{a_1}{a_2\sqrt{t_2}}\left[\ln\left(\frac{p_d}{p_1+p_s}\right) + \left(r - \frac{a_2^{2}}{2a_1^{2}}\right)t_2\right] \\
d_{11} &= \frac{1}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_1}{p_d}\right) + \left(r + \frac{a_1^{2}}{2}\right)t_2\right] \\
d_{12} &= \frac{1}{a_1\sqrt{t_2}}\left[\ln\left(\frac{p_1}{p_1+p_s+p_d}\right) + \left(r + \frac{a_1^{2}}{2}\right)t_2\right] \\
\end{align}
},
{
"math_input":
\begin{align}
\propto & \frac{\prod_{i\neq k}
\Gamma(n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i)}{\Gamma\bigl((\sum_{i=1}^K
n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i) +1\bigr)} \prod_{i\neq k} \frac{
\Gamma(n_{(\cdot),v}^{i,-(m,n)}+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{i,-(m,n)}+\beta_r \bigr)}\\
\times & \Gamma(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k + 1) \frac{
\Gamma(n_{(\cdot),v}^{k,-(m,n)}+\beta_v +
1)}{\Gamma\bigl((\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r)+1
\bigr)} \\
\propto & \frac{\Gamma(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k +
1)}{\Gamma\bigl((\sum_{i=1}^K n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i)+1
\bigr)} \frac{ \Gamma(n_{(\cdot),v}^{k,-(m,n)}+\beta_v +
1)}{\Gamma\bigl((\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r)+1
\bigr)}\\
= &
\frac{\Gamma(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k)\bigl(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k\bigr)}
{\Gamma\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i\bigr)\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i\bigr)} \frac{
\Gamma\bigl(n_{(\cdot),v}^{k,-(m,n)}+\beta_v\bigr)\bigl(n_{(\cdot),v}^{k,-(m,n)}+\beta_v\bigr)}
{\Gamma\bigl(\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r\bigr)
\bigl(\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r)} \\
\propto & \frac{\bigl(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k\bigr)}
{\bigl(\sum_{i=1}^K n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i\bigr)} \frac{
\bigl(n_{(\cdot),v}^{k,-(m,n)}+\beta_v\bigr)} {\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{k,-(m,n)}+\beta_r)}\\
\propto & \bigl(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k\bigr)\frac{
\bigl(n_{(\cdot),v}^{k,-(m,n)}+\beta_v\bigr)} {\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{k,-(m,n)}+\beta_r)}.
\end{align}
},
{
"math_input":
\begin{align}
\frac{d}{d\alpha}\,\textbf I(\alpha)\;
&=\;\int_0^{\frac{\pi}{2}}\,\frac{\partial}{\partial\alpha}\,\left(\frac{\ln\,(1\,+\,\cos\alpha\,\cos\,x)}{\cos\,x}\right)\,dx\,
\\
&=\;-\,\int_0^{\frac{\pi}{2}}\,\frac{\sin\alpha}{1+\cos\alpha\,\cos\,x}\,dx\,
\\
&=\;-\,\int_0^{\frac{\pi}{2}}\,\frac{\sin\alpha}{\left(\cos^2\,\frac{x}{2}+\sin^2\,\frac{x}{2}\right)\,+\,\cos\alpha\,\left(\cos^2\,\frac{x}{2}-\sin^2\,\frac{x}{2}\right)}\,dx\,
\\
&=\;-\,\frac{\sin\alpha}{1-\cos\alpha}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\cos^2\,\frac{x}{2}}\,\frac{1}{\left[\,\left(\frac{1+\cos\alpha}{1-\cos\alpha}\right)\,+\,\tan^2\,\frac{x}{2}\,\right]}\,dx\,
\\
&=\;-\,\frac{2\,\sin\alpha}{1-\cos\alpha}\,\int_0^{\frac{\pi}{2}}\,\frac{\frac{1}{2}\,\sec^2\,\frac{x}{2}}{\left[\,\left(\frac{2\,\cos^2\,\frac{\alpha}{2}}{2\,\sin^2\,\frac{\alpha}{2}}\right)\,+\,\tan^2\,\frac{x}{2}\,\right]}\,dx\,
\\
&=\;-\,\frac{2\left(2\,\sin\,\frac{\alpha}{2}\,\cos\,\frac{\alpha}{2}\right)}{2\,\sin^2\,\frac{\alpha}{2}}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\left[\,\left(\frac{\cos\,\frac{\alpha}{2}}{\sin\,\frac{\alpha}{2}}\right)^2\,+\,\tan^2\,\frac{x}{2}\,\right]}\,d\left(\tan\,\frac{x}{2}\right)\,
\\
&=\;-\,2\,\cot\,\frac{\alpha}{2}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\left[\,\cot^2\,\frac{\alpha}{2}\,+\,\tan^2\,\frac{x}{2}\,\right]}\,d\left(\tan\,\frac{x}{2}\right)\,
\\
&=\;-\,2\,\left(\tan^{-1}\,\left(\tan\,\frac{\alpha}{2}\,\tan\,\frac{x}{2}\,\right)\right)\,\bigg|_0^{\frac{\pi}{2}}\,
\\
&=\;-\,\alpha\,
\end{align}
},
{
"math_input":
\begin{align}
\overline{u}_S\,
&=\, \overline{u_x(\boldsymbol{\xi},t)}\, -\, \overline{u_x(\boldsymbol{x},t)}\,
\\
&=\, \overline{\left[
u_x(\boldsymbol{x},t)\,
+\, \left( \xi_x - x \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial x}\,
+\, \left( \xi_z - z \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial z}\,
+\, \cdots
\right] }
-\, \overline{u_x(\boldsymbol{x},t)}
\\
&\approx\, \overline{\left( \xi_x - x \right)\, \frac{\partial^2 \xi_x}{\partial x\, \partial t} }\,
+\, \overline{\left( \xi_z - z \right)\, \frac{\partial^2 \xi_x}{\partial z\, \partial t} }
\\
&=\, \overline{ \bigg[ - a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg]\,
\bigg[ -\omega\, k\, a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg] }\,
\\
&+\, \overline{ \bigg[ a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg]\,
\bigg[ \omega\, k\, a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg] }\,
\\
&=\, \overline{ \omega\, k\, a^2\, \text{e}^{2 k z}\,
\bigg[ \sin^2\, \left( k x - \omega t \right) + \cos^2\, \left( k x - \omega t \right) \bigg] }
\\
&=\, \omega\, k\, a^2\, \text{e}^{2 k z}.
\end{align}
},
{
"math_input": \begin{align}
\boldsymbol\tau &= \sum_{i=1}^n (\mathbf{r_i}-\mathbf{R})\times (m_i\mathbf{a}_i) \\
&= \sum_{i=1}^n \boldsymbol\Delta\mathbf{r}_i\times (m_i\mathbf{a}_i) \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times \mathbf{a}_i]\;\ldots\text{ cross-product scalar multiplication} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\mathbf{a}_{\text{tangential},i} + \mathbf{a}_{\text{centripetal},i} + \mathbf{A}_R)] \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\mathbf{a}_{\text{tangential},i} + \mathbf{a}_{\text{centripetal},i} + 0)] \\
&\;\;\;\;\;\ldots\;\mathbf{R}\text{ is either at rest or moving at a constant velocity but not accelerated, or } \\
&\;\;\;\;\;\;\;\;\;\;\;\text{the origin of the fixed (world) coordinate reference system is placed at the centre of mass }\mathbf{C} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times \mathbf{a}_{\text{tangential},i} + \boldsymbol\Delta\mathbf{r}_i\times \mathbf{a}_{\text{centripetal},i}]\;\ldots\text{ cross-product distributivity over addition} \\
&= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times\mathbf{v}_{\text{tangential},i})] \\
\boldsymbol\tau &= \sum_{i=1}^n m_i [\boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\alpha\times\boldsymbol\Delta\mathbf{r}_i) + \boldsymbol\Delta\mathbf{r}_i\times (\boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\Delta\mathbf{r}_i))] \\
\end{align}
},
{
"math_input":
\begin{align}
\overline{u}_S\,
&=\, \overline{u_x(\boldsymbol{\xi},t)}\, -\, \overline{u_x(\boldsymbol{x},t)}\,
\\
&=\, \overline{\left[
u_x(\boldsymbol{x},t)\,
+\, \left( \xi_x - x \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial x}\,
+\, \left( \xi_z - z \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial z}\,
+\, \cdots
\right] }
-\, \overline{u_x(\boldsymbol{x},t)}
\\
&\approx\, \overline{\left( \xi_x - x \right)\, \frac{\partial^2 \xi_x}{\partial x\, \partial t} }\,
+\, \overline{\left( \xi_z - z \right)\, \frac{\partial^2 \xi_x}{\partial z\, \partial t} }
\\
&=\, \overline{ \bigg[ - a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg]\,
\bigg[ -\omega\, k\, a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg] }\,
\\
&+\, \overline{ \bigg[ a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg]\,
\bigg[ \omega\, k\, a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg] }\,
\\
&=\, \overline{ \omega\, k\, a^2\, \text{e}^{2 k z}\,
\bigg[ \sin^2\, \left( k x - \omega t \right) + \cos^2\, \left( k x - \omega t \right) \bigg] }
\\
&=\, \omega\, k\, a^2\, \text{e}^{2 k z}.
\end{align}
},
{
"math_input": \begin{align}
\rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y} + u_z \frac{\partial u_x}{\partial z}\right)
&= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2} + \frac{\partial^2 u_x}{\partial z^2}\right) - \mu \frac{\partial}{\partial x} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \right) + \rho g_x \\
\rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}+ u_z \frac{\partial u_y}{\partial z}\right)
&= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_y}{\partial z^2}\right) - \mu \frac{\partial}{\partial y} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \right) + \rho g_y \\
\rho \left(\frac{\partial u_z}{\partial t} + u_x \frac{\partial u_z}{\partial x} + u_y \frac{\partial u_z}{\partial y}+ u_z \frac{\partial u_z}{\partial z}\right)
&= -\frac{\partial p}{\partial z} + \mu \left(\frac{\partial^2 u_z}{\partial x^2} + \frac{\partial^2 u_z}{\partial y^2} + \frac{\partial^2 u_z}{\partial z^2}\right) - \mu \frac{\partial}{\partial z} \left( \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z} \right) + \rho g_z.
\end{align}
},
{
"math_input": \begin{align}
j \left( \frac{5 \, i + 1}{2} \right) &= \left( \frac{2927 - 1323 \, \sqrt{5}}{2} \right)^3,\\
j \left( 5 \, i \right) &= \left( \frac{2927 + 1323 \, \sqrt{5}}{2} \right)^3,\\
j \left( \frac{5 \, i + 2}{4} \right) &= \Bigg( \frac{\left( 1 + \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 - 858585699 \, \sqrt{2} + 540309076 \, \sqrt{5} - 374537880 \, \sqrt{10} \, - \, \sqrt[4]{5} \left( 693172512 - 595746414 \, \sqrt{2} + 407357424 \, \sqrt{5} - 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3,\\
j \left( \frac{10 \, i + 1}{2} \right) &= \Bigg( \frac{\left( 1 + \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 - 858585699 \, \sqrt{2} + 540309076 \, \sqrt{5} - 374537880 \, \sqrt{10} \, + \, \sqrt[4]{5} \left( 693172512 - 595746414 \, \sqrt{2} + 407357424 \, \sqrt{5} - 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3,\\
j \left( \frac{5 \, i}{4} \right) &= \Bigg( \frac{\left( 1 + \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 + 858585699 \, \sqrt{2} + 540309076 \, \sqrt{5} + 374537880 \, \sqrt{10} \, - \, \sqrt[4]{5} \left( 693172512 + 595746414 \, \sqrt{2} + 407357424 \, \sqrt{5} + 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3,\\
j(20 \, i) &= \Bigg( \frac{\left( 1 + \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 + 858585699 \, \sqrt{2} + 540309076 \, \sqrt{5} + 374537880 \, \sqrt{10} \, + \, \sqrt[4]{5} \left( 693172512 + 595746414 \, \sqrt{2} + 407357424 \, \sqrt{5} + 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3.
\end{align}
},
{
"math_input": \begin{align}
\vec{\omega}&=\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \cos\psi & \sin\psi\\
0 & -\sin\psi & \cos\psi
\end{array}\right)\left(\begin{array}{c}
\dot{\psi}\\
0\\
0
\end{array}\right)+\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \cos\psi & \sin\psi\\
0 & -\sin\psi & \cos\psi
\end{array}\right)\left(\begin{array}{ccc}
\cos\alpha & \sin\alpha & 0\\
-\sin\alpha & \cos\alpha & 0\\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{c}
0\\
0\\
\dot{\alpha}
\end{array}\right)\\
&{}+\left(\begin{array}{ccc}
1 & 0 & 0\\
0 & \cos\psi & \sin\psi\\
0 & -\sin\psi & \cos\psi
\end{array}\right)\left(\begin{array}{ccc}
\cos\alpha & \sin\alpha & 0\\
-\sin\alpha & \cos\alpha & 0\\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{ccc}
\cos\delta & 0 & -\sin\delta\\
0 & 1 & 0\\
\sin\delta & 0 & \cos\delta
\end{array}\right)\left(\begin{array}{ccc}
\cos\Phi & \sin\Phi & 0\\
-\sin\Phi & \cos\Phi & 0\\
0 & 0 & 1
\end{array}\right)\\
&{}\times\left(\begin{array}{ccc}
\cos\Omega t & \sin\Omega t & 0\\
-\sin\Omega t & \cos\Omega t & 0\\
0 & 0 & 1
\end{array}\right)\left(\begin{array}{c}
0\\
0\\
\Omega
\end{array}\right)\\
&= \left(\begin{array}{c}
\dot{\psi}\\
0\\
0\\
\end{array}\right)+\left(\begin{array}{c}
0\\
\dot{\alpha}\sin\psi\\
\dot{\alpha}\cos\psi
\end{array}\right)+\left(\begin{array}{c}
-\Omega\sin\delta\cos\alpha\\
\Omega(\sin\delta\sin\alpha\cos\psi+\cos\delta\sin\psi)\\
\Omega(-\sin\delta\sin\alpha\sin\psi+\cos\delta\cos\psi)
\end{array}\right).\end{align}
},
{
"math_input":
\begin{align}
\ln q_\mu^*(\mu) &= \operatorname{E}_{\tau}[\ln p(\mathbf{X}|\mu,\tau) + \ln p(\mu|\tau) + \ln p(\tau)] + C \\
&= \operatorname{E}_{\tau}[\ln p(\mathbf{X}|\mu,\tau)] + \operatorname{E}_{\tau}[\ln p(\mu|\tau)] + \operatorname{E}_{\tau}[\ln p(\tau)] + C \\
&= \operatorname{E}_{\tau}[\ln \prod_{n=1}^N \mathcal{N}(x_n|\mu,\tau^{-1})] + \operatorname{E}_{\tau}[\ln \mathcal{N}(\mu|\mu_0, (\lambda_0 \tau)^{-1})] + C_2 \\
&= \operatorname{E}_{\tau}[\ln \prod_{n=1}^N \sqrt{\frac{\tau}{2\pi}} e^{-\frac{(x_n-\mu)^2\tau}{2}}] + \operatorname{E}_{\tau}[\ln \sqrt{\frac{\lambda_0 \tau}{2\pi}} e^{-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}}] + C_2 \\
&= \operatorname{E}_{\tau}[\sum_{n=1}^N \left(\frac{1}{2}(\ln\tau - \ln 2\pi) - \frac{(x_n-\mu)^2\tau}{2}\right)] + \operatorname{E}_{\tau}[\frac{1}{2}(\ln \lambda_0 + \ln \tau - \ln 2\pi) - \frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}] + C_2 \\
&= \operatorname{E}_{\tau}[\sum_{n=1}^N -\frac{(x_n-\mu)^2\tau}{2}] + \operatorname{E}_{\tau}[-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}] + \operatorname{E}_{\tau}[\sum_{n=1}^N \frac{1}{2}(\ln\tau - \ln 2\pi)] + \operatorname{E}_{\tau}[\frac{1}{2}(\ln \lambda_0 + \ln \tau - \ln 2\pi)] + C_2 \\
&= \operatorname{E}_{\tau}[\sum_{n=1}^N -\frac{(x_n-\mu)^2\tau}{2}] + \operatorname{E}_{\tau}[-\frac{(\mu-\mu_0)^2\lambda_0 \tau}{2}] + C_3 \\
&= - \frac{\operatorname{E}_{\tau}[\tau]}{2} \{ \sum_{n=1}^N (x_n-\mu)^2 + \lambda_0(\mu-\mu_0)^2 \} + C_3
\end{align}
},
{
"math_input":
\operatorname{adj}(\mathbf{A}) = \begin{pmatrix}
+\left| \begin{matrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{matrix} \right| &
-\left| \begin{matrix} A_{12} & A_{13} \\ A_{32} & A_{33} \end{matrix} \right| &
+\left| \begin{matrix} A_{12} & A_{13} \\ A_{22} & A_{23} \end{matrix} \right| \\
& & \\
-\left| \begin{matrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{matrix} \right| &
+\left| \begin{matrix} A_{11} & A_{13} \\ A_{31} & A_{33} \end{matrix} \right| &
-\left| \begin{matrix} A_{11} & A_{13} \\ A_{21} & A_{23} \end{matrix} \right| \\
& & \\
+\left| \begin{matrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{matrix} \right| &
-\left| \begin{matrix} A_{11} & A_{12} \\ A_{31} & A_{32} \end{matrix} \right| &
+\left| \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} \right|
\end{pmatrix} = \begin{pmatrix}
+\left| \begin{matrix} 5 & 6 \\ 8 & 9 \end{matrix} \right| &
-\left| \begin{matrix} 2 & 3 \\ 8 & 9 \end{matrix} \right| &
+\left| \begin{matrix} 2 & 3 \\ 5 & 6 \end{matrix} \right| \\
& & \\
-\left| \begin{matrix} 4 & 6 \\ 7 & 9 \end{matrix} \right| &
+\left| \begin{matrix} 1 & 3 \\ 7 & 9 \end{matrix} \right| &
-\left| \begin{matrix} 1 & 3 \\ 4 & 6 \end{matrix} \right| \\
& & \\
+\left| \begin{matrix} 4 & 5 \\ 7 & 8 \end{matrix} \right| &
-\left| \begin{matrix} 1 & 2 \\ 7 & 8 \end{matrix} \right| &
+\left| \begin{matrix} 1 & 2 \\ 4 & 5 \end{matrix} \right|
\end{pmatrix}
},
{
"math_input":
\begin{cases}
\begin{bmatrix}
\dot{e}_1\\
\dot{e}_2\\
\vdots\\
\dot{e}_i\\
\vdots\\
\dot{e}_{n-1}\\
\dot{e}_n
\end{bmatrix}
=
\mathord{\overbrace{
\begin{bmatrix}
\dot{h}_1(x)\\
\dot{h}_2(x)\\
\vdots\\
\dot{h}_i(x)\\
\vdots\\
\dot{h}_{n-1}(x)\\
\dot{h}_n(x)
\end{bmatrix}
}^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(x)}}
-
\mathord{\overbrace{
M(\hat{x}) \, \operatorname{sgn}( V(t) - H(\hat{x}(t)) )
}^{\tfrac{\operatorname{d}}{\operatorname{d}t} H(\hat{x})}}
=
\begin{bmatrix}
h_2(x)\\
h_3(x)\\
\vdots\\
h_{i+1}(x)\\
\vdots\\
h_n(x)\\
L_f^n h(x)
\end{bmatrix}
-
\begin{bmatrix}
m_1 \operatorname{sgn}( v_1(t) - h_1(\hat{x}(t)) )\\
m_2 \operatorname{sgn}( v_2(t) - h_2(\hat{x}(t)) )\\
\vdots\\
m_i \operatorname{sgn}( v_i(t) - h_i(\hat{x}(t)) )\\
\vdots\\
m_{n-1} \operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\hat{x}(t)) )\\
m_n \operatorname{sgn}( v_n(t) - h_n(\hat{x}(t)) )
\end{bmatrix}\\
=
\begin{bmatrix}
h_2(x) - m_1(\hat{x}) \operatorname{sgn}( \mathord{\overbrace{ \mathord{\overbrace{v_1(t)}^{v_1(t) = y(t) = h_1(x)}} - h_1(\hat{x}(t)) }^{e_1}} )\\
h_3(x) - m_2(\hat{x}) \operatorname{sgn}( v_2(t) - h_2(\hat{x}(t)) )\\
\vdots\\
h_{i+1}(x) - m_i(\hat{x}) \operatorname{sgn}( v_i(t) - h_i(\hat{x}(t)) )\\
\vdots\\
h_n(x) - m_{n-1}(\hat{x}) \operatorname{sgn}( v_{n-1}(t) - h_{n-1}(\hat{x}(t)) )\\
L_f^n h(x) - m_n(\hat{x}) \operatorname{sgn}( v_n(t) - h_n(\hat{x}(t)) )
\end{bmatrix}.
\end{cases}
},
{
"math_input":
\begin{array}{c}
\left[ {{\begin{array}{*{20}c}
{\Delta x_1 } \\
{\Delta x_2 } \\
{\Delta x_3 } \\
\vdots \\
{\Delta x_{n - 1} } \\
{\Delta T} \\
\end{array} }} \right] = \left[ {{\begin{array}{*{20}c}
{1 / x_1 } & 0 & 0 & 0 & 0 & { - \frac{H_1^\circ }{RT^{2}}} \\
0 & {1 / x_2 } & 0 & 0 & 0 & { - \frac{H_2^\circ }{RT^{2}}} \\
0 & 0 & {1 / x_3 } & 0 & 0 & { - \frac{H_3^\circ }{RT^{2}}} \\
0 & 0 & 0 & \ddots & 0 & { - \frac{H_4^\circ }{RT^{2}}} \\
0 & 0 & 0 & 0 & {1 / x_{n - 1} } & { - \frac{H_{n - 1}^\circ }{RT^{2}}}
\\
{\frac{ - 1}{1 - \sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{1 = 1}^{n - 1} {x_i } }} & {\frac{ - 1}{1 -
\sum\limits_{1 = 1}^{n - 1} {x_i } }} & { -
\frac{H_n^\circ }{RT^{2}}} \\
\end{array} }} \right]^{ - 1}
.\left[ {{\begin{array}{*{20}c}
{\ln x_1 + \frac{H_1 ^\circ }{RT} - \frac{H_1^\circ }{RT_1^\circ }}
\\
{\ln x_2 + \frac{H_2 ^\circ }{RT} - \frac{H_2^\circ }{RT_2^\circ }}
\\
{\ln x_3 + \frac{H_3 ^\circ }{RT} - \frac{H_3^\circ }{RT_3^\circ }}
\\
\vdots \\
{\ln x_{n - 1} + \frac{H_{n - 1} ^\circ }{RT} - \frac{H_{n - 1}^\circ
}{RT_{n - 1i}^\circ }} \\
{\ln \left( {1 - \sum\limits_{i = 1}^{n - 1} {x_i } } \right) + \frac{H_n
^\circ }{RT} - \frac{H_n^\circ }{RT_n^\circ }} \\
\end{array} }} \right]
\end{array}
},
{
"math_input": \begin{align}
K_\frac{1}{2} (z) &= \sqrt{\frac{\pi}{2}} e^{-z} z^{-\tfrac{1}{2}}, \qquad z>0 \\
I_{-\frac{1}{2}}(z) &= \sqrt{\frac{2}{\pi z}}\cosh(z) \\
I_{ \frac{1}{2}}(z) &= \sqrt{\frac{2}{\pi z}}\sinh(z) \\
I_\nu(z) &= \sum_{k=0} \frac{z^k}{k!} J_{\nu+k}(z) \\
J_\nu(z) &= \sum_{k=0} (-1)^k \frac{z^k}{k!} I_{\nu+k}(z) \\
I_\nu (\lambda z) &= \lambda^\nu \sum_{k=0} \frac{\left((\lambda^2-1)\frac z 2\right)^k}{k!} I_{\nu+k}(z) \\
I_\nu (z_1+z_2) &= \sum_{k=-\infty}^\infty I_{\nu-k}(z_1)I_k(z_2) \\
J_\nu (z_1\pm z_2) &= \sum_{k=-\infty}^\infty J_{\nu \mp k}(z_1)J_k(z_2) \\
I_\nu (z) &= \tfrac{z}{2 \nu} \left (I_{\nu-1}(z)-I_{\nu+1}(z) \right ) \\
J_\nu (z) &= \tfrac{z}{2 \nu} \left (J_{\nu-1}(z)+J_{\nu+1}(z) \right ) \\
J_\nu'(z) &= \begin{cases}\tfrac{1}{2} \left (J_{\nu-1}(z)-J_{\nu+1}(z) \right) & \nu \neq 0 \\ -J_1(z) & \nu =0 \end{cases} \\
I_\nu'(z) &= \begin{cases}\tfrac{1}{2} \left (I_{\nu-1}(z)+I_{\nu+1}(z) \right) & \nu \neq 0 \\ I_1(z) & \nu =0 \end{cases} \\
\left(\tfrac{z}{2}\right)^\nu &= \Gamma(\nu) \sum_{k=0} I_{\nu+2k}(z)(\nu+2k){-\nu\choose k} = \Gamma(\nu) \sum_{k=0}(-1)^k J_{\nu+2k}(z)(\nu+2k){-\nu \choose k} = \Gamma(\nu+1) \sum_{k=0}\frac{\left(\tfrac{z}{2}\right)^k}{k!} J_{\nu+k}(z)\\
1 &= \sum_{n=0}^\infty (2n+1) j_n(z)^2\\
\frac{\sin(2z)}{2z} &= \sum_{n=0}^\infty (-1)^n (2n+1) j_n(z)^2
\end{align}
},
{
"math_input": \begin{align}
\frac{d}{d\alpha} \textbf{I}(\alpha) &= \int_0^{\frac{\pi}{2}} \frac{\partial}{\partial\alpha} \left(\frac{\ln(1 + \cos\alpha \cos x)}{\cos x}\right)\,dx \\
&=-\int_0^{\frac{\pi}{2}}\frac{\sin \alpha}{1+\cos \alpha \cos x}\,dx \\
&=-\int_0^{\frac{\pi}{2}}\frac{\sin \alpha}{\left(\cos^2 \frac{x}{2}+\sin^2 \frac{x}{2}\right)+\cos \alpha\,\left(\cos^2\,\frac{x}{2}-\sin^2 \frac{x}{2}\right)}\,dx \\
&=-\frac{\sin\alpha}{1-\cos\alpha} \int_0^{\frac{\pi}{2}} \frac{1}{\cos^2\frac{x}{2}}\frac{1}{\left[\left(\frac{1+\cos \alpha}{1-\cos \alpha}\right) +\tan^2 \frac{x}{2} \right]}\,dx \\
&=-\frac{2\,\sin\alpha}{1-\cos\alpha} \int_0^{\frac{\pi}{2}}\,\frac{\frac{1}{2}\,\sec^2\,\frac{x}{2}}{\left[\,\left(\frac{2\,\cos^2\,\frac{\alpha}{2}}{2\,\sin^2\,\frac{\alpha}{2}}\right) + \tan^2\,\frac{x}{2} \right]} \,dx \\
&=-\frac{2\left(2\,\sin\,\frac{\alpha}{2}\,\cos\,\frac{\alpha}{2}\right)}{2\,\sin^2\,\frac{\alpha}{2}}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\left[\left(\frac{\cos \frac{\alpha}{2}}{\sin\,\frac{\alpha}{2}}\right)^2\,+\,\tan^2\,\frac{x}{2}\,\right]}\,d\left(\tan\,\frac{x}{2}\right)\\
&=-2\cot \frac{\alpha}{2}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\left[\,\cot^2\,\frac{\alpha}{2} + \tan^2\,\frac{x}{2}\,\right]}\,d\left(\tan \frac{x}{2}\right)\,\\
&=-2\left(\tan^{-1} \left(\tan \frac{\alpha}{2} \tan \frac{x}{2} \right)\right) \bigg|_0^{\frac{\pi}{2}}\\
&=-\alpha
\end{align}
},
{
"math_input":
\mathbf{C} = \begin{pmatrix}
+\left| \begin{matrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{matrix} \right| &
-\left| \begin{matrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{matrix} \right| &
+\left| \begin{matrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{matrix} \right| \\
& & \\
-\left| \begin{matrix} A_{12} & A_{13} \\ A_{32} & A_{33} \end{matrix} \right| &
+\left| \begin{matrix} A_{11} & A_{13} \\ A_{31} & A_{33} \end{matrix} \right| &
-\left| \begin{matrix} A_{11} & A_{12} \\ A_{31} & A_{32} \end{matrix} \right| \\
& & \\
+\left| \begin{matrix} A_{12} & A_{13} \\ A_{22} & A_{23} \end{matrix} \right| &
-\left| \begin{matrix} A_{11} & A_{13} \\ A_{21} & A_{23} \end{matrix} \right| &
+\left| \begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} \right|
\end{pmatrix} = \begin{pmatrix}
+\left| \begin{matrix} 5 & 6 \\ 8 & 9 \end{matrix} \right| &
-\left| \begin{matrix} 4 & 6 \\ 7 & 9 \end{matrix} \right| &
+\left| \begin{matrix} 4 & 5 \\ 7 & 8 \end{matrix} \right| \\
& & \\
-\left| \begin{matrix} 2 & 3 \\ 8 & 9 \end{matrix} \right| &
+\left| \begin{matrix} 1 & 3 \\ 7 & 9 \end{matrix} \right| &
-\left| \begin{matrix} 1 & 2 \\ 7 & 8 \end{matrix} \right| \\
& & \\
+\left| \begin{matrix} 2 & 3 \\ 5 & 6 \end{matrix} \right| &
-\left| \begin{matrix} 1 & 3 \\ 4 & 6 \end{matrix} \right| &
+\left| \begin{matrix} 1 & 2 \\ 4 & 5 \end{matrix} \right|
\end{pmatrix}
},
{
"math_input": \left\{\begin{array}{l}
4 x \left(\nu _2+\nu_1 x\right){}^2 f''(x)+f'(x) \left(-2 \nu _2^2 \nu _1+8 \nu _2^2+
16 \nu _1^2 x^2+4 \nu_2 \nu_1^2 x^2-2 \lambda \nu_2 \nu _1 x-2 \nu_2 \nu_1^2 x+4 \nu_2^2
\nu_1 x+24 \nu_2 \nu_1 x\right)+\nu_1 \left(\nu_2+2\right) f(x) \left(-\lambda
\nu_2-\nu_2 \nu_1+4 \nu_2+4 \nu_1 x+\nu_2 \nu_1 x\right)=0, \\[12pt]
f(1)=\frac{e^{-\lambda /2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu _2}{2}}
\left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu _1-\nu _2\right)} \,
_1F_1\left(\frac{1}{2} \left(\nu _1+\nu _2\right);\frac{\nu_1}{2};
\frac{\lambda \nu _1}{2 \left(\nu _1+\nu _2\right)}\right)}{B\left(\frac{\nu_1}{2},
\frac{\nu_2}{2}\right)}, \\[12pt]
f'(1)=\frac{e^{-\lambda/2} \nu_1^{\frac{\nu_1}{2}} \nu_2^{\frac{\nu_2}{2}}
\left(\nu _1+\nu _2\right){}^{\frac{1}{2} \left(-\nu_1-\nu_2-2\right)}
\left(\nu_2 \left(\lambda \, _1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2+2\right);
\frac{1}{2} \left(\nu_1+2\right);\frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)-2 \,
_1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2};
\frac{\lambda\nu_1}{2 \left(\nu_1+\nu _2\right)}\right)\right)-2 \nu _1 \,
_1F_1\left(\frac{1}{2} \left(\nu_1+\nu_2\right);\frac{\nu_1}{2};
\frac{\lambda \nu_1}{2 \left(\nu_1+\nu_2\right)}\right)\right)}
{2 B\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)}
\end{array}\right\}
},
{
"math_input": \begin{align}
\sum_{k=0}^\infty &(k+r)(k+r-1) A_kz^{k+r-2}-\frac{1}{z} \sum_{k=0}^\infty (k+r)A_kz^{k+r-1} + \left(\frac{1}{z^2} - \frac{1}{z}\right) \sum_{k=0}^\infty A_kz^{k+r} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1) A_kz^{k+r-2} -\frac{1}{z} \sum_{k=0}^\infty (k+r) A_kz^{k+r-1} +\frac{1}{z^2} \sum_{k=0}^\infty A_kz^{k+r} -\frac{1}{z} \sum_{k=0}^\infty A_kz^{k+r} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r)A_kz^{k+r-2}+\sum_{k=0}^\infty A_kz^{k+r-2}-\sum_{k=0}^\infty A_kz^{k+r-1} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r) A_kz^{k+r-2} + \sum_{k=0}^\infty A_kz^{k+r-2} - \sum_{k-1=0}^\infty A_{k-1}z^{k+r-2} \\
&= \sum_{k=0}^\infty (k+r)(k+r-1)A_kz^{k+r-2}-\sum_{k=0}^\infty (k+r)A_kz^{k+r-2}+\sum_{k=0}^\infty A_kz^{k+r-2}-\sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= \left \{ \sum_{k=0}^{\infty} \left ((k+r)(k+r-1) - (k+r) + 1\right ) A_kz^{k+r-2} \right \} -\sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= \left \{ \left ( r(r-1) - r +1 \right ) A_0 z^{r-2} + \sum_{k=1}^{\infty} \left ((k+r)(k+r-1) - (k+r) + 1\right ) A_kz^{k+r-2} \right \} - \sum_{k=1}^\infty A_{k-1}z^{k+r-2} \\
&= (r-1)^2 A_0 z^{r-2} + \left \{ \sum_{k=1}^{\infty} (k+r-1)^2 A_kz^{k+r-2} - \sum_{k=1}^\infty A_{k-1}z^{k+r-2} \right \} \\
&= (r-1)^2 A_0 z^{r-2} + \sum_{k=1}^{\infty} \left ( (k+r-1)^2 A_k - A_{k-1} \right ) z^{k+r-2}
\end{align}
},
{
"math_input":
\begin{align}
\propto & \prod_{i\neq k}
\Gamma(n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i)
\prod_{i\neq k} \frac{
\Gamma(n_{(\cdot),v}^{i,-(m,n)}+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{i,-(m,n)}+\beta_r \bigr)}\\
\times & \Gamma(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k + 1) \frac{
\Gamma(n_{(\cdot),v}^{k,-(m,n)}+\beta_v +
1)}{\Gamma\bigl((\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r)+1
\bigr)} \\
=& \prod_{i\neq k}
\Gamma(n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i)
\prod_{i\neq k} \frac{
\Gamma(n_{(\cdot),v}^{i,-(m,n)}+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{i,-(m,n)}+\beta_r \bigr)}\\
\times & \Gamma(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k)
\frac{\Gamma(n_{(\cdot),v}^{k,-(m,n)}+\beta_v )}
{\Gamma\bigl(\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r \bigr)} \\
\times & (n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k)
\frac{n_{(\cdot),v}^{k,-(m,n)}+\beta_v }
{\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r} \\
=& \prod_{i}
\Gamma(n_{m,(\cdot)}^{i,-(m,n)}+\alpha_i)
\prod_{i} \frac{
\Gamma(n_{(\cdot),v}^{i,-(m,n)}+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^{i,-(m,n)}+\beta_r \bigr)}\\
\times & (n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k)
\frac{n_{(\cdot),v}^{k,-(m,n)}+\beta_v }
{\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r} \\
\propto &
(n_{m,(\cdot)}^{k,-(m,n)}+\alpha_k)
\frac{n_{(\cdot),v}^{k,-(m,n)}+\beta_v }
{\sum_{r=1}^V n_{(\cdot),r}^{k,-(m,n)}+\beta_r}. \\
\end{align}
},
{
"math_input": {}_{
x_n=-e^{\frac{2n\pi{\rm{i}}}{N-2}}\sqrt[N-2]{\frac{b}{a}}{}_{N-1}F_{N-2}
\begin{bmatrix}
-\frac{1}{N\left(N-2\right)},-\frac{1}{N\left(N-2\right)}+\frac{1}{N},-\frac{1}{N\left(N-2\right)}+\frac{2}{N},\cdots,-\frac{1}{N\left(N-2\right)}+\frac{1}{N},\frac{N-5}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N-3}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N+1}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N+3}{2N},\cdots,-\frac{1}{N\left(N-2\right)}+\frac{N-1}{N},;\\[8pt]
\frac{1}{N-2},\frac{2}{N-2},\cdots,\frac{2N-5}{2N-4},;\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix}
+\sqrt[N-2]{\frac{b}{a}}\sum^{N-3}_{q=1}\frac{\Gamma\left(\frac{2q-1}{N-2}+q\right)}{\Gamma\left(\frac{2q-1}{N-2}+1\right)}\cdot\left(-\frac{c}{b}\sqrt[N-2]{\frac{a^2}{b^2}}\right)^q\cdot\frac{e^{\frac{2n\left(1-2q\right)}{N-2}\pi{\rm{i}}}}{q!}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{Nq-1}{N\left(N-2\right)},\frac{Nq-1}{N\left(N-2\right)}+\frac{1}{N},\frac{Nq-1}{N\left(N-2\right)}+\frac{2}{N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-3}{2N},\frac{Nq-1}{N\left(N-2\right)}+\frac{N+1}{2N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-1}{N};\\[8pt]
\frac{q+1}{N-2},\frac{q+2}{N-2},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{q+N-2}{N-2},\frac{2q+2N-5}{2N-4};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix},n=1,2,\cdots,N-2
}
},
{
"math_input": \begin{alignat}{3} F = \frac{1}{|\nabla \theta|}\cdot \frac{\partial \theta}{\partial x}\left \{ \frac{1}{C_p} \left ( \frac{p_\circ}{p} \right )^\kappa \left [ \frac{\partial}{\partial x} \left (\frac{dQ}{dt} \right ) \right ] - \left ( \frac{\partial u}{\partial x} \frac{\partial \theta}{\partial x} \right ) - \left ( \frac{\partial v}{\partial x} \frac{\partial \theta}{\partial y} \right ) - \left ( \frac{\partial w}{\partial x} \frac{\partial \theta}{\partial z} \right ) \right \} \\
+ \frac{\partial \theta}{\partial y}\left \{ \frac{1}{C_p} \left ( \frac{p_\circ}{p} \right )^\kappa \left [ \frac{\partial}{\partial y} \left (\frac{dQ}{dt} \right ) \right ] - \left ( \frac{\partial u}{\partial y} \frac{\partial \theta}{\partial x} \right ) - \left ( \frac{\partial v}{\partial y} \frac{\partial \theta}{\partial y} \right ) - \left ( \frac{\partial w}{\partial y} \frac{\partial \theta}{\partial z} \right ) \right \} \\
+ \frac{\partial \theta}{\partial z}\left \{ \frac{p_\circ^\kappa}{C_p} \left [ \frac{\partial}{\partial z} \left (p^{-\kappa} \frac{dQ}{dt} \right ) \right ] - \left ( \frac{\partial u}{\partial z} \frac{\partial \theta}{\partial x} \right ) - \left ( \frac{\partial v}{\partial z} \frac{\partial \theta}{\partial y} \right ) - \left ( \frac{\partial w}{\partial z} \frac{\partial \theta}{\partial z} \right ) \right \}\end{alignat}
},
{
"math_input":
\underbrace{ \frac{\partial k}{\partial t}}_{ \begin{smallmatrix}\text{Local}\\\text{derivative}\end{smallmatrix}}
+
\underbrace{\overline{u}_j \frac{\partial k}{\partial x_j}}_{ \begin{smallmatrix}\text{Advection}\end{smallmatrix}}
= -
\underbrace{ \frac{1}{\rho_o} \frac{\partial \overline{u'_i p'}}{\partial x_i} } _{ \begin{smallmatrix}\text{Pressure}\\\text{diffusion}\end{smallmatrix}}
-
\underbrace{ \frac{\partial \overline{k u_i}}{\partial x_j} }_{ \begin{smallmatrix} \text{Turbulent}\\ \text{transport} \\ \mathcal{T} \end{smallmatrix}}
+ \underbrace{ \nu\frac{\partial^2 k}{\partial x^2_j} }_{\begin{smallmatrix} \text{Molecular}\\ \text{viscous}\\ \text{transport} \end{smallmatrix}}
\underbrace{ - \overline{u'_i u'_j}\frac{\partial \overline{u_i}}{\partial x_j} }_{\begin{smallmatrix} \text{Production}\\ \mathcal{P} \end{smallmatrix}}
- \underbrace{ \nu \overline{\frac{\partial u'_i}{\partial x_j}\frac{\partial u'_i}{\partial x_j}} }_{\begin{smallmatrix} \text{Dissipation}\\ \epsilon_k \end{smallmatrix}}
- \underbrace{ \frac{g}{\rho_o} \overline{\rho' u'_i}\delta_{i3} }_{\begin{smallmatrix} \text{Buoyancy flux}\\ b \end{smallmatrix}}
},
{
"math_input":
\begin{align}
\frac{\partial G_{ij}}{\partial x^k} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^k}~\frac{\partial X^\beta}{\partial x^j} +
\frac{\partial^2 X^\alpha}{\partial x^j \partial x^k}~\frac{\partial X^\beta}{\partial x^i}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j}~\frac{\partial X^\gamma}{\partial x^k}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma} \\
\frac{\partial G_{ik}}{\partial x^j} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k} +
\frac{\partial^2 X^\alpha}{\partial x^j \partial x^k}~\frac{\partial X^\beta}{\partial x^i}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^k}~\frac{\partial X^\gamma}{\partial x^j}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma} \\
\frac{\partial G_{jk}}{\partial x^i} & = \left(\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j}~\frac{\partial X^\beta}{\partial x^k} +
\frac{\partial^2 X^\alpha}{\partial x^i \partial x^k}~\frac{\partial X^\beta}{\partial x^j}\right)~g_{\alpha\beta} +
\frac{\partial X^\alpha}{\partial x^j}~\frac{\partial X^\beta}{\partial x^k}~\frac{\partial X^\gamma}{\partial x^i}~\frac{\partial g_{\alpha\beta}}{\partial X^\gamma}
\end{align}
},
{
"math_input":
\begin{array}{lclcl}
() & & &=& \emptyset \\
& & & & \\
(1) &=& () \rightarrow 1 &=& \{\{()\},\{(),1\}\} \\
& & &=& \{\{\emptyset\},\{\emptyset,1\}\} \\
& & & & \\
(1,2) &=& (1) \rightarrow 2 &=& \{\{(1)\},\{(1),2\}\} \\
& & &=& \{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\
& & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\} \\
& & & & \\
(1,2,3) &=& (1,2) \rightarrow 3 &=& \{\{(1,2)\},\{(1,2),3\}\} \\
& & &=& \{\{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\
& & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\}\}, \\
& & & & \{\{\{\{\{\emptyset\},\{\emptyset,1\}\}\}, \\
& & & & \{\{\{\emptyset\},\{\emptyset,1\}\},2\}\},3\}\} \\
\end{array}
},
{
"math_input":
\begin{bmatrix}
\boldsymbol{I}_m & \boldsymbol{0} & \boldsymbol{V}_1^{(t)}\\
\boldsymbol{0} & \boldsymbol{I}_m & \boldsymbol{V}_1^{(b)} & \boldsymbol{0}\\
\boldsymbol{0} & \boldsymbol{W}_2^{(t)} & \boldsymbol{I}_m & \boldsymbol{0} & \boldsymbol{V}_2^{(t)}\\
& \boldsymbol{W}_2^{(b)} & \boldsymbol{0} & \boldsymbol{I}_m & \boldsymbol{V}_2^{(b)} & \boldsymbol{0} \\
& & \ddots & \ddots & \ddots & \ddots & \ddots\\
& & & \boldsymbol{0} & \boldsymbol{W}_{p-1}^{(t)} & \boldsymbol{I}_m & \boldsymbol{0} & \boldsymbol{V}_{p-1}^{(t)}\\
& & & & \boldsymbol{W}_{p-1}^{(b)} & \boldsymbol{0} & \boldsymbol{I}_m & \boldsymbol{V}_{p-1}^{(b)} & \boldsymbol{0}\\
& & & & & \boldsymbol{0} & \boldsymbol{W}_p^{(t)} & \boldsymbol{I}_m & \boldsymbol{0}\\
& & & & & & \boldsymbol{W}_p^{(b)} & \boldsymbol{0} & \boldsymbol{I}_m
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{X}_1^{(t)}\\
\boldsymbol{X}_1^{(b)}\\
\boldsymbol{X}_2^{(t)}\\
\boldsymbol{X}_2^{(b)}\\
\vdots\\
\boldsymbol{X}_{p-1}^{(t)}\\
\boldsymbol{X}_{p-1}^{(b)}\\
\boldsymbol{X}_p^{(t)}\\
\boldsymbol{X}_p^{(b)}
\end{bmatrix}
=
\begin{bmatrix}
\boldsymbol{G}_1^{(t)}\\
\boldsymbol{G}_1^{(b)}\\
\boldsymbol{G}_2^{(t)}\\
\boldsymbol{G}_2^{(b)}\\
\vdots\\
\boldsymbol{G}_{p-1}^{(t)}\\
\boldsymbol{G}_{p-1}^{(b)}\\
\boldsymbol{G}_p^{(t)}\\
\boldsymbol{G}_p^{(b)}
\end{bmatrix}\text{,}
},
{
"math_input": {\mathbf{A}\otimes\mathbf{B}} = \begin{bmatrix}
a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} &
\cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\
a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} &
\cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\
\vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\
a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} &
\cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\
\vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\
\vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\
a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} &
\cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\
a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} &
\cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\
\vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\
a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} &
\cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq}
\end{bmatrix}.
},
{
"math_input":
\begin{align}
&-10\sin^2 i \ \left(-\hat{g}\ \left(\frac{3}{8}\ +\ \frac{3}{16}\ {e_g}^2\ +\ \frac{15}{16}\ {e_h}^2\right)
+\hat{h}\ \left(\frac{3}{8}\ e_g\ e_h\right)\right) \\
&+6\ \left(-\hat{g}\ \left(\frac{1}{2}\ +\ \frac{3}{8}\ {e_g}^2\ +\ \frac{9}{8}\ {e_h}^2\right)
+\hat{h}\ \left(\frac{3}{4}\ e_g\ e_h\right)\right) \\
&-15\sin^2 i \ \left(\hat{g}\ \left(\frac{1}{8}\ +\ \frac{3}{16}\ {e_g}^2\ +\ \frac{3}{16}\ {e_h}^2\right)
+\hat{h}\ \left(\frac{3}{8}\ e_g\ e_h\right)\right) \\
&+3\left(\hat{g}\ \left(\frac{1}{2}\ +\ \frac{9}{8}\ {e_g}^2\ +\ \frac{3}{8}\ {e_h}^2\right)
+\hat{h}\ \left(\frac{3}{4}\ e_g\ e_h\right)\right) \\
&+\frac{15}{2}\sin^2 i\ e_g \ \left(-\hat{g}\ \left(\frac{1}{8}\ e_g\right) + \hat{h}\ \left( \frac{1}{8}\ e_h\right)\right) \\
&-\frac{15}{2}\sin^2 i\ e_h \ \left(-\hat{g}\ \left(\frac{1}{8}\ e_h\right) + \hat{h}\ \left( \frac{1}{8}\ e_g\right)\right) \\
&-\frac {3}{2}\ e_g \ \left(-\hat{g}\ \left(\frac{1}{4}\ e_g\right) + \hat{h}\ \left(\frac{1}{4}\ e_h\right)\right) \\
&+\frac{3}{2}\ e_h \ \left(-\hat{g}\ \left(\frac{1}{4}\ e_h\right) + \hat{h}\ \left(\frac{3}{4}\ e_g\right)\right) = \\
&\frac{3}{2}\ \left(\frac{5}{4}\ \sin^2 i\ -\ 1\right)\left((1-{e_g}^2\ +\ 4\ {e_h}^2)\hat{g}\ -\ 5\ e_g\ e_h\ \hat{h}\right)
\end{align}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of states} \\
T &=& \text{number of observations} \\
\phi_{i=1 \dots N, j=1 \dots N} &=& \text{probability of transition from state } i \text{ to state } j \\
\boldsymbol\phi_{i=1 \dots N} &=& N\text{-dimensional vector, composed of } \phi_{i,1 \dots N} \text{; must sum to 1} \\
V &=& \text{dimension of categorical observations, e.g. size of word vocabulary} \\
\theta_{i=1 \dots N, j=1 \dots V} &=& \text{probability for state } i \text{ of observing the } j\text{th item} \\
\boldsymbol\theta_{i=1 \dots N} &=& V\text{-dimensional vector, composed of }\theta_{i,1 \dots V} \text{; must sum to 1} \\
x_{t=1 \dots T} &=& \text{state of observation at time } t \\
y_{t=1 \dots T} &=& \text{observation at time } t \\
\alpha &=& \text{shared concentration hyperparameter of } \boldsymbol\theta \text{ for each state} \\
\beta &=& \text{concentration hyperparameter controlling the density of the transition matrix} \\
\boldsymbol\phi_{i=1 \dots N} &\sim& \operatorname{Symmetric-Dirichlet}_N(\beta) \\
\boldsymbol\theta_{1 \dots V} &\sim& \text{Symmetric-Dirichlet}_V(\alpha) \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{t-1}}) \\
y_{t=1 \dots T} &\sim& \text{Categorical}(\boldsymbol\theta_{x_t})
\end{array}
},
{
"math_input": \begin{align}
& {} \operatorname{E}[X_1 X_2 X_3 X_4 X_5 X_6] \\
&{} = \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_4 ]\operatorname{E}[X_5 X_6 ] + \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_5 ]\operatorname{E}[X_4 X_6] + \operatorname{E}[X_1 X_2 ]\operatorname{E}[X_3 X_6 ]\operatorname{E}[X_4 X_5] \\
&{} + \operatorname{E}[X_1 X_3 ]\operatorname{E}[X_2 X_4 ]\operatorname{E}[X_5 X_6 ] + \operatorname{E}[X_1 X_3 ]\operatorname{E}[X_2 X_5 ]\operatorname{E}[X_4 X_6 ] + \operatorname{E}[X_1 X_3]\operatorname{E}[X_2 X_6]\operatorname{E}[X_4 X_5] \\
&+ \operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_3]\operatorname{E}[X_5 X_6]+\operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_5]\operatorname{E}[X_3 X_6]+\operatorname{E}[X_1 X_4]\operatorname{E}[X_2 X_6]\operatorname{E}[X_3 X_5] \\
& + \operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_3]\operatorname{E}[X_4 X_6]+\operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_4]\operatorname{E}[X_3 X_6]+\operatorname{E}[X_1 X_5]\operatorname{E}[X_2 X_6]\operatorname{E}[X_3 X_4] \\
& + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_3]\operatorname{E}[X_4 X_5 ] + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_4 ]\operatorname{E}[X_3 X_5] + \operatorname{E}[X_1 X_6]\operatorname{E}[X_2 X_5]\operatorname{E}[X_3 X_4].
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\boldsymbol{T}\cdot\mathbf{c} & = \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(x_1+\alpha c_1, x_2 + \alpha c_2, x_3 + \alpha c_3)\right|_{\alpha=0} \equiv \left.\cfrac{d}{d\alpha}~\boldsymbol{T}(y_1, y_2, y_3)\right|_{\alpha=0} \\
& = \left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~\cfrac{\partial y_1}{\partial \alpha} +
\cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~\cfrac{\partial y_2}{\partial \alpha} +
\cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~\cfrac{\partial y_3}{\partial \alpha}
\right]_{\alpha=0} =
\left [\cfrac{\partial{\boldsymbol{T}}}{\partial y_1}~c_1 +
\cfrac{\partial{\boldsymbol{T}}}{\partial y_2}~c_2 +
\cfrac{\partial{\boldsymbol{T}}}{\partial y_3}~c_3
\right]_{\alpha=0} \\
& = \cfrac{\partial{\boldsymbol{T}}}{\partial x_1}~c_1 +
\cfrac{\partial{\boldsymbol{T}}}{\partial x_2}~c_2 +
\cfrac{\partial{\boldsymbol{T}}}{\partial x_3}~c_3 \equiv \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~c_i = \cfrac{\partial{\boldsymbol{T}}}{\partial x_i}~(\mathbf{e}_i\cdot\mathbf{c})
= \left[\cfrac{\partial{\boldsymbol{T}}}{\partial x_i}\otimes\mathbf{e}_i\right]\cdot\mathbf{c} \qquad \square
\end{align}
},
{
"math_input": \begin{align}
\delta_\Phi\mathcal{L}\, =\,
&-\, \rho\, \int_{t_0}^{t_1} \iint
\left\{
\frac{\partial}{\partial t} \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\; \text{d}z\;
+\, \boldsymbol{\nabla} \cdot \int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\, \boldsymbol{\nabla}\Phi\; \text{d}z\,
\right\}\; \text{d}\boldsymbol{x}\; \text{d}t
\\
&+\, \rho\, \int_{t_0}^{t_1} \iint
\left\{
\int_{-h(\boldsymbol{x})}^{\eta(\boldsymbol{x},t)} \delta\Phi\;
\left( \boldsymbol{\nabla} \cdot \boldsymbol{\nabla}\Phi\, +\, \frac{\partial^2\Phi}{\partial z^2} \right)\; \text{d}z\,
\right\}\; \text{d}\boldsymbol{x}\; \text{d}t
\\
&+\, \rho\, \int_{t_0}^{t_1} \iint
\left[
\left( \frac{\partial\eta}{\partial t}\, +\, \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} \eta\, -\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi
\right]_{z=\eta(\boldsymbol{x},t)}\; \text{d}\boldsymbol{x}\; \text{d}t
\\
&-\, \rho\, \int_{t_0}^{t_1} \iint
\left[
\left( \boldsymbol{\nabla}\Phi \cdot \boldsymbol{\nabla} h\, +\, \frac{\partial\Phi}{\partial z} \right)\, \delta\Phi
\right]_{z=-h(\boldsymbol{x})}\; \text{d}\boldsymbol{x}\; \text{d}t
\\
=\, &0.
\end{align}
},
{
"math_input":
\begin{align}
W_3 &= \beta-t^2 \\
W_5 &= 4\beta^3(1-6t^2)+\beta^2(1+8t^2) -2\beta t^2 +t^4 \\
W_7 &= 61-479t^2+179t^4-t^6+O(e^2) \\
W_4 &= 4\beta^2+\beta-t^2 \\
W_6 &= 8\beta^4(11{-}24t^2)-28\beta^3(1{-}6t^2) +\beta^2(1{-}32t^2)
-2\beta t^2 +t^4 \\
W_8 &= 1385-3111t^2+543t^4-t^6 +O(e^2)\\
V_3&= \beta_1+2t_1^2 \\
V_5&= 4\beta_1^3(1-6t_1^2)-\beta_1^2(9-68t_1^2)
-72\beta_1 t_1^2 -24t_1^4\\
V_7&= 61+662t_1^2+1320t_1^4+720t_1^6 \\
U_4&= 4\beta_1^2-9\beta_1(1-t_1^2)-12t_1^2\\
U_6&= 8\beta_1^4(11-24t_1^2)-12\beta_1^3(21-71t_1^2)
+15\beta_1^2(15-98t_1^2+15t_1^4) \\
&\qquad\qquad +180\beta_1(5t_1^2-3t_1^4)+360t_1^4\\
U_8&=-1385-3633t_1^2-4095t_1^4-1575t_1^6\\
H_2&= \beta\\
H_4&= 4\beta^3(1-6t^2)+\beta^2(1+24t^2)-4\beta t^2\\
H_6&=61-148t^2+16t^4\\
H_3&=2\beta^2-\beta\\
H_5&=\beta^4(11-24t^2)-\beta^3(11-36t^2)
+\beta^2(2-14t^2)+\beta t^2\\
H_7&=17-26t^2+2t^4\\
K_2&=\beta_1 \\
K_4&=4\beta_1^3(1-6t_1^2)-3\beta_1^2(1-16t_1^2)
-24\beta_1 t_1^2\\
K_6&=1\\
K_3&=2\beta_1^2-3\beta_1 -t_1^2\\
K_5&=\beta_1^4(11-24t_1^2)-3\beta_1^3(8-23t_1^2)
+5\beta_1^2(3-14t_1^2)+30\beta_1 t_1^2+3t_1^4\\
K_7&=-17-77t_1^2-105t_1^4-45t_1^6
\end{align}
},
{
"math_input":
\begin{align}
{n \choose k}\, p^k q^{n-k} & = \frac{n!}{k!\left(n-k\right)!} p^k q^{n-k} \\
& \simeq \frac{n^ne^{-n}\sqrt{2\pi n} }{k^ke^{-k}\sqrt{2\pi k} {(n-k)}^{n-k}e^{-(n-k)}\sqrt{2\pi (n-k)} }p^k q^{n-k}\\
& =\left[\frac{\sqrt{2\pi n} }{\sqrt{2\pi k} \sqrt{2\pi (n-k)} }\right]\left[\frac{n^n }{k^k {(n-k)}^{n-k} }\right]\left[\frac{e^{-n}}{e^{-k}e^{-(n-k)} }\right]p^k q^{n-k}\\
& =\left[\frac{\sqrt{n} }{\sqrt{k} \sqrt{2\pi (n-k)} }\right]\left[\frac{n^n }{k^k {(n-k)}^{n-k} }\right]\left[\frac{e^{-n}}{e^{-k} e^{-n}{ e}^k }\right]p^ kq^{n-k}\\
& =\left[\sqrt{\frac{n}{2\pi k(n-k)}}\right]\left[n^n{\left(\frac{p }{k }\right)}^k{\left(\frac{q }{n-k }\right)}^{(n- k)}\right] e^{-n+k+n-k}\\
& =\left[\sqrt{\frac{n}{2\pi k(n-k)}}\right]\left[n^{n-k+k}{\left(\frac{p }{k }\right)}^k{\left(\frac{q }{n-k }\right)}^{(n- k)}\right]\\
& =\left[\sqrt{\frac{n}{2\pi k(n-k)}}\right] \left[ n^{n-k} n^k {\left(\frac{p}{k}\right)}^k {\left(\frac{q}{n-k}\right)}^{(n-k)}\right]\\
& =\left[\sqrt{\frac{n}{2\pi k(n-k)}}\right]\left[{\left(\frac{np }{k }\right)}^k{\left(\frac{nq }{n-k }\right)}^{(n- k)} \right]\\
& =\left[\sqrt{\frac{n}{2\pi k(n-k)}}\right]\left[{\left(\frac{k }{np }\right)}^{-k}{\left(\frac{n-k}{nq }\right)}^{-(n- k)}\right]\\
\end{align}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of states} \\
T &=& \text{number of observations} \\
\phi_{i=1 \dots N, j=1 \dots N} &=& \text{probability of transition from state } i \text{ to state } j \\
\boldsymbol\phi_{i=1 \dots N} &=& N\text{-dimensional vector, composed of } \phi_{i,1 \dots N} \text{; must sum to 1} \\
\mu_{i=1 \dots N} &=& \text{mean of observations associated with state } i \\
\sigma^2_{i=1 \dots N} &=& \text{variance of observations associated with state } i \\
x_{t=1 \dots T} &=& \text{state of observation at time } t \\
y_{t=1 \dots T} &=& \text{observation at time } t \\
\beta &=& \text{concentration hyperparameter controlling the density of the transition matrix} \\
\mu_0, \lambda &=& \text{shared hyperparameters of the means for each state} \\
\nu, \sigma_0^2 &=& \text{shared hyperparameters of the variances for each state} \\
\boldsymbol\phi_{i=1 \dots N} &\sim& \operatorname{Symmetric-Dirichlet}_N(\beta) \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{t-1}}) \\
\mu_{i=1 \dots N} &\sim& \mathcal{N}(\mu_0, \lambda\sigma_i^2) \\
\sigma_{i=1 \dots N}^2 &\sim& \operatorname{Inverse-Gamma}(\nu, \sigma_0^2) \\
y_{t=1 \dots T} &\sim& \mathcal{N}(\mu_{x_t}, \sigma_{x_t}^2)
\end{array}
},
{
"math_input":
\begin{align}
S_0(s) & = 0 \\[10pt]
S_1(s) & = s \\[10pt]
S_2(s) & = 4s-4s^2 \\
& = 4s(1-s) \\[10pt]
S_3(s) & = 9s-24s^2+16s^3 \\
& = s(3-4s)^2 \\[10pt]
S_4(s) & = 16s-80s^2+128s^3-64s^4 \\
& = 16s(1-s)(1-2s)^2 \\[10pt]
S_5(s) & = 25s-200s^2+560s^3-640s^4+256s^5 \\
& = s(5-20s+16s^2)^2 \\[10pt]
S_6(s) & = 36s-420s^2+1792s^3-3456s^4+3072s^5-1024s^6 \\
& = 4s(1-s)(1-4s)^2(3-4s)^2 \\[10pt]
S_7(s) & = 49s-784s^2+4704s^3-13440s^4+19712s^5-14336s^6+4096s^7 \\
& = s(7-56s+112s^2-64s^3)^2 \\[10pt]
S_8(s) & = 64s-1344s^2+10752s^3-42240s^4+90112s^5-106496s^6 \\
& {} \qquad + 65536s^7-16384s^8 \\
& = 64s(s-1)(1-2s)^2(1-8s+8s^2)^2 \\[10pt]
S_9(s) & = 81s - 2160s^2 + 22176s^3 - 114048s^4 + 329472s^5 - 559104s^6 \\
& {} \qquad + 552960s^7 - 294912s^8 + 65536s^9 \\
& = s(-3+4s)^2(-3+36s-96s^2+64s^3)^2 \\[10pt]
S_{10}(s) & = 100s - 3300s^2 + 42240s^3 - 274560s^4 + 1025024s^5 \\
{} & \qquad - 2329600s^6 + 3276800s^7 - 2785280s^8 + 1310720s^9 - 262144s^{10} \\
& = 4s(1-s)(5 - 20s+16s^2)^2(1-12s+16s^2)^2\\[10pt]
S_{11}(s) & = 121s - 4840s^2 + 75504s^3 - 604032s^4 + 2818816s^5 \\
{} & \qquad -8200192s^6 + 15319040s^7 - 18382848s^8 + 13697024s^9 -5767168s^{10} + 1048576s^{11}\\
& = s(11 -220s + 1232s^2 -2816s^3 +2816s^4 -1024s^5)^2
\end{align}
},
{
"math_input":
\begin{bmatrix}
a_{1,1} & a_{1,2} \\
a_{2,1} & a_{2,2} \\
\end{bmatrix}
\otimes
\begin{bmatrix}
b_{1,1} & b_{1,2} \\
b_{2,1} & b_{2,2} \\
\end{bmatrix}
=
\begin{bmatrix}
a_{1,1} \begin{bmatrix}
b_{1,1} & b_{1,2} \\
b_{2,1} & b_{2,2} \\
\end{bmatrix} & a_{1,2} \begin{bmatrix}
b_{1,1} & b_{1,2} \\
b_{2,1} & b_{2,2} \\
\end{bmatrix} \\
& \\
a_{2,1} \begin{bmatrix}
b_{1,1} & b_{1,2} \\
b_{2,1} & b_{2,2} \\
\end{bmatrix} & a_{2,2} \begin{bmatrix}
b_{1,1} & b_{1,2} \\
b_{2,1} & b_{2,2} \\
\end{bmatrix} \\
\end{bmatrix}
=
\begin{bmatrix}
a_{1,1} b_{1,1} & a_{1,1} b_{1,2} & a_{1,2} b_{1,1} & a_{1,2} b_{1,2} \\
a_{1,1} b_{2,1} & a_{1,1} b_{2,2} & a_{1,2} b_{2,1} & a_{1,2} b_{2,2} \\
a_{2,1} b_{1,1} & a_{2,1} b_{1,2} & a_{2,2} b_{1,1} & a_{2,2} b_{1,2} \\
a_{2,1} b_{2,1} & a_{2,1} b_{2,2} & a_{2,2} b_{2,1} & a_{2,2} b_{2,2} \\
\end{bmatrix}.
},
{
"math_input":
\begin{align}
& P(Z_{(m,n)}=k|\boldsymbol{Z_{-(m,n)}}, \boldsymbol{W};\alpha,\beta) \\
\propto &
P(Z_{(m,n)}=k,\boldsymbol{Z_{-(m,n)}},\boldsymbol{W};\alpha,\beta) \\
= & \left(\frac{\Gamma\left(\sum_{i=1}^K \alpha_i
\right)}{\prod_{i=1}^K \Gamma(\alpha_i)}\right)^M \prod_{j\neq m}
\frac{\prod_{i=1}^K
\Gamma(n_{j,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{j,(\cdot)}^i+\alpha_i \bigr)} \\
& \times \left( \frac{\Gamma\bigl(\sum_{r=1}^V \beta_r
\bigr)}{\prod_{r=1}^V \Gamma(\beta_r)}\right)^K \prod_{i=1}^K
\prod_{r\neq v}
\Gamma(n_{(\cdot),r}^i+\beta_r) \\
& \times \frac{\prod_{i=1}^K
\Gamma(n_{m,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^i+\alpha_i \bigr)} \prod_{i=1}^K \frac{
\Gamma(n_{(\cdot),v}^i+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)} \\
\propto & \frac{\prod_{i=1}^K
\Gamma(n_{m,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^i+\alpha_i \bigr)} \prod_{i=1}^K \frac{
\Gamma(n_{(\cdot),v}^i+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)}\\
\propto & \prod_{i=1}^K
\Gamma(n_{m,(\cdot)}^i+\alpha_i) \prod_{i=1}^K \frac{
\Gamma(n_{(\cdot),v}^i+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)}.
.
\end{align}
},
{
"math_input":
\begin{align}
\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial H_\lambda}{\partial\lambda}\bigg|\Psi_\lambda(t)\bigg\rangle &=
\frac{\partial}{\partial\lambda}\langle\Psi_\lambda(t)|H_\lambda|\Psi_\lambda(t)\rangle
- \bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg|H_\lambda\bigg|\Psi_\lambda(t)\bigg\rangle
- \bigg\langle\Psi_\lambda(t)\bigg|H_\lambda\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
&= i\hbar \frac{\partial}{\partial\lambda}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
- i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg\rangle
+ i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
&= i\hbar \bigg\langle\Psi_\lambda(t)\bigg| \frac{\partial^2\Psi_\lambda(t)}{\partial\lambda \partial t}\bigg\rangle
+ i\hbar\bigg\langle\frac{\partial\Psi_\lambda(t)}{\partial t}\bigg|\frac{\partial\Psi_\lambda(t)}{\partial\lambda}\bigg\rangle \\
&= i \hbar \frac{\partial}{\partial t}\bigg\langle\Psi_\lambda(t)\bigg|\frac{\partial \Psi_\lambda(t)}{\partial \lambda}\bigg\rangle
\end{align}
},
{
"math_input":
\begin{align}
p(\mu,\sigma^2|\mathbf{X}) & \propto p(\mu,\sigma^2) \, p(\mathbf{X}|\mu,\sigma^2) \\
& \propto (\sigma^2)^{-(\nu_0+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + n_0(\mu-\mu_0)^2\right)\right] {\sigma^2}^{-n/2} \exp\left[-\frac{1}{2\sigma^2} \left(S + n(\bar{x} -\mu)^2\right)\right] \\
&= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + n_0(\mu-\mu_0)^2 + n(\bar{x} -\mu)^2\right)\right] \\
&= (\sigma^2)^{-(\nu_0+n+3)/2} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2 + (n_0+n)\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right)\right] \\
& \propto (\sigma^2)^{-1/2} \exp\left[-\frac{n_0+n}{2\sigma^2}\left(\mu-\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}\right)^2\right] \\
& \quad\times (\sigma^2)^{-(\nu_0/2+n/2+1)} \exp\left[-\frac{1}{2\sigma^2}\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right] \\
& = \mathcal{N}_{\mu|\sigma^2}\left(\frac{n_0\mu_0 + n\bar{x}}{n_0 + n}, \frac{\sigma^2}{n_0+n}\right) \cdot {\rm IG}_{\sigma^2}\left(\frac12(\nu_0+n), \frac12\left(\nu_0\sigma_0^2 + S + \frac{n_0 n}{n_0+n}(\mu_0-\bar{x})^2\right)\right) . \\
\end{align}
},
{
"math_input":
\begin{align}
0 &= \int_0^{\lambda} \eta(\xi)\; \text{d}\xi
= 2\, \int_0^{\tfrac12\lambda} \left[ \eta_2 + \left( \eta_1 - \eta_2 \right)\,
\operatorname{cn}^2\, \left( \begin{array}{c|c} \displaystyle \frac{\xi}{\Delta} & m \end{array} \right) \right]\; \text{d}\xi
\\
&= 2\, \int_0^{\tfrac12\pi} \Bigl[ \eta_2 + \left( \eta_1 - \eta_2 \right)\, \cos^2\, \psi \Bigr]\, \frac{\text{d}\xi}{\text{d}\psi}\; \text{d}\psi
= 2\, \Delta\, \int_0^{\tfrac12\pi} \frac{\eta_1 - \left( \eta_1 - \eta_2 \right)\, \sin^2\, \psi}{\sqrt{1 - m\, \sin^2\, \psi}}\; \text{d}\psi
\\
&= 2\, \Delta\, \int_0^{\tfrac12\pi} \frac{\eta_1 - m\, \left( \eta_1 - \eta_3 \right)\, \sin^2\, \psi}{\sqrt{1 - m\, \sin^2\, \psi}}\; \text{d}\psi
= 2\, \Delta\, \int_0^{\tfrac12\pi} \left[ \frac{\eta_3}{\sqrt{1 - m\, \sin^2\, \psi}}
+ \left( \eta_1 - \eta_3 \right)\, \sqrt{1 - m\, \sin^2\, \psi} \right]\; \text{d}\psi
\\
&= 2\, \Delta\, \Bigl[ \eta_3\, K(m) + \left( \eta_1 - \eta_3 \right)\, E(m) \Bigr]
= 2\, \Delta\, \Bigl[ \eta_3\, K(m) + \frac{H}{m}\, E(m) \Bigr],
\end{align}
},
{
"math_input":
\begin{alignat}{2}
\epsilon(0,\omega) & \simeq 1 + V_q \sum_{k,i}{ \frac{q_i \frac{\partial f_k}{\partial k_i}}{\hbar \omega_0 - \frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}} }\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}(1+\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0})\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0}\\
& = 1 + \frac{V_q}{\hbar \omega_0} 2 \int d^2 k (\frac{L}{2 \pi})^2 \sum_{i,j}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar k_j q_j}{m \omega_0}\\
& = 1 + \frac{V_q L^2}{m \omega_0^2} 2 \int \frac{d^2 k}{(2 \pi)^2} \sum_{i,j}{q_i q_j k_j \frac{\partial f_k}{\partial k_i}}\\
& = 1 + \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j 2 \int \frac{d^2 k}{(2 \pi)^2} k_j \frac{\partial f_k}{\partial k_i}}\\
& = 1 - \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j 2 \int \frac{d^2 k}{(2 \pi)^2} k_k \frac{\partial f_j}{\partial k_i}}\\
& = 1 - \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j n \delta_{ij}}\\
& = 1 - \frac{2 \pi e^2}{\epsilon q L^2} \frac{L^2}{m \omega_0^2} q^2 n\\
& = 1 - \frac{\omega_{pl}^2(q)}{\omega_0^2},
\end{alignat}
},
{
"math_input":
\begin{align}
\langle \partial_{t} u , e^{i k x}\rangle &= \langle \partial_{t} \sum_{l} \hat{u}_{l} e^{i l x} , e^{i k x} \rangle = \langle \sum_{l} \partial_{t} \hat{u}_{l} e^{i l x} , e^{i k x} \rangle = 2 \pi \partial_t \hat{u}_k,
\\
\langle f , e^{i k x} \rangle &= \langle \sum_{l} \hat{f}_{l} e^{i l x} , e^{i k x}\rangle= 2 \pi \hat{f}_k, \text{ and}
\\
\langle
\frac{1}{2} u^2 - \rho \partial_{x} u
,
\partial_x e^{i k x}
\rangle
&=
\langle
\frac{1}{2}
\left(\sum_{p} \hat{u}_p e^{i p x}\right)
\left(\sum_{q} \hat{u}_q e^{i q x}\right)
- \rho \partial_x \sum_{l} \hat{u}_l e^{i l x}
,
\partial_x e^{i k x}
\rangle
\\
&=
\langle
\frac{1}{2}
\sum_{p} \sum_{q} \hat{u}_p \hat{u}_q e^{i \left(p+q\right) x}
,
i k e^{i k x}
\rangle
-
\langle
\rho i \sum_{l} l \hat{u}_l e^{i l x}
,
i k e^{i k x}
\rangle
\\
&=
-\frac{i k}{2}
\langle
\sum_{p} \sum_{q} \hat{u}_p \hat{u}_q e^{i \left(p+q\right) x}
,
e^{i k x}
\rangle
- \rho k
\langle
\sum_{l} l \hat{u}_l e^{i l x}
,
e^{i k x}
\rangle
\\
&=
- i \pi k \sum_{p+q=k} \hat{u}_p \hat{u}_q - 2\pi\rho{}k^2\hat{u}_k.
\end{align}
},
{
"math_input":
\begin{align}
M_{xx} & = -D\left(\frac{\partial^2 w}{\partial x^2}+\nu\,\frac{\partial^2 w}{\partial y^2}\right) \\
& = \frac{2M_0(1-\nu)}{\pi}\sum_{m=1}^\infty\frac{1}{(2m-1)\cosh\alpha_m}\,
\sin\frac{(2m-1)\pi x}{a}
\left[
-\frac{(2m-1)\pi y}{a}\sinh\frac{(2m-1)\pi y}{a} + \right. \\
& \qquad \qquad \qquad \qquad
\left. \left\{\frac{2\nu}{1-\nu} + \alpha_m\tanh\alpha_m\right\}\cosh\frac{(2m-1)\pi y}{a}
\right] \\
M_{xy} & = (1-\nu)D\frac{\partial^2 w}{\partial x \partial y} \\
& = -\frac{2M_0(1-\nu)}{\pi}\sum_{m=1}^\infty\frac{1}{(2m-1)
\cosh\alpha_m}\,\cos\frac{(2m-1)\pi x}{a}
\left[\frac{(2m-1)\pi y}{a}\cosh\frac{(2m-1)\pi y}{a} + \right. \\
& \qquad \qquad \qquad \qquad
\left. (1-\alpha_m\tanh\alpha_m)\sinh\frac{(2m-1)\pi y}{a}\right] \\
Q_{zx} & = \frac{\partial M_{xx}}{\partial x}-\frac{\partial M_{xy}}{\partial y} \\
& = \frac{4M_0}{a}\sum_{m=1}^\infty \frac{1}{\cosh\alpha_m}\,
\cos\frac{(2m-1)\pi x}{a}\cosh\frac{(2m-1)\pi y}{a}\,.
\end{align}
},
{
"math_input":
\begin{bmatrix}
0&0&0&0&0&0&0&0&0\\
1&0&0&0&0&0&0&0&0\\
1&1&0&0&0&0&0&0&0\\
1&1&1&0&0&0&0&0&0\\
1&0&0&1&0&0&0&0&0\\
1&1&0&1&0&0&0&0&0\\
1&1&1&1&0&0&0&0&0\\
1&0&0&1&1&0&0&0&0\\
1&1&0&1&1&0&0&0&0\\
1&1&1&1&1&0&0&0&0\\
1&0&0&1&1&1&0&0&0\\
1&1&0&1&1&1&0&0&0\\
1&1&0&1&1&1&0&0&0\\
1&0&0&1&0&0&1&0&0\\
1&1&0&1&0&0&1&0&0\\
1&1&1&1&0&0&1&0&0\\
1&0&0&1&1&0&1&0&0\\
1&1&0&1&1&0&1&0&0\\
1&1&1&1&1&0&1&0&0\\
1&0&0&1&1&1&1&0&0\\
1&1&0&1&1&1&1&0&0\\
1&1&1&1&1&1&1&0&0\\
1&0&0&1&0&0&1&1&0\\
1&1&0&1&0&0&1&1&0\\
1&1&1&1&1&0&1&0&0\\
1&0&0&1&1&0&1&1&0\\
1&1&0&1&1&0&1&1&0\\
1&1&1&1&1&0&1&1&0\\
1&0&0&1&1&1&1&1&0\\
1&1&0&1&1&1&1&1&0\\
1&1&1&1&1&1&1&1&0\\
1&0&0&1&0&0&0&0&1\\
1&1&0&1&0&0&0&0&1\\
1&1&1&1&0&0&0&0&1\\
1&0&0&1&1&0&0&0&1\\
1&1&0&1&1&0&0&0&1\\
1&1&1&1&1&0&0&0&1\\
1&0&0&1&1&1&0&0&1\\
1&1&0&1&1&1&0&0&1\\
1&1&1&1&1&1&0&0&1\\
1&0&0&1&0&0&1&0&1\\
1&1&0&1&0&0&1&0&1\\
1&1&1&1&0&0&1&0&1\\
1&0&0&1&1&0&1&0&1\\
1&1&0&1&1&0&1&0&1\\
1&1&1&1&1&0&1&0&1\\
1&0&0&1&1&1&1&0&1\\
1&1&0&1&1&1&1&0&1\\
1&1&1&1&1&1&1&0&1\\
1&0&0&1&0&0&1&1&1\\
1&1&0&1&0&0&1&1&1\\
1&1&1&1&0&0&1&1&1\\
1&0&0&1&1&0&1&1&1\\
1&1&0&1&1&0&1&1&1\\
1&1&1&1&1&0&1&1&1\\
1&0&0&1&1&1&1&1&1\\
1&1&0&1&1&1&1&1&1\\
1&1&1&1&1&1&1&1&1
\end{bmatrix}
},
{
"math_input": \begin{align}
AA^{T} &= \begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}\cdot
\begin{bmatrix}
1 & 4\\
2 & 5\\
3 & 6
\end{bmatrix} =
\begin{bmatrix}
14 & 32\\
32 & 77
\end{bmatrix} \\
(AA^{T})^{-1} &= \begin{bmatrix}
14 & 32\\
32 & 77
\end{bmatrix}^{-1} = \frac{1}{54}
\begin{bmatrix}
77 & -32\\
-32 & 14
\end{bmatrix} \\
A^{T}(AA^{T})^{-1} &= \frac{1}{54}
\begin{bmatrix}
1 & 4\\
2 & 5\\
3 & 6
\end{bmatrix}\cdot
\begin{bmatrix}
77 & -32\\
-32 & 14
\end{bmatrix} = \frac{1}{18}
\begin{bmatrix}
-17 & 8\\
-2 & 2\\
13 & -4
\end{bmatrix} = A^{-1}_\text{right}
\end{align}
},
{
"math_input": A^{-1}=\left[\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
-3 & 3 & 0 & 0 & -2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
2 & -2 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 3 & 0 & 0 & -2 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & -2 & 0 & 0 & 1 & 1 & 0 & 0 \\
-3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -3 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & -1 & 0 \\
9 & -9 & -9 & 9 & 6 & 3 & -6 & -3 & 6 & -6 & 3 & -3 & 4 & 2 & 2 & 1 \\
-6 & 6 & 6 & -6 & -3 & -3 & 3 & 3 & -4 & 4 & -2 & 2 & -2 & -2 & -1 & -1 \\
2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\
-6 & 6 & 6 & -6 & -4 & -2 & 4 & 2 & -3 & 3 & -3 & 3 & -2 & -1 & -2 & -1 \\
4 & -4 & -4 & 4 & 2 & 2 & -2 & -2 & 2 & -2 & 2 & -2 & 1 & 1 & 1 & 1
\end{smallmatrix}\right]
},
{
"math_input": \begin{align}
\operatorname{H}[\mathbf{X}] &= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \ln\left (\Gamma_p(\tfrac{n}{2}) \right ) -\tfrac{1}{2}(n-p-1) \operatorname{E}[\ln|\mathbf{X}|] + \tfrac{np}{2} \\
&= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \tfrac{1}{4} p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) \\
&\qquad \qquad -\tfrac{1}{2}(n-p-1)\left(\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + p\ln(2) + \ln|\mathbf{V}|\right) + \tfrac{np}{2} \\
&= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \tfrac{1}{4} p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) \\
&\qquad \qquad - \left ( \tfrac{1}{2}(n-p-1)\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + \tfrac{1}{2}(n-p-1)p\ln(2) + \tfrac{1}{2}(n-p-1)\ln|\mathbf{V}|\right) + \tfrac{np}{2} \\
&= \tfrac{p+1}{2}\ln|\mathbf{V}| +\tfrac{1}{2}p(p+1)\ln(2) + \tfrac{1}{4}p(p-1) \ln(\pi) + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n}{2}+\tfrac{1-i}{2}\right ) \right ) -\tfrac{1}{2}(n-p-1)\sum_{i=1}^p \psi\left(\tfrac{1}{2}(n+1-i)\right) + \tfrac{np}{2}
\end{align}
},
{
"math_input":
F_{2^3} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\
1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\
1&\omega^2&\omega^4&\omega^6&\omega^8&\omega^{10}&\omega^{12}&\omega^{14} \\
1&\omega^3&\omega^6&\omega^9&\omega^{12}&\omega^{15}&\omega^{18}&\omega^{21} \\
1&\omega^4&\omega^8&\omega^{12}&\omega^{16}&\omega^{20}&\omega^{24}&\omega^{28} \\
1&\omega^5&\omega^{10}&\omega^{15}&\omega^{20}&\omega^{25}&\omega^{30}&\omega^{35} \\
1&\omega^6&\omega^{12}&\omega^{18}&\omega^{24}&\omega^{30}&\omega^{36}&\omega^{42} \\
1&\omega^7&\omega^{14}&\omega^{21}&\omega^{28}&\omega^{35}&\omega^{42}&\omega^{49} \\
\end{bmatrix} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\
1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\
1&\omega^2&\omega^4&\omega^6&1&\omega^2&\omega^4&\omega^6 \\
1&\omega^3&\omega^6&\omega&\omega^4&\omega^7&\omega^2&\omega^5 \\
1&\omega^4&1&\omega^4&1&\omega^4&1&\omega^4 \\
1&\omega^5&\omega^2&\omega^7&\omega^4&\omega&\omega^6&\omega^3 \\
1&\omega^6&\omega^4&\omega^2&1&\omega^6&\omega^4&\omega^2 \\
1&\omega^7&\omega^6&\omega^5&\omega^4&\omega^3&\omega^2&\omega \\
\end{bmatrix}.
},
{
"math_input":
F_{2^3} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\
1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\
1&\omega^2&\omega^4&\omega^6&\omega^8&\omega^{10}&\omega^{12}&\omega^{14} \\
1&\omega^3&\omega^6&\omega^9&\omega^{12}&\omega^{15}&\omega^{18}&\omega^{21} \\
1&\omega^4&\omega^8&\omega^{12}&\omega^{16}&\omega^{20}&\omega^{24}&\omega^{28} \\
1&\omega^5&\omega^{10}&\omega^{15}&\omega^{20}&\omega^{25}&\omega^{30}&\omega^{35} \\
1&\omega^6&\omega^{12}&\omega^{18}&\omega^{24}&\omega^{30}&\omega^{36}&\omega^{42} \\
1&\omega^7&\omega^{14}&\omega^{21}&\omega^{28}&\omega^{35}&\omega^{42}&\omega^{49} \\
\end{bmatrix} = \frac{1}{\sqrt{2^3}} \begin{bmatrix} 1&1&1&1&1&1&1&1 \\
1&\omega&\omega^2&\omega^3&\omega^4&\omega^5&\omega^6&\omega^7 \\
1&\omega^2&\omega^4&\omega^6&1&\omega^2&\omega^4&\omega^6 \\
1&\omega^3&\omega^6&\omega&\omega^4&\omega^7&\omega^2&\omega^5 \\
1&\omega^4&1&\omega^4&1&\omega^4&1&\omega^4 \\
1&\omega^5&\omega^2&\omega^7&\omega^4&\omega&\omega^6&\omega^3 \\
1&\omega^6&\omega^4&\omega^2&1&\omega^6&\omega^4&\omega^2 \\
1&\omega^7&\omega^6&\omega^5&\omega^4&\omega^3&\omega^2&\omega \\
\end{bmatrix}
},
{
"math_input":
\begin{align}
\sum_{i = 1}^n (y_i - \overline{y})^2 &= \sum_{i = 1}^n (y_i - \overline{y} + \hat{y}_i - \hat{y}_i)^2
= \sum_{i = 1}^n ((\hat{y}_i - \bar{y}) + \underbrace{(y_i - \hat{y}_i)}_{\hat{\varepsilon}_i})^2 \\
&= \sum_{i = 1}^n ((\hat{y}_i - \bar{y})^2 + 2 \hat{\varepsilon}_i (\hat{y}_i - \bar{y}) + \hat{\varepsilon}_i^2) \\
&= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 + 2 \sum_{i = 1}^n \hat{\varepsilon}_i (\hat{y}_i - \bar{y}) \\
&= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 + 2 \sum_{i = 1}^n \hat{\varepsilon}_i(\hat{\beta}_0 + \hat{\beta}_1 x_{i1} + \cdots + \hat{\beta}_p x_{ip} - \overline{y}) \\
&= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 + 2 (\hat{\beta}_0 - \overline{y}) \underbrace{\sum_{i = 1}^n \hat{\varepsilon}_i}_0 + 2 \hat{\beta}_1 \underbrace{\sum_{i = 1}^n \hat{\varepsilon}_i x_{i1}}_0 + \cdots + 2 \hat{\beta}_p \underbrace{\sum_{i = 1}^n \hat{\varepsilon}_i x_{ip}}_0 \\
&= \sum_{i = 1}^n (\hat{y}_i - \bar{y})^2 + \sum_{i = 1}^n \hat{\varepsilon}_i^2 = \mathrm{ESS} + \mathrm{RSS} \\
\end{align}
},
{
"math_input":
\begin{align}
&\int\limits_{0}^{2\pi}\left(-\hat{t}\ {\left(\frac{p}{r}\right)}^3\ 2 \ \sin u\,\ \left(5\sin^2 i \ \sin^2 u\ -\ 3\right)\ - \ \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)\ {\left(\frac{p}{r}\right)}^3\ \frac{3}{2}\ \left(5\ \sin^2 i \ \sin^2 u\ -1\right) \ \cos u\right)\ du\ = \\
&-10\sin^2 i \ \int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^3\ \sin^3 u\ du \\
&+6\ \int\limits_{0}^{2\pi}\hat{t}\ {\left(\frac{p}{r}\right)}^3\ \sin u\ du \\
&-15\ \sin^2 i \int\limits_{0}^{2\pi} \hat{r}\ {\left(\frac{p}{r}\right)}^3\ \sin^2 u \ \cos u\ du \\
&+3\ \int\limits_{0}^{2\pi} \hat{r}\ {\left(\frac{p}{r}\right)}^3\ \cos u\ du \\
&+\frac{15}{2}\sin^2 i\ e_g\int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^2\ \ \ \sin^3 u \ \cos u\ du \\
&-\frac{15}{2}\sin^2 i\ e_h \ \int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^2\ \ \ \sin^2 u \ \cos^2 u\ du \\
&-\frac{3}{2}\ e_g \ \int\limits_{0}^{2\pi} \ \hat{t}\ {\left(\frac{p}{r}\right)}^2\ \ \sin u \ \cos u\ du \\
&+\frac{3}{2}\ e_h \ \int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^2\ \ \cos^2 u\ du
\end{align}
},
{
"math_input": \begin{align}
\frac{\partial y}{\partial c}
&= b_0 x^c \ln(x) \sum_{r = 0}^\infty \frac{(c + \gamma - 1) (c + \alpha)_r (c + \beta)_r}
{(c + 1)_r (c + \gamma)_r} x^r +\\
&\qquad+ b_0 x^c \sum_{r = 0}^\infty \frac{(c + \gamma - 1) (c + \alpha)_r (c + \beta)_r}
{(c + 1)_r (c + \gamma)_r}
\Biggl(\frac{1}{c + \gamma - 1} + \\
&\qquad\qquad+ \sum_{k = 0}^{r - 1} \left(\frac{1}{c + \alpha + k} + \frac{1}{c + \beta + k} - \frac{1}{c + 1 + k}
- \frac{1}{c + \gamma + k} \right) \Biggr)x^r \\
&= b_0 x^c \sum_{r = 0}^\infty \qquad\frac{(c + \gamma - 1) (c + \alpha)_r (c + \beta)_r}
{(c + 1)_r (c + \gamma)_r}
\Biggl(\ln x + \frac{1}{c + \gamma - 1} + \\
&\qquad+ \sum_{k = 0}^{r - 1} \left(\frac{1}{c + \alpha + k} + \frac{1}{c +\beta + k}
- \frac{1}{c + 1 + k} - \frac{1}{c + \gamma + k}\right) \Biggr) x^r.
\end{align}
},
{
"math_input": \begin{smallmatrix}
&\text{H}& &\text{H}& & & &\text{H}& &\text{H}& & &\text{H}& & & \\
& | & & | & &\mathsf{ADH} & & | & & | &\mathsf{ALDH} & & | & & & \\
\text{H}\,-\!&\text{C}&\!-\!&\text{C}&\!-\,\text{O}\,-\,\text{H}&\xrightarrow{\qquad}&\text{H}\,-\!&\text{C}&\!-\!&\text{C}&\xrightarrow{\qquad\ }&\text{H}\,-\!&\text{C}&\!-\!&\text{C}&\!-\,\text{O}\,-\,\text{H}\\
& | & & | & & & & | & & \| & & & | & & \| & \\
&\text{H}& &\text{H}& & & &\text{H}& &\text{O}& & &\text{H}& &\text{O}& \\
\end{smallmatrix}
},
{
"math_input": \begin{align} & \frac{D^1(f\circ{}g)}{1!} & = \left(f^{(1)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!} }{1!} \\[8pt]
& \frac{D^2(f\circ g)}{2!} & = \left(f^{(1)}\circ{}g\right)\frac{\frac{g^{(2)} }{2!} }{1!} & {} + \left(f^{(2)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!} }{2!} \\[8pt]
& \frac{D^3(f\circ g)}{3!} & = \left(f^{(1)}\circ{}g\right)\frac{\frac{g^{(3)} }{3!} }{1!} & {} + \left(f^{(2)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!} }{1!}\frac{\frac{g^{(2)} }{2!} }{1!} & {} + \left(f^{(3)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!} }{3!} \\[8pt]
& \frac{D^4(f\circ g)}{4!} & = \left(f^{(1)}\circ{}g\right)\frac{\frac{g^{(4)} }{4!} }{1!} & {} + \left(f^{(2)}\circ{}g\right)\left(\frac{\frac{g^{(1)} }{1!} }{1!}\frac{\frac{g^{(3)} }{3!} }{1!}+\frac{\frac{g^{(2)} }{2!}\frac{g^{(2)} }{2!} }{2!}\right) & {} + \left(f^{(3)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!} }{2!}\frac{\frac{g^{(2)} }{2!} }{1!} & {} + \left(f^{(4)}\circ{}g\right)\frac{\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!}\frac{g^{(1)} }{1!} }{4!}
\end{align}
},
{
"math_input": \begin{align}
&\frac{\partial}{\partial x}\left(2 \mu \frac{\partial u}{\partial x} + \lambda \nabla \cdot \mathbf{v}\right) +
\frac{\partial}{\partial y}\left(\mu\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)\right) +
\frac{\partial}{\partial z}\left(\mu\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}\right)\right) \\ \\
& =
2 \mu \frac{\partial^2 u}{\partial x^2} +
\mu \frac{\partial^2 u}{\partial y^2} + \mu \frac{\partial^2 v}{\partial y \, \partial x} +
\mu \frac{\partial^2 u}{\partial z^2} + \mu \frac{\partial^2 w}{\partial z \, \partial x} \\ \\
& =
\mu \frac{\partial^2 u}{\partial x^2} +
\mu \frac{\partial^2 u}{\partial y^2} +
\mu \frac{\partial^2 u}{\partial z^2} +
\mu \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 v}{\partial y \, \partial x} + \mu \frac{\partial^2 w}{\partial z \, \partial x} \\ \\
& = \mu \nabla^2 u + \mu \frac{\partial}{\partial x} \cancelto{0}{\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right)} = \mu \nabla^2 u
\end{align}
},
{
"math_input":
\begin{align}
\bar{\Pi}^0_0 & = 1 &
\bar{\Pi}^1_3 & = \frac{1}{4}\sqrt{6}(5z^2-r^2) &
\bar{\Pi}^4_4 & = \frac{1}{8}\sqrt{35} \\
\bar{\Pi}^0_1 & = z &
\bar{\Pi}^2_3 & = \frac{1}{2}\sqrt{15}\; z &
\bar{\Pi}^0_5 & = \frac{1}{8}z(63z^4-70z^2r^2+15r^4) \\
\bar{\Pi}^1_1 & = 1 &
\bar{\Pi}^3_3 & = \frac{1}{4}\sqrt{10} &
\bar{\Pi}^1_5 & = \frac{1}{8}\sqrt{15} (21z^4-14z^2r^2+r^4) \\
\bar{\Pi}^0_2 & = \frac{1}{2}(3z^2-r^2) &
\bar{\Pi}^0_4 & = \frac{1}{8}(35 z^4-30 r^2 z^2 +3r^4 ) &
\bar{\Pi}^2_5 & = \frac{1}{4}\sqrt{105}(3z^2-r^2)z \\
\bar{\Pi}^1_2 & = \sqrt{3}z &
\bar{\Pi}^1_4 & = \frac{\sqrt{10}}{4} z(7z^2-3r^2) &
\bar{\Pi}^3_5 & = \frac{1}{16}\sqrt{70} (9z^2-r^2) \\
\bar{\Pi}^2_2 & = \frac{1}{2}\sqrt{3} &
\bar{\Pi}^2_4 & = \frac{1}{4}\sqrt{5}(7z^2-r^2) &
\bar{\Pi}^4_5 & = \frac{3}{8}\sqrt{35} z \\
\bar{\Pi}^0_3 & = \frac{1}{2} z(5z^2-3r^2) &
\bar{\Pi}^3_4 & = \frac{1}{4}\sqrt{70}\;z &
\bar{\Pi}^5_5 & = \frac{3}{16}\sqrt{14} \\
\end{align}
},
{
"math_input":
\begin{align}
& \Pr(Y_i=1\mid\mathbf{X}_i) \\[4pt]
= {} & \Pr(Y_i^{1\ast} > Y_i^{0\ast}\mid\mathbf{X}_i) & \\
= {} & \Pr(Y_i^{1\ast} - Y_i^{0\ast} > 0\mid\mathbf{X}_i) & \\
= {} & \Pr(\boldsymbol\beta_1 \cdot \mathbf{X}_i + \varepsilon_1 - (\boldsymbol\beta_0 \cdot \mathbf{X}_i + \varepsilon_0) > 0) & \\
= {} & \Pr((\boldsymbol\beta_1 \cdot \mathbf{X}_i - \boldsymbol\beta_0 \cdot \mathbf{X}_i) + (\varepsilon_1 - \varepsilon_0) > 0) & \\
= {} & \Pr((\boldsymbol\beta_1 - \boldsymbol\beta_0) \cdot \mathbf{X}_i + (\varepsilon_1 - \varepsilon_0) > 0) & \\
= {} & \Pr((\boldsymbol\beta_1 - \boldsymbol\beta_0) \cdot \mathbf{X}_i + \varepsilon > 0) & & \text{(substitute }\varepsilon\text{ as above)} \\
= {} & \Pr(\boldsymbol\beta \cdot \mathbf{X}_i + \varepsilon > 0) & & \text{(substitute }\boldsymbol\beta\text{ as above)} \\
= {} & \Pr(\varepsilon > -\boldsymbol\beta \cdot \mathbf{X}_i) & & \text{(now, same as above model)}\\
= {} & \Pr(\varepsilon < \boldsymbol\beta \cdot \mathbf{X}_i) & \\
= {} & \operatorname{logit}^{-1}(\boldsymbol\beta \cdot \mathbf{X}_i) & \\
= {} & p_i
\end{align}
},
{
"math_input": \begin{matrix} {\rm GPN's} \\ 0 \\1\\2\\~\\~\\5\\~\\ 7\\ ~ \\ ~\\ \vdots \\ ~ \\ ~ \end{matrix}
~~~ p(n) = \begin{vmatrix} ~~1 & -1~ & ~& ~ & ~ &~&~&~ \\
~~1 & ~1 & -1~ & ~ \\
~~0 & ~1 & ~1 & -1~ & ~ \\
~~0 & ~0 & ~1 & ~1 &-1~ & ~ \\
-1 &~0 & ~0 & ~1 & ~1 &-1~ & ~ \\
~~0 & -1~ & ~0 & ~0 & ~1 & ~1 & -1~ & ~ \\
-1 & ~0& -1~ & ~0 & ~0 & ~1 & ~1 & -1~ &~ \\
~~0 & -1~ &~0& -1~ & ~0 & ~0 & ~1 & ~1 & -1~ &~ \\
~~0 & ~0 & -1~ &~0& -1~ & ~0 & ~0 & ~1 & ~1 & ~ \\
~~ \vdots & ~ & ~ & ~ & ~ & ~ &~ & ~ & ~ & \ddots \\
\end{vmatrix} _{ n \times n} .
},
{
"math_input":
\begin{align}
\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \right]
&+ \frac{\partial}{\partial x}
\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_x + S_{xx} + \frac12 \rho g (h+\overline{\eta})^2 \right]
+ \frac{\partial}{\partial y}
\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_x \overline{u}_y + S_{xy} \right]
\\
&= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial x} h
+ \tau_{w,x} - \tau_{b,x},
\\
\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \right]
&+ \frac{\partial}{\partial x}
\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_x + S_{yx} \right]
+ \frac{\partial}{\partial y}
\left[ \rho \left( h + \overline{\eta} \right) \overline{u}_y \overline{u}_y + S_{yy} + \frac12 \rho g (h+\overline{\eta})^2 \right]
\\
&= \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial y} h
+ \tau_{w,y} - \tau_{b,y}.
\end{align}
},
{
"math_input": \begin{align}\mathbf{S} & = \mathbf{E} \times \mathbf{H}\\
& = \mathrm{Re}\left(\mathbf{\widetilde E}\right) \times \mathrm{Re}\left(\mathbf{\widetilde H} \right)\\
& = \mathrm{Re}\left(\mathbf{E_c} e^{j\omega t}\right) \times \mathrm{Re}\left(\mathbf{H_c} e^{j\omega t}\right)\\
& = \frac{1}{2}\left(\mathbf{E_c} e^{j\omega t} + \mathbf{E_c}^* e^{-j\omega t}\right) \times \frac{1}{2}\left(\mathbf{H_c} e^{j\omega t} + \mathbf{H_c}^* e^{-j\omega t}\right)\\
& = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \mathbf{E_c}^* \times \mathbf{H_c} + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \mathbf{E_c}^* \times \mathbf{H_c}^* e^{-2j\omega t}\right)\\
& = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \left(\mathbf{E_c} \times \mathbf{H_c}^*\right)^* + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right)^*\right)\\
& = \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c}^*\right) + \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right).
\end{align}
},
{
"math_input": \begin{align}\mathbf{S} & = \mathbf{E} \times \mathbf{H}\\
& = \mathrm{Re}\left(\mathbf{\widetilde E}\right) \times \mathrm{Re}\left(\mathbf{\widetilde H} \right)\\
& = \mathrm{Re}\left(\mathbf{E_c} e^{j\omega t}\right) \times \mathrm{Re}\left(\mathbf{H_c} e^{j\omega t}\right)\\
& = \frac{1}{2}\left(\mathbf{E_c} e^{j\omega t} + \mathbf{E_c}^* e^{-j\omega t}\right) \times \frac{1}{2}\left(\mathbf{H_c} e^{j\omega t} + \mathbf{H_c}^* e^{-j\omega t}\right)\\
& = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \mathbf{E_c}^* \times \mathbf{H_c} + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \mathbf{E_c}^* \times \mathbf{H_c}^* e^{-2j\omega t}\right)\\
& = \frac{1}{4}\left(\mathbf{E_c} \times \mathbf{H_c}^* + \left(\mathbf{E_c} \times \mathbf{H_c}^*\right)^* + \mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t} + \left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right)^*\right)\\
& = \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c}^*\right) + \frac{1}{2}\mathrm{Re}\left(\mathbf{E_c} \times \mathbf{H_c} e^{2j\omega t}\right)
\end{align}
},
{
"math_input": \begin{align} A(4, 3) & = A(3, A(4, 2)) \\
& = A(3, A(3, A(4, 1))) \\
& = A(3, A(3, A(3, A(4, 0)))) \\
& = A(3, A(3, A(3, A(3, 1)))) \\
& = A(3, A(3, A(3, A(2, A(3, 0))))) \\
& = A(3, A(3, A(3, A(2, A(2, 1))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(2, 0)))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(1, 1)))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0))))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1))))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, 2)))))) \\
& = A(3, A(3, A(3, A(2, A(1, 3))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(1, 2)))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0)))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1)))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, 3)))))) \\
& = A(3, A(3, A(3, A(2, A(0, 4))))) \\
& = A(3, A(3, A(3, A(2, 5)))) \\
& = ... \\
& = A(3, A(3, A(3, 13))) \\
& = ... \\
& = A(3, A(3, 65533)) \\
& = ... \\
& = A(3, 2^{65536} - 3) \\
& = ... \\
& = 2^{2^{ \overset{65536}{} }} - 3. \\
\end{align}
},
{
"math_input":
\begin{array}{ccc} \pi\varepsilon\varrho\iota\varphi\varepsilon\varrho\varepsilon\iota\tilde\omega\nu & \varepsilon\overset{\text{'}}\nu\vartheta\varepsilon\iota\tilde\omega\nu & \overset{\text{`}}\varepsilon\xi\eta\kappa\omicron\sigma\tau\tilde\omega\nu \\
\begin{array}{|l|} \hline \pi\delta\angle' \\ \pi\varepsilon \\ \pi\varepsilon\angle' \\ \hline \pi\stigma \\ \pi\stigma\angle' \\ \pi\zeta \\ \hline \end{array} & \begin{array}{|r|r|r|} \hline \pi & \mu\alpha & \gamma \\ \pi\alpha & \delta & \iota\varepsilon \\ \pi\alpha & \kappa\zeta & \kappa\beta \\ \hline \pi\alpha & \nu & \kappa\delta \\ \pi\beta & \iota\gamma & \iota\vartheta \\ \pi\beta & \lambda\stigma & \vartheta \\ \hline \end{array} & \begin{array}{|r|r|r|r|} \hline \circ & \circ & \mu\stigma & \kappa\varepsilon \\ \circ & \circ & \mu\stigma & \iota\delta \\ \circ & \circ & \mu\stigma & \gamma \\ \hline \circ & \circ & \mu\varepsilon & \nu\beta \\ \circ & \circ & \mu\varepsilon & \mu \\ \circ & \circ & \mu\varepsilon & \kappa\vartheta \\ \hline \end{array}
\end{array}
},
{
"math_input": \begin{align}
x:\;\; \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y} + u_z \frac{\partial u_x}{\partial z}\right)
&= -\frac{\partial P}{\partial x} + \frac{\partial \tau_{xx}}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} + \frac{\partial \tau_{xz}}{\partial z} + \rho g_x
\\
y:\;\; \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}+ u_z \frac{\partial u_y}{\partial z}\right)
&= -\frac{\partial P}{\partial y} + \frac{\partial \tau_{yx}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{yz}}{\partial z} + \rho g_y
\\
z:\;\; \rho \left(\frac{\partial u_z}{\partial t} + u_x \frac{\partial u_z}{\partial x} + u_y \frac{\partial u_z}{\partial y}+ u_z \frac{\partial u_z}{\partial z}\right)
&= -\frac{\partial P}{\partial z} + \frac{\partial \tau_{zx}}{\partial x} + \frac{\partial \tau_{zy}}{\partial y} + \frac{\partial \tau_{zz}}{\partial z} + \rho g_z.
\end{align}
},
{
"math_input":
\begin{align}
d\, {\rm tr}(\mathbf{AXBX^{\rm T}C}) &= d\, {\rm tr}(\mathbf{CAXBX^{\rm T}}) = {\rm tr}(d(\mathbf{CAXBX^{\rm T}})) \\
&= {\rm tr}(\mathbf{CAX} d(\mathbf{BX^{\rm T}}) + d(\mathbf{CAX})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}(\mathbf{CAX} d(\mathbf{BX^{\rm T}})) + {\rm tr}(d(\mathbf{CAX})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}(\mathbf{CAXB} d(\mathbf{X^{\rm T}})) + {\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}(\mathbf{CAXB} (d\mathbf{X})^{\rm T}) + {\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}\left((\mathbf{CAXB} (d\mathbf{X})^{\rm T})^{\rm T}\right) + {\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}((d\mathbf{X})\mathbf{B^{\rm T}X^{\rm T}A^{\rm T}C^{\rm T}}) + {\rm tr}(\mathbf{CA}(d\mathbf{X})\mathbf{BX^{\rm T}}) \\
&= {\rm tr}(\mathbf{B^{\rm T}X^{\rm T}A^{\rm T}C^{\rm T}}(d\mathbf{X})) + {\rm tr}(\mathbf{BX^{\rm T}}\mathbf{CA}(d\mathbf{X})) \\
&= {\rm tr}\left((\mathbf{B^{\rm T}X^{\rm T}A^{\rm T}C^{\rm T}} + \mathbf{BX^{\rm T}}\mathbf{CA})d\mathbf{X}\right)
\end{align}
},
{
"math_input":
\begin{array}{rlll}
\Pr(Y_i=1\mid\mathbf{X}_i) &=& \Pr(Y_i^{1\ast} > Y_i^{0\ast}\mid\mathbf{X}_i) & \\
&=& \Pr(Y_i^{1\ast} - Y_i^{0\ast} > 0\mid\mathbf{X}_i) & \\
&=& \Pr(\boldsymbol\beta_1 \cdot \mathbf{X}_i + \varepsilon_1 - (\boldsymbol\beta_0 \cdot \mathbf{X}_i + \varepsilon_0) > 0) & \\
&=& \Pr((\boldsymbol\beta_1 \cdot \mathbf{X}_i - \boldsymbol\beta_0 \cdot \mathbf{X}_i) + (\varepsilon_1 - \varepsilon_0) > 0) & \\
&=& \Pr((\boldsymbol\beta_1 - \boldsymbol\beta_0) \cdot \mathbf{X}_i + (\varepsilon_1 - \varepsilon_0) > 0) & \\
&=& \Pr((\boldsymbol\beta_1 - \boldsymbol\beta_0) \cdot \mathbf{X}_i + \varepsilon > 0) & \text{(substitute }\varepsilon\text{ as above)} \\
&=& \Pr(\boldsymbol\beta \cdot \mathbf{X}_i + \varepsilon > 0) & \text{(substitute }\boldsymbol\beta\text{ as above)} \\
&=& \Pr(\varepsilon > -\boldsymbol\beta \cdot \mathbf{X}_i) & \text{(now, same as above model)}\\
&=& \Pr(\varepsilon < \boldsymbol\beta \cdot \mathbf{X}_i) & \\
&=& \operatorname{logit}^{-1}(\boldsymbol\beta \cdot \mathbf{X}_i) & \\
&=& p_i &
\end{array}
},
{
"math_input": \begin{align}
\varepsilon_{ij} &= \frac{1}{2}\left(u_{i,j}+u_{j,i}\right) \\
&=
\left[\begin{matrix}
\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \\
\end{matrix}\right] \\
&=
\left[\begin{matrix}
\frac{\partial u_1}{\partial x_1} & \frac{1}{2} \left(\frac{\partial u_1}{\partial x_2}+\frac{\partial u_2}{\partial x_1}\right) & \frac{1}{2} \left(\frac{\partial u_1}{\partial x_3}+\frac{\partial u_3}{\partial x_1}\right) \\
\frac{1}{2} \left(\frac{\partial u_2}{\partial x_1}+\frac{\partial u_1}{\partial x_2}\right) & \frac{\partial u_2}{\partial x_2} & \frac{1}{2} \left(\frac{\partial u_2}{\partial x_3}+\frac{\partial u_3}{\partial x_2}\right) \\
\frac{1}{2} \left(\frac{\partial u_3}{\partial x_1}+\frac{\partial u_1}{\partial x_3}\right) & \frac{1}{2} \left(\frac{\partial u_3}{\partial x_2}+\frac{\partial u_2}{\partial x_3}\right) & \frac{\partial u_3}{\partial x_3} \\
\end{matrix}\right] \end{align}
},
{
"math_input": {\underbrace{\partial \overline{hu} \over \partial t}}_{
\begin{smallmatrix}
\text{Change in}\\
\text{x mass flux}\\
\text{over time}
\end{smallmatrix}}
+ \underbrace{{\partial \over \partial x} \left( \overline{hu^2}+{1 \over 2}{k_{ap}g_zh^2}\right) + {\partial \overline{huv} \over \partial y}}_{
\begin{smallmatrix}
\text{Total spatial variation}\\
\text{of x,y momentum fluxes}\\
\text{in x-direction}
\end{smallmatrix}}
= \underbrace{-hk_{ap} \sgn \left({\partial u \over \partial y}\right){\partial hg_z \over \partial y}\sin \phi_{int}}_{
\begin{smallmatrix}
\text{Dissipative internal}\\
\text{friction force}\\
\text{in x-direction}
\end{smallmatrix}}
- \underbrace{{u \over \sqrt{u^2+v^2}}\left[ g_zh \left(1+{u \over r_xg_x}\right) \right]\tan \phi_{bed}}_{
\begin{smallmatrix}
\text{Dissipative basal}\\
\text{friction force}\\
\text{in x-direction}
\end{smallmatrix}}
+ \underbrace{g_xh}_{
\begin{smallmatrix}
\text{Driving}\\
\text{gravitational}\\
\text{force in}\\
\text{x-direction}
\end{smallmatrix}}
},
{
"math_input": {\underbrace{\partial \overline{hv} \over \partial t}}_{
\begin{smallmatrix}
\text{Change in}\\
\text{y mass flux}\\
\text{over time}
\end{smallmatrix}}
+ \underbrace{{\partial \overline{huv} \over \partial x} + {\partial \over \partial y} \left( \overline{hv^2}+{1 \over 2}{k_{ap}g_zh^2}\right)}_{
\begin{smallmatrix}
\text{Total spatial variation}\\
\text{of x,y momentum fluxes}\\
\text{in y-direction}
\end{smallmatrix}}
= \underbrace{-hk_{ap} \sgn \left({\partial v \over \partial x}\right){\partial hg_z \over \partial x}\sin \phi_{int}}_{
\begin{smallmatrix}
\text{Dissipative internal}\\
\text{friction force}\\
\text{in y-direction}
\end{smallmatrix}}
- \underbrace{{v \over \sqrt{u^2+v^2}}\left[ g_zh \left(1+{v \over r_yg_y}\right) \right]\tan \phi_{bed}}_{
\begin{smallmatrix}
\text{Dissipative basal}\\
\text{friction force}\\
\text{in y-direction}
\end{smallmatrix}}
+ \underbrace{g_yh}_{
\begin{smallmatrix}
\text{Driving}\\
\text{gravitational}\\
\text{force in}\\
\text{y-direction}
\end{smallmatrix}}
},
{
"math_input":
\begin{align}
& P(Z_{(m,n)}=k|\boldsymbol{Z_{-(m,n)}}, \boldsymbol{W};\alpha,\beta) \\
\propto &
P(Z_{(m,n)}=k,\boldsymbol{Z_{-(m,n)}},\boldsymbol{W};\alpha,\beta) \\
= & \left(\frac{\Gamma\left(\sum_{i=1}^K \alpha_i
\right)}{\prod_{i=1}^K \Gamma(\alpha_i)}\right)^M \prod_{j\neq m}
\frac{\prod_{i=1}^K
\Gamma(n_{j,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{j,(\cdot)}^i+\alpha_i \bigr)} \\
& \times \left( \frac{\Gamma\bigl(\sum_{r=1}^V \beta_r
\bigr)}{\prod_{r=1}^V \Gamma(\beta_r)}\right)^K \prod_{i=1}^K
\prod_{r\neq v}
\Gamma(n_{(\cdot),r}^i+\beta_r) \\
& \times \frac{\prod_{i=1}^K
\Gamma(n_{m,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^i+\alpha_i \bigr)} \prod_{i=1}^K \frac{
\Gamma(n_{(\cdot),v}^i+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)} \\
\propto & \frac{\prod_{i=1}^K
\Gamma(n_{m,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K
n_{m,(\cdot)}^i+\alpha_i \bigr)} \prod_{i=1}^K \frac{
\Gamma(n_{(\cdot),v}^i+\beta_v)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)}.
\end{align}
},
{
"math_input": \begin{equation} A(4, 3) & = A(3, A(4, 2)) \\ & = A(3, A(3, A(4, 1))) \\ & = A(3, A(3, A(3, A(4, 0)))) \\ & = A(3, A(3, A(3, A(3, 1)))) \\ & = A(3, A(3, A(3, A(2, A(3, 0))))) \\ & = A(3, A(3, A(3, A(2, A(2, 1))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(2, 0)))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(1, 1)))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0))))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1))))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(1, 3))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(1, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0)))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1)))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, 3))))) \\ & = A(3, A(3, A(3, A(2, A(0, 4))))) \\ & = A(3, A(3, A(3, A(2, 5)))) \\ & = ... \\ & = A(3, A(3, A(3, 13))) \\ & = ... \\ & = A(3, A(3, 65533)) \\ & = ... \\ & = A(3, 2^{65536} - 3) \\ & = ... \\ & = 2^{2^{ \overset{65536}{} }} - 3. \\ \end{equation}
},
{
"math_input": \displaystyle
\begin{array}{lcl}
b_{k} &=& \frac{A^{k}b_{0}}{\| A^{k} b_{0} \|} \\
&=& \frac{\left( VJV^{-1} \right)^{k} b_{0}}{\|\left( VJV^{-1} \right)^{k}b_{0}\|} \\
&=& \frac{ VJ^{k}V^{-1} b_{0}}{\| V J^{k} V^{-1} b_{0}\|} \\
&=& \frac{ VJ^{k}V^{-1} \left( c_{1}v_{1} + c_{2}v_{2} + \cdots + c_{n}v_{n} \right)}
{\| V J^{k} V^{-1} \left( c_{1}v_{1} + c_{2}v_{2} + \cdots + c_{n}v_{n} \right)\|} \\
&=& \frac{ VJ^{k}\left( c_{1}e_{1} + c_{2}e_{2} + \cdots + c_{n}e_{n} \right)}
{\| V J^{k} \left( c_{1}e_{1} + c_{2}e_{2} + \cdots + c_{n}e_{n} \right) \|} \\
&=& \left( \frac{\lambda_{1}}{|\lambda_{1}|} \right)^{k} \frac{c_{1}}{|c_{1}|}
\frac{ v_{1} + \frac{1}{c_{1}} V \left( \frac{1}{\lambda_1} J \right)^{k}
\left( c_{2}e_{2} + \cdots + c_{n}e_{n} \right)}
{\| v_{1} + \frac{1}{c_{1}} V \left( \frac{1}{\lambda_1} J \right)^{k}
\left( c_{2}e_{2} + \cdots + c_{n}e_{n} \right) \| }
\end{array}
},
{
"math_input":
\begin{align}
\frac{\partial f}{\partial \boldsymbol{A}}:\boldsymbol{T}
& = \left.\cfrac{d}{d\alpha} \left[\alpha^3~\det(\boldsymbol{A})~
\left(\cfrac{1}{\alpha^3} + I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\cfrac{1}{\alpha^2} +
I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\cfrac{1}{\alpha} + I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})\right)
\right] \right|_{\alpha=0} \\
& = \left.\det(\boldsymbol{A})~\cfrac{d}{d\alpha} \left[
1 + I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha +
I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^2 + I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^3
\right] \right|_{\alpha=0} \\
& = \left.\det(\boldsymbol{A})~\left[I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) +
2~I_2(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha + 3~I_3(\boldsymbol{A}^{-1}\cdot\boldsymbol{T})~\alpha^2
\right] \right|_{\alpha=0} \\
& = \det(\boldsymbol{A})~I_1(\boldsymbol{A}^{-1}\cdot\boldsymbol{T}) ~.
\end{align}
},
{
"math_input": \begin{align}
A(4, 3) & = A(3, A(4, 2)) \\
& = A(3, A(3, A(4, 1))) \\
& = A(3, A(3, A(3, A(4, 0)))) \\
& = A(3, A(3, A(3, A(3, 1)))) \\
& = A(3, A(3, A(3, A(2, A(3, 0))))) \\
& = A(3, A(3, A(3, A(2, A(2, 1))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(2, 0)))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(1, 1)))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0))))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1))))))) \\
& = A(3, A(3, A(3, A(2, A(1, A(0, 2)))))) \\
& = A(3, A(3, A(3, A(2, A(1, 3))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(1, 2)))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0)))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1)))))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2)))))) \\
& = A(3, A(3, A(3, A(2, A(0, A(0, 3))))) \\
& = A(3, A(3, A(3, A(2, A(0, 4))))) \\
& = A(3, A(3, A(3, A(2, 5)))) \\
& = ... \\
& = A(3, A(3, A(3, 13))) \\
& = ... \\
& = A(3, A(3, 65533)) \\
& = ... \\
& = A(3, 2^{65536} - 3) \\
& = ... \\
& = 2^{2^{ \overset{65536}{} }} - 3. \\
\end{align}
},
{
"math_input": \begin{align} A(4, 3) & = A(3, A(4, 2)) \\ & = A(3, A(3, A(4, 1))) \\ & = A(3, A(3, A(3, A(4, 0)))) \\ & = A(3, A(3, A(3, A(3, 1)))) \\ & = A(3, A(3, A(3, A(2, A(3, 0))))) \\ & = A(3, A(3, A(3, A(2, A(2, 1))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(2, 0)))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(1, 1)))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, A(1, 0))))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, A(0, 1))))))) \\ & = A(3, A(3, A(3, A(2, A(1, A(0, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(1, 3))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(1, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(1, 1))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(1, 0)))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, A(0, 1)))))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, A(0, 2)))))) \\ & = A(3, A(3, A(3, A(2, A(0, A(0, 3))))) \\ & = A(3, A(3, A(3, A(2, A(0, 4))))) \\ & = A(3, A(3, A(3, A(2, 5)))) \\ & = ... \\ & = A(3, A(3, A(3, 13))) \\ & = ... \\ & = A(3, A(3, 65533)) \\ & = ... \\ & = A(3, 2^{65536} - 3) \\ & = ... \\ & = 2^{2^{ \overset{65536}{} }} - 3. \\ \end{align}
},
{
"math_input":
\begin{align}
M_{11} & = D\left[\mathcal{A}\left(\frac{\partial \varphi_1}{\partial x_1}+\nu\frac{\partial \varphi_2}{\partial x_2}\right)
- (1-\mathcal{A})\left(\frac{\partial^2 w^0}{\partial x_1^2} + \nu\frac{\partial^2 w^0}{\partial x_2^2}\right)\right]
+ \frac{q}{1-\nu}\,\mathcal{B}\\
M_{22} & = D\left[\mathcal{A}\left(\frac{\partial \varphi_2}{\partial x_2}+\nu\frac{\partial \varphi_1}{\partial x_1}\right)
- (1-\mathcal{A})\left(\frac{\partial^2 w^0}{\partial x_2^2} + \nu\frac{\partial^2 w^0}{\partial x_1^2}\right)\right]
+ \frac{q}{1-\nu}\,\mathcal{B}\\
M_{12} & = \frac{D(1-\nu)}{2}\left[\mathcal{A}\left(\frac{\partial \varphi_1}{\partial x_2}+\frac{\partial \varphi_2}{\partial x_1}\right)
- 2(1-\mathcal{A})\,\frac{\partial^2 w^0}{\partial x_1 \partial x_2}\right] \\
Q_1 & = \mathcal{A} \kappa G h\left(\varphi_1 + \frac{\partial w^0}{\partial x_1}\right) \\
Q_2 & = \mathcal{A} \kappa G h\left(\varphi_2 + \frac{\partial w^0}{\partial x_2}\right) \,.
\end{align}
},
{
"math_input": \begin{align}
\operatorname{H}[\mathbf{X}] &= \tfrac{n}{2}\ln|\mathbf{V}| +\tfrac{np}{2}\ln(2) + \ln\left (\Gamma_p(\tfrac{n}{2}) \right ) -\frac{n-p-1}{2} \operatorname{E}[\ln|\mathbf{X}|] + \tfrac{np}{2} \\
&= \tfrac{n}{2}\ln|\mathbf{V}| + \tfrac{np}{2}\ln(2) + \frac{p(p-1) \ln(\pi)}{4} + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n+1-i}{2}\right ) \right ) -\frac{n-p-1}{2}\left(\sum_{i=1}^p \psi\left(\tfrac{n+1-i}{2}\right) + p\ln(2) + \ln|\mathbf{V}|\right) + \tfrac{np}{2} \\
&= \tfrac{n}{2}\ln|\mathbf{V}| + \tfrac{np}{2}\ln(2) + \frac{p(p-1) \ln(\pi)}{4} + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n+1-i}{2}\right ) \right) - \frac{n-p-1}{2}\sum_{i=1}^p \psi\left(\tfrac{n+1-i}{2}\right) - \frac{n-p-1}{2} \left(p\ln(2) +\ln|\mathbf{V}| \right ) + \tfrac{np}{2} \\
&= \tfrac{p+1}{2}\ln|\mathbf{V}| + \tfrac{1}{2}p(p+1)\ln(2) + \frac{p(p-1) \ln(\pi)}{4} + \sum_{i=1}^p \ln \left (\Gamma\left ( \tfrac{n+1-i}{2} \right) \right ) - \frac{n-p-1}{2}\sum_{i=1}^p \psi\left(\tfrac{n+1-i}{2}\right) + \tfrac{np}{2}
\end{align}
},
{
"math_input": \begin{align}
&\quad \mathbf{C}(A(a), B(b)) + \mathbf{C}(A(a), B(b')) +
\mathbf{C}(A(a'), B(b)) - \mathbf{C}(A(a'), B(b'))&\\
&= \int_\Lambda A(a, \lambda) B(b, \lambda) \rho(\lambda) d \lambda +
\int_\Lambda A(a, \lambda) B(b', \lambda) \rho(\lambda) d \lambda +
\int_\Lambda A(a', \lambda) B(b, \lambda) \rho(\lambda) d \lambda -
\int_\Lambda A(a', \lambda) B(b', \lambda) \rho(\lambda) d \lambda&\\
&= \int_\Lambda \big\{
A(a, \lambda) B(b, \lambda) +
A(a, \lambda) B(b', \lambda) +
A(a', \lambda) B(b, \lambda) -
A(a', \lambda) B(b', \lambda)
\big\} \rho(\lambda) d \lambda&\\
&= \int_\Lambda \big\{
A(a, \lambda) \left[
B(b, \lambda) + B(b', \lambda)
\right] + A(a', \lambda) \left[
B(b, \lambda) - B(b', \lambda)
\right]
\big\} \rho(\lambda) d \lambda\\
&\leq 2
\end{align}
},
{
"math_input":
\begin{align}
\mathcal{M}(n-1,n)&\leq\mathcal{M}(n,n+x)\leq\mathcal{M}(n,n+1)\;\;\mathrm{when}\;0< x\leq 1\\
\frac{\log\left(\Gamma(n)\right)-\log\left(\Gamma(n-1)\right)}{n-(n-1)}&\leq
\frac{\log\left(\Gamma(n)\right)-\log\left(\Gamma(n+x)\right)}{n-(n+x)}\leq
\frac{\log\left(\Gamma(n)\right)-\log\left(\Gamma(n+1)\right)}{n-(n+1)}\\
\frac{\log\left((n-1)!\right)-\log\left((n-2)!\right)}{1}&\leq
\frac{\log\left(\Gamma(n+x)\right)-\log\left((n-1)!\right)}{x}\leq
\frac{\log\left(n!\right)-\log\left((n-1)!\right)}{1}\\
\log\left(\frac{(n-1)!}{(n-2)!}\right)&\leq
\frac{\log\left(\Gamma(n+x)\right)-\log\left((n-1)!\right)}{x}\leq
\log\left(\frac{n!}{(n-1)!}\right)\\
\log\left(n-1\right)&\leq
\frac{\log\left(\Gamma(n+x)\right)-\log\left((n-1)!\right)}{x}\leq
\log\left(n\right)\\
x\cdot\log\left(n-1\right)+\log\left((n-1)!\right)&\leq
\log\left(\Gamma(n+x)\right)\leq
x\cdot\log\left(n\right)+\log\left((n-1)!\right)\\
\log\left((n-1)^x(n-1)!\right)&\leq
\log\left(\Gamma(n+x)\right)\leq
\log\left(n^x(n-1)!\right)
\end{align}
},
{
"math_input":
\begin{align}
f_Z(z) &= \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sigma_Y} e^{-{(z-x-\mu_Y)^2 \over 2\sigma_Y^2}} \frac{1}{\sqrt{2\pi}\sigma_X} e^{-{(x-\mu_X)^2 \over 2\sigma_X^2}} dx \\
&= \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\sqrt{\sigma_X^2+\sigma_Y^2}} \exp \left[ - { (z-(\mu_X+\mu_Y))^2 \over 2(\sigma_X^2+\sigma_Y^2) } \right] \frac{1}{\sqrt{2\pi}\frac{\sigma_X\sigma_Y}{\sqrt{\sigma_X^2+\sigma_Y^2}}} \exp \left[ - \frac{\left(x-\frac{\sigma_X^2(z-\mu_Y)+\sigma_Y^2\mu_X}{\sigma_X^2+\sigma_Y^2}\right)^2}{2\left(\frac{\sigma_X\sigma_Y}{\sqrt{\sigma_X^2+\sigma_Y^2}}\right)^2} \right] dx \\
&= \frac{1}{\sqrt{2\pi(\sigma_X^2+\sigma_Y^2)}} \exp \left[ - { (z-(\mu_X+\mu_Y))^2 \over 2(\sigma_X^2+\sigma_Y^2) } \right] \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}\frac{\sigma_X\sigma_Y}{\sqrt{\sigma_X^2+\sigma_Y^2}}} \exp \left[ - \frac{\left(x-\frac{\sigma_X^2(z-\mu_Y)+\sigma_Y^2\mu_X}{\sigma_X^2+\sigma_Y^2}\right)^2}{2\left(\frac{\sigma_X\sigma_Y}{\sqrt{\sigma_X^2+\sigma_Y^2}}\right)^2} \right] dx \\
\end{align}
},
{
"math_input": \begin{align}
f_1 (T) f_2(T) &= \left (\frac{1}{2\pi i}\int_{\Gamma_1}\frac{f_1(\zeta)}{\zeta-T} d \zeta \right ) \left (\frac{1}{2 \pi i} \int_{\Gamma_2}\frac{f_2(\omega)}{\omega-T}\, d \omega \right )\\
&= \frac{1}{(2\pi i)^2} \int_{\Gamma_1} \int_{\Gamma_2} \frac{f_1(\zeta)f_2(\omega)}{(\zeta-T)(\omega-T)}\; d \omega \, d \zeta \\
&= \frac{1}{(2\pi i)^2} \int_{\Gamma_1} \int_{\Gamma_2} f_1(\zeta) f_2 (\omega) \left ( \frac{(\zeta - T)^{-1} - (\omega - T)^{-1}}{\omega - \zeta} \right ) d \omega \, d \zeta && \text{First Resolvent Formula}\\
&= \frac{1}{(2 \pi i)^2}\left \{\left (\int _{\Gamma_1} \frac{f_1(\zeta)}{\zeta-T}\left[\int_{\Gamma_2}\frac{f_2(\omega)}{\omega - \zeta} d\omega\right] d \zeta \right )- \left (\int_{\Gamma_2} \frac{f_2(\omega)}{\omega-T}\left[\int_{\Gamma_1}\frac{f_1(\zeta)}{\omega - \zeta}d\zeta\right] d \omega\right)\right \} \\
&= \frac{1}{(2 \pi i)^2} \int _{\Gamma_1} \frac{f_1(\zeta)}{\zeta-T}\left[\int_{\Gamma_2}\frac{f_2(\omega)}{\omega - \zeta} d\omega\right] d \zeta
\end{align}
},
{
"math_input": \begin{align}
& z\left( {x_1 \,\,x_2 } \right)\,\,\,\, \approx \,\,\,z\left( {\bar x_1 \,\,\bar x_2 } \right)\,\,\, + \,\,\,\,{{\partial z} \over {\partial x_1 }}\left( {x_1 - \,\,\bar x_1 } \right)\,\,\, + \,\,\,{{\partial z} \over {\partial x_2 }}\left( {x_2 - \,\,\bar x_2 } \right)\,\,\, \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,{1 \over 2}{{\partial ^2 z} \over {\partial x_1 \partial x_2 }}\left( {x_1 - \,\,\bar x_1 } \right)\left( {x_2 - \,\,\bar x_2 } \right)\,\,\, + \,\,\,{1 \over 2}{{\partial ^2 z} \over {\partial x_2 \partial x_1 }}\left( {x_2 - \,\,\bar x_2 } \right)\left( {x_1 - \,\,\bar x_1 } \right) \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,{1 \over 2}{{\partial ^2 z} \over {\partial x_1 \partial x_1 }}\left( {x_1 - \,\,\bar x_1 } \right)\left( {x_1 - \,\,\bar x_1 } \right)\,\,\, + \,\,\,{1 \over 2}{{\partial ^2 z} \over {\partial x_2 \partial x_2 }}\left( {x_2 - \,\,\bar x_2 } \right)\left( {x_2 - \,\,\bar x_2 } \right)\end{align}
},
{
"math_input":
\begin{pmatrix}
\boldsymbol{\alpha^3}'\\
\boldsymbol{\alpha^2\beta}'\\
\boldsymbol{\alpha\beta^2}'\\
\boldsymbol{\beta^3}'\\
\boldsymbol{\alpha^2\gamma}'\\
\boldsymbol{\alpha\beta\gamma}'\\
\boldsymbol{\beta^2\gamma}'\\
\boldsymbol{\alpha\gamma^2}'\\
\boldsymbol{\beta\gamma^2}'\\
\boldsymbol{\gamma^3}'
\end{pmatrix}=\begin{pmatrix}
1&0&0&0&0&0&0&0&0&0\\
{1\over 2}&{1\over 2}&0&0&0&0&0&0&0&0\\
{1\over 4}&{2\over 4}&{1\over 4}&0&0&0&0&0&0&0\\
{1\over 8}&{3\over 8}&{3\over 8}&{1\over 8}&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0&0&0\\
0&0&0&0&{1\over 2}&{1\over 2}&0&0&0&0\\
0&0&0&0&{1\over 4}&{2\over 4}&{1\over 4}&0&0&0\\
0&0&0&0&0&0&0&1&0&0\\
0&0&0&0&0&0&0&{1\over 2}&{1\over 2}&0\\
0&0&0&0&0&0&0&0&0&1
\end{pmatrix}\cdot\begin{pmatrix}
\boldsymbol{\alpha^3}\\
\boldsymbol{\alpha^2\beta}\\
\boldsymbol{\alpha\beta^2}\\
\boldsymbol{\beta^3}\\
\boldsymbol{\alpha^2\gamma}\\
\boldsymbol{\alpha\beta\gamma}\\
\boldsymbol{\beta^2\gamma}\\
\boldsymbol{\alpha\gamma^2}\\
\boldsymbol{\beta\gamma^2}\\
\boldsymbol{\gamma^3}
\end{pmatrix}
},
{
"math_input":
\begin{align}
\frac{x}{R} &= \int_{0}^{\lambda'} \frac{H-S^2}{\left(1+S^2\right)^{1/2}}d\lambda' - \frac{S}{\left(1+S^2\right)^{1/2}}\ln\tan\left(\frac{\pi}{4}+\frac{\phi'}{2}\right) \\
\frac{y}{R} &= \left(H+1\right) \int_{0}^{\lambda'} \frac{S}{\left(1+S^2\right)^{1/2}}d\lambda' + \frac{1}{\left(1+S^2\right)^{1/2}}\ln\tan\left(\frac{\pi}{4}+\frac{\phi'}{2}\right) \\
S &= \left(P_{2}/P_{1}\right) \sin i \cos \lambda' \\
H &= 1 - \left(P_{2}/P_{1}\right) \cos i \\
\tan\lambda' &= \cos i \tan \lambda_{t} + \sin i \tan \phi / \cos \lambda_{t} \\
\sin\phi' &= \cos i \sin \phi - \sin i \cos \phi \sin \lambda_{t} \\
\lambda_{t} &= \lambda + \left(P_{2}/P_{1}\right) \lambda'. \\
\phi &= \text{geodetic (or geographic) latitude.} \\
\lambda &= \text{geodetic (or geographic) longitude.} \\
P_{2} &= \text{time required for revolution of satellite.} \\
P_{1} &= \text{length of Earth rotation.} \\
i &= \text{angle of inclation.} \\
R &= \text{radius of Earth.} \\
x,y &= \text{rectangular map coordinates.}
\end{align}
},
{
"math_input": \begin{cases}
\begin{cases}
\begin{cases}
\begin{cases}
\begin{cases}
\begin{cases}
\begin{cases}
\begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1 &\qquad \text{ ( by Lyapunov function } V_x, \text{ subsystem stabilized by } u_x(\textbf{x}) \text{ )}\\
\dot{z}_1 = f_1( \mathbf{x}, z_1 ) + g_1( \mathbf{x}, z_1 ) z_2
\end{cases}\\
\dot{z}_2 = f_2( \mathbf{x}, z_1, z_2 ) + g_2( \mathbf{x}, z_1, z_2 ) z_3
\end{cases}\\
\vdots\\
\end{cases}\\
\dot{z}_i = f_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) + g_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) z_{i+1}
\end{cases}\\
\vdots
\end{cases}\\
\dot{z}_{k-2} = f_{k-2}( \mathbf{x}, z_1, z_2, \ldots z_{k-2} ) + g_{k-2}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2} ) z_{k-1}
\end{cases}\\
\dot{z}_{k-1} = f_{k-1}( \mathbf{x}, z_1, z_2, \ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2}, z_{k-1} ) z_k
\end{cases}\\
\dot{z}_k = f_k( \mathbf{x}, z_1, z_2, \ldots z_{k-1}, z_k ) + g_k( \mathbf{x}, z_1, z_2, \ldots, z_{k-1}, z_k ) u
\end{cases}
},
{
"math_input":
\begin{align}
\frac{d E_{\lambda}}{d\lambda} &= \frac{d}{d\lambda}\langle\psi(\lambda)|\hat{H}_{\lambda}|\psi(\lambda)\rangle \\
&=\bigg\langle\frac{d\psi(\lambda)}{d\lambda}\bigg|\hat{H}_{\lambda}\bigg|\psi(\lambda)\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\hat{H}_{\lambda}\bigg|\frac{d\psi(\lambda)}{d\lambda}\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
&=E_{\lambda}\bigg\langle\frac{d\psi(\lambda)}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle + E_{\lambda}\bigg\langle\psi(\lambda)\bigg|\frac{d\psi(\lambda)}{d\lambda}\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
&=E_{\lambda}\frac{d}{d\lambda}\bigg\langle\psi(\lambda)\bigg|\psi(\lambda)\bigg\rangle + \bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle \\
&=\bigg\langle\psi(\lambda)\bigg|\frac{d\hat{H}_{\lambda}}{d\lambda}\bigg|\psi(\lambda)\bigg\rangle.
\end{align}
},
{
"math_input":
\begin{align}
& \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{r\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{rr}-\sigma_{\theta\theta}-\sigma_{\phi\phi}+\sigma_{r\theta}\cot\theta) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
& \frac{\partial \sigma_{r\theta}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\theta \phi}}{\partial \phi} + \cfrac{1}{r}[(\sigma_{\theta\theta}-\sigma_{\phi\phi})\cot\theta + 3\sigma_{r\theta}] + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
& \frac{\partial \sigma_{r\phi}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta \phi}}{\partial \theta} + \cfrac{1}{r\sin\theta}\frac{\partial \sigma_{\phi\phi}}{\partial \phi} + \cfrac{1}{r}(2\sigma_{\theta\phi}\cot\theta+3\sigma_{r\phi}) + F_\phi = \rho~\frac{\partial^2 u_\phi}{\partial t^2}
\end{align}
},
{
"math_input":
\begin{align}
c^k(\ell,m,\ell',m') &= c^k(\ell,-m,\ell',-m')\\
&=(-1)^{m-m'}c^k(\ell',m',\ell,m)\\
&=(-1)^{m-m'}\sqrt{\frac{2\ell+1}{2k+1}}c^\ell(\ell',m',k,m'-m)\\
& = (-1)^{m'}\sqrt{\frac{2\ell'+1}{2k+1}}c^{\ell'}(k,m-m',\ell,m).\\
\sum_{m=-\ell}^{\ell} c^k(\ell,m,\ell,m) &= (2\ell+1)\delta_{k,0}.\\
\sum_{m=-\ell}^\ell \sum_{m'=-\ell'}^{\ell'} c^k(\ell,m,\ell',m')^2 &= \sqrt{(2\ell+1)(2\ell'+1)}\cdot c^k(\ell,0,\ell',0).\\
\sum_{m=-\ell}^\ell c^k(\ell,m,\ell',m')^2 & = \sqrt{\frac{2\ell+1}{2\ell'+1}}\cdot c^k(\ell,0,\ell',0).\\
\sum_{m=-\ell}^\ell c^k(\ell,m,\ell',m')c^k(\ell,m,\tilde\ell,m') &= \delta_{\ell',\tilde\ell}\cdot\sqrt{\frac{2\ell+1}{2\ell'+1}}\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m) c^k(\ell,m+r,\tilde\ell,m) &= \delta_{\ell,\tilde\ell} \cdot \frac{\sqrt{(2\ell+1)(2\ell'+1)}}{2k+1}\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m)c^q(\ell,m+r,\ell',m) &= \delta_{k,q}\cdot\frac{\sqrt{(2\ell+1)(2\ell'+1)}}{2k+1}\cdot c^k(\ell,0,\ell',0).
\end{align}
},
{
"math_input": \begin{align}
\rho \left(\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z}\right)
&= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) + \rho g_x \\
\rho \left(\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y}+ w \frac{\partial v}{\partial z}\right)
&= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}\right) + \rho g_y \\
\rho \left(\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial x} + v \frac{\partial w}{\partial y}+ w \frac{\partial w}{\partial z}\right)
&= -\frac{\partial p}{\partial z} + \mu \left(\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}\right) + \rho g_z.
\end{align}
},
{
"math_input":
\begin{align}
\mathcal{I}(\theta)
& =
-\operatorname{E}
\left[ \left.
\frac{\partial^2}{\partial\theta^2} \log(f(A;\theta))
\right| \theta \right] \qquad (1) \\
& =
-\operatorname{E}
\left[ \left.
\frac{\partial^2}{\partial\theta^2} \log
\left(
\theta^A(1-\theta)^B\frac{(A+B)!}{A!B!}
\right)
\right| \theta \right] \qquad (2) \\
& =
-\operatorname{E}
\left[ \left.
\frac{\partial^2}{\partial\theta^2}
\left(
A \log (\theta) + B \log(1-\theta)
\right)
\right| \theta \right] \qquad (3) \\
& =
-\operatorname{E}
\left[ \left.
\frac{\partial}{\partial\theta}
\left(
\frac{A}{\theta} - \frac{B}{1-\theta}
\right)
\right| \theta \right] \qquad (4) \\
& =
+\operatorname{E}
\left[ \left.
\frac{A}{\theta^2} + \frac{B}{(1-\theta)^2}
\right| \theta \right] \qquad (5) \\
& =
\frac{n\theta}{\theta^2} + \frac{n(1-\theta)}{(1-\theta)^2} \qquad (6) \\
& \text{since the expected value of }A\text{ given }\theta\text{ is }n\theta,\text{ etc.} \\
& = \frac{n}{\theta(1-\theta)} \qquad (7)
\end{align}
},
{
"math_input":
\boldsymbol{\mu}_a = \left[ \begin{array}{cccccc}
0 & 0 & 0 & 0 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right] , \boldsymbol{\mu}_b = \left[ \begin{array}{cccccc}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]
},
{
"math_input": \begin{align}
x_1 =
&-\frac{b}{3 a}\\
&-\frac{1}{3 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\
&-\frac{1}{3 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\
x_2 =
&-\frac{b}{3 a}\\
&+\frac{1+i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\
&+\frac{1-i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\
x_3 =
&-\frac{b}{3 a}\\
&+\frac{1-i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d+\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}\\
&+\frac{1+i \sqrt{3}}{6 a} \sqrt[3]{\tfrac12\left[2 b^3-9 a b c+27 a^2 d-\sqrt{\left(2 b^3-9 a b c+27 a^2 d\right)^2-4 \left(b^2-3 a c\right)^3}\right]}
\end{align}
},
{
"math_input":
\begin{align}
\frac{\partial^2 \Phi_3}{\partial t^2}
+ g\, \frac{\partial \Phi_3}{\partial z}
= & - \eta_1\, \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi_2}{\partial t^2}
+ g\, \frac{\partial \Phi_2}{\partial z}
\right)
- \eta_2\, \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right)
\\ &
- 2\, \frac{\partial}{\partial t} \left( \mathbf{u}_1 \cdot \mathbf{u}_2 \right)
- \tfrac12\, \eta_1^2\,
\frac{\partial^2}{\partial z^2}
\left(
\frac{\partial^2 \Phi_1}{\partial t^2}
+ g\, \frac{\partial \Phi_1}{\partial z}
\right)
\\ &
- \eta_1\, \frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}_1|^2 \right)
- \tfrac12\, \mathbf{u}_1 \cdot \boldsymbol{\nabla} \left( |\mathbf{u}_1|^2 \right).
\end{align}
},
{
"math_input":
\begin{align}
M_{xx} & = -D\left(\frac{\partial^2 w}{\partial x^2}+\nu\,\frac{\partial^2 w}{\partial y^2}\right) \\
& = q_{x1}\left(\frac{x-a}{b}\right) - \left[\frac{3yq_{x2}}{b^3\nu_b\cosh^3[\nu_b(x-a)]}\right]
\times \\
& \quad \left[6\sinh(\nu_b a) - \sinh[\nu_b(2x-a)] +
\sinh[\nu_b(2x-3a)] + 8\sinh[\nu_b(x-a)]\right] \\
M_{xy} & = (1-\nu)D\frac{\partial^2 w}{\partial x \partial y} \\
& = \frac{q_{x2}}{2b}\left[1 -
\frac{2+\cosh[\nu_b(x-2a)] - \cosh[\nu_b x]}{2\cosh^2[\nu_b(x-a)]}\right] \\
Q_{zx} & = \frac{\partial M_{xx}}{\partial x}-\frac{\partial M_{xy}}{\partial y} \\
& = \frac{q_{x1}}{b} - \left(\frac{3yq_{x2}}{2b^3\cosh^4[\nu_b(x-a)]}\right)\times
\left[32 + \cosh[\nu_b(3x-2a)] - \cosh[\nu_b(3x-4a)]\right. \\
& \qquad \left. - 16\cosh[2\nu_b(x-a)] +
23\cosh[\nu_b(x-2a)] - 23\cosh(\nu_b x)\right]\,.
\end{align}
},
{
"math_input":
\mathbf{H}_3=
\begin{pmatrix}
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \\
1 \; 0 \; 0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\
1 \; 1 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\
0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \\
1 \; 0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \\
0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \\
0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1
\end{pmatrix}.
},
{
"math_input":
\Pr
\begin{cases}
Ds
\begin{cases}
Sp (\pi)
\begin{cases}
Va: \text{Spam},W_0,W_1 \ldots W_{N-1} \\
Dc:
\begin{cases}
P(\text{Spam} \land W_0 \land \ldots \land W_n \land \ldots \land W_{N-1})\\
= P(\text{Spam})\prod_{n=0}^{N-1}P(W_n\mid\text{Spam})
\end{cases}\\
Fo:
\begin{cases}
P(\text{Spam}):
\begin{cases}
P([\text{Spam}=\text{false}])=0.25 \\
P([\text{Spam}=\text{true}])=0.75
\end{cases}\\
P(W_n\mid\text{Spam}):
\begin{cases}
P(W_n\mid[\text{Spam}=\text{false}])\\
=\frac{1+a^n_f}{2+a_f} \\
P(W_n\mid[\text{Spam}=\text{true}])\\
=\frac{1+a^n_t}{2+a_t}
\end{cases} \\
\end{cases}\\
\end{cases}\\
\text{Identification (based on }\delta)
\end{cases}\\
Qu: P(\text{Spam}\mid w_0 \land \ldots \land w_n \land \ldots \land w_{N-1})
\end{cases}
},
{
"math_input":
\begin{align}
\frac{d^\acute{n}F(P_0)}{dP^\acute{n}} & =\frac{d^{\acute{n}-1}F'(P_0)}{dP^{\acute{n}-1}}
=\frac{d^{\acute{n}-2}F''(P_0)}{dP^{\acute{n}-2}}
=\frac{d^{\acute{n}-3}F'''(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}F^{(r)}(P_0)}{dP^{\acute{n}-r}},
\\[10pt]
& =\frac{d^{\acute{n}-1}G(P_0)}{dP^{\acute{n}-1}} \\[10pt]
& =\frac{d^{\acute{n}-2}G'(P_0)}{dP^{\acute{n}-2}}=\ \frac{d^{\acute{n}-3}G''(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}G^{(r-1)}(P_0)}{dP^{\acute{n}-r}}, \\[10pt]
& {\color{white}.}\qquad\qquad\qquad=\frac{d^{\acute{n}-2}H(P_0)}{dP^{\acute{n}-2}}
=\ \frac{d^{\acute{n}-3}H'(P_0)}{dP^{\acute{n}-3}}=\cdots=\frac{d^{\acute{n}-r}H^{(r-2)}(P_0)}{dP^{\acute{n}-r}}, \\
& {\color{white}.}\qquad\qquad\qquad\qquad\qquad\qquad\ =\ \frac{d^{\acute{n}-3}I(P_0)}{dP^{\acute{n}-3}}
=\cdots=\frac{d^{\acute{n}-r}I^{(r-3)}(P_0)}{dP^{\acute{n}-r}}, \\[10pt]
& =F^{(\acute{n})}(P)=G^{(\acute{n}-1)}(P)=H^{(\acute{n}-2)}(P)=I^{(\acute{n}-3)}(P)=\cdots
\end{align}
},
{
"math_input":
\mathbf{H}_2=
\begin{pmatrix}
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \\
0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \\
1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \\
0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \\
1 \; 0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \\
0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \\
0 \; 0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0
\end{pmatrix},
},
{
"math_input": \begin{align}du_{\pm}& =d(e^{t\mp x}\cdot(1\pm \operatorname{erf}(z_\pm)))
\\& =d(e^{t\mp x})\cdot(1\pm \operatorname{erf}(z_\pm)) +e^{t\mp x}\cdot d(1\pm \operatorname{erf}(z_\pm))
\\& =e^{t\mp x}\cdot(1\pm \operatorname{erf}(z_\pm))\cdot d(t\mp x) \pm e^{t\mp x}\cdot 2\cdot\pi^{-\frac 1 2}\cdot e^{-z_\pm^2}\cdot dz_\pm
\\& =u_{\pm}\cdot(dt\mp dx) \pm e^{t\mp x}\cdot 2\pi^{-\frac 1 2}e^{-\frac {x^2}{4t}-t\pm x}\cdot( \frac 1 2 t^{-\frac 1 2}dx
- \frac 1 4 x t^{-\frac 3 2}dt \mp \frac 1 2t^{-\frac 1 2}dt)
\\& =u_{\pm}dt\mp u_{\pm}dx\pm \pi^{-\frac 1 2}e^{-\frac {x^2}{4t}}
\cdot (t^{-\frac 1 2}dx - \frac 1 2 x t^{-\frac 3 2}dt \mp t^{-\frac 1 2}dt)
\\& =(\mp u_{\pm} \pm \pi^{-\frac 1 2}e^{-\frac {x^2}{4t}}t^{-\frac 1 2})dx
+(u_{\pm} \pm \pi^{-\frac 1 2}e^{-\frac {x^2}{4t}}t^{-\frac 3 2}\cdot(-\frac x 2\mp t))dt
\\& =\mp(u_{\pm}-\pi^{-\frac 1 2}e^{-\frac{x^2}{4t}}t^{-\frac 1 2})dx
+(u_{\pm}+\pi^{-\frac 1 2}e^{-\tfrac{x^2}{4t}}t^{-\frac 3 2}\cdot(\mp\frac x 2-t))dt
\end{align}
},
{
"math_input":
\begin{align}
U &= \begin{bmatrix}
U_{e 1} & U_{e 2} & U_{e 3} \\
U_{\mu 1} & U_{\mu 2} & U_{\mu 3} \\
U_{\tau 1} & U_{\tau 2} & U_{\tau 3}
\end{bmatrix} \\
&= \begin{bmatrix}
1 & 0 & 0 \\
0 & c_{23} & s_{23} \\
0 & -s_{23} & c_{23}
\end{bmatrix}
\begin{bmatrix}
c_{13} & 0 & s_{13} e^{-i\delta} \\
0 & 1 & 0 \\
-s_{13} e^{i\delta} & 0 & c_{13}
\end{bmatrix}
\begin{bmatrix}
c_{12} & s_{12} & 0 \\
-s_{12} & c_{12} & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
e^{i\alpha_1 / 2} & 0 & 0 \\
0 & e^{i\alpha_2 / 2} & 0 \\
0 & 0 & 1
\end{bmatrix} \\
&= \begin{bmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i\delta} \\
- s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta} & c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta} & s_{23} c_{13}\\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta} & - c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta} & c_{23} c_{13}
\end{bmatrix}
\begin{bmatrix}
e^{i\alpha_1 / 2} & 0 & 0 \\
0 & e^{i\alpha_2 / 2} & 0 \\
0 & 0 & 1
\end{bmatrix} \\
\end{align}
},
{
"math_input":
X=\begin{vmatrix}
x_{11} & x_{12} & x_{13} &\cdots & x_{1n} \\
x_{12} & x_{22} & x_{23} &\cdots & x_{2n} \\
x_{13} & x_{23} & x_{33} &\cdots & x_{3n} \\
\vdots& \vdots & \vdots &\ddots & \vdots \\
x_{1n} & x_{2n} & x_{3n} &\cdots & x_{nn}
\end{vmatrix},
D=\begin{vmatrix}
2 \frac{\partial} { \partial x_{11} } & \frac{\partial} {\partial x_{12}} & \frac{\partial} { \partial x_{13}} &\cdots & \frac{\partial}{\partial x_{1n} } \\[6pt]
\frac{\partial} {\partial x_{12} } & 2 \frac{\partial} {\partial x_{22}} & \frac{\partial} { \partial x_{23}} &\cdots & \frac{\partial}{\partial x_{2n} } \\[6pt]
\frac{\partial} {\partial x_{13} } & \frac{\partial} {\partial x_{23}} & 2\frac{\partial} { \partial x_{33}} &\cdots & \frac{\partial}{\partial x_{3n} } \\[6pt]
\vdots& \vdots & \vdots &\ddots & \vdots \\
\frac{\partial} {\partial x_{1n} } & \frac{\partial} {\partial x_{2n}} & \frac{\partial} { \partial x_{3n}} &\cdots & 2 \frac{\partial}{\partial x_{nn} }
\end{vmatrix}
},
{
"math_input":
\mathbf{H}_1 =
\begin{pmatrix}
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \\
0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \\
1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \\
0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \\
0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \\
0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \\
0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \\
0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1
\end{pmatrix},
},
{
"math_input": \begin{align}
\ \varepsilon'_{ij} &= \varepsilon_{ij} - \frac{\varepsilon_{kk}}{3}\delta_{ij} \\
\left[{\begin{matrix}
\varepsilon'_{11} & \varepsilon'_{12} & \varepsilon'_{13} \\
\varepsilon'_{21} & \varepsilon'_{22} & \varepsilon'_{23} \\
\varepsilon'_{31} & \varepsilon'_{32} & \varepsilon'_{33} \\
\end{matrix}}\right]
&=\left[{\begin{matrix}
\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \\
\end{matrix}}\right]-\left[{\begin{matrix}
\varepsilon_M & 0 & 0 \\
0 & \varepsilon_M & 0 \\
0 & 0 & \varepsilon_M \\
\end{matrix}}\right] \\
&=\left[{\begin{matrix}
\varepsilon_{11}-\varepsilon_M & \varepsilon_{12} & \varepsilon_{13} \\
\varepsilon_{21} & \varepsilon_{22}-\varepsilon_M & \varepsilon_{23} \\
\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}-\varepsilon_M \\
\end{matrix}}\right] \\
\end{align}\,\!
},
{
"math_input": \begin{align}
y_{i+1}^{j} &= y_i^{j} -\frac{\delta_{ij}}{q_i} \\
\sum\limits_{i=1}^m y_i^{j} &= (k_{1}^j- k_{0}^j) + (k_{2}^j- k_{1}^j) + \cdots + (k_{m-1}^j- k_{m-2}^j) + (k_{m}^j- k_{m-1}^j) \\
&= k_{m}^j - k_{0}^j \\
\sum\limits_{i=1}^m y_i^{j} &= k_{m}^j \\ \\
y_1^{j} &= (k_{1}^j- k_{0}^j) = k_{1}^j \\
y_2^{j} &= y_1^{j} -\frac{\delta_{1j}}{q_1} = k_1^{j} -\frac{\delta_{1j}}{ q_1 } \\
y_3^{j} &= k_1^{j} -\frac{\delta_{1j}}{q_1} -\frac{\delta_{2j}}{ q_2 } \\
& \; \vdots \\
y_i^{j} &= k_1^{j} -\sum\limits_{r=1}^{i-1} \frac{\delta_{rj}}{ q_r}
\quad = \quad
\left\{
\begin{array}{lcr}
k_1^j & \text{for} & j \geq i\\
k_1^j - \frac{1}{q_j} & \text{for} & j \leq i
\end{array} \right. \\
k_i^j &= \quad \quad \; \sum\limits_{m=1}^i y_m^{j} \quad = \quad
\left\{
\begin{array}{lcr}
i \cdot k_1^j & \text{for} & j \geq i\\
i \cdot k_1^j - \frac{i-j}{q_j} & \text{for} & j \leq i
\end{array} \right.
\end{align}
},
{
"math_input":
\mathbf{A} \circ \mathbf{B} =
\left[
\begin{array} {c | c}
\mathbf{A}_{11} \circ \mathbf{B} & \mathbf{A}_{12} \circ \mathbf{B} \\
\hline
\mathbf{A}_{21} \circ \mathbf{B} & \mathbf{A}_{22} \circ \mathbf{B}
\end{array}
\right]
=
\left[
\begin{array} {c | c | c | c }
\mathbf{A}_{11} \otimes \mathbf{B}_{11} & \mathbf{A}_{11} \otimes \mathbf{B}_{12} & \mathbf{A}_{12} \otimes \mathbf{B}_{11} & \mathbf{A}_{12} \otimes \mathbf{B}_{12} \\
\hline
\mathbf{A}_{11} \otimes \mathbf{B}_{21} & \mathbf{A}_{11} \otimes \mathbf{B}_{22} & \mathbf{A}_{12} \otimes \mathbf{B}_{21} & \mathbf{A}_{12} \otimes \mathbf{B}_{22} \\
\hline
\mathbf{A}_{21} \otimes \mathbf{B}_{11} & \mathbf{A}_{21} \otimes \mathbf{B}_{12} & \mathbf{A}_{22} \otimes \mathbf{B}_{11} & \mathbf{A}_{22} \otimes \mathbf{B}_{12} \\
\hline
\mathbf{A}_{21} \otimes \mathbf{B}_{21} & \mathbf{A}_{21} \otimes \mathbf{B}_{22} & \mathbf{A}_{22} \otimes \mathbf{B}_{21} & \mathbf{A}_{22} \otimes \mathbf{B}_{22}
\end{array}
\right]
},
{
"math_input": \begin{align}\int |\alpha\rangle\langle\alpha| \, d^2\alpha
&= \int \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{|\alpha|^2}} \cdot \frac{\alpha^n (\alpha^*)^k}{\sqrt{n!k!}} |n\rangle \langle k| \, d^2\alpha \\
&= \int_0^{\infty} \int_0^{2\pi} \sum_{n=0}^{\infty}\sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \, d\theta dr \\
&= \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} \int_0^{2\pi} e^{-{r^2}} \cdot \frac{r^{n+k+1}e^{i(n-k)\theta}}{\sqrt{n!k!}} |n\rangle \langle k| \, d\theta dr \\
&= 2\pi \sum_{n=0}^{\infty} \int_0^{\infty} \sum_{k=0}^{\infty} e^{-{r^2}} \cdot \frac{r^{n+k+1}\delta(n-k)}{\sqrt{n!k!}} |n\rangle \langle k| \, dr \\
&= 2\pi \sum_{n=0}^{\infty} \int e^{-{r^2}} \cdot \frac{r^{2n+1}}{n!} |n\rangle \langle n| \, dr \\
&= \pi \sum_{n=0}^{\infty} \int e^{-u} \cdot \frac{u^n}{n!} |n\rangle \langle n| \, du \\
&= \pi \sum_{n=0}^{\infty} |n\rangle \langle n| \\
&= \pi \hat{I}.\end{align}
},
{
"math_input": \boldsymbol{\mu}_c = \left[ \begin{array}{cccccc}
0 & 0 & 0& 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right] , \boldsymbol{\mu}_d = \left[ \begin{array}{cccccc}
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{array}
\right]
},
{
"math_input":
\begin{align}
J_2^0 := & \cfrac{1}{6}\left[a_1(\sigma_{22}-\sigma_{33})^2+a_2(\sigma_{33}-\sigma_{11})^2 +a_3(\sigma_{11}-\sigma_{22})^2\right] + a_4\sigma_{23}^2 + a_5\sigma_{31}^2 + a_6\sigma_{12}^2 \\
J_3^0 := & \cfrac{1}{27}\left[(b_1+b_2)\sigma_{11}^3 +(b_3+b_4)\sigma_{22}^3 + \{2(b_1+b_4)-(b_2+b_3)\}\sigma_{33}^3\right] \\
& -\cfrac{1}{9}\left[(b_1\sigma_{22}+b_2\sigma_{33})\sigma_{11}^2+(b_3\sigma_{33}+b_4\sigma_{11})\sigma_{22}^2
+ \{(b_1-b_2+b_4)\sigma_{11}+(b_1-b_3+b_4)\sigma_{22}\}\sigma_{33}^2\right] \\
& + \cfrac{2}{9}(b_1+b_4)\sigma_{11}\sigma_{22}\sigma_{33} + 2 b_{11}\sigma_{12}\sigma_{23}\sigma_{31}\\
& - \cfrac{1}{3}\left[\{2b_9\sigma_{22}-b_8\sigma_{33}-(2b_9-b_8)\sigma_{11}\}\sigma_{31}^2+
\{2b_{10}\sigma_{33}-b_5\sigma_{22}-(2b_{10}-b_5)\sigma_{11}\}\sigma_{12}^2 \right.\\
& \qquad \qquad\left. \{(b_6+b_7)\sigma_{11} - b_6\sigma_{22}-b_7\sigma_{33}\}\sigma_{23}^2
\right]
\end{align}
},
{
"math_input":
\begin{matrix}
1 5 7 9 9 \rightarrow &
4 2 2 0 8 \rightarrow &
2 0 2 8 4 \rightarrow &
2 2 6 4 2 \rightarrow &
0 4 2 2 0 \rightarrow &
4 2 0 2 0 \rightarrow \\
2 2 2 2 4 \rightarrow &
0 0 0 2 2 \rightarrow &
0 0 2 0 2 \rightarrow &
0 2 2 2 2 \rightarrow &
2 0 0 0 2 \rightarrow &
2 0 0 2 0 \rightarrow \\
2 0 2 2 2 \rightarrow &
2 2 0 0 0 \rightarrow &
0 2 0 0 2 \rightarrow &
2 2 0 2 2 \rightarrow &
0 2 2 0 0 \rightarrow &
2 0 2 0 0 \rightarrow \\
2 2 2 0 2 \rightarrow &
0 0 2 2 0 \rightarrow &
0 2 0 2 0 \rightarrow &
2 2 2 2 0 \rightarrow &
0 0 0 2 2 \rightarrow &
\cdots \quad \quad \\
\end{matrix}
},
{
"math_input": \begin{align}
d\tilde{t}'= & \tilde{\gamma}\left[1+\kappa\cdot\tilde{\mathbf{v}}/c-\kappa'\cdot\tilde{\mathbf{v}}'/c\right]d\tilde{t}-\left(\kappa'+\tilde{\gamma}\tilde{\mathbf{v}}'\right)\cdot d\tilde{\mathbf{x}}/c\\
& -\left[\tilde{\gamma}\left(1+\kappa\cdot\tilde{\mathbf{v}}/c\right)-1\right]\frac{\kappa'\cdot\tilde{\mathbf{v}}}{\tilde{\mathbf{v}}^{2}c}\tilde{\mathbf{v}}\cdot d\tilde{\mathbf{x}}+\tilde{\gamma}\kappa\cdot\tilde{\mathbf{v}}\left(\kappa\cdot d\tilde{\mathbf{x}}\right)/c,\\
d\tilde{\mathbf{x}}'= & -\tilde{\gamma}\tilde{\mathbf{v}}d\tilde{t}+d\tilde{\mathbf{x}}+\left[\tilde{\gamma}\left(1+\kappa\cdot\tilde{\mathbf{v}}/c\right)-1\right]\frac{\tilde{\mathbf{v}}\cdot d\mathbf{x}}{\tilde{\mathbf{v}}^{2}}\tilde{\mathbf{v}}-\tilde{\gamma}\tilde{\mathbf{v}}\left(\kappa\cdot d\tilde{\mathbf{x}}\right)/c,\\
\tilde{\gamma}= & \gamma\left(1-\kappa\cdot \mathbf{v}/c\right),\\
\tilde{\mathbf{v}}= & \frac{\mathbf{v}}{1-\kappa\cdot \mathbf{v}/c},
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\cdot\mathbf{v} &
= \cfrac{\partial v_r}{\partial r} +
\cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} + v_r \right)
+ \cfrac{\partial v_z}{\partial z}\\
\boldsymbol{\nabla}\cdot\boldsymbol{S} &
= \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r
+ \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_\theta
+ \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_z \\
& +
\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta +\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_z \\
& +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_r +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z
\end{align}
},
{
"math_input":
\begin{align}
\mathrm{ad}_x(y) & = d (\mathrm{Ad}_{e})_{x}(y) \\
& = \lim_{\varepsilon \to 0}\frac{(I+\varepsilon x)y(I+\varepsilon x)^{-1}-y}{\varepsilon} \\
& = \lim_{\varepsilon \to 0}\frac{(I+\varepsilon x)y(I-\varepsilon x +(\varepsilon x)^2+O(\varepsilon^3))-y}{\varepsilon} \\
& = \lim_{\varepsilon \to 0}\frac{((I+\varepsilon x)yI- (I+\varepsilon x)y\varepsilon x +(I+\varepsilon x)y(\varepsilon x)^2 +O(\varepsilon^3))-y}{\varepsilon} \\
& = \lim_{\varepsilon \to 0}\frac{(I y I+\varepsilon x y I- I y \varepsilon x-\varepsilon x y \varepsilon x +Iy(\varepsilon x)^2+\varepsilon xy(\varepsilon x)^2 +O(\varepsilon^3))-y}{\varepsilon} \\
& = \lim_{\varepsilon \to 0}\frac{y+ x y \varepsilon - y x \varepsilon- x y x \varepsilon^{2} +y x^{2}\varepsilon^2 + x y x^{2}\varepsilon^2 +O(\varepsilon^3) -y}{\varepsilon} \\
& = \lim_{\varepsilon \to 0}x y - y x - x y x \varepsilon +y x^{2}\varepsilon + x y x^{2}\varepsilon +O(\varepsilon^2) \\
& = [x,y]
\end{align}
},
{
"math_input": \begin{align}
& a=\left( \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right)\quad b=\frac{1}{\sqrt{5}}\left( \begin{matrix}
0 \\
1 \\
2 \\
\end{matrix} \right) \\
& I=\left( \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right)\quad G=\frac{1}{\sqrt{5}}\left( \begin{matrix}
0 & -1 & -2 \\
1 & 0 & 0 \\
2 & 0 & 0 \\
\end{matrix} \right) \\
& P=-{{G}^{2}}=\frac{1}{5}\left( \begin{matrix}
5 & 0 & 0 \\
0 & 1 & 2 \\
0 & 2 & 4 \\
\end{matrix} \right)\quad P\left( \begin{matrix}
1 \\
2 \\
3 \\
\end{matrix} \right)=\frac{1}{5}\left( \begin{matrix}
5 \\
8 \\
16 \\
\end{matrix} \right)=a+\frac{8}{\sqrt{5}}b \\
& \theta =\frac{\pi}{6} \quad \Rightarrow \quad R=\frac{1}{10}\left( \begin{matrix}
5\sqrt{3} & -\sqrt{5} & -2\sqrt{5} \\
\sqrt{5} & 8+\sqrt{3} & -4+2\sqrt{3} \\
2\sqrt{5} & -4+2\sqrt{3} & 2+4\sqrt{3} \\
\end{matrix} \right) \\
\end{align}
},
{
"math_input": \begin{align}
&\lim_{\alpha\to 0}\text{excess kurtosis} =\lim_{\beta \to 0} \text{excess kurtosis} = \lim_{\mu \to 0}\text{excess kurtosis} = \lim_{\mu \to 1}\text{excess kurtosis} =\infty\\
&\lim_{\alpha \to \infty}\text{excess kurtosis} = \frac{6}{\beta},\text{ } \lim_{\beta \to 0}(\lim_{\alpha\to \infty} \text{excess kurtosis}) = \infty,\text{ } \lim_{\beta \to \infty}(\lim_{\alpha\to \infty} \text{excess kurtosis}) = 0\\
&\lim_{\beta \to \infty}\text{excess kurtosis} = \frac{6}{\alpha},\text{ } \lim_{\alpha \to 0}(\lim_{\beta \to \infty} \text{excess kurtosis}) = \infty,\text{ } \lim_{\alpha \to \infty}(\lim_{\beta \to \infty} \text{excess kurtosis}) = 0\\
&\lim_{\nu \to 0} \text{excess kurtosis} = - 6 + \frac{1}{\mu (1 - \mu)},\text{ } \lim_{\mu \to 0}(\lim_{\nu \to 0} \text{excess kurtosis}) = \infty,\text{ } \lim_{\mu \to 1}(\lim_{\nu \to 0} \text{excess kurtosis}) = \infty\\
\end{align}
},
{
"math_input": \begin{array}{lll}
\vec{v'} &=& \vec{v} \cos^2 \frac{\alpha}{2} + (\vec{u}\vec{v} - \vec{v}\vec{u}) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - \vec{u}\vec{v}\vec{u} \sin^2 \frac{\alpha}{2} \\
&=& \vec{v} \cos^2 \frac{\alpha}{2} + 2 (\vec{u} \times \vec{v}) \sin \frac{\alpha}{2} \cos \frac{\alpha}{2} - (\vec{v} (\vec{u} \cdot \vec{u}) - 2 \vec{u} (\vec{u} \cdot \vec{v})) \sin^2 \frac{\alpha}{2} \\
&=& \vec{v} (\cos^2 \frac{\alpha}{2} - \sin^2 \frac{\alpha}{2}) + (\vec{u} \times \vec{v}) (2 \sin \frac{\alpha}{2} \cos \frac{\alpha}{2}) + \vec{u} (\vec{u} \cdot \vec{v}) (2 \sin^2 \frac{\alpha}{2}) \\
&=& \vec{v} \cos \alpha + (\vec{u} \times \vec{v}) \sin \alpha + \vec{u} (\vec{u} \cdot \vec{v}) (1 - \cos \alpha) \\
&=& (\vec{v} - \vec{u} (\vec{u} \cdot \vec{v})) \cos \alpha + (\vec{u} \times \vec{v}) \sin \alpha + \vec{u} (\vec{u} \cdot \vec{v}) \\
&=& \vec{v}_{\bot} \cos \alpha + (\vec{u} \times \vec{v}_{\bot}) \sin \alpha + \vec{v}_{\|}
\end{array}
},
{
"math_input": \begin{align}
\mathrm{d}{\omega} &= \mathrm{d} \left (f_I \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\
&= \mathrm{d}f_I \wedge \left (\mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) + f_I \mathrm{d} \left (
\mathrm{d} x^{i_1}\wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\
&= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} + \sum_{p=1}^k (-1)^{(p-1)} f_I \mathrm{d} x^{i_1}
\wedge \cdots \wedge \mathrm{d}x^{i_{p-1}} \wedge \mathrm{d}^2x^{i_p} \wedge \mathrm{d}x^{i_{p+1}} \wedge \cdots \wedge\mathrm{d}
x^{i_k} \\
&= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\
&= \sum_{i=1}^n \frac{\partial f_I}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\
\end{align}
},
{
"math_input":
P_x = \frac{\begin{vmatrix} \begin{vmatrix} x_1 & y_1\\x_2 & y_2\end{vmatrix} & \begin{vmatrix} x_1 & 1\\x_2 & 1\end{vmatrix} \\\\ \begin{vmatrix} x_3 & y_3\\x_4 & y_4\end{vmatrix} & \begin{vmatrix} x_3 & 1\\x_4 & 1\end{vmatrix} \end{vmatrix} }
{\begin{vmatrix} \begin{vmatrix} x_1 & 1\\x_2 & 1\end{vmatrix} & \begin{vmatrix} y_1 & 1\\y_2 & 1\end{vmatrix} \\\\ \begin{vmatrix} x_3 & 1\\x_4 & 1\end{vmatrix} & \begin{vmatrix} y_3 & 1\\y_4 & 1\end{vmatrix} \end{vmatrix}}\,\!
\qquad
P_y = \frac{\begin{vmatrix} \begin{vmatrix} x_1 & y_1\\x_2 & y_2\end{vmatrix} & \begin{vmatrix} y_1 & 1\\y_2 & 1\end{vmatrix} \\\\ \begin{vmatrix} x_3 & y_3\\x_4 & y_4\end{vmatrix} & \begin{vmatrix} y_3 & 1\\y_4 & 1\end{vmatrix} \end{vmatrix} }
{\begin{vmatrix} \begin{vmatrix} x_1 & 1\\x_2 & 1\end{vmatrix} & \begin{vmatrix} y_1 & 1\\y_2 & 1\end{vmatrix} \\\\ \begin{vmatrix} x_3 & 1\\x_4 & 1\end{vmatrix} & \begin{vmatrix} y_3 & 1\\y_4 & 1\end{vmatrix} \end{vmatrix}}\,\!
},
{
"math_input": \begin{align}
k_2 &= f\left(y^1_{t+h/2}, t + \frac{h}{2}\right) = f\left(y_t + \frac{h}{2} k_1, t + \frac{h}{2}\right) \\
&= f\left(y_t, t\right) + \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \\
k_3 &= f\left(y^2_{t+h/2}, t + \frac{h}{2}\right) = f\left(y_t + \frac{h}{2} f\left(y_t + \frac{h}{2} k_1, t + \frac{h}{2}\right), t + \frac{h}{2}\right) \\
&= f\left(y_t, t\right) + \frac{h}{2} \frac{d}{dt} \left[ f\left(y_t,t\right) + \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \right] \\
k_4 &= f\left(y^3_{t+h}, t + h\right) = f\left(y_t + h f\left(y_t + \frac{h}{2} k_2, t + \frac{h}{2}\right), t + h\right) \\
&= f\left(y_t + h f\left(y_t + \frac{h}{2} f\left(y_t + \frac{h}{2} f\left(y_t, t\right), t + \frac{h}{2}\right), t + \frac{h}{2}\right), t + h\right) \\
&= f\left(y_t, t\right) + h \frac{d}{dt} \left[ f\left(y_t,t\right) + \frac{h}{2} \frac{d}{dt}\left[ f\left(y_t,t\right) + \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \right]\right]
\end{align}
},
{
"math_input": \begin{align}
|n\rangle = \left |n^{(0)} \right \rangle &+ \lambda\sum_{k \ne n} \left |k^{(0)}\right\rangle \frac{\left\langle k^{(0)}\right|V\left|n^{(0)}\right\rangle}{E_n^{(0)}-E_k^{(0)}} + \lambda^2\sum_{k\neq n}\sum_{\ell \neq n} \left |k^{(0)}\right\rangle \frac{\left \langle k^{(0)} \right |V \left |\ell^{(0)} \right \rangle \left \langle \ell^{(0)} \right |V \left |n^{(0)} \right \rangle}{\left (E_n^{(0)}-E_k^{(0)}\right ) \left (E_n^{(0)}-E_\ell^{(0)} \right )} \\
& -\lambda^2 \sum_{k\neq n}\left |k^{(0)}\right\rangle \frac{\left \langle n^{(0)} \right |V\left |n^{(0)} \right \rang \left \langle k^{(0)} \right |V\left |n^{(0)} \right \rang}{\left (E_n^{(0)}-E_k^{(0)} \right )^2} - \frac{1}{2} \lambda^2 \left |n^{(0)} \right \rangle\sum_{k \ne n} \frac{\left \langle n^{(0)} \right |V\left |k^{(0)} \right \rang \left \langle k^{(0)} \right |V\left |n^{(0)} \right \rang}{\left (E_n^{(0)}-E_k^{(0)} \right )^2} + O(\lambda^3).
\end{align}
},
{
"math_input":
\begin{align}
\operatorname{H}[\mathbf{X}] &= \frac{n}{2}\ln|\mathbf{V}| +\frac{np}{2}\ln 2 + \ln\Gamma_p(\frac{n}{2}) -\frac{(n-p-1)}{2} \operatorname{E}[\ln|\mathbf{X}|] + \frac{np}{2} \\
&= \frac{n}{2}\ln|\mathbf{V}| +\frac{np}{2}\ln 2 + \frac{p(p-1)}{4}\ln\pi + \sum_{i=1}^p \ln
\Gamma\left[ n/2+(1-j)/2\right] \\
&\quad-\frac{(n-p-1)}{2}\left(\sum_{i=1}^p \psi\left(\frac{n+1-i}{2}\right) + p\ln 2 + \ln|\mathbf{V}|\right) + \frac{np}{2} \\
&= \frac{n}{2}\ln|\mathbf{V}| - \frac{(n-p-1)}{2}\ln|\mathbf{V}| +\frac{np}{2}\ln 2 -\frac{(n-p-1)}{2}p\ln 2 + \frac{p(p-1)}{4}\ln\pi \\
&\quad+ \sum_{i=1}^p \ln\Gamma\left[ n/2+(1-j)/2\right] -\frac{(n-p-1)}{2}\sum_{i=1}^p \psi\left(\frac{n+1-i}{2}\right) + \frac{np}{2} \\
&= \frac{p+1}{2}\ln|\mathbf{V}| +\frac{p(p+1)}{2}\ln 2 + \frac{p(p-1)}{4}\ln\pi \\
&\quad+ \sum_{i=1}^p \ln\Gamma\left[ n/2+(1-j)/2\right] -\frac{(n-p-1)}{2}\sum_{i=1}^p \psi\left(\frac{n+1-i}{2}\right) + \frac{np}{2} \\
\end{align}
},
{
"math_input":
=
\begin{bmatrix}
1 & \frac{1000}{877}\cdot\frac{\cos 33}{\cos 33^2+\sin 33^2} & \frac{1000}{877}\cdot\frac{\sin 33}{\cos 33^2+\sin 33^2} \\
1 & \frac{500}{21106759}\cdot\frac{16663\sin 33-24518\cos 33}{\cos 33^2+\sin 33^2} & \frac{-500}{21106759}\cdot\frac{16663\cos 33+24518\sin 33}{\cos 33^2+\sin 33^2} \\
1 & \frac{250}{123}\cdot\frac{\sin 33}{\cos 33^2+\sin 33^2} & \frac{250}{123}\cdot\frac{\cos 33}{\cos 33^2+\sin 33^2}
\end{bmatrix}
=
\begin{bmatrix}
1 & \frac{1000}{877}\cos 33 & \frac{1000}{877}\sin 33 \\
1 & \frac{8331500\sin 33-12259000\cos 33}{21106759} & -\frac{8331500\cos 33+12259000\sin 33}{21106759} \\
1 & -\frac{1000}{492}\sin 33 & \frac{1000}{492}\cos 33
\end{bmatrix}
=
\begin{bmatrix}
1 & \frac{1000}{877}\cos 33 & \frac{1000}{877}\sin 33 \\
1 & \frac{9500}{24067}\sin 33-\frac{299000}{514799}\cos 33 & -\frac{9500}{24067}\cos 33-\frac{299000}{514799}\sin 33 \\
1 & -\frac{250}{123}\sin 33 & \frac{250}{123}\cos 33
\end{bmatrix}
},
{
"math_input":
\begin{align}
F(A) & = \sum_{\sigma\in S_{n},\sigma(j_{1})<\sigma(j_{2})}\left[\sgn(\sigma)\left(\prod_{i = 1, i \neq j_1, i\neq j_2}^na_{\sigma(i)}^{i}\right)a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(j_{2})}^{j_{2}}+\sgn(\sigma')\left(\prod_{i = 1, i \neq j_1, i\neq j_2}^na_{\sigma'(i)}^{i}\right)a_{\sigma'(j_{1})}^{j_{1}}a_{\sigma'(j_{2})}^{j_{2}}\right]\\
& =\sum_{\sigma\in S_{n},\sigma(j_{1})<\sigma(j_{2})}\left[\sgn(\sigma)\left(\prod_{i = 1, i \neq j_1, i\neq j_2}^na_{\sigma(i)}^{i}\right)a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(j_{2})}^{j_{2}}-\sgn(\sigma)\left(\prod_{i = 1, i \neq j_1, i\neq j_2}^na_{\sigma(i)}^{i}\right)a_{\sigma(j_{2})}^{j_{1}}a_{\sigma(j_{1})}^{j_{2}}\right]\\
& =\sum_{\sigma\in S_{n},\sigma(j_{1})<\sigma(j_{2})}\sgn(\sigma)\left(\prod_{i = 1, i \neq j_1, i\neq j_2}^na_{\sigma(i)}^{i}\right)\left(a_{\sigma(j_{1})}^{j_{1}}a_{\sigma(j_{2})}^{j_{2}}-a_{\sigma(j_{1})}^{j_{2}}a_{\sigma(j_{2})}^{j_{_{1}}}\right)\\
\\
\end{align}
},
{
"math_input": \begin{align}
a' & = \bigg|\bigg| \Big(\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}\Big) - \bigg( \Big(\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}\Big) \cdot \Big(\begin{smallmatrix}0\\\;\;1/\sqrt{2}\\-1/\sqrt{2}\end{smallmatrix}\Big) \bigg) \Big(\begin{smallmatrix}0\\\;\;1/\sqrt{2}\\-1/\sqrt{2}\end{smallmatrix}\Big) \bigg|\bigg| \\
& = \bigg|\bigg| \Big(\begin{smallmatrix}-a\\K-b\\-c\end{smallmatrix}\Big) - \Big( 0 + \tfrac{K-b}{\sqrt{2}} + \tfrac{c}{\sqrt{2}} \Big) \Big(\begin{smallmatrix}0\\\;\;1/\sqrt{2}\\-1/\sqrt{2}\end{smallmatrix}\Big) \bigg|\bigg| \\
& = \bigg|\bigg| \bigg(\begin{smallmatrix}-a\\K-b-\tfrac{K-b+c}{2}\\-c+\tfrac{K-b+c}{2}\end{smallmatrix}\bigg) \bigg|\bigg| = \bigg|\bigg| \bigg(\begin{smallmatrix}-a\\\tfrac{K-b-c}{2}\\\tfrac{K-b-c}{2}\end{smallmatrix}\bigg) \bigg|\bigg| \\
& = \sqrt{(-a)^2 + \big(\tfrac{K-b-c}{2}\big)^2 + \big(\tfrac{K-b-c}{2}\big)^2} = \sqrt{a^2 + \tfrac{(K-b-c)^2}{2}} \\
\end{align}
},
{
"math_input": \begin{align}
&\,\,\,\,\,\,\, x^{71} && &&- x^{69} &&- 2x^{68} &&- x^{67} &&+ 2x^{66} &&+ 2x^{65} &&+ x^{64} &&- x^{63} \\
&- x^{62} &&- x^{61} &&- x^{60} &&- x^{59} &&+ 2x^{58} &&+ 5x^{57} &&+ 3x^{56} &&- 2x^{55} &&- 10x^{54} \\
&- 3x^{53} &&- 2x^{52} &&+ 6x^{51} &&+ 6x^{50} &&+ x^{49} &&+ 9x^{48} &&- 3x^{47} &&- 7x^{46} &&- 8x^{45} \\
&- 8x^{44} &&+ 10x^{43} &&+ 6x^{42} &&+ 8x^{41} &&- 5x^{40} &&- 12x^{39} &&+ 7x^{38} &&- 7x^{37} &&+ 7x^{36} \\
&+ x^{35} &&- 3x^{34} &&+ 10x^{33} &&+ x^{32} &&- 6x^{31} &&- 2x^{30} &&- 10x^{29} &&- 3x^{28} &&+ 2x^{27} \\
&+ 9x^{26} &&- 3x^{25} &&+ 14x^{24} &&- 8x^{23} && &&- 7x^{21} &&+ 9x^{20} &&+ 3x^{19} &&- 4x^{18} \\
&- 10x^{17} &&- 7x^{16} &&+ 12x^{15} &&+ 7x^{14} &&+ 2x^{13} &&- 12x^{12} &&- 4x^{11} &&- 2x^{10} &&+ 5x^9 \\
& &&+ x^7 &&- 7x^6 &&+ 7x^5 &&- 4x^4 &&+ 12x^3 &&- 6x^2 &&+ 3x &&- 6
\end{align}
},
{
"math_input":
\begin{align}
& \left[
\frac{\partial^2 \Phi}{\partial t^2}
+ g\, \frac{\partial \Phi}{\partial z}
\right]_0
+ \eta \left[ \frac{\partial}{\partial z}
\left(
\frac{\partial^2 \Phi}{\partial t^2}
+ g\, \frac{\partial \Phi}{\partial z}
\right)
\right]_0
+ \left[
\frac{\partial}{\partial t} \left( |\mathbf{u}|^2 \right)
\right]_0
\\ & \quad
+ \tfrac12\, \eta^2
\left[ \frac{\partial^2}{\partial z^2}
\left(
\frac{\partial^2 \Phi}{\partial t^2}
+ g\, \frac{\partial \Phi}{\partial z}
\right)
\right]_0
+ \eta \left[
\frac{\partial^2}{\partial t\, \partial z} \left( |\mathbf{u}|^2 \right)
\right]_0
+ \biggl[
\tfrac12\, \mathbf{u} \cdot \boldsymbol{\nabla} \left( |\mathbf{u}|^2 \right)
\biggr]_0
\\ & \quad
+ \cdots
= 0,
\end{align}
},
{
"math_input": \begin{align}
{n + 1 \choose i}(1 - t)\mathbf{b}_{i, n} &= {n \choose i} \mathbf{b}_{i, n + 1} \Rightarrow (1 - t)\mathbf{b}_{i, n} = \frac{n + 1 - i}{n + 1} \mathbf{b}_{i, n + 1} \\
{n + 1 \choose i + 1} t\mathbf{b}_{i, n} &= {n \choose i} \mathbf{b}_{i + 1, n + 1} \Rightarrow t\mathbf{b}_{i, n} = \frac{i + 1}{n + 1} \mathbf{b}_{i + 1, n + 1} \\
\mathbf{B}(t) &= (1 - t)\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i
+ t\sum_{i=0}^n \mathbf{b}_{i, n}(t)\mathbf{P}_i \\
&= \sum_{i=0}^n \frac{n + 1 - i}{n + 1}\mathbf{b}_{i, n + 1}(t)\mathbf{P}_i
+ \sum_{i=0}^n \frac{i + 1}{n + 1}\mathbf{b}_{i + 1, n + 1}(t)\mathbf{P}_i \\
&= \sum_{i=0}^{n + 1} \left(\frac{i}{n + 1}\mathbf{P}_{i - 1}
+ \frac{n + 1 - i}{n + 1}\mathbf{P}_i\right) \mathbf{b}_{i, n + 1}(t) \\
&= \sum_{i=0}^{n+1} \mathbf{b}_{i, n + 1}(t)\mathbf{P'}_i
\end{align}
},
{
"math_input": \begin{align}
Q_{\bold{u}}(\theta)
&{}=
\begin{bmatrix}
0&-z&y\\
z&0&-x\\
-y&x&0
\end{bmatrix} \sin \theta + (I - \bold{u}\bold{u}^T) \cos \theta + \bold{u}\bold{u}^T \\
&{}=
\begin{bmatrix}
(1-x^2) c_{\theta} + x^2 & - z s_{\theta} - x y c_{\theta} + x y & y s_{\theta} - x z c_{\theta} + x z \\
z s_{\theta} - x y c_{\theta} + x y & (1-y^2) c_{\theta} + y^2 & -x s_{\theta} - y z c_{\theta} + y z \\
-y s_{\theta} - x z c_{\theta} + x z & x s_{\theta} - y z c_{\theta} + y z & (1-z^2) c_{\theta} + z^2
\end{bmatrix} \\
&{}=
\begin{bmatrix}
x^2 (1-c_{\theta}) + c_{\theta} & x y (1-c_{\theta}) - z s_{\theta} & x z (1-c_{\theta}) + y s_{\theta} \\
x y (1-c_{\theta}) + z s_{\theta} & y^2 (1-c_{\theta}) + c_{\theta} & y z (1-c_{\theta}) - x s_{\theta} \\
x z (1-c_{\theta}) - y s_{\theta} & y z (1-c_{\theta}) + x s_{\theta} & z^2 (1-c_{\theta}) + c_{\theta}
\end{bmatrix} ,
\end{align}
},
{
"math_input": \begin{array}{ll} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \,&= \,\mathopen{:} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \, \mathclose{:} \\ & - \,\mathopen{:} \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, \mathclose{:} + \,\mathopen{:} \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:} +\,\mathopen{:} \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, \mathclose{:} - \mathopen{:} \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:} \\ & -\mathopen{:} \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet\bullet} \,\hat{f}_2^{\dagger\bullet} \, \mathclose{:}+\mathopen{:} \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^{\dagger\bullet\bullet}\mathclose{:} \end{array}
},
{
"math_input":
\begin{align}
\frac{1}{\lambda}\, \int_0^\lambda \eta^2\; \text{d}x
&= \frac{1}{\lambda} \int_0^\lambda
\left\{ \eta_2 + H\, \operatorname{cn}^2 \left( \begin{array}{c|c} \displaystyle \frac{\xi}{\Delta} & m\end{array} \right) \right\}^2\;
\text{d}\xi
= \frac{H^2}{\lambda} \int_0^\lambda
\operatorname{cn}^4 \left( \begin{array}{c|c} \displaystyle \frac{\xi}{\Delta} & m\end{array} \right)\; \text{d}\xi
- \eta_2^2
\\
&= \frac{\Delta\, H^2}{\lambda} \int_0^{\pi} \cos^4\, \psi\, \frac{\text{d}\xi}{\text{d}\psi}\; \text{d}\psi - \eta_2^2
= \frac{H^2}{2\, K(m)} \int_0^{\pi} \frac{\cos^4\, \psi}{\sqrt{1 - m\, \sin^2\, \psi}}\; \text{d}\psi - \eta_2^2
\\
&= \frac13\, \frac{H^2}{m^2}\, \left[ \left( 2 - 5\, m + 3\, m^2 \right) + \left( 4\, m - 2 \right)\, \frac{E(m)}{K(m)} \right]
- \frac{H^2}{m^2}\, \left( 1 - m - \frac{E(m)}{K(m)} \right)^2
\end{align}
},
{
"math_input":
\begin{align}
\delta U & = \int_{\Omega^0} \int_{-h}^h \boldsymbol{\sigma}:\delta\boldsymbol{\epsilon}~dx_3~d\Omega
= \int_{\Omega^0} \int_{-h}^h \left[\sigma_{\alpha\beta}~\delta\varepsilon_{\alpha\beta} + 2~\kappa~\sigma_{\alpha 3}~\delta\varepsilon_{\alpha 3}\right]~dx_3~d\Omega \\
& = \int_{\Omega^0} \int_{-h}^h \left[\frac{1}{2}~\sigma_{\alpha\beta}~(\delta u^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{x_3}{2}~\sigma_{\alpha\beta}~(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + \kappa~\sigma_{\alpha 3}\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]~dx_3~d\Omega \\
& = \int_{\Omega^0} \left[\frac{1}{2}~N_{\alpha\beta}~(\delta u^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{1}{2}M_{\alpha\beta}~(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + Q_\alpha\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]~d\Omega
\end{align}
},
{
"math_input": \left[\begin{matrix}
\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
\varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
\varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
\end{matrix}\right]
=
\left[\begin{matrix}
\frac{\partial u_x}{\partial x} & \frac{1}{2} \left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right) & \frac{1}{2} \left(\frac{\partial u_x}{\partial z}+\frac{\partial u_z}{\partial x}\right) \\
\frac{1}{2} \left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right) & \frac{\partial u_y}{\partial y} & \frac{1}{2} \left(\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y}\right) \\
\frac{1}{2} \left(\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right) & \frac{1}{2} \left(\frac{\partial u_z}{\partial y}+\frac{\partial u_y}{\partial z}\right) & \frac{\partial u_z}{\partial z} \\
\end{matrix}\right] \,\!
},
{
"math_input":
\begin{align}
\delta U & = \int_{\Omega^0} \int_{-h}^h \boldsymbol{\sigma}:\delta\boldsymbol{\epsilon}~dx_3~d\Omega
= \int_{\Omega^0} \int_{-h}^h \left[\sigma_{\alpha\beta}~\delta\varepsilon_{\alpha\beta} + 2~\sigma_{\alpha 3}~\delta\varepsilon_{\alpha 3}\right]~dx_3~d\Omega \\
& = \int_{\Omega^0} \int_{-h}^h \left[\frac{1}{2}~\sigma_{\alpha\beta}~(\delta u^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{x_3}{2}~\sigma_{\alpha\beta}~(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + \kappa~\sigma_{\alpha 3}\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]~dx_3~d\Omega \\
& = \int_{\Omega^0} \left[\frac{1}{2}~N_{\alpha\beta}~(\delta u^0_{\alpha,\beta}+\delta u^0_{\beta,\alpha}) - \frac{1}{2}M_{\alpha\beta}~(\delta \varphi_{\alpha,\beta}+\delta\varphi_{\beta,\alpha}) + Q_\alpha\left(\delta w^0_{,\alpha} - \delta \varphi_\alpha\right)\right]~d\Omega
\end{align}
},
{
"math_input":
\begin{array}{l}
(\forall L\subseteq \Sigma^*) \\
\quad (\mbox{regular}(L) \Rightarrow \\
\quad ((\exists p\geq 1) ( (\forall w\in L) ((|w|\geq p) \Rightarrow \\
\quad\quad ((\exists x,y,z \in \Sigma^*) (w=xyz \land (|y|\geq 1 \land |xy|\leq p \land
(\forall i\geq 0)(xy^iz\in L))))))))
\end{array}
},
{
"math_input":
\begin{pmatrix}
{\color{BrickRed}1} & {\color{BurntOrange}2} & {\color{Violet}3} \\
{\color{BrickRed}4} & {\color{BurntOrange}5} & {\color{Violet}6} \\
{\color{BrickRed}7} & {\color{BurntOrange}8} & {\color{Violet}9} \\
\end{pmatrix}
\begin{pmatrix}
{\color{BrickRed}a} & {\color{BrickRed}d} \\
{\color{BurntOrange}b} & {\color{BurntOrange}e} \\
{\color{Violet}c} & {\color{Violet}f} \\
\end{pmatrix}
=
\begin{pmatrix}
{\color{BrickRed}1a} & {\color{BrickRed}1d} \\
{\color{BrickRed}4a} & {\color{BrickRed}4d} \\
{\color{BrickRed}7a} & {\color{BrickRed}7d} \\
\end{pmatrix}+
\begin{pmatrix}
{\color{BurntOrange}2b} & {\color{BurntOrange}2e} \\
{\color{BurntOrange}5b} & {\color{BurntOrange}5e} \\
{\color{BurntOrange}8b} & {\color{BurntOrange}8e} \\
\end{pmatrix}+
\begin{pmatrix}
{\color{Violet}3c} & {\color{Violet}3f} \\
{\color{Violet}6c} & {\color{Violet}6f} \\
{\color{Violet}9c} & {\color{Violet}9f} \\
\end{pmatrix}.
},
{
"math_input": \beta_k (\tilde{T}) =
\begin{bmatrix}
0 & \; & & & & \; & & z \\
\frac{1}{2} & \ddots & & & & & & 0 \\
\; & \ddots & \ddots & & & & & \vdots \\
\; & \; & \frac{1}{2} & 0 & & \; & & \\
& \; & & 1 & 0 & & & \\
& & & \; &\frac{1}{2} & \ddots & & \; \\
\; & & & &\; & \ddots & \ddots & \vdots \\
\; & \; & & &\; & \; & \frac{1}{2} & 0
\end{bmatrix}
.
},
{
"math_input": \begin{align}
y_{t+h} &= y_t + h \left\lbrace a \cdot f(y_t, t) + b \cdot \left[ f\left(y_t, t\right) + \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \right] \right.+ \\
&+ c \cdot \left[ f\left(y_t, t\right) + \frac{h}{2} \frac{d}{dt} \left[ f\left(y_t,t\right) + \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \right] \right] + \\
&+ d \cdot \left[f\left(y_t, t\right) + h \frac{d}{dt} \left[ f\left(y_t,t\right) + \frac{h}{2} \frac{d}{dt}\left[ f\left(y_t,t\right)
+ \left. \frac{h}{2} \frac{d}{dt}f\left(y_t,t\right) \right]\right]\right]\right\rbrace + \mathcal{O}(h^5) \\
&= y_t + a \cdot h f_t + b \cdot h f_t + b \cdot \frac{h^2}{2} \frac{df_t}{dt} + c \cdot h f_t+ c \cdot \frac{h^2}{2} \frac{df_t}{dt} + \\
&+ c \cdot \frac{h^3}{4} \frac{d^2f_t}{dt^2} + d \cdot h f_t + d \cdot h^2 \frac{df_t}{dt} + d \cdot \frac{h^3}{2} \frac{d^2f_t}{dt^2} + d \cdot \frac{h^4}{4} \frac{d^3f_t}{dt^3} + \mathcal{O}(h^5)
\end{align}
},
{
"math_input":
\begin{bmatrix}
\boldsymbol{I}_m\\
& \boldsymbol{I}_m & \boldsymbol{V}_1^{(b)}\\
& \boldsymbol{W}_2^{(t)} & \boldsymbol{I}_m\\
& & & \boldsymbol{I}_m & \boldsymbol{V}_2^{(b)}\\
& & & \ddots & \ddots & \ddots\\
& & & & \boldsymbol{W}_{p-1}^{(t)} & \boldsymbol{I}_m\\
& & & & & & \boldsymbol{I}_m & \boldsymbol{V}_{p-1}^{(b)}\\
& & & & & & \boldsymbol{W}_p^{(t)} & \boldsymbol{I}_m\\
& & & & & & & & \boldsymbol{I}_m
\end{bmatrix}
\begin{bmatrix}
\boldsymbol{X}_1^{(t)}\\
\boldsymbol{X}_1^{(b)}\\
\boldsymbol{X}_2^{(t)}\\
\boldsymbol{X}_2^{(b)}\\
\vdots\\
\boldsymbol{X}_{p-1}^{(t)}\\
\boldsymbol{X}_{p-1}^{(b)}\\
\boldsymbol{X}_p^{(t)}\\
\boldsymbol{X}_p^{(b)}
\end{bmatrix}
=
\begin{bmatrix}
\boldsymbol{G}_1^{(t)}\\
\boldsymbol{G}_1^{(b)}\\
\boldsymbol{G}_2^{(t)}\\
\boldsymbol{G}_2^{(b)}\\
\vdots\\
\boldsymbol{G}_{p-1}^{(t)}\\
\boldsymbol{G}_{p-1}^{(b)}\\
\boldsymbol{G}_p^{(t)}\\
\boldsymbol{G}_p^{(b)}
\end{bmatrix}
},
{
"math_input":
\begin{align}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \phi}{\partial \mathbf{q}} \dot{\mathbf{q}} T \right)
& = \left( \frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) \frac{\partial \phi}{\partial \mathbf{q}} \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \left( \frac{d}{dt} \frac{\partial \phi}{\partial \mathbf{q}} \right) \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \phi}{\partial \mathbf{q}} \ddot{\mathbf{q}} \, T \\[6pt]
& = \frac{\partial L}{\partial \mathbf{q}} \frac{\partial \phi}{\partial \mathbf{q}} \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \left( \frac{\partial^2 \phi}{(\partial \mathbf{q})^2} \dot{\mathbf{q}} \right) \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \phi}{\partial \mathbf{q}} \ddot{\mathbf{q}} \, T.
\end{align}
},
{
"math_input": T \mapsto
\begin{bmatrix}
0 & \; & & & & \; & & T_z \\
\frac{1}{2}I & \ddots & & & & & & 0 \\
\; & \ddots & \ddots & & & & & \vdots \\
\; & \; & \frac{1}{2}I & 0 & & \; & & \\
& \; & & I & 0 & & & \\
& & & \; &\frac{1}{2}I& \ddots & & \; \\
\; & & & &\; & \ddots & \ddots & \vdots \\
\; & \; & & &\; & \; & \frac{1}{2}I & 0
\end{bmatrix}
.
},
{
"math_input":
\begin{align}
x_1(t) &= \log\frac{(\lambda_1-\lambda_2)^2 (\lambda_1-\lambda_3)^2 (\lambda_2-\lambda_3)^2 a_1 a_2 a_3}{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}
\\
x_2(t) &= \log\frac{\sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3}
\\
x_3(t) &= \log(a_1+a_2+a_3)
\\
m_1(t) &= \frac{\sum_{j<k} \lambda_j^2 \lambda_k^2 (\lambda_j-\lambda_k)^2 a_j a_k}{\lambda_1 \lambda_2 \lambda_3 \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
\\
m_2(t) &= \frac{ \left( \lambda_1^2 a_1 + \lambda_2^2 a_2 + \lambda_3^2 a_3 \right) \sum_{j<k} (\lambda_j-\lambda_k)^2 a_j a_k}{ \left( \lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3 \right) \sum_{j<k} \lambda_j \lambda_k (\lambda_j-\lambda_k)^2 a_j a_k}
\\
m_3(t) &= \frac{a_1+a_2+a_3}{\lambda_1 a_1 + \lambda_2 a_2 + \lambda_3 a_3}
\end{align}
},
{
"math_input":
\begin{align}
& \int_{\boldsymbol{\varphi}} \prod_{i=1}^K P(\varphi_i;\beta) \prod_{j=1}^M \prod_{t=1}^N P(W_{j,t}|\varphi_{Z_{j,t}}) \, d\boldsymbol{\varphi} \\
= & \prod_{i=1}^K \int_{\varphi_i} P(\varphi_i;\beta) \prod_{j=1}^M \prod_{t=1}^N P(W_{j,t}|\varphi_{Z_{j,t}}) \, d\varphi_i\\
= & \prod_{i=1}^K \int_{\varphi_i} \frac{\Gamma\bigl(\sum_{r=1}^V \beta_r \bigr)}{\prod_{r=1}^V \Gamma(\beta_r)} \prod_{r=1}^V \varphi_{i,r}^{\beta_r - 1} \prod_{r=1}^V \varphi_{i,r}^{n_{(\cdot),r}^i} \, d\varphi_i \\
= & \prod_{i=1}^K \int_{\varphi_i} \frac{\Gamma\bigl(\sum_{r=1}^V \beta_r \bigr)}{\prod_{r=1}^V \Gamma(\beta_r)} \prod_{r=1}^V \varphi_{i,r}^{n_{(\cdot),r}^i+\beta_r - 1} \, d\varphi_i \\
= & \prod_{i=1}^K \frac{\Gamma\bigl(\sum_{r=1}^V \beta_r
\bigr)}{\prod_{r=1}^V \Gamma(\beta_r)}\frac{\prod_{r=1}^V
\Gamma(n_{(\cdot),r}^i+\beta_r)}{\Gamma\bigl(\sum_{r=1}^V
n_{(\cdot),r}^i+\beta_r \bigr)} .
\end{align}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of states} \\
T &=& \text{number of observations} \\
\phi_{i=1 \dots N, j=1 \dots N} &=& \text{probability of transition from state } i \text{ to state } j \\
\boldsymbol\phi_{i=1 \dots N} &=& N\text{-dimensional vector, composed of } \phi_{i,1 \dots N} \text{; must sum to 1} \\
V &=& \text{dimension of categorical observations, e.g. size of word vocabulary} \\
\theta_{i=1 \dots N, j=1 \dots V} &=& \text{probability for state } i \text{ of observing the } j\text{th item} \\
\boldsymbol\theta_{i=1 \dots N} &=& V\text{-dimensional vector, composed of }\theta_{i,1 \dots V} \text{; must sum to 1} \\
x_{t=1 \dots T} &=& \text{state of observation at time } t \\
y_{t=1 \dots T} &=& \text{observation at time } t \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{t-1}}) \\
y_{t=1 \dots T} &\sim& \text{Categorical}(\boldsymbol\theta_{x_t})
\end{array}
},
{
"math_input":
\begin{align}
M_{xx} & = -D\left(\frac{\partial^2 w}{\partial x^2} + \nu \frac{\partial^2 w}{\partial y^2}\right) \\
& = \sum_{m=1}^\infty \sum_{n=1}^\infty\frac{16 q_0}{(2m-1)(2n-1)\pi^4}\,
\left[\frac{(2m-1)^2}{a^2}+\nu\frac{(2n-1)^2}{b^2}\right] \,\times\\
& \qquad \qquad \left[\frac{(2m-1)^2}{a^2}+\frac{(2n-1)^2}{b^2}\right]^{-2}
\sin\frac{(2m-1) \pi x}{a}\sin\frac{(2n-1) \pi y}{b} \\
M_{yy} & = -D\left(\frac{\partial^2 w}{\partial y^2} + \nu \frac{\partial^2 w}{\partial x^2}\right) \\
& = \sum_{m=1}^\infty \sum_{n=1}^\infty\frac{16 q_0}{(2m-1)(2n-1)\pi^4}\,
\left[\frac{(2n-1)^2}{b^2}+\nu\frac{(2m-1)^2}{a^2}\right] \,\times\\
& \qquad \qquad \left[\frac{(2m-1)^2}{a^2}+\frac{(2n-1)^2}{b^2}\right]^{-2}
\sin\frac{(2m-1) \pi x}{a}\sin\frac{(2n-1) \pi y}{b} \,.
\end{align}
},
{
"math_input":
\begin{align}
\lambda^3&\left(I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -
\boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}\right) +
\lambda^2\left(I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -
\boldsymbol{A}^T\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}\right) + \\
&\qquad \qquad\lambda\left(I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -
\boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}\right) +
\left(I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -
\boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}}\right) = 0 ~.
\end{align}
},
{
"math_input":
\begin{array}{|l|rrr|rrr|}
\hline
\text{arc} & \text{chord} & & & \text{sixtieths} & & \\
\hline
{}\,\,\,\,\,\,\,\,\,\, \tfrac12 & 0 & 31 & 25 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, 1 & 1 & 2 & 50 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, 1\tfrac12 & 1 & 34 & 15 & 1 & 2 & 50 \\
{}\,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
109 & 97 & 41 & 38 & 0 & 36 & 23 \\
109\tfrac12 & 97 & 59 & 49 & 0 & 36 & 9 \\
110 & 98 & 17 & 54 & 0 & 35 & 56 \\
110\tfrac12 & 98 & 35 & 52 & 0 & 35 & 42\\
111 & 98 & 53 & 43 & 0 & 35 & 29 \\
111\tfrac12 & 99 & 11 & 27 & 0 & 35 & 15 \\
112 & 99 & 29 & 5 & 0 & 35 & 1\\
112\tfrac12 & 99 & 46 & 35 & 0 & 34 & 48 \\
113 & 100 & 3 & 59 & 0 & 34 & 34 \\
{}\,\,\,\,\,\,\, \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
179 & 119 & 59 & 44 & 0 & 0 & 25 \\
179\frac12 & 119 & 59 & 56 & 0 & 0 & 9 \\
180 & 120 & 0 & 0 & 0 & 0 & 0 \\
\hline
\end{array}
},
{
"math_input": \begin{align}
y_N - y(t_N) &{}=
y_N - \underbrace{\prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) + \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0)}_{=0} - y(t_N) \\
&{}= y_N - \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) + \underbrace{\sum_{n=0}^{N-1}\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n) - \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n)}_{=\prod_{n=0}^{N-1} \Phi(h_n)\ y(t_n)-\sum_{n=N}^{N}\left[\prod_{j=n}^{N-1} \Phi(h_j)\right]\ y(t_n) = \prod_{j=0}^{N-1} \Phi(h_j)\ y(t_0) - y(t_N) } \\
&{}= \prod_{j=0}^{N-1}\Phi(h_j)\ y_0 - \prod_{j=0}^{N-1}\Phi(h_j)\ y(t_0) + \sum_{n=1}^N\ \prod_{j=n-1}^{N-1} \Phi(h_j)\ y(t_{n-1}) - \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ y(t_n) \\
&{}= \prod_{j=0}^{N-1}\Phi(h_j)\ (y_0-y(t_0)) + \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j) \left[ \Phi(h_{n-1}) - E(h_{n-1}) \right] \ y(t_{n-1}) \\
&{}= \prod_{j=0}^{N-1}\Phi(h_j)\ (y_0-y(t_0)) + \sum_{n=1}^N\ \prod_{j=n}^{N-1} \Phi(h_j)\ d_n
\end{align}
},
{
"math_input":
\begin{matrix}
Proc & ::= & \textit{STOP} & \; \\
&|& \textit{SKIP} & \; \\
&|& e \rightarrow \textit{Proc} & (\textit{prefixing})\\
&|& \textit{Proc} \;\Box\; \textit{Proc} & (\textit{external} \; \textit{choice})\\
&|& \textit{Proc} \;\sqcap\; \textit{Proc} & (\textit{nondeterministic} \; \textit{choice})\\
&|& \textit{Proc} \;\vert\vert\vert\; \textit{Proc} & (\textit{interleaving}) \\
&|& \textit{Proc} \;|[ \{ X \} ]| \;\textit{Proc} & (\textit{interface} \; \textit{parallel})\\
&|& \textit{Proc} \setminus X & (\textit{hiding})\\
&|& \textit{Proc} ; \textit{Proc} & (\textit{sequential} \; \textit{composition})\\
&|& \mathrm{if} \; b \; \mathrm{then} \; \textit{Proc}\; \mathrm{else}\; Proc & (\textit{boolean} \; \textit{conditional})\\
&|& \textit{Proc} \;\triangleright\; \textit{Proc} & (\textit{timeout})\\
&|& \textit{Proc} \;\triangle\; \textit{Proc} & (\textit{interrupt})
\end{matrix}
},
{
"math_input":
\begin{bmatrix}
\mathbf{d}_x \\
\mathbf{d}_y \\
\mathbf{d}_z \\
\end{bmatrix}=\begin{bmatrix}
1 & 0 & 0 \\
0 & {\cos ( \mathbf{- \theta}_x ) } & { - \sin ( \mathbf{- \theta}_x ) } \\
0 & { \sin ( \mathbf{- \theta}_x ) } & { \cos ( \mathbf{- \theta}_x ) } \\
\end{bmatrix}\begin{bmatrix}
{ \cos ( \mathbf{- \theta}_y ) } & 0 & { \sin ( \mathbf{- \theta}_y ) } \\
0 & 1 & 0 \\
{ - \sin ( \mathbf{- \theta}_y ) } & 0 & { \cos ( \mathbf{- \theta}_y ) } \\
\end{bmatrix}\begin{bmatrix}
{ \cos ( \mathbf{- \theta}_z ) } & { - \sin ( \mathbf{- \theta}_z ) } & 0 \\
{ \sin ( \mathbf{- \theta}_z ) } & { \cos ( \mathbf{- \theta}_z ) } & 0 \\
0 & 0 & 1 \\
\end{bmatrix}\left( {\begin{bmatrix}
\mathbf{a}_x \\
\mathbf{a}_y \\
\mathbf{a}_z \\
\end{bmatrix} - \begin{bmatrix}
\mathbf{c}_x \\
\mathbf{c}_y \\
\mathbf{c}_z \\
\end{bmatrix}} \right)
},
{
"math_input": \begin{align}
0 & = \begin{matrix} \frac{2}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha }) \\
& = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (2F_{ \alpha \beta }) + \partial_\alpha (2F_{ \beta \gamma }) + \partial_\beta (2F_{ \gamma \alpha }) \} \\
& = \begin{matrix} \frac{1}{6} \end{matrix} \{ \partial_\gamma (F_{ \alpha \beta } - F_{ \beta \alpha}) + \partial_\alpha (F_{ \beta \gamma } - F_{ \gamma \beta}) + \partial_\beta (F_{ \gamma \alpha } - F_{ \alpha \gamma}) \} \\
& = \begin{matrix} \frac{1}{6} \end{matrix} ( \partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta \gamma } + \partial_\beta F_{ \gamma \alpha } - \partial_\gamma F_{ \beta \alpha} - \partial_\alpha F_{ \gamma \beta} - \partial_\beta F_{ \alpha \gamma} ) \\
& = \partial_{[ \gamma} F_{ \alpha \beta ]}
\end{align}
},
{
"math_input":
\begin{align}
SS_{{\text{total}}} = \left( {{\mathbf{y}} - \bar y{\mathbf{1}}} \right)^2 & = \left( {{\mathbf{y}} - \bar y{\mathbf{1}} + {\mathbf{\hat y}} - {\mathbf{\hat y}}} \right)^2 , \\
& = \left( {\left( {{\mathbf{\hat y}} - \bar y{\mathbf{1}}} \right) + \left( {{\mathbf{y}} - {\mathbf{\hat y}}} \right)} \right)^2 , \\
& = \left( {{\mathbf{\hat y}} - \bar y{\mathbf{1}}} \right)^2 + {\hat \varepsilon }^2 + 2{\hat \varepsilon }^T \left( {{\mathbf{\hat y}} - \bar y{\mathbf{1}}} \right) , \\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} + 2{\hat \varepsilon }^T \left( {X{\hat \beta } - \bar y{\mathbf{1}}} \right) ,\\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} + 2\left( {\hat \varepsilon ^T X} \right){\hat \beta - }2\bar y{\hat \varepsilon }^T { \mathbf{1} } , \\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} .\\
\end{align}
},
{
"math_input":
\begin{array}{rl}
\mbox{Ax. 1.} & P(\varphi) \land \Box\; \forall x [\varphi(x) \rightarrow \psi(x)] \rightarrow P(\psi)\\
\mbox{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi)\\
\mbox{Th. 1.} & P(\varphi) \rightarrow \Diamond\; \exists x\; [\varphi(x)]\\
\mbox{Df. 1.} & G(x) \iff \forall \varphi[P(\varphi) \rightarrow \varphi(x)]\\
\mbox{Ax. 3.} & P(G)\\
\mbox{Th. 2.} & \Diamond\; \exists x\; G(x)\\
\mbox{Df. 2.} & \varphi\;\operatorname{ess}\;x \iff \varphi(x) \land \forall\psi\lbrace\psi(x) \rightarrow \Box\; \forall x[\varphi(x) \rightarrow \psi(x)]\rbrace\\
\mbox{Ax. 4.} & P(\varphi) \rightarrow \Box\; P(\varphi)\\
\mbox{Th. 3.} & G(x) \rightarrow G\;\operatorname{ess}\;x\\
\mbox{Df. 3.} & E(x) \iff \forall \varphi[\varphi\;\operatorname{ess}\;x \rightarrow \Box\; \exists x\; \varphi(x)]\\
\mbox{Ax. 5.} & P(E)\\
\mbox{Th. 4.} & \Box\; \exists x\; G(x)
\end{array}
},
{
"math_input":
\begin{align}
I_1 =\ & \eta_{20} + \eta_{02} \\
I_2 =\ & (\eta_{20} - \eta_{02})^2 + (2\eta_{11})^2 \\
I_3 =\ & (\eta_{30} - 3\eta_{12})^2 + (3\eta_{21} - \eta_{03})^2 \\
I_4 =\ & (\eta_{30} + \eta_{12})^2 + (\eta_{21} + \eta_{03})^2 \\
I_5 =\ & (\eta_{30} - 3\eta_{12}) (\eta_{30} + \eta_{12})[ (\eta_{30} + \eta_{12})^2 - 3 (\eta_{21} + \eta_{03})^2] + \\
\ & (3\eta_{21} - \eta_{03}) (\eta_{21} + \eta_{03})[ 3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2] \\
I_6 =\ & (\eta_{20} - \eta_{02})[(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2] + 4\eta_{11}(\eta_{30} + \eta_{12})(\eta_{21} + \eta_{03}) \\
I_7 =\ & (3\eta_{21} - \eta_{03})(\eta_{30} + \eta_{12})[(\eta_{30} + \eta_{12})^2 - 3(\eta_{21} + \eta_{03})^2] - \\
\ & (\eta_{30} - 3\eta_{12})(\eta_{21} + \eta_{03})[3(\eta_{30} + \eta_{12})^2 - (\eta_{21} + \eta_{03})^2].
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\sigma} & = -p~\boldsymbol{\mathit{1}} +
2\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\
& = - p~\boldsymbol{\mathit{1}} + 2\left[\left(\cfrac{\partial W}{\partial \bar{I}_1} +
I_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -
\cfrac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\
& = - p~\boldsymbol{\mathit{1}} + \lambda_1~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
\lambda_2~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\cfrac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3
\end{align}
},
{
"math_input":
\begin{align}
\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) U_x \right]
&+ \frac{\partial}{\partial x}
\left[ \rho \left( h + \overline{\eta} \right) U_x U_x + S_{xx} \right]
+ \frac{\partial}{\partial y}
\left[ \rho \left( h + \overline{\eta} \right) U_x U_y + S_{xy} \right]
\\
&= - \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial x} h
+ \tau_{w,x} - \tau_{b,x},
\\
\frac{\partial}{\partial t}\left[ \rho \left( h + \overline{\eta} \right) U_y \right]
&+ \frac{\partial}{\partial x}
\left[ \rho \left( h + \overline{\eta} \right) U_y U_x + S_{yx} \right]
+ \frac{\partial}{\partial y}
\left[ \rho \left( h + \overline{\eta} \right) U_y U_y + S_{yy} \right]
\\
&= - \rho g \left( h + \overline{\eta} \right) \frac{\partial}{\partial y} h
+ \tau_{w,y} - \tau_{b,y}.
\end{align}
},
{
"math_input":
\begin{align}
SS_{{\text{total}}} = \Vert {{\mathbf{y}} - \bar {\mathbf{y}}} \Vert^2 & = \Vert {{\mathbf{y}} - \bar {\mathbf{y}} + {\mathbf{\hat y}} - {\mathbf{\hat y}}} \Vert^2 , \\
& = \Vert {\left( {{\mathbf{\hat y}} - \bar {\mathbf{y}}} \right) + \left( {{\mathbf{y}} - {\mathbf{\hat y}}} \right)} \Vert^2 , \\
& = \Vert {{\mathbf{\hat y}} - \bar {\mathbf{y}}} \Vert^2 + \Vert{\hat \varepsilon }\Vert^2 + 2{\hat \varepsilon }^T \left( {{\mathbf{\hat y}} - \bar {\mathbf{y}}} \right) , \\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} + 2{\hat \varepsilon }^T \left( {X{\hat \beta } - \bar {\mathbf{y}}} \right) ,\\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} + 2\left( {\hat \varepsilon ^T X} \right){\hat \beta - }2 {\hat \varepsilon }^T {\bar {\mathbf{y}}} , \\
& = SS_{{\text{regression}}} + SS_{{\text{error}}} .\\
\end{align}
},
{
"math_input":
\begin{bmatrix}
\boldsymbol{I}_{3m} & \boldsymbol{0} & \boldsymbol{V}_1^{[2](t)}\\
\boldsymbol{0} & \boldsymbol{I}_m & \boldsymbol{V}_1^{[2](b)} & \boldsymbol{0}\\
\boldsymbol{0} & \boldsymbol{W}_2^{[2](t)} & \boldsymbol{I}_m & \boldsymbol{0} & \boldsymbol{V}_2^{[2](t)}\\
& \boldsymbol{W}_2^{[2](b)} & \boldsymbol{0} & \boldsymbol{I}_{3m} & \boldsymbol{V}_2^{[2](b)} & \boldsymbol{0} \\
& & \ddots & \ddots & \ddots & \ddots & \ddots\\
& & & \boldsymbol{0} & \boldsymbol{W}_{p/2-1}^{[2](t)} & \boldsymbol{I}_{3m} & \boldsymbol{0} & \boldsymbol{V}_{p/2-1}^{[2](t)}\\
& & & & \boldsymbol{W}_{p/2-1}^{[2](b)} & \boldsymbol{0} & \boldsymbol{I}_m & \boldsymbol{V}_{p/2-1}^{[2](b)} & \boldsymbol{0}\\
& & & & & \boldsymbol{0} & \boldsymbol{W}_{p/2}^{[2](t)} & \boldsymbol{I}_m & \boldsymbol{0}\\
& & & & & & \boldsymbol{W}_{p/2}^{[2](b)} & \boldsymbol{0} & \boldsymbol{I}_{3m}
\end{bmatrix}\text{.}
},
{
"math_input":
\begin{align}
\left\langle \frac{\delta F[\rho]}{\delta\rho}, \phi \right\rangle
& {} = \frac{d}{d\varepsilon} \left. \int f( \mathbf{r}, \rho + \varepsilon \phi, \nabla\rho+\varepsilon\nabla\phi )\, d\mathbf{r} \right|_{\varepsilon=0} \\
& {} = \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d\mathbf{r} \\
& {} = \int \left[ \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \left( \frac{\partial f}{\partial\nabla\rho} \phi \right) - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\mathbf{r} \\
& {} = \int \left[ \frac{\partial f}{\partial\rho} \phi - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\mathbf{r} \\
& {} = \left\langle \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}\,, \phi \right\rangle,
\end{align}
},
{
"math_input":
\begin{align}
F(A^1, \dots, cA^j, \dots) & = \sum_{\sigma \in S_n} \sgn(\sigma) ca_{\sigma(j)}^j\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\\
& = c \sum_{\sigma \in S_n} \sgn(\sigma) a_{\sigma(j)}^j\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\\
&=c F(A^1, \dots, A^j, \dots)\\
\\
F(A^1, \dots, b+A^j, \dots) & = \sum_{\sigma \in S_n} \sgn(\sigma)\left(b_{\sigma(j)} + a_{\sigma(j)}^j\right)\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\\
& = \sum_{\sigma \in S_n} \sgn(\sigma)
\left( \left(b_{\sigma(j)}\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right) + \left(a_{\sigma(j)}^j\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right)\right)\\
& = \left(\sum_{\sigma \in S_n} \sgn(\sigma) b_{\sigma(j)}\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right)
+ \left(\sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i = 1}^n a_{\sigma(i)}^i\right)\\
&= F(A^1, \dots, b, \dots) + F(A^1, \dots, A^j, \dots)\\
\\
\end{align}
},
{
"math_input":
exp\begin{pmatrix}
. & . & . & . & . & . & . & . & . & . \\
-4 & . & . & . & . & . & . & . & . & . \\
. & -3 & . & . & . & . & . & . & . & . \\
. & . & -2 & . & . & . & . & . & . & . \\
. & . & . & -1 & . & . & . & . & . & . \\
. & . & . & . & 0 & . & . & . & . & . \\
. & . & . & . & . & 1 & . & . & . & . \\
. & . & . & . & . & . & 2 & . & . & . \\
. & . & . & . & . & . & . & 3 & . & . \\
. & . & . & . & . & . & . & . & 4 & .
\end{pmatrix} =
\begin{pmatrix}
1 & . & . & . & . & . & . & . & . & . \\
-4 & 1 & . & . & . & . & . & . & . & . \\
6 & -3 & 1 & . & . & . & . & . & . & . \\
-4 & 3 & -2 & 1 & . & . & . & . & . & . \\
1 & -1 & 1 & -1 & 1 & . & . & . & . & . \\
. & . & . & . & . & 1 & . & . & . & . \\
. & . & . & . & . & 1 & 1 & . & . & . \\
. & . & . & . & . & 1 & 2 & 1 & . & . \\
. & . & . & . & . & 1 & 3 & 3 & 1 & . \\
. & . & . & . & . & 1 & 4 & 6 & 4 & 1
\end{pmatrix}
},
{
"math_input": \begin{align}
a_0 &= 1 ,\, &&a_1 = \frac12 ,\, &&a_2 = \frac14 ,\, &&a_3 = \frac18 ,\, &&a_4 = \frac1{16} ,\, &&a_5 = \frac1{32} ,\, &&\ldots ,\, &&a_k = \frac1{2^k} ,\, &&\ldots \\
b_0 &= 1 ,\, &&b_1 = 1 ,\, &&b_2 = \frac14 ,\, &&b_3 = \frac14 ,\, &&b_4 = \frac1{16} ,\, &&b_5 = \frac1{16} ,\, &&\ldots ,\, &&b_k = \frac1{4^{\left\lfloor \frac{k}{2} \right\rfloor}} ,\, &&\ldots \\
c_0 &= \frac12 ,\, &&c_1 = \frac14 ,\, &&c_2 = \frac1{16} ,\, &&c_3 = \frac1{256} ,\, &&c_4 = \frac1{65\,536} ,\, &&&&\ldots ,\, &&c_k = \frac1{2^{2^k}} ,\, &&\ldots \\
d_0 &= 1 ,\, &&d_1 = \frac12 ,\, &&d_2 = \frac13 ,\, &&d_3 = \frac14 ,\, &&d_4 = \frac15 ,\, &&d_5 = \frac16 ,\, &&\ldots ,\, &&d_k = \frac1{k+1} ,\, &&\ldots
\end{align}
},
{
"math_input":
\begin{align}
\mathbf{A}(\mathbf{r}) &= \sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}}
\left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} +
\bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\
\mathbf{E}(\mathbf{r}) &= i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar\omega}{2 V\epsilon_0}}
\left(\mathbf{e}^{(\mu)} a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} -
\bar{\mathbf{e}}^{(\mu)} {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right) \\
\mathbf{B}(\mathbf{r}) &= i\sum_{\mathbf{k},\mu} \sqrt{\frac{\hbar}{2 \omega V\epsilon_0}}
\left((\mathbf{k}\times\mathbf{e}^{(\mu)}) a^{(\mu)}(\mathbf{k}) e^{i\mathbf{k}\cdot\mathbf{r}} -
(\mathbf{k}\times\bar{\mathbf{e}}^{(\mu)}) {a^\dagger}^{(\mu)}(\mathbf{k}) e^{-i\mathbf{k}\cdot\mathbf{r}} \right), \\
\end{align}
},
{
"math_input":
\begin{align}
\operatorname{E}(s^2) & = \operatorname{E}\left(\sum_{i=1}^n \frac{(x_i-\overline{x})^2}{n-1} \right)\\
& = \frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^n(x_i-\mu+\mu-\overline{x})^2 \right) \\
& = \frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^n(x_i-\mu)^2 - 2(\overline{x}-\mu)\sum_{i=1}^n(x_i-\mu) + \sum_{i=1}^n(\overline{x}-\mu)^2\right) \\
& = \frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^n(x_i-\mu)^2 - 2(\overline{x}-\mu)n \left( \frac{\sum_{i=1}^n x_i}{n}-\mu \right) + n(\overline{x}-\mu)^2\right) \\
& = \frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^n(x_i-\mu)^2 - 2n(\overline{x}-\mu)^2 + n(\overline{x}-\mu)^2\right) \\
& = \frac{1}{n-1}\operatorname{E}\left(\sum_{i=1}^n(x_i-\mu)^2 - n(\overline{x}-\mu)^2\right) \\
& = \frac{1}{n-1}\left(\sum_{i=1}^n\operatorname{E}((x_i-\mu)^2) - n\operatorname{E}((\overline{x}-\mu)^2) \right) \\
\end{align}
},
{
"math_input": \begin{align}
\frac{\partial}{\partial b} \left (\int_a^b f(x)\; \mathrm{d}x \right ) &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ \int_a^{b+\Delta b} f(x)\,\mathrm{d}x - \int_a^b f(x)\,\mathrm{d}x \right] \\
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \int_b^{b+\Delta b} f(x)\,\mathrm{d}x \\
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ f(b) \Delta b + \mathcal{O}\left(\Delta b^2\right) \right] \\
&= f(b) \\
\frac{\partial}{\partial a} \left (\int_a^b f(x)\; \mathrm{d}x \right )&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ \int_{a+\Delta a}^b f(x)\,\mathrm{d}x - \int_a^b f(x)\,\mathrm{d}x \right] \\
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \int_{a+\Delta a}^a f(x)\,\mathrm{d}x \\
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ -f(a)\, \Delta a + \mathcal{O}\left(\Delta a^2\right) \right]\\
&= -f(a).
\end{align}
},
{
"math_input":
\begin{align}
& \int_{\theta_j} P(\theta_j;\alpha) \prod_{t=1}^N P(Z_{j,t}|\theta_j) \, d\theta_j = \int_{\theta_j} \frac{\Gamma\bigl(\sum_{i=1}^K \alpha_i \bigr)}{\prod_{i=1}^K \Gamma(\alpha_i)} \prod_{i=1}^K \theta_{j,i}^{n_{j,(\cdot)}^i+\alpha_i - 1} \, d\theta_j \\
= & \frac{\Gamma\bigl(\sum_{i=1}^K \alpha_i \bigr)}{\prod_{i=1}^K \Gamma(\alpha_i)}\frac{\prod_{i=1}^K \Gamma(n_{j,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K n_{j,(\cdot)}^i+\alpha_i \bigr)} \int_{\theta_j} \frac{\Gamma\bigl(\sum_{i=1}^K n_{j,(\cdot)}^i+\alpha_i \bigr)}{\prod_{i=1}^K \Gamma(n_{j,(\cdot)}^i+\alpha_i)} \prod_{i=1}^K \theta_{j,i}^{n_{j,(\cdot)}^i+\alpha_i - 1} \, d\theta_j \\
= & \frac{\Gamma\bigl(\sum_{i=1}^K \alpha_i \bigr)}{\prod_{i=1}^K \Gamma(\alpha_i)}\frac{\prod_{i=1}^K \Gamma(n_{j,(\cdot)}^i+\alpha_i)}{\Gamma\bigl(\sum_{i=1}^K n_{j,(\cdot)}^i+\alpha_i \bigr)}.
\end{align}
},
{
"math_input":
\begin{align}
{\mathbf{A=LDL}^\mathrm{T}} & =
\begin{pmatrix}
\mathbf I & 0 & 0 \\
\mathbf L_{21} & \mathbf I & 0 \\
\mathbf L_{31} & \mathbf L_{32} & \mathbf I\\
\end{pmatrix}
\begin{pmatrix}
\mathbf D_1 & 0 & 0 \\
0 & \mathbf D_2 & 0 \\
0 & 0 & \mathbf D_3\\
\end{pmatrix}
\begin{pmatrix}
\mathbf I & \mathbf L_{21}^\mathrm T & \mathbf L_{31}^\mathrm T \\
0 & \mathbf I & \mathbf L_{32}^\mathrm T \\
0 & 0 & \mathbf I\\
\end{pmatrix} \\
& = \begin{pmatrix}
\mathbf D_1 & &(\mathrm{symmetric}) \\
\mathbf L_{21} \mathbf D_1 & \mathbf L_{21} \mathbf D_1 \mathbf L_{21}^\mathrm T + \mathbf D_2& \\
\mathbf L_{31} \mathbf D_1 & \mathbf L_{31} \mathbf D_{1} \mathbf L_{21}^\mathrm T + \mathbf L_{32} \mathbf D_2 & \mathbf L_{31} \mathbf D_1 \mathbf L_{31}^\mathrm T + \mathbf L_{32} \mathbf D_2 \mathbf L_{32}^\mathrm T + \mathbf D_3
\end{pmatrix}
\end{align}
},
{
"math_input":
\begin{align}
\varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
\varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
\varepsilon_{\phi\phi} & = \cfrac{1}{r\sin\theta}\left(\cfrac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
\varepsilon_{r\theta} & = \cfrac{1}{2}\left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) \\
\varepsilon_{\theta \phi} & = \cfrac{1}{2r}\left(\cfrac{1}{\sin\theta}\cfrac{\partial u_\theta}{\partial \phi} + \cfrac{\partial u_\phi}{\partial \theta} - u_\phi\cot\theta\right) \\
\varepsilon_{\phi r} & = \cfrac{1}{2}\left(\cfrac{1}{r\sin\theta}\cfrac{\partial u_r}{\partial \phi} + \cfrac{\partial u_\phi}{\partial r} - \cfrac{u_\phi}{r}\right)
\end{align}
},
{
"math_input":
\begin{align}
P_0^0(\cos\theta) & = 1 \\[8pt]
P_1^0(\cos\theta) & = \cos\theta \\[8pt]
P_1^1(\cos\theta) & = -\sin\theta \\[8pt]
P_2^0(\cos\theta) & = \tfrac{1}{2} (3\cos^2\theta-1) \\[8pt]
P_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\[8pt]
P_2^2(\cos\theta) & = 3\sin^2\theta \\[8pt]
P_3^0(\cos\theta) & = \tfrac{1}{2} (5\cos^3\theta-3\cos\theta) \\[8pt]
P_3^1(\cos\theta) & = -\tfrac{3}{2} (5\cos^2\theta-1)\sin\theta \\[8pt]
P_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\[8pt]
P_3^3(\cos\theta) & = -15\sin^3\theta \\[8pt]
P_4^0(\cos\theta) & = \tfrac{1}{8} (35\cos^4\theta-30\cos^2\theta+3) \\[8pt]
P_4^1(\cos\theta) & = - \tfrac{5}{2} (7\cos^3\theta-3\cos\theta)\sin\theta \\[8pt]
P_4^2(\cos\theta) & = \tfrac{15}{2} (7\cos^2\theta-1)\sin^2\theta \\[8pt]
P_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\[8pt]
P_4^4(\cos\theta) & = 105\sin^4\theta
\end{align}
},
{
"math_input":
S_k(A_n)
= \frac{
\begin{vmatrix}
A_{n-k} & \cdots & A_{n-1} & A_n \\
\Delta A_{n-k} & \cdots & \Delta A_{n-1} & \Delta A_{n} \\
\Delta A_{n-k+1} & \cdots & \Delta A_{n} & \Delta A_{n+1} \\
\vdots & & \vdots & \vdots \\
\Delta A_{n-1} & \cdots & \Delta A_{n+k-2} & \Delta A_{n+k-1} \\
\end{vmatrix}
}{
\begin{vmatrix}
1 & \cdots & 1 & 1 \\
\Delta A_{n-k} & \cdots & \Delta A_{n-1} & \Delta A_{n} \\
\Delta A_{n-k+1} & \cdots & \Delta A_{n} & \Delta A_{n+1} \\
\vdots & & \vdots & \vdots \\
\Delta A_{n-1} & \cdots & \Delta A_{n+k-2} & \Delta A_{n+k-1} \\
\end{vmatrix}
},
},
{
"math_input":
\begin{align}
\Rightarrow {\ln \left[{\left(1+x\sqrt{\frac{q}{np}}\right)}^{-k}{\left(1-x\sqrt{\frac{p}{nq}}\right)}^{-\left(n-k\right)}\right]}& =-\left(np\times x\sqrt{\frac{q}{np}}-np\times \frac{x^2q}{2np}+x\sqrt{npq}\times x\sqrt{\frac{q}{np}}-x\sqrt{npq}\times \frac{x^2q}{2np}+\cdots \right)\\
& -\left(-nq\times x\sqrt{\frac{p}{nq}}-nq\times \frac{x^2p}{2nq}+x\sqrt{npq}\times x\sqrt{\frac{p}{nq}}+x\sqrt{npq}\times \frac{x^2p}{2nq}+\cdots \right)\\
& =-\left(x\sqrt{npq}-\frac{x^2q}{2}+x^2q+\cdots \right)-\left(-x\sqrt{npq}-\frac{x^2p}{2}+x^2p+\cdots \right)\\
& =-\left(x\sqrt{npq}+\frac{x^2q}{2}+\cdots \right)-\left(-x\sqrt{npq}+\frac{x^2p}{2}+\cdots \right)\\
& =-x\sqrt{npq}-\frac{x^2q}{2}+x\sqrt{npq}-\frac{x^2p}{2}-\cdots \\
& =-\frac{x^2q}{2}-\frac{x^2p}{2}-\cdots \\
& =-\frac{x^2}{2}\left(q+p\right)-\cdots \\
& =-\frac{x^2}{2}-\cdots \\
\end{align}
},
{
"math_input": \begin{align}
y &= \frac{G}{(1 + \gamma - \alpha - \beta)_{\alpha + \beta - \gamma - 1}} (1 - x)^{\gamma - \alpha - \beta}
\sum_{r = \alpha + \beta - \gamma}^\infty \frac{(\gamma - \beta )_r (\gamma - \alpha)_r}
{(1)_r (1)_{r + \gamma - \alpha - \beta}} (1 - x)^r + {}\\
&\quad
{} + H \sum_{r = 0}^\infty \frac{(\gamma - \alpha - \beta)(\gamma - \beta)_r (\gamma - \alpha)_r}
{(1 + \gamma - \alpha - \beta)_r (1)_r}
\Biggl(\ln(1 - x) + \frac{1}{\alpha + \beta - \gamma} + {} \\
&\qquad + \sum_{k = 0}^{r - 1} \left(\frac{1}{\alpha + k} + \frac{1}{\beta + k}
- \frac{1}{1 + k} - \frac{1}{\alpha + \beta - \gamma + 1 + k} \right)
\Biggr) (1 - x)^r
\end{align}
},
{
"math_input":
X=\begin{vmatrix}
0 & x_{12} & x_{13} &\cdots & x_{1n} \\
-x_{12} & 0 & x_{23} &\cdots & x_{2n} \\
-x_{13} & -x_{23} & 0 &\cdots & x_{3n} \\
\vdots& \vdots & \vdots &\ddots & \vdots \\
-x_{1n} & -x_{2n} & -x_{3n} &\cdots & 0
\end{vmatrix},
D=\begin{vmatrix}
0 & \frac{\partial} {\partial x_{12}} & \frac{\partial} {\partial x_{13}} &\cdots & \frac{\partial}{\partial x_{1n} } \\[6pt]
-\frac{\partial} { \partial x_{12} } & 0 & \frac{\partial} { \partial x_{23}} &\cdots & \frac{\partial}{\partial x_{2n} } \\[6pt]
-\frac{\partial} {\partial x_{13} } & -\frac{\partial} {\partial x_{23}} & 0 &\cdots & \frac{\partial}{\partial x_{3n} } \\[6pt]
\vdots& \vdots & \vdots &\ddots & \vdots \\[6pt]
-\frac{\partial} {\partial x_{1n} } & -\frac{\partial} {\partial x_{2n}} & -\frac{\partial} {\partial x_{3n}} &\cdots & 0
\end{vmatrix}.
},
{
"math_input": \begin{align}
\lim_{x \rightarrow x_0} \operatorname{ap} \ a\cdot f(x_0) & =a \cdot \lim_{x \rightarrow x_0} \operatorname{ap} \ f(x_0) \\
\lim_{x \rightarrow x_0} \operatorname{ap} \ (f(x_{0})+g(x_0)) & = \lim_{x \rightarrow x_0} \operatorname{ap} \ f(x_0) + \lim_{x \rightarrow x_0} \operatorname{ap} \ g(x_0) \\
\lim_{x \rightarrow x_0} \operatorname{ap} \ (f(x_{0})-g(x_0)) & = \lim_{x \rightarrow x_0} \operatorname{ap} \ f(x_0)-\lim_{x \rightarrow x_0} \operatorname{ap} \ g(x_{0}) \\
\lim_{x \rightarrow x_0} \operatorname{ap} \ (f(x_0)\cdot g(x_0)) & = \lim_{x \rightarrow x_0} \operatorname{ap} \ f(x_0) \cdot \lim_{x \rightarrow x_{0}} \operatorname{ap} \ g(x_0) \\
\lim_{x \rightarrow x_0} \operatorname{ap} \ (f(x_0)/g(x_0)) & = \lim_{x \rightarrow x_0} \operatorname{ap} \ f(x_0) / \lim\limits_{x \rightarrow x_0} \operatorname{ap} \ g(x_0)
\end{align}
},
{
"math_input":
\begin{align}
\varepsilon_{rr} & = \frac{\partial u_r}{\partial r}\\
\varepsilon_{\theta\theta}& = \frac{1}{r}\left(\frac{\partial u_\theta}{\partial \theta} + u_r\right)\\
\varepsilon_{\phi\phi} & = \frac{1}{r\sin\theta}\left(\frac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
\varepsilon_{r\theta} & = \frac{1}{2}\left(\frac{1}{r}\frac{\partial u_r}{\partial \theta} + \frac{\partial u_\theta}{\partial r}- \frac{u_\theta}{r}\right) \\
\varepsilon_{\theta \phi} & = \frac{1}{2r}\left(\frac{1}{\sin\theta}\frac{\partial u_\theta}{\partial \phi}
+\left(\frac{\partial u_\phi}{\partial \theta}-u_\phi \cot(\theta)\right)\right)\\
\varepsilon_{r \phi} & = \frac{1}{2} \left(\frac{1}{r \sin \theta} \frac{\partial u_r}{\partial \phi} + \frac{\partial u_\phi}{\partial r} - \frac{u_\phi}{r}\right)
\end{align}
},
{
"math_input": {}_{\rm{erfi}\left(\rm{erfi}\frac{\sqrt3}{3}\right)=\frac{2}{\sqrt\pi}\int_0^{\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t} e^{u^2} \rm{d} u
=\frac{2}{\sqrt\pi}e^{\left(\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right)^2}\int_0^{\infty}\frac{\sin\left[\frac{4u\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t\right]}{e^{u^2}}{\rm{d}}u
=\frac{2}{\sqrt\pi}\int_0^{{}_{\frac{2\sqrt[3]e}{\sqrt\pi}\int_0^{\infty}\frac{\sin\left(\frac{2}{3}\sqrt3t\right)}{e^{t^2}}{\rm{d}}t}} e^{u^2} {\rm{d}} u
=\frac{2}{\sqrt\pi}e^{\left(\frac{2}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)^2}\int_0^{\infty}\frac{\sin\left(\frac{4u}{\sqrt\pi}\int_0^{\frac{\sqrt3}{3}} e^{t^2} \rm{d} t\right)}{e^{u^2}} {\rm{d}} u\approx 1.00002087363809430195879}
},
{
"math_input": \begin{align}
\left(
A_r \frac{\partial B_r}{\partial r}
+ \frac{A_\theta}{r} \frac{\partial B_r}{\partial \theta}
+ \frac{A_\phi}{r\sin\theta} \frac{\partial B_r}{\partial \phi}
- \frac{A_\theta B_\theta + A_\phi B_\phi}{r}
\right) &\boldsymbol{\hat{r}} \\
+ \left(
A_r \frac{\partial B_\theta}{\partial r}
+ \frac{A_\theta}{r} \frac{\partial B_\theta}{\partial \theta}
+ \frac{A_\phi}{r\sin\theta} \frac{\partial B_\theta}{\partial \phi}
+ \frac{A_\theta B_r}{r} - \frac{A_\phi B_\phi\cot\theta}{r}
\right) &\boldsymbol{\hat{\theta}} \\
+ \left(
A_r \frac{\partial B_\phi}{\partial r}
+ \frac{A_\theta}{r} \frac{\partial B_\phi}{\partial \theta}
+ \frac{A_\phi}{r\sin\theta} \frac{\partial B_\phi}{\partial \phi}
+ \frac{A_\phi B_r}{r}
+ \frac{A_\phi B_\theta \cot\theta}{r}
\right) &\boldsymbol{\hat{\phi}}
\end{align}
},
{
"math_input": q(n) = \begin{vmatrix} ~1& ~ & ~&~&~&~&~&~&~1~\\
-1& ~1& ~ & ~&~&~&~&~&~0~\\
-1& -1& ~1& ~ & ~&~&~&~&-1~\\
~0& -1& -1& ~1 & ~ & ~&~&~&~0~\\
~0 & ~0 & -1& -1&~1 & ~&~&~&-1~\\
~1& ~0 & ~0& -1& -1&~1&~&~& ~0~\\
~0 & ~1& ~0 & ~0& -1& -1&~1 & ~ &~0~\\
~1 & ~0& ~1 & ~0&~0& -1& -1 &~&~0~\\
~ \vdots & ~&~&~&~&~& ~& \ddots & ~\vdots~ \end{vmatrix}_{(n+1) \times (n+1)} ,
},
{
"math_input":
\begin{align}
&\left[ f_{t+1}(\text{AA}), f_{t+1}(\text{Aa}), f_{t+1}(\text{aa})\right] \\
&\quad=
f_t(\text{AA}) f_t(\text{AA}) \left[ 1, 0, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{Aa}) \left[ 1/2, 1/2, 0 \right]
+ 2 f_t(\text{AA}) f_t(\text{aa}) \left[ 0, 1, 0 \right] \\
&\quad\quad+
f_t(\text{Aa}) f_t(\text{Aa}) \left[ 1/4, 1/2, 1/4 \right]
+ 2 f_t(\text{Aa}) f_t(\text{aa}) \left[ 0, 1/2, 1/2 \right]
+ f_t(\text{aa}) f_t(\text{aa}) \left[ 0, 0, 1 \right] \\
&\quad=
\left[ \left(f_t(\text{AA}) + \frac{f_t(\text{Aa})}{2} \right)^2,
2 \left(f_t(\text{AA}) + \frac{f_t(\text{Aa})}{2} \right)
\left(f_t(\text{aa}) + \frac{f_t(\text{Aa})}{2} \right),
\left(f_t(\text{aa}) + \frac{f_t(\text{Aa})}{2} \right)^2
\right]\\
&\quad=
\left[ f_t(\text{A})^2, 2 f_t(\text{A}) f_t(\text{a}), f_t(\text{a})^2 \right]
\end{align}
},
{
"math_input":
\begin{array}{rl}
\text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\
\text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\
\text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\
\text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\
\text{Ax. 3.} & P(G) \\
\text{Th. 2.} & \Diamond \; \exists x \; G(x) \\
\text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \\
\text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\
\text{Th. 3.} & G(x) \to G \text{ ess } x \\
\text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists x \; \varphi(x)] \\
\text{Ax. 5.} & P(E) \\
\text{Th. 4.} & \Box \; \exists x \; G(x)
\end{array}
},
{
"math_input":
\begin{pmatrix}
{\color{BrickRed}1} & {\color{BurntOrange}2} & {\color{Violet}3} \\
{\color{BrickRed}4} & {\color{BurntOrange}5} & {\color{Violet}6} \\
{\color{BrickRed}7} & {\color{BurntOrange}8} & {\color{Violet}9} \\
\end{pmatrix}
\begin{pmatrix}
{\color{BrickRed}a} & {\color{BrickRed}d} \\
{\color{BurntOrange}b} & {\color{BurntOrange}e} \\
{\color{Violet}c} & {\color{Violet}f} \\
\end{pmatrix}
=
\begin{pmatrix}
{\color{BrickRed}1a} + {\color{BurntOrange}2b} + {\color{Violet}3c} & {\color{BrickRed}1d} + {\color{BurntOrange}2e} + {\color{Violet}3f} \\
{\color{BrickRed}4a} + {\color{BurntOrange}5b} + {\color{Violet}6c} & {\color{BrickRed}4d} + {\color{BurntOrange}5e} + {\color{Violet}6f} \\
{\color{BrickRed}7a} + {\color{BurntOrange}8b} + {\color{Violet}9c} & {\color{BrickRed}7d} + {\color{BurntOrange}8e} + {\color{Violet}9f} \\
\end{pmatrix}
},
{
"math_input":
\begin{array}{l}
\left( {z - {\rm E}[z]} \right)^2 \approx \,\,\,\left[ \begin{array}{l}
\left\{ {\frac{{\partial z}}{{\partial x_1 }}\left( {x_1 - \,\,\bar x_1 } \right)\,\, + \,\,\,\frac{{\partial z}}{{\partial x_2 }}\left( {x_2 - \,\,\bar x_2 } \right)} \right\}\,\,\, + \\
\,\,\,\frac{{\partial ^2 z}}{{\partial x_1 \partial x_2 }}\left[ {\left( {x_1 - \,\,\bar x_1 } \right)\left( {x_2 - \,\,\bar x_2 } \right)\,\, - \,\,\sigma _{1,2} } \right]\,\,\, + \\
\,\,\,\frac{1}{2}\frac{{\partial ^2 z}}{{\partial x_1^2 }}\left[ {\left( {x_1 - \,\,\bar x_1 } \right)^2 - \,\,\sigma _1^2 } \right]\,\,\, + \,\,\,\frac{1}{2}\frac{{\partial ^2 z}}{{\partial x_2^2 }}\left[ {\left( {x_2 - \,\,\bar x_2 } \right)^2 - \,\,\sigma _2^2 } \right] \\
\end{array} \right]^2 \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\
\end{array}
},
{
"math_input": \left\{\begin{matrix}
1+1+1 & = & 3 \\
1+1+2 & = & 4 \\
1+1+3 & = & 5 \\
1+2+1 & = & 4 \\
1+2+2 & = & 5 \\
1+2+3 & = & 6 \\
1+3+1 & = & 5 \\
1+3+2 & = & 6 \\
1+3+3 & = & 7 \\
2+1+1 & = & 4 \\
2+1+2 & = & 5 \\
2+1+3 & = & 6 \\
2+2+1 & = & 5 \\
2+2+2 & = & 6 \\
2+2+3 & = & 7 \\
2+3+1 & = & 6 \\
2+3+2 & = & 7 \\
2+3+3 & = & 8 \\
3+1+1 & = & 5 \\
3+1+2 & = & 6 \\
3+1+3 & = & 7 \\
3+2+1 & = & 6 \\
3+2+2 & = & 7 \\
3+2+3 & = & 8 \\
3+3+1 & = & 7 \\
3+3+2 & = & 8 \\
3+3+3 & = & 9
\end{matrix}\right\}
=\left\{\begin{matrix}
3 & \mbox{with}\ \mbox{probability}\ 1/27 \\
4 & \mbox{with}\ \mbox{probability}\ 3/27 \\
5 & \mbox{with}\ \mbox{probability}\ 6/27 \\
6 & \mbox{with}\ \mbox{probability}\ 7/27 \\
7 & \mbox{with}\ \mbox{probability}\ 6/27 \\
8 & \mbox{with}\ \mbox{probability}\ 3/27 \\
9 & \mbox{with}\ \mbox{probability}\ 1/27
\end{matrix}\right\}
},
{
"math_input": \begin{matrix}
\left(A_r \frac{\partial B_r}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_r}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_r}{\partial \phi}\!-\!\frac{A_\theta B_\theta\!+\!A_\phi B_\phi}{r}\right) \boldsymbol{\hat r} \!+\!\\
\left(A_r \frac{\partial B_\theta}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_\theta}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_\theta}{\partial \phi}\!+\!\frac{A_\theta B_r}{r}-\frac{A_\phi B_\phi\cot(\theta)}{r}\right) \boldsymbol{\hat\theta} \!+\!\\
\left(A_r \frac{\partial B_\phi}{\partial r}\!+\!\frac{A_\theta}{r}\frac{\partial B_\phi}{\partial \theta}\!+\!\frac{A_\phi}{r\sin(\theta)}\frac{\partial B_\phi}{\partial \phi}\!+\!\frac{A_\phi B_r}{r}\!+\!\frac{A_\phi B_\theta \cot(\theta)}{r}\right)\boldsymbol{\hat\phi}
\end{matrix}
},
{
"math_input":
\begin{align}
\frac{\partial}{\partial x_1}& \left( \frac{\partial^2 \varepsilon_{22}}{\partial x_3^2} + \frac{\partial^2 \varepsilon_{33}}{\partial x_2^2} -
2 \frac{\partial^2 \varepsilon_{23}}{\partial x_2 \partial x_3}\right) -
\frac{\partial}{\partial x_2}\left[ \frac{\partial^2 \varepsilon_{22}}{\partial x_1 \partial x_3} -
\frac{\partial}{\partial x_2} \left ( \frac{\partial \varepsilon_{23}}{\partial x_1} - \frac{\partial \varepsilon_{13}}{\partial x_2} + \frac{\partial \varepsilon_{12}}{\partial x_3}\right) \right] \\
& -
\frac{\partial}{\partial x_3}\left[ \frac{\partial^2 \varepsilon_{33}}{\partial x_1 \partial x_2} -
\frac{\partial}{\partial x_3} \left ( \frac{\partial \varepsilon_{23}}{\partial x_1} + \frac{\partial \varepsilon_{13}}{\partial x_2} - \frac{\partial \varepsilon_{12}}{\partial x_3}\right)\right]=0
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial v_r}{\partial r}~\mathbf{e}_r\otimes\mathbf{e}_r +
\cfrac{1}{r}\left(\cfrac{\partial v_r}{\partial \theta} - v_\theta\right)~\mathbf{e}_r\otimes\mathbf{e}_\theta + \cfrac{\partial v_r}{\partial z}~\mathbf{e}_r\otimes\mathbf{e}_z \\[8pt]
& + \cfrac{\partial v_\theta}{\partial r}~\mathbf{e}_\theta\otimes\mathbf{e}_r +
\cfrac{1}{r}\left(\cfrac{\partial v_\theta}{\partial \theta} + v_r \right)~\mathbf{e}_\theta\otimes\mathbf{e}_\theta + \cfrac{\partial v_\theta}{\partial z}~\mathbf{e}_\theta\otimes\mathbf{e}_z \\[8pt]
& + \cfrac{\partial v_z}{\partial r}~\mathbf{e}_z\otimes\mathbf{e}_r +
\cfrac{1}{r}\cfrac{\partial v_z}{\partial \theta}~\mathbf{e}_z\otimes\mathbf{e}_\theta + \cfrac{\partial v_z}{\partial z}~\mathbf{e}_z\otimes\mathbf{e}_z
\end{align}
},
{
"math_input":
\begin{bmatrix}
\epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
\end{bmatrix}
= \begin{bmatrix}
\tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xy}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm y}} & 0 & 0 & 0 \\
0 & 0 & 0 & \tfrac{2(1+\nu_{\rm yz})}{E_{\rm y}} & 0 & 0 \\
0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
\end{bmatrix}
\begin{bmatrix}
\sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
\end{bmatrix}
},
{
"math_input":
\begin{array}{lcl}
\mathit{F}_{11} = \cfrac{1}{\mathit{X}_\mathit{f}\mathit{X}^\prime_\mathit{f}}\ ,\ \mathit{F}_{22} = \mathit{F}_{33} = \cfrac{1}{\mathit{Y}_\mathit{f}\mathit{Y}^\prime_\mathit{f}}\\
\mathit{F}_{44} = \cfrac{1}{\mathit{S}_{\mathit{f}4}^2}\ ,\ \mathit{F}_{55} = \mathit{F}_{66} = \cfrac{1}{\mathit{S}_{\mathit{f}6}^2}\\
\mathit{F}_{1} = \cfrac{1}{\mathit{X}_\mathit{f}} - \cfrac{1}{\mathit{X}_\mathit{f}^\prime}\ ,\ \mathit{F}_{2} = \mathit{F}_{3} = \cfrac{1}{\mathit{Y}_\mathit{f}} - \cfrac{1}{\mathit{Y}_\mathit{f}^\prime}\\
\mathit{F}_{12} = \mathit{F}_{21} = \mathit{F}_{13} = \mathit{F}_{31} = -\cfrac{1}{2\sqrt{\mathit{X}_\mathit{f}\mathit{X}_\mathit{f}^\prime\mathit{Y}_\mathit{f}\mathit{Y}_\mathit{f}^\prime}}\ ,\ \mathit{F}_{23} = \mathit{F}_{32} = -\cfrac{1}{2\mathit{Y}_\mathit{f}\mathit{Y}_\mathit{f}^\prime}
\end{array}
},
{
"math_input": \begin{align}
&{} D(X, Z) = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-z|f(x)h(z) \, dx\, dz\ = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-z|f(x)h(z) \, dx\, dz \int_{-\infty}^\infty g(y) dy\ \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |(x-y)+(y-z)|f(x)g(y)h(z) \, dx\, dy\, dz\ \\
&{} \le \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty (|x-y|+|y-z|)f(x)g(y)h(z) \, dx\, dy\, dz\ \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|f(x)g(y)h(z) \, dx\, dy\, dz\ + \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty |y-z|f(x)g(y)h(z) \, dx\, dy\, dz\ \\
&{} = \int_{-\infty}^\infty \int_{-\infty}^\infty |x-y|f(x)g(y) \, dx\, dy\ + \int_{-\infty}^\infty \int_{-\infty}^\infty |y-z|g(y)h(z) \, dy\, dz\ \\
&{} = D(X, Y) + D(Y, Z)
\end{align}
},
{
"math_input":
\begin{align}
\mathbf{u}_1 & = \mathbf{v}_1, & \mathbf{e}_1 & = {\mathbf{u}_1 \over \|\mathbf{u}_1\|} \\
\mathbf{u}_2 & = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2),
& \mathbf{e}_2 & = {\mathbf{u}_2 \over \|\mathbf{u}_2\|} \\
\mathbf{u}_3 & = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3), & \mathbf{e}_3 & = {\mathbf{u}_3 \over \|\mathbf{u}_3\|} \\
\mathbf{u}_4 & = \mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_3}\,(\mathbf{v}_4), & \mathbf{e}_4 & = {\mathbf{u}_4 \over \|\mathbf{u}_4\|} \\
& {}\ \ \vdots & & {}\ \ \vdots \\
\mathbf{u}_k & = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k), & \mathbf{e}_k & = {\mathbf{u}_k\over \|\mathbf{u}_k \|}.
\end{align}
},
{
"math_input":
\begin{align}
p_H(x|\boldsymbol{\chi},\nu) &= {\displaystyle \int\limits_\boldsymbol{\eta} p_F(x|\boldsymbol{\eta}) p_G(\boldsymbol{\eta}|\boldsymbol{\chi},\nu) \,\operatorname{d}\boldsymbol{\eta}} \\
&= {\displaystyle \int\limits_\boldsymbol{\eta} h(x)g(\boldsymbol{\eta})e^{\boldsymbol{\eta}^{\rm T}\mathbf{T}(x)} f(\boldsymbol{\chi},\nu)g(\boldsymbol{\eta})^\nu e^{\boldsymbol{\eta}^{\rm T}\boldsymbol{\chi}} \,\operatorname{d}\boldsymbol{\eta}} \\
&= {\displaystyle h(x) f(\boldsymbol{\chi},\nu) \int\limits_\boldsymbol{\eta} g(\boldsymbol{\eta})^{\nu+1} e^{\boldsymbol{\eta}^{\rm T}(\boldsymbol{\chi} + \mathbf{T}(x))} \,\operatorname{d}\boldsymbol{\eta}} \\
&= h(x) \dfrac{f(\boldsymbol{\chi},\nu)}{f(\boldsymbol{\chi} + \mathbf{T}(x), \nu+1)}
\end{align}
},
{
"math_input":
\begin{array}{cl}
\displaystyle\frac{x:\sigma \in \Gamma}{\Gamma \vdash x:\sigma}&[\texttt{Var}]\\ \\
\displaystyle\frac{\Gamma \vdash e_0:\tau \rightarrow \tau' \quad\quad \Gamma \vdash e_1 : \tau }{\Gamma \vdash e_0\ e_1 : \tau'}&[\texttt{App}]\\ \\
\displaystyle\frac{\Gamma,\;x:\tau\vdash e:\tau'}{\Gamma \vdash \lambda\ x\ .\ e : \tau \rightarrow \tau'}&[\texttt{Abs}]\\ \\
\displaystyle\frac{\Gamma \vdash e_0:\sigma \quad\quad \Gamma,\,x:\sigma \vdash e_1:\tau}{\Gamma \vdash \texttt{let}\ x = e_0\ \texttt{in}\ e_1 : \tau} &[\texttt{Let}]\\ \\ \\
\displaystyle\frac{\Gamma \vdash e:\sigma' \quad \sigma' \sqsubseteq \sigma}{\Gamma \vdash e:\sigma}&[\texttt{Inst}]\\ \\
\displaystyle\frac{\Gamma \vdash e:\sigma \quad \alpha \notin \text{free}(\Gamma)}{\Gamma \vdash e:\forall\ \alpha\ .\ \sigma}&[\texttt{Gen}]\\ \\
\end{array}
},
{
"math_input":
[\mathsf{S}] = \begin{bmatrix}
s_{1111} & s_{1122} & s_{1133} & 2s_{1123} & 2s_{1131} & 2s_{1112} \\
s_{2211} & s_{2222} & s_{2233} & 2s_{2223} & 2s_{2231} & 2s_{2212} \\
s_{3311} & s_{3322} & s_{3333} & 2s_{3323} & 2s_{3331} & 2s_{3312} \\
2s_{2311} & 2s_{2322} & 2s_{2333} & 4s_{2323} & 4s_{2331} & 4s_{2312} \\
2s_{3111} & 2s_{3122} & 2s_{3133} & 4s_{3123} & 4s_{3131} & 4s_{3112} \\
2s_{1211} & 2s_{1222} & 2s_{1233} & 4s_{1223} & 4s_{1231} & 4s_{1212}
\end{bmatrix} \equiv \begin{bmatrix}
S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\
S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\
S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36} \\
S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46} \\
S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56} \\
S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix}
},
{
"math_input": \begin{align}
\tanh x &= x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} \\
\coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi \\
\operatorname {sech}\, x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2} \\
\operatorname {csch}\, x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi
\end{align}
},
{
"math_input": \begin{array}{cccccccccccccccccc}
& & & & & & & & & 1 & & & & & & & &\\
& & & & & & & & \frac{1}{2} & & \frac{1}{2} & & & & & & &\\
& & & & & & & \frac{1}{3} & & \frac{1}{6} & & \frac{1}{3} & & & & & &\\
& & & & & & \frac{1}{4} & & \frac{1}{12} & & \frac{1}{12} & & \frac{1}{4} & & & & &\\
& & & & & \frac{1}{5} & & \frac{1}{20} & & \frac{1}{30} & & \frac{1}{20} & & \frac{1}{5} & & & &\\
& & & & \frac{1}{6} & & \frac{1}{30} & & \frac{1}{60} & & \frac{1}{60} & & \frac{1}{30} & & \frac{1}{6} & & &\\
& & & \frac{1}{7} & & \frac{1}{42} & & \frac{1}{105} & & \frac{1}{140} & & \frac{1}{105} & & \frac{1}{42} & & \frac{1}{7} & &\\
& & \frac{1}{8} & & \frac{1}{56} & & \frac{1}{168} & & \frac{1}{280} & & \frac{1}{280} & & \frac{1}{168} & & \frac{1}{56} & & \frac{1}{8} &\\
& & & & &\vdots & & & & \vdots & & & & \vdots& & & & \\
\end{array}
},
{
"math_input":
\begin{bmatrix}
\epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
\end{bmatrix}
= \begin{bmatrix}
\tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm zx}}{E_{\rm z}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm z}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xz}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm z}} & 0 & 0 & 0 \\
0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\
0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm zx}} & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
\end{bmatrix}
\begin{bmatrix}
\sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
\end{bmatrix}
},
{
"math_input":
\begin{bmatrix}
\epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
\end{bmatrix}
= \begin{bmatrix}
\tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm y}} & 0 & 0 & 0 \\
-\tfrac{\nu_{\rm xy}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm y}} & 0 & 0 & 0 \\
0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\
0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm zx}} & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
\end{bmatrix}
\begin{bmatrix}
\sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
\end{bmatrix}
},
{
"math_input": \begin{align}
\prod_{i=0}^9\frac{n-i}{30-i}
&=\prod_{i=0}^9\frac{(n-\frac 9 2)-(i-\frac 9 2)}{(30-\frac 9 2)-(i-\frac 9 2)}
=\prod_{i=0}^9\frac{(2n-9)-(2i-9)}{(2\cdot 30-9)-(2i-9)}
=\prod_{i=0}^9\frac{r_n-r_i}{r_{30}-r_i}\\
&=\left(\prod_{i=0}^4\frac{r_n -r_i}{r_{30}-r_i}\right)\cdot\left(\prod_{i=5}^9\frac{r_n-r_{i}}{r_{30}-r_{i}}\right)\\
& =\left(\prod_{i=0}^4\frac{r_n -r_i}{r_{30}-r_i}\right)\cdot\left(\prod_{i=0}^4\frac{r_n-r_{9-i}}{r_{30}-r_{9-i}}\right)\\
& =\left(\prod_{i=0}^4\frac{r_n -r_i}{r_{30}-r_i}\right)\cdot\left(\prod_{i=0}^4\frac{r_n+r_{i}}{r_{30}+r_{i}}\right)\\
&=\prod_{i=0}^4\frac{(r_n-r_i)(r_n +r_{i})}{(r_{30}-r_i)(r_{30}+r_{i})}
=\prod_{i=0}^4\frac{r_n ^2-r_i^2}{r_{30}^2-r_i^2}\\
&=\frac{(r_n ^2-81)(r_n ^2-49)(r_n ^2-25)(r_n ^2-9)(r_n ^2-1)}{2520\cdot 2552\cdot 2576\cdot 2592\cdot 2600}
.
\end{align}
},
{
"math_input": \mathbf{K}_{\text{Gauss}} = \frac{\sigma^2}{\pi \delta_x \delta_y Q^2} \begin{pmatrix} \frac{2}{\sigma_x \sigma_y} &0 &0 &\frac{-1}{A \sigma_y} &\frac{-1}{A \sigma_x} \\ 0
&\frac{2 \sigma_x}{A^2 \sigma_y} &0 &0 &0 \\ 0 &0 &\frac{2 \sigma_y}{A^2 \sigma_x} &0 &0 \\ \frac{-1}{A \sigma_y} &0 &0 &\frac{2 \sigma_x}{A^2 \sigma_y} &0 \\
\frac{-1}{A \sigma_x} &0 &0 &0 &\frac{2 \sigma_y}{A^2 \sigma_x} \end{pmatrix} \ , \qquad \mathbf{K}_{\text{Poiss}} = \frac{1}{2 \pi} \begin{pmatrix} \frac{3A}{\sigma_x \sigma_y} &0 &0 &\frac{-1}{\sigma_y} &\frac{-1}{\sigma_x} \\ 0
&\frac{\sigma_x}{A \sigma_y} &0 &0 &0 \\ 0 &0 &\frac{\sigma_y}{A \sigma_x} &0 &0 \\ \frac{-1}{\sigma_y} &0 &0 &\frac{2 \sigma_x}{3A \sigma_y} &\frac{1}{3A} \\
\frac{-1}{\sigma_x} &0 &0 &\frac{1}{3A} &\frac{2 \sigma_y}{3A \sigma_x} \end{pmatrix} \ .
},
{
"math_input":
\begin{align}
\eta(x,t) =&
a\, \left\{
\cos\, \theta
+ ka\, \frac{3 - \sigma^2}{4\, \sigma^3}\, \cos\, 2\theta
\right\}
\\ &
+ \mathcal{O} \left( (ka)^3 \right),
\\
\Phi(x,z,t) =&
a\, \frac{\omega}{k}\, \frac{\cosh\, k(z+h)}{\sinh\, kh}
\\ & \times
\left\{
\sin\, \theta
+ ka\, \frac{3 \cosh\, 2k(z+h)}{8\, \sinh^3\, kh}\, \sin\, 2\theta
\right\}
\\ &
- (ka)^2\, \frac{1}{2\, \sinh\, 2kh}\, \frac{g\, t}{k}
+ \mathcal{O} \left( (ka)^3 \right),
\\
c =& \frac{\omega}{k} = \sqrt{\frac{g}{k}\, \sigma}
+ \mathcal{O} \left( (ka)^2 \right),
\\
\sigma =& \tanh\, kh
\quad \text{and} \quad
\theta(x,t) = k x - \omega t.
\end{align}
},
{
"math_input":
\begin{align}
\mathbb{E}[\ln |\mathbf{X}|] &= \frac{ \partial A(\ldots,\eta_2) }{ \partial \eta_2 } = \frac{ \partial }{ \partial \eta_2 } \left[ -\left(\eta_2+\frac{p+1}{2}\right)(p\ln 2 + \ln|\mathbf{V}|) + \ln\Gamma_p\left(\eta_2+\frac{p+1}{2}\right) \right] \\
&= \frac{ \partial }{ \partial \eta_2 } \left[ \left(\eta_2+\frac{p+1}{2}\right)(p\ln 2 + \ln|\mathbf{V}|) + p(p-1)/4 \ln \pi + \sum_{j=1}^p \ln \Gamma\left[\eta_2+\frac{p+1}{2}+(1-j)/2\right] \right] \\
&= p\ln 2 + \ln|\mathbf{V}| + \sum_{j=1}^p \psi\left[\eta_2+\frac{p+1}{2}+(1-j)/2\right] \\
&= p\ln 2 + \ln|\mathbf{V}| + \sum_{j=1}^p \psi\left[\frac{n-p-1}{2}+\frac{p+1}{2}+(1-j)/2\right] \\
&= p\ln 2 + \ln|\mathbf{V}| + \sum_{j=1}^p \psi\left[\frac{n}{2}+(1-j)/2\right] \\
&= p\ln 2 + \ln|\mathbf{V}| + \sum_{j=1}^p \psi\left(\frac{n+1-j}{2}\right)
\end{align}
},
{
"math_input": \begin{cases}
\dot{\mathbf{x}} = f_x(\mathbf{x}) + g_x(\mathbf{x}) z_1 &\qquad \text{ ( by Lyapunov function } V_x, \text{ subsystem stabilized by } u_x(\textbf{x}) \text{ )}\\
\dot{z}_1 = f_1( \mathbf{x}, z_1 ) + g_1( \mathbf{x}, z_1 ) z_2\\
\dot{z}_2 = f_2( \mathbf{x}, z_1, z_2 ) + g_2( \mathbf{x}, z_1, z_2 ) z_3\\
\vdots\\
\dot{z}_i = f_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) + g_i( \mathbf{x}, z_1, z_2, \ldots, z_i ) z_{i+1}\\
\vdots\\
\dot{z}_{k-2} = f_{k-2}( \mathbf{x}, z_1, z_2, \ldots z_{k-2} ) + g_{k-2}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2} ) z_{k-1}\\
\dot{z}_{k-1} = f_{k-1}( \mathbf{x}, z_1, z_2, \ldots z_{k-2}, z_{k-1} ) + g_{k-1}( \mathbf{x}, z_1, z_2, \ldots, z_{k-2}, z_{k-1} ) z_k\\
\dot{z}_k = f_k( \mathbf{x}, z_1, z_2, \ldots z_{k-1}, z_k ) + g_k( \mathbf{x}, z_1, z_2, \ldots, z_{k-1}, z_k ) u
\end{cases}
},
{
"math_input": f(J_{\lambda,n})=\left(\begin{matrix}
f(\lambda) & f^\prime (\lambda) & \frac{f^{\prime\prime}(\lambda)}{2} & \cdots & \frac{f^{(n-2)}(\lambda)}{(n-2)!} & \frac{f^{(n-1)}(\lambda)}{(n-1)!} \\
0 & f(\lambda) & f^\prime (\lambda) & \cdots & \frac{f^{(n-3)}(\lambda)}{(n-3)!} & \frac{f^{(n-2)}(\lambda)}{(n-2)!} \\
0 & 0 & f(\lambda) & \cdots & \frac{f^{(n-4)}(\lambda)}{(n-4)!} & \frac{f^{(n-3)}(\lambda)}{(n-3)!} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & f(\lambda) & f^\prime (\lambda) \\
0 & 0 & 0 & \cdots & 0 & f(\lambda) \\
\end{matrix}\right)=\left(\begin{matrix}
a_0 & a_1 & a_2 & \cdots & a_{n-1} \\
0 & a_0 & a_1 & \cdots & a_{n-2} \\
0 & 0 & a_0 & \cdots & a_{n-3} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_1 \\
0 & 0 & 0 & \cdots & a_0
\end{matrix}\right).
},
{
"math_input":
\begin{align}
p(\vec\varphi) & = p(\vec\theta) \left|\det\frac{\partial\theta_i}{\partial\varphi_j}\right| \\
& \propto \sqrt{\det I(\vec\theta)\, {\det}^2\frac{\partial\theta_i}{\partial\varphi_j}} \\
& = \sqrt{\det \frac{\partial\theta_k}{\partial\varphi_i}\, \det \operatorname{E}\!\left[\frac{\partial \ln L}{\partial\theta_k} \frac{\partial \ln L}{\partial\theta_l} \right]\, \det \frac{\partial\theta_l}{\partial\varphi_j}} \\
& = \sqrt{\det \operatorname{E}\!\left[\sum_{k,l} \frac{\partial\theta_k}{\partial\varphi_i} \frac{\partial \ln L}{\partial\theta_k} \frac{\partial \ln L}{\partial\theta_l} \frac{\partial\theta_l}{\partial\varphi_j} \right]} \\
& = \sqrt{\det \operatorname{E}\!\left[\frac{\partial \ln L}{\partial\varphi_i} \frac{\partial \ln L}{\partial\varphi_j}\right]}
= \sqrt{\det I(\vec\varphi)}.
\end{align}
},
{
"math_input":
\begin{matrix}
\left\langle \delta F[\rho], \phi \right\rangle
& = & \frac{d}{d\epsilon} \left. \int f( \mathbf{r}, \rho + \epsilon \phi, \nabla\rho+\epsilon\nabla\phi )\, d^3r \right|_{\epsilon=0} \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d^3r \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi + \nabla \cdot \left[ \frac{\partial f}{\partial\nabla\rho} \phi \right] - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\
& = & \int \left( \frac{\partial f}{\partial\rho} \phi - \left[ \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right] \phi \right) d^3r \\
& = & \left\langle \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}\,, \phi \right\rangle
\end{matrix}
},
{
"math_input":
\begin{bmatrix}
\mathbf{d}_x \\
\mathbf{d}_y \\
\mathbf{d}_z \\
\end{bmatrix}=\begin{bmatrix}
1 & 0 & 0 \\
0 & {\cos \mathbf{\theta}_x } & { - \sin \mathbf{\theta}_x } \\
0 & { \sin \mathbf{\theta}_x } & { \cos \mathbf{\theta}_x } \\
\end{bmatrix}\begin{bmatrix}
{ \cos \mathbf{\theta}_y } & 0 & { \sin \mathbf{\theta}_y } \\
0 & 1 & 0 \\
{ - \sin \mathbf{\theta}_y } & 0 & { \cos \mathbf{\theta}_y } \\
\end{bmatrix}\begin{bmatrix}
{ \cos \mathbf{\theta}_z } & { - \sin \mathbf{\theta}_z } & 0 \\
{ \sin \mathbf{\theta}_z } & { \cos \mathbf{\theta}_z } & 0 \\
0 & 0 & 1 \\
\end{bmatrix}\left( {\begin{bmatrix}
\mathbf{a}_x \\
\mathbf{a}_y \\
\mathbf{a}_z \\
\end{bmatrix} - \begin{bmatrix}
\mathbf{c}_x \\
\mathbf{c}_y \\
\mathbf{c}_z \\
\end{bmatrix}} \right)
},
{
"math_input":
\begin{align}
\sigma &=
-\begin{pmatrix}
p&0&0\\
0&p&0\\
0&0&p
\end{pmatrix} +
\mu \begin{pmatrix}
2 \displaystyle{\frac{\partial u}{\partial x}} & \displaystyle{\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}} &\displaystyle{ \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}} \\
\displaystyle{\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}} & 2 \displaystyle{\frac{\partial v}{\partial y}} & \displaystyle{\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}} \\
\displaystyle{\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z}} & \displaystyle{\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z}} & 2\displaystyle{\frac{\partial w}{\partial z}}
\end{pmatrix} \\
&= -p I + \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T) = -p I + 2 \mu e\\
\end{align}
},
{
"math_input": \begin{align}
\mathbf{r} & =\mathbf{r}\left ( t \right ) = r \mathbf{\hat{e}}_r\\
\mathbf{v} & = v \mathbf{\hat{e}}_r + r\,\frac{{\rm d}\theta}{{\rm d}t}\mathbf{\hat{e}}_\theta + r\,\frac{{\rm d}\phi}{{\rm d}t}\,\sin\theta \mathbf{\hat{e}}_\phi \\
\mathbf{a} & = \left( a - r\left(\frac{{\rm d}\theta}{{\rm d}t}\right)^2 - r\left(\frac{{\rm d}\phi}{{\rm d}t}\right)^2\sin^2\theta \right)\mathbf{\hat{e}}_r \\
& + \left( r \frac{{\rm d}^2 \theta}{{\rm d}t^2 } + 2v\frac{{\rm d}\theta}{{\rm d}t} - r\left(\frac{{\rm d}\phi}{{\rm d}t}\right)^2\sin\theta\cos\theta \right) \mathbf{\hat{e}}_\theta \\
& + \left( r\frac{{\rm d}^2 \phi}{{\rm d}t^2 }\,\sin\theta + 2v\,\frac{{\rm d}\phi}{{\rm d}t}\,\sin\theta + 2 r\,\frac{{\rm d}\theta}{{\rm d}t}\,\frac{{\rm d}\phi}{{\rm d}t}\,\cos\theta \right) \mathbf{\hat{e}}_\phi
\end{align} \,\!
},
{
"math_input": \begin{align}\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
3 & 5 & 6 & 7 \\
2 & 4 & 3 & 8
\end{array}\right]&\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & -4 & 12 & -8 \\
2 & 4 & 3 & 8
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & -4 & 12 & -8 \\
0 & -2 & 7 & -2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & -3 & 2 \\
0 & -2 & 7 & -2
\end{array}\right]
\\
&\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & -3 & 2 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & -2 & 5 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 3 & 0 & 9 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right]\sim
\left[\begin{array}{rrr|r}
1 & 0 & 0 & -15 \\
0 & 1 & 0 & 8 \\
0 & 0 & 1 & 2
\end{array}\right].\end{align}
},
{
"math_input":
\begin{align}
\Pr(Y_i=1) &= \frac{e^{(\boldsymbol\beta_1 +\mathbf{C}) \cdot \mathbf{X}_i}}{e^{(\boldsymbol\beta_0 +\mathbf{C})\cdot \mathbf{X}_i} + e^{(\boldsymbol\beta_1 +\mathbf{C}) \cdot \mathbf{X}_i}} \, \\
&= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i} e^{\mathbf{C} \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} e^{\mathbf{C} \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i} e^{\mathbf{C} \cdot \mathbf{X}_i}} \, \\
&= \frac{e^{\mathbf{C} \cdot \mathbf{X}_i}e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{e^{\mathbf{C} \cdot \mathbf{X}_i}(e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i})} \, \\
&= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}} \, \\
\end{align}
},
{
"math_input":
\begin{align}
p(Y_i=1) &= \frac{e^{(\boldsymbol\beta_1 +\mathbf{C}) \cdot \mathbf{X}_i}}{e^{(\boldsymbol\beta_0 +\mathbf{C})\cdot \mathbf{X}_i} + e^{(\boldsymbol\beta_1 +\mathbf{C}) \cdot \mathbf{X}_i}} \, \\
&= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i} e^{-\mathbf{C} \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} e^{\mathbf{C} \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i} e^{\mathbf{C} \cdot \mathbf{X}_i}} \, \\
&= \frac{e^{\mathbf{C} \cdot \mathbf{X}_i}e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{e^{\mathbf{C} \cdot \mathbf{X}_i}(e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i})} \, \\
&= \frac{e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}}{e^{\boldsymbol\beta_0 \cdot \mathbf{X}_i} + e^{\boldsymbol\beta_1 \cdot \mathbf{X}_i}} \, \\
\end{align}
},
{
"math_input":
\begin{align}
P(W_{n+1}) & = {\sum_{k=0}^n}P(W_{n+1}|k)f(k;N,m,n)\\
& = {\sum_{k=0}^n}\frac{m-k}{N-n}f(k;N,m,n) \\
& = {\sum_{k=0}^n}\frac{m-k}{N-n}\frac{\binom mk \binom {N-m} {n-k}}{\binom Nn} \\
& = \frac{1}{(N-n)\binom Nn} \left \{ m\sum_{k=0}^n \binom mk \binom {N-m} {n-k} - \sum_{k=0}^n k\binom mk \binom {N-m} {n-k}\right \} \\
& = \frac{1}{(N-n)\binom Nn}\left\{ m\binom Nn - \sum_{k=1}^n k\frac{m}{k} \binom {m-1}{k-1} \binom {N-m} {n-k}\right \} \\
& = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \sum_{k=1}^n \binom {m-1}{k-1} \binom {N-1-(m-1)} {n-1-(k-1)}\right \} \\
& = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \binom {N-1}{n-1}\right \} \\
& = \frac{m}{(N-n)\binom Nn}\left\{ \binom Nn - \frac{n}{N}\binom Nn\right \} \\
& = \frac{m}{(N-n)}\left\{ 1 - \frac{n}{N} \right\} = \frac{m}{N}
\end{align}
},
{
"math_input":
\begin{align}
\arcsin x &{}= -i\,\ln\left(i\,x+\sqrt{1-x^2}\right) &{}= \arccsc \frac{1}{x}\\[10pt]
\arccos x &{}= -i\,\ln\left(x+i\,\sqrt{1-x^2}\right) = \frac{\pi}{2}\,+i\ln\left(i\,x+\sqrt{1-x^2}\right) = \frac{\pi}{2}-\arcsin x &{}= \arcsec \frac{1}{x}\\[10pt]
\arctan x &{}= \tfrac{1}{2}i\left(\ln\left(1-i\,x\right)-\ln\left(1+i\,x\right)\right) &{}= \arccot \frac{1}{x}\\[10pt]
\arccot x &{}= \tfrac{1}{2}i\left(\ln\left(1-\frac{i}{x}\right)-\ln\left(1+\frac{i}{x}\right)\right) &{}= \arctan \frac{1}{x}\\[10pt]
\arcsec x &{}= -i\,\ln\left(i\,\sqrt{1-\frac{1}{x^2}}+\frac{1}{x}\right) = i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right)+\frac{\pi}{2} = \frac{\pi}{2}-\arccsc x &{}= \arccos \frac{1}{x}\\[10pt]
\arccsc x &{}= -i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right) &{}= \arcsin \frac{1}{x}
\end{align}
},
{
"math_input": \begin{align} A & = T \cdot \bar{P2} \cdot \bar{P1} \cdot \bar{P0} \\
& = \bar{T \cdot \bar{P2}} + P1 + P0 \\
B & = T \cdot \bar{P2} \cdot \bar{P1} \cdot P0 \\
& = \bar{T \cdot \bar{P2}} + P1 + \bar{P0} \\
C & = T \cdot \bar{P2} \cdot P1 \cdot \bar{P0}\\
& = \bar{T \cdot \bar{P2}} + \bar{P1} + P0 \\
D & = T \cdot \bar{P2} \cdot P1 \cdot P0 \\
& = \bar{T \cdot \bar{P2}} + \bar{P1} + \bar{P0} \\
E & = T \cdot P2 \cdot \bar{P1} \cdot \bar{P0} \\
F & = T \cdot P2 \cdot \bar{P1} \cdot P0 \\
G & = T \cdot P2 \cdot P1 \cdot \bar{P0} \\
H & = T \cdot P2 \cdot P1 \cdot P0 \\
\end{align}
},
{
"math_input":
[\mathsf{C}] = \begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\
c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\
c_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\
c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\
c_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\
c_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212}
\end{bmatrix} \equiv \begin{bmatrix}
C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix}
},
{
"math_input":
\begin{align}
p(\mu|\mathbf{X}) \propto p(\mathbf{X}|\mu) p(\mu) & = \left(\frac{\tau}{2\pi}\right)^{n/2} \exp\left[-\frac{1}{2}\tau \left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right)\right] \sqrt{\frac{\tau_0}{2\pi}} \exp\left(-\frac{1}{2}\tau_0(\mu-\mu_0)^2\right) \\
&\propto \exp\left(-\frac{1}{2}\left(\tau\left(\sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2\right) + \tau_0(\mu-\mu_0)^2\right)\right) \\
&\propto \exp\left(-\frac{1}{2}(n\tau(\bar{x}-\mu)^2 + \tau_0(\mu-\mu_0)^2)\right) \\
&= \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2 + \frac{n\tau\tau_0}{n\tau+\tau_0}(\bar{x} - \mu_0)^2\right) \\
&\propto \exp\left(-\frac{1}{2}(n\tau + \tau_0)\left(\mu - \dfrac{n\tau \bar{x} + \tau_0\mu_0}{n\tau + \tau_0}\right)^2\right)
\end{align}
},
{
"math_input":
\begin{align}
D^2 & {} = \sum_\text{cyclic} a^2S_A\left(\frac {aS_A} {2S^2} - \frac {S_BS_C} {aS^2}\right)^2 \\
& {} = \frac {1} {4S^4} \sum_\text{cyclic} a^4S_A^3 - \frac {S_AS_BS_C} {S^4} \sum_\text{cyclic} a^2S_A + \frac {S_AS_BS_C} {S^4} \sum_\text{cyclic} S_BS_C \\
& {} = \frac {1} {4S^4} \sum_\text{cyclic} a^2S_A^2(S^2-S_BS_C) - 2(S_\omega-4R^2) + (S_\omega-4R^2) \\
& {} = \frac {1} {4S^2} \sum_\text{cyclic} a^2S_A^2 - \frac {S_AS_BS_C} {S^4} \sum_\text{cyclic} a^2S_A - (S_\omega-4R^2) \\
& {} = \frac {1} {4S^2} \sum_\text{cyclic} a^2(b^2c^2-S^2) - \frac {1} {2}(S_\omega-4R^2) -(S_\omega-4R^2) \\
& {} = \frac {3a^2b^2c^2} {4S^2} - \frac {1} {4} \sum_\text{cyclic} a^2 - \frac {3} {2}(S_\omega-4R^2) \\
& {} = 3R^2- \frac {1} {2} S_\omega - \frac {3} {2} S_\omega + 6R^2 \\
& {} = 9R^2- 2S_\omega.
\end{align}
},
{
"math_input":
\begin{align}
d^4\sigma &=
\frac{Z^2\alpha_\text{fine}^3\hbar^2}{(2\pi)^2}\frac{|\mathbf{p}_f|}{|\mathbf{p}_i|}
\frac{d\omega}{\omega}\frac{d\Omega_i \, d\Omega_f \, d\Phi}{|\mathbf{q}|^4}\times
\\
& {}\times \left[
\frac{\mathbf{p}_f^2\sin^2\Theta_f}{(E_f-c|\mathbf{p}_f|\cos\Theta_f)^2}\left
(4E_i^2-c^2\mathbf{q}^2\right)\right. \\
& {}+ \frac{\mathbf{p}_i^2\sin^2\Theta_i}{(E_i-c|\mathbf{p}_i|\cos\Theta_i)^2}\left
(4E_f^2-c^2\mathbf{q}^2\right) \\
& {}+ 2\hbar^2\omega^2\frac{\mathbf{p}_i^2\sin^2\Theta_i+\mathbf{p}_f^2\sin^2\Theta_f}{(E_f-c|\mathbf{p}_f|\cos\Theta_f)(E_i-c|\mathbf{p}_i|\cos\Theta_i)}
\\
& {}- 2\left.\frac{|\mathbf{p}_i||\mathbf{p}_f|\sin\Theta_i\sin\Theta_f\cos\Phi}{(E_f-c|\mathbf{p}_f|\cos\Theta_f)(E_i-c|\mathbf{p}_i|\cos\Theta_i)}\left(2E_i^2+2E_f^2-c^2\mathbf{q}^2\right)\right].
\end{align}
},
{
"math_input": \Sigma=\begin{bmatrix}
{\color{Red}Var \left (X_{1(1)} \right)} & {\color{OliveGreen}Cov \left (X_{1(1)},X_{1(2)} \right)} & Cov \left (X_{1(1)},X_{1(3)} \right) & \cdots & Cov \left (X_{1(1)},X_{1(n)} \right) \\
{\color{OliveGreen}Cov \left (X_{1(2)},X_{1(1)} \right)} & {\color{Turquoise}Var \left (X_{1(2)} \right)} & {\color{RubineRed}Cov \left (X_{1(2)},X_{1(3)} \right)} & \cdots & Cov \left (X_{1(2)},X_{1(k)} \right) \\
Cov \left (X_{1(3)},X_{1(1)} \right) & {\color{RubineRed}Cov \left (X_{1(3)},X_{1(2)} \right)} & Var \left (X_{1(3)} \right) & \cdots & Cov \left (X_{1(3)},X_{1(k)} \right) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
Cov \left (X_{1(k)},X_{1(1)} \right) & Cov \left (X_{1(k)},X_{1(2)} \right) & Cov \left (X_{1(k)},X_{1(3)} \right) & \cdots & Var \left (X_{1(k)} \right) \\
\end{bmatrix}.
},
{
"math_input":
\mathbf{J} = \begin{bmatrix}
\cfrac{\partial x_1}{\partial q^1} & \cfrac{\partial x_1}{\partial q^2} & \cfrac{\partial x_1}{\partial q^3} \\
\cfrac{\partial x_2}{\partial q^1} & \cfrac{\partial x_2}{\partial q^2} & \cfrac{\partial x_2}{\partial q^3} \\
\cfrac{\partial x_3}{\partial q^1} & \cfrac{\partial x_3}{\partial q^2} & \cfrac{\partial x_3}{\partial q^3} \\
\end{bmatrix},\quad
\mathbf{J}^{-1} = \begin{bmatrix}
\cfrac{\partial q^1}{\partial x_1} & \cfrac{\partial q^1}{\partial x_2} & \cfrac{\partial q^1}{\partial x_3} \\
\cfrac{\partial q^2}{\partial x_1} & \cfrac{\partial q^2}{\partial x_2} & \cfrac{\partial q^2}{\partial x_3} \\
\cfrac{\partial q^3}{\partial x_1} & \cfrac{\partial q^3}{\partial x_2} & \cfrac{\partial q^3}{\partial x_3} \\
\end{bmatrix}
},
{
"math_input":
\begin{align}
\frac{\partial}{\partial b}\int_a^b f(x)\; dx
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ \int_a^{b+\Delta b} f(x)\,dx - \int_a^b f(x)\,dx \right]
\\
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \int_b^{b+\Delta b} f(x)\,dx
= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ f(b)\, \Delta b + \mathcal{O}\left(\Delta b^2\right) \right]
\\
&= f(b)
\qquad \text{and}
\\
\frac{\partial}{\partial a}\int_a^b f(x)\; dx
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ \int_{a+\Delta a}^b f(x)\,dx - \int_a^b f(x)\,dx \right]
\\
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \int_{a+\Delta a}^a f(x)\,dx
= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ -f(a)\, \Delta a + \mathcal{O}\left(\Delta a^2\right) \right]
\\
&= -f(a).
\end{align}
},
{
"math_input":
\begin{align}
& \mathbf x_0 := \text{Some initial guess} \\
& \mathbf r_0 := \mathbf M^{-1}(\mathbf b - \mathbf{A x}_0) \\
& \mathbf p_0 := \mathbf r_0 \\
& \text{Iterate, with } k \text{ starting at } 0: \\
& \qquad \alpha_k := \frac{\mathbf r_k^\mathrm{T} \mathbf A \mathbf r_k}{(\mathbf{A p}_k)^\mathrm{T} \mathbf M^{-1} \mathbf{A p}_k} \\
& \qquad \mathbf x_{k+1} := \mathbf x_k + \alpha_k \mathbf{p}_k \\
& \qquad \mathbf r_{k+1} := \mathbf r_k - \alpha_k \mathbf M^{-1} \mathbf{A p}_k \\
& \qquad \beta_k := \frac{\mathbf r_{k + 1}^\mathrm{T} \mathbf A \mathbf r_{k + 1}}{\mathbf r_k^\mathrm{T} \mathbf A \mathbf r_k} \\
& \qquad \mathbf p_{k+1} := \mathbf r_{k+1} + \beta_k \mathbf{p}_k \\
& \qquad \mathbf{A p}_{k + 1} := \mathbf A \mathbf r_{k+1} + \beta_k \mathbf{A p}_k \\
& \qquad k := k + 1 \\
\end{align}
},
{
"math_input":
\begin{align}
&\eta_i=\sum F_i\ell^{(-\alpha)}\\
&\partial_j\eta_i=g_{ij}=\sum{\partial_i\ell^{(\alpha)}\partial_j\ell^{(-\alpha)}}=\sum F_i\partial_j\ell^{(-\alpha)}\\
&\Psi^{(\alpha\neq -1)}(\theta)=\frac{2}{1+\alpha}\sum p\\
&\Psi^{(\alpha=-1)}(\theta)=\sum p(\log p-1)\\
&\psi(\theta)=\Psi^{(\alpha)}\\
&\phi(\theta)=\Psi^{(-\alpha)}-\sum C(x)\ell^{(-\alpha)}\\
&D^{\alpha}(p||q)=\Psi^{(\alpha)}+\Psi^{(-\alpha)}-\sum\ell_p^{(\alpha)}\ell_q^{(-\alpha)}\\
&D^{\alpha\neq\pm 1}(p||q)=\frac{4}{1-\alpha^2}\sum\{\frac{1-\alpha}{2}p+\frac{1+\alpha}{2}q-p^{\frac{1-\alpha}{2}}q^{\frac{1+\alpha}{2}}\}\\
&D^{\alpha=\pm 1}(p||q)=\sum \{p-q+p\log\frac{p}{q}\}\\
&\theta^i\eta'_i=\sum\{\ell^{(\alpha)}(v;\theta)-C(v)\}\ell^{(-\alpha)}(v;\theta')\\
&D(\theta||\theta')=\psi(\theta)+\phi(\theta)-\theta^i\eta'_i
\end{align}
},
{
"math_input": \begin{align}
\mathbf{r} & =\mathbf{r}\left ( t \right ) = r \bold{\hat{e}}_r\\
\mathbf{v} & = v \bold{\hat{e}}_r + r\,\frac{{\rm d}\theta}{{\rm d}t}\bold{\hat{e}}_\theta + r\,\frac{{\rm d}\phi}{{\rm d}t}\,\sin\theta \bold{\hat{e}}_\phi \\
\mathbf{a} & = \left( a - r\left(\frac{{\rm d}\theta}{{\rm d}t}\right)^2 - r\left(\frac{{\rm d}\phi}{{\rm d}t}\right)^2\sin^2\theta \right)\bold{\hat{e}}_r \\
& + \left( r \frac{{\rm d}^2 \theta}{{\rm d}t^2 } + 2v\frac{{\rm d}\theta}{{\rm d}t} - r\left(\frac{{\rm d}\phi}{{\rm d}t}\right)^2\sin\theta\cos\theta \right) \bold{\hat{e}}_\theta \\
& + \left( r\frac{{\rm d}^2 \phi}{{\rm d}t^2 }\,\sin\theta + 2v\,\frac{{\rm d}\phi}{{\rm d}t}\,\sin\theta + 2 r\,\frac{{\rm d}\theta}{{\rm d}t}\,\frac{{\rm d}\phi}{{\rm d}t}\,\cos\theta \right) \bold{\hat{e}}_\phi
\end{align} \,\!
},
{
"math_input": \begin{matrix}
F_{\mathbf{K}} & = & f \left[
\begin{matrix}
e^{-i\mathbf{K}\cdot\vec{0}} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{y})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{y} + \hat{z})} + e^{-i\mathbf{K}\cdot(a/2)(\hat{x} + \hat{z})} + \\
e^{-i\mathbf{K}\cdot(a/4)(\hat{x} + \hat{y} + \hat{z})} +
e^{-i\mathbf{K}\cdot(a/4)(3\hat{x} + \hat{y} + 3\hat{z})} +
e^{-i\mathbf{K}\cdot(a/4)(3\hat{x} + 3\hat{y} + \hat{z})} +
e^{-i\mathbf{K}\cdot(a/4)(\hat{x} + 3\hat{y} + 3\hat{z})}
\end{matrix}
\right] \\
& = & f \left[
\begin{matrix}
1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} + \\
(-i)^{h + k + l} + (-i)^{3h + k + 3l} + (-i)^{3h + 3k + l} + (-i)^{h + 3k + 3l}
\end{matrix}
\right] \\
& = & f \left[ 1 + (-1)^{h + k} + (-1)^{k + l} + (-1)^{h + l} \right] \cdot \left[ 1 + (-i)^{h + k + l} \right]\\
\end{matrix}
},
{
"math_input": \overset{{{A}'}}{\mathop{\left[ \begin{matrix}
2 & 1 & -1 \\
-2 & 2 & 2 \\
-2 & 1 & 3
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c
\end{matrix} \right]=\left[ \begin{matrix}
a_1 \\
b_1 \\
c_1
\end{matrix} \right],\quad \text{ }\overset{{{B}'}}{\mathop{\left[ \begin{matrix}
2 & 1 & 1 \\
2 & -2 & 2 \\
2 & -1 & 3
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c \\
\end{matrix} \right]=\left[ \begin{matrix}
a_2 \\
b_2 \\
c_2
\end{matrix} \right],\quad \text{ }\overset{{{C}'}}{\mathop{\left[ \begin{matrix}
2 & -1 & 1 \\
2 & 2 & 2 \\
2 & 1 & 3 \\
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c \\
\end{matrix} \right]=\left[ \begin{matrix}
a_3 \\
b_3 \\
c_3
\end{matrix} \right]
},
{
"math_input": \begin{align}W' & =
\begin{vmatrix}
y'_1 & y'_2 & \cdots & y'_n\\
y'_1 & y'_2 & \cdots & y'_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y'''_1 & y'''_2 & \cdots & y'''_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)}
\end{vmatrix}
+
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y''_1 & y''_2 & \cdots & y''_n\\
y'''_1 & y'''_2 & \cdots & y'''_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)}
\end{vmatrix}\\
&\qquad+\ \cdots\ +
\begin{vmatrix}
y_1 & y_2 & \cdots & y_n\\
y'_1 & y'_2 & \cdots & y'_n\\
\vdots & \vdots & \ddots & \vdots\\
y_1^{(n-3)} & y_2^{(n-3)} & \cdots & y_n^{(n-3)}\\
y_1^{(n-2)} & y_2^{(n-2)} & \cdots & y_n^{(n-2)}\\
y_1^{(n)} & y_2^{(n)} & \cdots & y_n^{(n)}
\end{vmatrix}.\end{align}
},
{
"math_input": \begin{align}
E_{KL}&=\frac{1}{2}\left( \frac{\partial x_j}{\partial X_K}\frac{\partial x_j}{\partial X_L}-\delta_{KL}\right) \\
&=\frac{1}{2}\left[\delta_{jM}\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\delta_{jN}\left(\frac{\partial U_N}{\partial X_L}+\delta_{NL}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left[\delta_{MN}\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\left(\frac{\partial U_N}{\partial X_L}+\delta_{NL}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left[\left(\frac{\partial U_M}{\partial X_K}+\delta_{MK}\right)\left(\frac{\partial U_M}{\partial X_L}+\delta_{ML}\right)-\delta_{KL}\right] \\
&=\frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}+\frac{\partial U_M}{\partial X_K}\frac{\partial U_M}{\partial X_L}\right)
\end{align}\,\!
},
{
"math_input": \overset{A}{\mathop{\left[ \begin{matrix}
-1 & 2 & 2 \\
-2 & 1 & 2 \\
-2 & 2 & 3 \\
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c \\
\end{matrix} \right]=\left[ \begin{matrix}
a_1 \\
b_1 \\
c_1 \\
\end{matrix} \right],\quad \text{ }\overset{B}{\mathop{\left[ \begin{matrix}
1 & 2 & 2 \\
2 & 1 & 2 \\
2 & 2 & 3 \\
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c \\
\end{matrix} \right]=\left[ \begin{matrix}
a_2 \\
b_2 \\
c_2
\end{matrix} \right],\quad \text{ }\overset{C}{\mathop{\left[ \begin{matrix}
1 & -2 & 2 \\
2 & -1 & 2 \\
2 & -2 & 3
\end{matrix} \right]}} \left[ \begin{matrix}
a \\
b \\
c
\end{matrix} \right]=\left[ \begin{matrix}
a_3 \\
b_3 \\
c_3
\end{matrix} \right]
},
{
"math_input":
\begin{align}
\cfrac{\mathrm{d}}{\mathrm{d}t}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & =
\int_{\Omega_0} \left(
\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right) ~\text{dV}_0 \\
& =
\int_{\Omega_0}
\left(\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]+
\hat{\mathbf{f}}(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~J(\mathbf{X},t) ~\text{dV}_0 \\
& =
\int_{\Omega(t)}
\left(\dot{\mathbf{f}}(\mathbf{x},t)+
\mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV}
\end{align}
},
{
"math_input":
\begin{align}
\omega_\phi &= {1 \over r}\left(-\frac{1}{\sin\theta}\left({\partial \over \partial r} \left(\frac{\partial\Psi}{\partial r}\right)\right) -
\frac{1}{r^2 }{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right) \boldsymbol{\hat \phi} \\
&= {1 \over r}\left(-\frac{1}{\sin\theta}\left(\frac{\partial^2\Psi}{\partial r^2}\right) -
\frac{\sin\theta}{r^2 \sin\theta}{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right) \boldsymbol{\hat \phi} \\
&= -\frac{1}{r\sin\theta} \left(\frac{\partial^2\Psi}{\partial r^2} + \frac{\sin\theta}{r^2}{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right) \boldsymbol{\hat \phi} \\
\end{align}
},
{
"math_input":
Y_{V} = \begin{pmatrix} \begin{pmatrix}
q_{1,1,1} d_{1,1} v_{1,1}^{\alpha_{1,1,1}-1} x_{1}^{\beta_{1,1,1}} \\ \vdots \\ q_{1,1,i} d_{1,1} v_{1,1}^{\alpha_{1,1,i}-1} x_{i}^{\beta_{1,1,i}}
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{1,n,1} d_{1,n} v_{1,n}^{\alpha_{1,n,1}-1} x_{1}^{\beta_{1,n,1}} \\ \vdots \\ q_{1,n,i} d_{1,n} v_{1,n}^{\alpha_{1,n,i}-1} x_{i}^{\beta_{1,n,i}}
\end{pmatrix} \\ \vdots & \ddots & \vdots \\ \begin{pmatrix}
q_{j,1,1} d_{j,1} v_{j,1}^{\alpha_{j,1,1}-1} x_{1}^{\beta_{j,1,1}} \\ \vdots \\ q_{j,1,i} d_{j,1} v_{j,1}^{\alpha_{j,1,i}-1} x_{i}^{\beta_{j,1,i}}
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{j,n,1} d_{j,n} v_{j,n}^{\alpha_{j,n,1}-1} x_{1}^{\beta_{j,n,1}} \\ \vdots \\ q_{j,n,i} d_{j,n} v_{j,n}^{\alpha_{j,n,i}-1} x_{i}^{\beta_{j,n,i}}
\end{pmatrix} \end{pmatrix}
},
{
"math_input": \begin{align}
\Pr\left( \|X - \mu\|_\alpha \ge k \sigma_\alpha \right) &= \int_\Omega \mathbf{1}_{\|X - \mu\|_\alpha \ge k \sigma_\alpha} \, \mathrm d\Pr \\
& = \int_\Omega \left ( \frac{\|X - \mu\|_\alpha^2}{\|X - \mu\|_\alpha^2} \right ) \cdot \mathbf{1}_{\|X - \mu\|_\alpha \ge k \sigma_\alpha} \, \mathrm d\Pr \\
&\le \int_\Omega \left (\frac{\|X - \mu\|_\alpha^2}{(k\sigma_\alpha)^2} \right ) \cdot \mathbf{1}_{\|X - \mu\|_\alpha \ge k \sigma_\alpha} \, \mathrm d\Pr \\
&\le \frac{1}{k^2 \sigma_\alpha^2} \int_\Omega \|X - \mu\|_\alpha^2 \, \mathrm d\Pr && \mathbf{1}_{\|X - \mu\|_\alpha \ge k \sigma_\alpha} \le 1\\
&= \frac{1}{k^2 \sigma_\alpha^2} \left (\operatorname{E}\|X - \mu\|_\alpha^2 \right )\\
&= \frac{1}{k^2 \sigma_\alpha^2} \left (\sigma_\alpha^2 \right )\\
&= \frac{1}{k^2}
\end{align}
},
{
"math_input":
\begin{matrix}
& X & & & U & & \Sigma & & V^T \\
& (\textbf{d}_j) & & & & & & & (\hat{\textbf{d}}_j) \\
& \downarrow & & & & & & & \downarrow \\
(\textbf{t}_i^T) \rightarrow
&
\begin{bmatrix}
x_{1,1} & \dots & x_{1,n} \\
\\
\vdots & \ddots & \vdots \\
\\
x_{m,1} & \dots & x_{m,n} \\
\end{bmatrix}
&
=
&
(\hat{\textbf{t}}_i^T) \rightarrow
&
\begin{bmatrix}
\begin{bmatrix} \, \\ \, \\ \textbf{u}_1 \\ \, \\ \,\end{bmatrix}
\dots
\begin{bmatrix} \, \\ \, \\ \textbf{u}_l \\ \, \\ \, \end{bmatrix}
\end{bmatrix}
&
\cdot
&
\begin{bmatrix}
\sigma_1 & \dots & 0 \\
\vdots & \ddots & \vdots \\
0 & \dots & \sigma_l \\
\end{bmatrix}
&
\cdot
&
\begin{bmatrix}
\begin{bmatrix} & & \textbf{v}_1 & & \end{bmatrix} \\
\vdots \\
\begin{bmatrix} & & \textbf{v}_l & & \end{bmatrix}
\end{bmatrix}
\end{matrix}
},
{
"math_input": \begin{array}{ll}
{\rm (MTL1)}\colon & (A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C)) \\
{\rm (MTL2)}\colon & A \otimes B \rightarrow A\\
{\rm (MTL3)}\colon & A \otimes B \rightarrow B \otimes A\\
{\rm (MTL4a)}\colon & A \wedge B \rightarrow A\\
{\rm (MTL4b)}\colon & A \wedge B \rightarrow B \wedge A\\
{\rm (MTL4c)}\colon & A \otimes (A \rightarrow B) \rightarrow A \wedge B\\
{\rm (MTL5a)}\colon & (A \rightarrow (B \rightarrow C)) \rightarrow (A \otimes B \rightarrow C)\\
{\rm (MTL5b)}\colon & (A \otimes B \rightarrow C) \rightarrow (A \rightarrow (B \rightarrow C))\\
{\rm (MTL6)}\colon & ((A \rightarrow B) \rightarrow C) \rightarrow (((B \rightarrow A) \rightarrow C) \rightarrow C)\\
{\rm (MTL7)}\colon & \bot \rightarrow A
\end{array}
},
{
"math_input": \begin{align}
\frac{\partial}{\partial b} \left (\int_a^b f(x)\; dx \right ) &= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ \int_a^{b+\Delta b} f(x)\,dx - \int_a^b f(x)\,dx \right] \\
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \int_b^{b+\Delta b} f(x)\,dx \\
&= \lim_{\Delta b \to 0} \frac{1}{\Delta b} \left[ f(b) \Delta b + \mathcal{O}\left(\Delta b^2\right) \right] \\
&= f(b) \\
\frac{\partial}{\partial a} \left (\int_a^b f(x)\; dx \right )&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ \int_{a+\Delta a}^b f(x)\,dx - \int_a^b f(x)\,dx \right] \\
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \int_{a+\Delta a}^a f(x)\,dx \\
&= \lim_{\Delta a \to 0} \frac{1}{\Delta a} \left[ -f(a)\, \Delta a + \mathcal{O}\left(\Delta a^2\right) \right]\\
&= -f(a).
\end{align}
},
{
"math_input":
\Pr\begin{cases}
Ds\begin{cases}
Sp(\pi)\begin{cases}
Va:\\
S^{0},\ldots,S^{T},O^{0},\ldots,O^{T}\\
Dc:\\
\begin{cases}
& P\left(S^{0}\wedge\cdots\wedge O^{T}\mid\pi\right)\\
= & \left[\begin{array}{c}
P\left(S^{0}\wedge O^{0}\mid\pi\right)\\
\prod_{t=1}^{T}\left[P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\times P\left(O^{t}\mid S^{t}\wedge\pi\right)\right]\end{array}\right]\end{cases}\\
Fo:\\
\begin{cases}
P\left(S^{0}\wedge O^{0}\mid\pi\right)\equiv \text{Matrix}\\
P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\equiv \text{Matrix}\\
P\left(O^{t}\mid S^{t}\wedge\pi\right)\equiv \text{Matrix}\end{cases}\end{cases}\\
Id\end{cases}\\
Qu:\\
\max_{S^{1}\wedge\cdots\wedge S^{T-1}}\left[P\left(S^{1}\wedge\cdots\wedge S^{T-1}\mid S^{T}\wedge O^{0}\wedge\cdots\wedge O^{T}\wedge\pi\right)\right]\end{cases}
},
{
"math_input": \begin{align}
\operatorname{E}\biggl[\frac{\operatorname{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]}{UV}1_G\biggr]
&=\operatorname{E}\biggl[\operatorname{E}\biggl[\frac{|XY|}{UV}1_G\bigg|\,\mathcal{G}\biggr]\biggr]\\
&=\operatorname{E}\biggl[\frac{|X|}{U}1_G\cdot\frac{|Y|}{V}1_G\biggr]\\
&\le\biggl(\operatorname{E}\biggl[\frac{|X|^p}{U^p}1_G\biggr]\biggr)^{\!1/p\;}
\biggl(\operatorname{E}\biggl[\frac{|Y|^q}{V^q}1_G\biggr]\biggr)^{\!1/q}\\
&=\biggl(\operatorname{E}\biggl[\underbrace{\frac{\operatorname{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]}{U^p}}_{=\,1\text{ a.s. on }G}1_G\biggr]\biggr)^{\!1/p\;}
\biggl(\operatorname{E}\biggl[\underbrace{\frac{\operatorname{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]}{V^p}}_{=\,1\text{ a.s. on }G}1_G\biggr]\biggr)^{\!1/q}\\
&=\operatorname{E}\bigl[1_G\bigr].
\end{align}
},
{
"math_input":
\begin{align}
\frac{\partial X^\mu}{\partial x^m}\,_{(x)}\Gamma^m_{ij} & = \frac{\partial X^\mu}{\partial x^m}~\frac{\partial x^m}{\partial X^\nu}~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\frac{\partial X^\mu}{\partial x^m}~\frac{\partial x^m}{\partial X^\alpha}~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j} \\
& = \delta^\mu_\nu~\frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\nu_{\alpha\beta} +
\delta^\mu_\alpha~\frac{\partial^2 X^\alpha}{\partial x^i \partial x^j} \\
& = \frac{\partial X^\alpha}{\partial x^i}~\frac{\partial X^\beta}{\partial x^j} \,_{(X)}\Gamma^\mu_{\alpha\beta} + \frac{\partial^2 X^\mu}{\partial x^i \partial x^j}
\end{align}
},
{
"math_input": \begin{align} g(x) &= \frac{1}{\sqrt{2 \pi \hbar}} \cdot \int_{-\infty}^{\infty} \tilde{g}(p) \cdot e^{ipx/\hbar} \, dp \\
&= \frac{1}{\sqrt{2 \pi \hbar}} \int_{-\infty}^{\infty} p \cdot \phi(p) \cdot e^{ipx/\hbar} \, dp \\
&= \frac{1}{2 \pi \hbar} \int_{-\infty}^{\infty} \left[ p \cdot \int_{-\infty}^{\infty} \psi(x) e^{-ipx/\hbar} \, dx \right] \cdot e^{ipx/\hbar} \, dp \\
&= \frac{i}{2 \pi} \int_{-\infty}^{\infty} \left[ \cancel{ \left. \psi(x) e^{-ipx/\hbar} \right|_{-\infty}^{\infty} } - \int_{-\infty}^{\infty} \frac{d\psi(x)}{dx} e^{-ipx/\hbar} \, dx \right] \cdot e^{ipx/\hbar} \, dp \\
&= \frac{-i}{2 \pi} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{d\psi(x)}{dx} e^{-ipx/\hbar} \, dx \, e^{ipx/\hbar} \, dp \\
&= \left( -i \hbar \frac{d}{dx} \right) \cdot \psi(x) ,\end{align}
},
{
"math_input": \begin{align}
\rho \left(\frac{\partial u_r}{\partial t} + u_r \frac{\partial u_r}{\partial r} + u_z \frac{\partial u_r}{\partial z}\right)
&= -\frac{\partial p}{\partial r} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_r}{\partial r}\right) +
\frac{\partial^2 u_r}{\partial z^2} - \frac{u_r}{r^2}\right] + \rho g_r \\
\rho \left(\frac{\partial u_z}{\partial t} + u_r \frac{\partial u_z}{\partial r} + u_z \frac{\partial u_z}{\partial z}\right)
&= -\frac{\partial p}{\partial z} + \mu \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial u_z}{\partial r}\right) +
\frac{\partial^2 u_z}{\partial z^2}\right] + \rho g_z \\
\frac{1}{r}\frac{\partial}{\partial r}\left(r u_r\right) + \frac{\partial u_z}{\partial z} &= 0.
\end{align}
},
{
"math_input":
\begin{align}
\varphi\, & =\,
\left\{\,
\varphi_b\,
-\, \frac{1}{2}\, z^2\, \frac{\partial^2 \varphi_b}{\partial x^2}\,
+\, \frac{1}{24}\, z^4\, \frac{\partial^4 \varphi_b}{\partial x^4}\,
+\, \cdots\,
\right\}\,
+\,
\left\{\,
z\, \left[ \frac{\partial \varphi}{\partial z} \right]_{z=-h}\,
-\, \frac{1}{6}\, z^3\, \frac{\partial^2}{\partial x^2}\, \left[ \frac{\partial \varphi}{\partial z} \right]_{z=-h}\,
+\, \cdots\,
\right\}
\\
& =\,
\left\{\,
\varphi_b\,
-\, \frac{1}{2}\, z^2\, \frac{\partial^2 \varphi_b}{\partial x^2}\,
+\, \frac{1}{24}\, z^4\, \frac{\partial^4 \varphi_b}{\partial x^4}\,
+\, \cdots\,
\right\},
\end{align}
},
{
"math_input":
\begin{align}
\nabla \times \mathbf F & =
\frac{\hat{ \mathbf e}_1}{h_2 h_3}
\left[
\frac{\partial}{\partial q^2} \left( h_3 F_3 \right) -
\frac{\partial}{\partial q^3} \left( h_2 F_2 \right)
\right] +
\frac{\hat{ \mathbf e}_2}{h_3 h_1}
\left[
\frac{\partial}{\partial q^3} \left( h_1 F_1 \right) -
\frac{\partial}{\partial q^1} \left( h_3 F_3 \right)
\right] \\[10pt]
& + \frac{\hat{ \mathbf e}_3}{h_1 h_2}
\left[
\frac{\partial}{\partial q^1} \left( h_2 F_2 \right) -
\frac{\partial}{\partial q^2} \left( h_1 F_1 \right)
\right]
=\frac{1}{h_1 h_2 h_3}
\begin{vmatrix}
h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3 \\
\dfrac{\partial}{\partial q^1} & \dfrac{\partial}{\partial q^2} & \dfrac{\partial}{\partial q^3} \\
h_1 F_1 & h_2 F_2 & h_3 F_3
\end{vmatrix}
\end{align}
},
{
"math_input": \begin{array}{rll}
\text{As found above,} &\log_{10}0.012\approx\bar{2}.079181 \\
\text{Since}\;\;\log_{10}0.85&=\log_{10}(10^{-1}\times 8.5)=-1+\log_{10}8.5&\approx-1+0.929419=\bar{1}.929419\;, \\
\log_{10}(0.012\times 0.85) &=\log_{10}0.012+\log_{10}0.85 &\approx\bar{2}.079181+\bar{1}.929419 \\
&=(-2+0.079181)+(-1+0.929419) &=-(2+1)+(0.079181+0.929419) \\
&=-3+1.008600 &=-2+0.008600\;^* \\
&\approx\log_{10}(10^{-2})+\log_{10}(1.02) &=\log_{10}(0.01\times 1.02) \\
&=\log_{10}(0.0102)
\end{array}
},
{
"math_input":
\begin{array}{lll}
& LAG_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 4 & . & . & . & . & . \\
. & . & 9 & . & . & . & . \\
. & . & . & 16 & . & . & . \\
. & . & . & . & 25 & . & . \\
. & . & . & . & . & 36 & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & . & . & . & . & . & . \\
1 & 1 & . & . & . & . & . \\
2 & 4 & 1 & . & . & . & . \\
6 & 18 & 9 & 1 & . & . & . \\
24 & 96 & 72 & 16 & 1 & . & . \\
120 & 600 & 600 & 200 & 25 & 1 & . \\
720 & 4320 & 5400 & 2400 & 450 & 36 & 1
\end{smallmatrix}
\right ]
;\quad
\end{array}
},
{
"math_input":
\begin{array}{c|c|c|c}
\textit{Model} & \textit{Info-Gap\ Format} & \textit{MP\ Format} & \textit{Classical\ Format} \\
\hline
\textit{Robustness} &\displaystyle \max\{\alpha: r_{c}\le \min_{u\in \mathcal{U}(\alpha,\tilde{u})} R(q,u)\} &\displaystyle \displaystyle \max\{\alpha: \alpha \le \min_{u\in \mathcal{U}(\alpha,\tilde{u})}\varphi(q,\alpha,u)\} & \displaystyle \max_{\alpha\ge 0}\ \min_{u\in \mathcal{U}(\alpha,\tilde{u})}\ \varphi(q,\alpha,u) \\
\textit{Opportuneness} &\displaystyle \min\{\alpha: r_{c}\le \max_{u\in \mathcal{U}(\alpha,\tilde{u})} R(q,u)\} &\displaystyle \displaystyle \min\{\alpha: \alpha \ge \min_{u\in \mathcal{U}(\alpha,\tilde{u})}\psi(q,\alpha,u)\} & \displaystyle \min_{\alpha\ge 0}\ \min_{u\in \mathcal{U}(\alpha,\tilde{u})}\ \psi(q,\alpha,u)
\end{array}
},
{
"math_input": \begin{align}
&\lim_{\alpha\to 0} \gamma_1 =\lim_{\mu\to 0} \gamma_1 = \infty\\
&\lim_{\beta\to 0} \gamma_1 =\lim_{\mu\to 1} \gamma_1= - \infty\\
&\lim_{\alpha\to \infty} \gamma_1 = - \frac{2}{\beta},\text{ } \lim_{\beta \to 0}(\lim_{\alpha\to \infty} \gamma_1) = - \infty,\text{ } \lim_{\beta \to \infty}(\lim_{\alpha\to \infty} \gamma_1) = 0\\
&\lim_{\beta\to \infty} \gamma_1 = \frac{2}{\alpha},\text{ } \lim_{\alpha \to 0}(\lim_{\beta \to \infty} \gamma_1) = \infty,\text{ } \lim_{\alpha \to \infty}(\lim_{\beta \to \infty} \gamma_1) = 0\\
&\lim_{\nu \to 0} \gamma_1 = \frac{1 - 2 \mu}{\sqrt{\mu (1-\mu)}},\text{ } \lim_{\mu \to 0}(\lim_{\nu \to 0} \gamma_1) = \infty,\text{ } \lim_{\mu \to 1}(\lim_{\nu \to 0} \gamma_1) = - \infty\\
\end{align}
},
{
"math_input": \begin{align} \langle \chi | \psi \rangle
& = \left ( \int \langle \chi | \phi \rangle \langle \phi | \mathrm{d}\phi \right ) \left ( \int | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi \right ) \\
& = \iint \langle \chi | \phi \rangle | \phi \rangle \langle \phi | \langle \phi | \psi \rangle \mathrm{d}^2\phi \\
& = \left ( \int | \phi \rangle \langle \phi | \mathrm{d}\phi \right ) \left ( \int \langle \chi | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi \right ) \\
& = 1 \left ( \int \langle \chi | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi \right ) \\
& = \int \langle \chi | \phi \rangle \langle \phi | \psi \rangle \mathrm{d}\phi \\
& = \int z(\phi)^{*} c(\phi) \mathrm{d}\phi \\
& = \int \chi(\phi)^{*} \psi(\phi) \mathrm{d}\phi \\
\end{align} \,\!
},
{
"math_input":
Z=
\begin{bmatrix}
1 & 1 & \cdots & 1 \\
y_{p-1} & y_{p} & \cdots & y_{T-1}\\
y_{p-2} & y_{p-1} & \cdots & y_{T-2}\\
\vdots & \vdots & \ddots & \vdots\\
y_{0} & y_{1} & \cdots & y_{T-p}
\end{bmatrix} =
\begin{bmatrix}
1 & 1 & \cdots & 1 \\
y_{1,p-1} & y_{1,p} & \cdots & y_{1,T-1} \\
y_{2,p-1} & y_{2,p} & \cdots & y_{2,T-1} \\
\vdots & \vdots & \ddots & \vdots\\
y_{k,p-1} & y_{k,p} & \cdots & y_{k,T-1} \\
y_{1,p-2} & y_{1,p-1} & \cdots & y_{1,T-2} \\
y_{2,p-2} & y_{2,p-1} & \cdots & y_{2,T-2} \\
\vdots & \vdots & \ddots & \vdots\\
y_{k,p-2} & y_{k,p-1} & \cdots & y_{k,T-2} \\
\vdots & \vdots & \ddots & \vdots\\
y_{1,0} & y_{1,1} & \cdots & y_{1,T-p} \\
y_{2,0} & y_{2,1} & \cdots & y_{2,T-p} \\
\vdots & \vdots & \ddots & \vdots\\
y_{k,0} & y_{k,1} & \cdots & y_{k,T-p}
\end{bmatrix}
},
{
"math_input":
\sigma _{\hat g}^2 \,\,\, \approx \,\,\,\,
\begin{pmatrix}
{{\partial \hat g} \over {\partial L}} & {{\partial \hat g} \over {\partial T}} & {{\partial \hat g} \over {\partial \theta }}
\end{pmatrix}
\begin{pmatrix}
{\sigma _L^2 } & 0 & 0 \\
0 & {\sigma _T^2 } & 0 \\
0 & 0 & {\sigma _\theta ^2 }
\end{pmatrix}
\begin{pmatrix}
{{{\partial \hat g} \over {\partial L}}} \\
{{{\partial \hat g} \over {\partial T}}} \\
{{{\partial \hat g} \over {\partial \theta }}}
\end{pmatrix}\,=
\,\left( {{{\partial \hat g} \over {\partial L}}} \right)^2 \sigma _L^2 \,\,\, + \,\,\,\left( {{{\partial \hat g} \over {\partial T}}} \right)^2 \sigma _T^2 \,\,\, + \,\,\,\left( {{{\partial \hat g} \over {\partial \theta }}} \right)^2 \sigma _\theta ^2
{\mathbf{\,\,\,\,\,\,\,\,Eq(17)}}
},
{
"math_input":
\begin{align}
\sigma_{xx}^{\mathrm{topface}} & = -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z - h - \tfrac{f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}
& = & -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} + \left(\tfrac{2h+f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}\\
\sigma_{xx}^{\mathrm{botface}} & = -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_b}{\mathrm{d} x^2} -\left(z + h + \tfrac{f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}
& = & -z~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} - \left(\tfrac{2h+f}{2}\right)~C_{11}^{\mathrm{face}}~\cfrac{\mathrm{d}^2 w_s}{\mathrm{d} x^2}
\end{align}
},
{
"math_input": {}_{
+\sqrt[N-2]{\frac{b}{a}}\sum^{N-3}_{q=1}\frac{\Gamma\left(\frac{2q-1}{N-2}+q\right)}{\Gamma\left(\frac{2q-1}{N-2}+1\right)}\cdot\left(-\frac{c}{b}\sqrt[N-2]{\frac{a^2}{b^2}}\right)^q\cdot\frac{e^{\frac{2n\left(1-2q\right)}{N-2}\pi{\rm{i}}}}{q!}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{Nq-1}{N\left(N-2\right)},\frac{Nq-1}{N\left(N-2\right)}+\frac{1}{N},\frac{Nq-1}{N\left(N-2\right)}+\frac{2}{N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-3}{2N},\frac{Nq-1}{N\left(N-2\right)}+\frac{N+1}{2N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-1}{N};\\[8pt]
\frac{q+1}{N-2},\frac{q+2}{N-2},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{q+N-2}{N-2},\frac{2q+2N-5}{2N-4};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix},n=1,2,\cdots,N-2
}
},
{
"math_input":
\begin{align}
d^4\sigma &=
\frac{Z^2\alpha_{fine}^3c^2}{(2\pi)^2\hbar}|\mathbf{p}_+||\mathbf{p}_-|
\frac{dE_+}{\omega^3}\frac{d\Omega_+ d\Omega_- d\Phi}{|\mathbf{q}|^4}\times \\
&\times\left[-
\frac{\mathbf{p}_-^2\sin^2\Theta_-}{(E_--c|\mathbf{p}_-|\cos\Theta_-)^2}\left
(4E_+^2-c^2\mathbf{q}^2\right)\right.\\
&-\frac{\mathbf{p}_+^2\sin^2\Theta_+}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)^2}\left
(4E_-^2-c^2\mathbf{q}^2\right) \\
&+2\hbar^2\omega^2\frac{\mathbf{p}_+^2\sin^2\Theta_++\mathbf{p}_-^2\sin^2\Theta_-}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)} \\
&+2\left.\frac{|\mathbf{p}_+||\mathbf{p}_-|\sin\Theta_+\sin\Theta_-\cos\Phi}{(E_+-c|\mathbf{p}_+|\cos\Theta_+)(E_--c|\mathbf{p}_-|\cos\Theta_-)}\left(2E_+^2+2E_-^2-c^2\mathbf{q}^2\right)\right]. \\
\end{align}
},
{
"math_input":
\begin{align}
\cfrac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & =
\int_{\Omega_0} \left(
\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right) ~\text{dV}_0 \\
& =
\int_{\Omega_0}
\left(\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]+
\hat{\mathbf{f}}(\mathbf{X},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~J(\mathbf{X},t) ~\text{dV}_0 \\
& =
\int_{\Omega(t)}
\left(\dot{\mathbf{f}}(\mathbf{x},t)+
\mathbf{f}(\mathbf{x},t)~\boldsymbol{\nabla} \cdot \mathbf{v}(\mathbf{x},t)\right)~\text{dV}
\end{align}
},
{
"math_input":
\begin{align}
w & = w^K + \frac{\mathcal{M}^K}{\kappa G h}\left(1 - \frac{\mathcal{B} c^2}{2}\right)
- \Phi + \Psi \\
\varphi_1 & = - \frac{\partial w^K}{\partial x_1}
- \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_1^K
+ \frac{\partial }{\partial x_1}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
+ \frac{1}{c^2}\frac{\partial \Omega}{\partial x_2} \\
\varphi_2 & = - \frac{\partial w^K}{\partial x_2}
- \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_2^K
+ \frac{\partial }{\partial x_2}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
+ \frac{1}{c^2}\frac{\partial \Omega}{\partial x_1}
\end{align}
},
{
"math_input":
\begin{align}
\Omega_1(t) &= \int_0^t A(t_1)\,dt_1, \\
\Omega_2(t) &= \frac{1}{2}\int_0^t dt_1 \int_0^{t_1} dt_2\ \left[ A(t_1),A(t_2)\right], \\
\Omega_3(t) &= \frac{1}{6} \int_0^t dt_1 \int_0^{t_1}d t_2 \int_0^{t_2} dt_3
\Bigl(\left[A(t_1),\left[A(t_2),A(t_3)\right]\right]+\left[A(t_3),\left[ A(t_2),A(t_{1})\right]\right]\Bigr), \\
\Omega_4(t) &= \frac{1}{12} \int_0^t dt_1 \int_0^{t_1}d t_2 \int_0^{t_2} dt_3 \int_0^{t_3} dt_4
\Bigl(\left[\left[\left[A_1,A_2\right],A_3\right],A_4\right] \\
&\quad+\left[A_1,\left[\left[A_2,A_3\right],A_4\right]\right]
+\left[A_1,\left[A_2,\left[A_3,A_4\right]\right]\right]
+\left[A_2,\left[A_3,\left[A_4,A_1\right]\right]\right]\Bigr)
\end{align}
},
{
"math_input":
\begin{array}{lllll}
A_2=\begin{pmatrix}
1 & 1/2 \\
1/2 & 1
\end{pmatrix};
&
A_2^{-1}=\begin{pmatrix}
4/3 & -2/3 \\
-2/3 & {\color{BrickRed}\mathbf{4/3}}
\end{pmatrix};
\\
\\
A_3=\begin{pmatrix}
1 & 1/2 & 1/3 \\
1/2 & 1 & 2/3 \\
1/3 & 2/3 & 1
\end{pmatrix};
&
A_3^{-1}=\begin{pmatrix}
4/3 & -2/3 & \\
-2/3 & 32/15 & -6/5 \\
& -6/5 & {\color{BrickRed}\mathbf{9/5}}
\end{pmatrix};
\\
\\
A_4=\begin{pmatrix}
1 & 1/2 & 1/3 & 1/4 \\
1/2 & 1 & 2/3 & 1/2 \\
1/3 & 2/3 & 1 & 3/4 \\
1/4 & 1/2 & 3/4 & 1
\end{pmatrix};
&
A_4^{-1}=\begin{pmatrix}
4/3 & -2/3 & & \\
-2/3 & 32/15 & -6/5 & \\
& -6/5 & 108/35 & -12/7 \\
& & -12/7 & {\color{BrickRed}\mathbf{16/7}}
\end{pmatrix}.
\\
\end{array}
},
{
"math_input": \begin{align}
y_{i+1}^{j} &= y_i^{j} -\frac{\delta_{ij}}{q_i} \\ \\
\sum_{i=1}^m y_i^{j} &= (k_{1}^j- k_{0}^j) + (k_{2}^j- k_{1}^j) + \cdots + (k_{m-1}^j- k_{m-2}^j) + (k_{m}^j- k_{m-1}^j) \\
&= k_{m}^j - k_{0}^j \\
\sum_{i=1}^m y_i^{j} &= k_{m}^j \\ \\
y_1^{j} &= (k_{1}^j- k_{0}^j) = k_{1}^j \\
y_2^{j} &= y_1^{j} -\frac{\delta_{1j}}{q_1} = k_1^{j} -\frac{\delta_{1j}}{ q_1 } \\
y_3^{j} &= k_1^{j} -\frac{\delta_{1j}}{q_1} -\frac{\delta_{2j}}{ q_2 } \\
& \vdots \\
y_i^{j} &= k_1^{j} -\sum_{r=1}^{i-1} \frac{\delta_{rj}}{q_r} = \begin{cases} k_1^j & j \geq i\\ k_1^j - \frac{1}{q_j} & j \leq i \end{cases} \\ \\
k_i^j &= \sum_{m=1}^i y_m^{j} = \begin{cases} i \cdot k_1^j & j \geq i\\ i \cdot k_1^j - \frac{i-j}{q_j} & j \leq i \end{cases}
\end{align}
},
{
"math_input":
\begin{align}
I(F_a;C) & = \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i,c_j) \log \frac{p(v_i|c_j)}{p(v_i)} \\
& = \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \left [\log p(v_i|c_j)- \log p(v_i) \right ] \\
& = \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \log p(v_i|c_j)- \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \log p(v_i) \\
& = \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \log p(v_i|c_j)- \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i,c_j) \log p(v_i) \\
& = \sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \log p(v_i|c_j)- \sum_{v_i \in F_a} \log p(v_i) \sum_{c_j \in C} p(v_i,c_j) \\
& = {\color{Blue}\sum_{v_i \in F_a} \sum_{c_j \in C} p(v_i|c_j)p(c_j) \log p(v_i|c_j)- \sum_{v_i \in F_a} p(v_i) \log p(v_i)} \\
\end{align}
},
{
"math_input": \begin{align} &{}\quad {\partial^3 \over \partial x_1\,\partial x_2\,\partial x_3} (uv) \\ \\
&{}= u \cdot{\partial^3 v \over \partial x_1\,\partial x_2\,\partial x_3} + {\partial u \over \partial x_1}\cdot{\partial^2 v \over \partial x_2\,\partial x_3} + {\partial u \over \partial x_2}\cdot{\partial^2 v \over \partial x_1\,\partial x_3} + {\partial u \over \partial x_3}\cdot{\partial^2 v \over \partial x_1\,\partial x_2} \\ \\
&{}\qquad + {\partial^2 u \over \partial x_1\,\partial x_2}\cdot{\partial v \over \partial x_3}
+ {\partial^2 u \over \partial x_1\,\partial x_3}\cdot{\partial v \over \partial x_2}
+ {\partial^2 u \over \partial x_2\,\partial x_3}\cdot{\partial v \over \partial x_1}
+ {\partial^3 u \over \partial x_1\,\partial x_2\,\partial x_3}\cdot v. \end{align}
},
{
"math_input": \begin{align}
& \mathbb{E} ( Y - h_1(X) )^2 = \int_0^1 \Big( y - h_1 ( f_1(x) ) \Big)^2 \, \mathrm{d}y = \\
& \int_0^{1/3} (y-h_1(3y))^2 \, \mathrm{d}y + \int_{1/3}^{2/3} \Big( y - h_1( 1.5(1-y) ) \Big)^2 \, \mathrm{d}y + \int_{2/3}^1 \Big( y - h_1(0.5) \Big)^2 \, \mathrm{d}y = \\
& \int_0^1 \Big( \frac x 3 - h_1(x) \Big)^2 \frac{ \mathrm{d}x }{ 3 } + \int_{0.5}^1 \Big( 1 - \frac{x}{1.5} - h_1(x) \Big)^2 \frac{ \mathrm{d} x }{ 1.5 } + \frac13 h_1^2(0.5) - \frac 5 9 h_1(0.5) + \frac{19}{81} = \\
& \frac13 \int_0^{0.5} \Big( h_1(x) - \frac x 3 \Big)^2 \, \mathrm{d}x + \frac13 h_1^2(0.5) - \frac 5 9 h_1(0.5) + \frac{19}{81} + \\
& \quad \frac13 \int_{0.5}^1 \bigg( \Big( h_1(x) - \frac x 3 \Big)^2 + 2 \Big( h_1(x) - 1 + \frac{2x}3 \Big)^2 \bigg) \, \mathrm{d}x \, ;
\end{align}
},
{
"math_input": H(f,g) = \begin{bmatrix}
0 & \dfrac{\partial g}{\partial x_1} & \dfrac{\partial g}{\partial x_2} & \cdots & \dfrac{\partial g}{\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_1} & \dfrac{\partial^2 f}{\partial x_1^2} & \dfrac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_1\,\partial x_n} \\[2.2ex]
\dfrac{\partial g}{\partial x_2} & \dfrac{\partial^2 f}{\partial x_2\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_2^2} & \cdots & \dfrac{\partial^2 f}{\partial x_2\,\partial x_n} \\[2.2ex]
\vdots & \vdots & \vdots & \ddots & \vdots \\[2.2ex]
\dfrac{\partial g}{\partial x_n} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_1} & \dfrac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \dfrac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}
},
{
"math_input":
Y = \begin{pmatrix} \begin{pmatrix}
q_{1,1,1} d_{1,1} v_{1,1}^{\alpha_{1,1,1}} x_{1}^{\beta_{1,1,1}} \\ \vdots \\ q_{1,1,i} d_{1,1} v_{1,1}^{\alpha_{1,1,i}} x_{i}^{\beta_{1,1,i}}
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{1,n,1} d_{1,n} v_{1,n}^{\alpha_{1,n,1}} x_{1}^{\beta_{1,n,1}} \\ \vdots \\ q_{1,n,i} d_{1,n} v_{1,n}^{\alpha_{1,n,i}} x_{i}^{\beta_{1,n,i}}
\end{pmatrix} \\ \vdots & \ddots & \vdots \\ \begin{pmatrix}
q_{j,1,1} d_{j,1} v_{j,1}^{\alpha_{j,1,1}} x_{1}^{\beta_{j,1,1}} \\ \vdots \\ q_{j,1,i} d_{j,1} v_{j,1}^{\alpha_{j,1,i}} x_{i}^{\beta_{j,1,i}}
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{j,n,1} d_{j,n} v_{j,n}^{\alpha_{j,n,1}} x_{1}^{\beta_{j,n,1}} \\ \vdots \\ q_{j,n,i} d_{j,n} v_{j,n}^{\alpha_{j,n,i}} x_{i}^{\beta_{j,n,i}}
\end{pmatrix} \end{pmatrix}
},
{
"math_input": \begin{align}
\Delta K = W &= \int_{\mathbf{r}_0}^{\mathbf{r}_1} \mathbf{F} \cdot d\mathbf{r} \\
&= \int_{t_0}^{t_1} \frac{d}{dt}(\gamma m_0 \mathbf{v})\cdot\mathbf{v}dt \\
&= \left. \gamma m_0 \mathbf{v} \cdot \mathbf{v} \right|^{t_1}_{t_0} - \int_{t_0}^{t_1} \gamma m_0\mathbf{v} \cdot \frac{d\mathbf{v}}{dt} dt \\
&= \left. \gamma m_0 v^2 \right|^{t_1}_{t_0} - m_0\int_{v_0}^{v_1} \gamma v\,dv \\
&= m_0 \left( \left. \gamma v^2 \right|^{t_1}_{t_0} - c^2\int_{v_0}^{v_1} \frac{2v/c^2}{2\sqrt{1-v^2/c^2}}\,dv \right) \\
&= \left. m_0\left(\frac {v^2}{\sqrt{1-v^2/c^2}} + c^2 \sqrt{1-v^2/c^2} \right) \right|^{t_1}_{t_0} \\
&= \left. \frac {m_0c^2}{\sqrt{1-v^2/c^2}} \right|^{t_1}_{t_0} \\
&= \left. {\gamma m_0c^2}\right|^{t_1}_{t_0} \\
&= \gamma_1 m_0c^2 - \gamma_0 mc^2.\end{align}
},
{
"math_input":
\mathbf{M} = \begin{bmatrix}
m_1 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & m_1 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
0 & \cdots & 0 & m_2 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & m_2 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & m_n & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & m_n\\
\end{bmatrix}
},
{
"math_input":
\begin{alignat}{2}
Q_\mathbf{x} &= \ell_\mathbf{x}&+&\mathbf{f}_\mathbf{x}^\mathsf{T} V'_\mathbf{x} \\
Q_\mathbf{u} &= \ell_\mathbf{u}&+&\mathbf{f}_\mathbf{u}^\mathsf{T} V'_\mathbf{x} \\
Q_{\mathbf{x}\mathbf{x}} &= \ell_{\mathbf{x}\mathbf{x}}&+&\mathbf{f}_\mathbf{x}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{x}+V_\mathbf{x}'\cdot\mathbf{f}_{\mathbf{x}\mathbf{x}}\\
Q_{\mathbf{u}\mathbf{u}} &= \ell_{\mathbf{u}\mathbf{u}}&+&\mathbf{f}_\mathbf{u}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{u}+{V'_\mathbf{x}}\cdot\mathbf{f}_{\mathbf{u}\mathbf{u}}\\
Q_{\mathbf{u}\mathbf{x}} &= \ell_{\mathbf{u}\mathbf{x}}&+&\mathbf{f}_\mathbf{u}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{x}+{V'_\mathbf{x}}\cdot\mathbf{f}_{\mathbf{u}\mathbf{x}}.
\end{alignat}
},
{
"math_input": \begin{align}
\left ( \frac{12345}{331}\right )&=\left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{823}{331}\right ) \\
&= \left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{161}{331}\right ) \\
&= \left ( \frac{3}{331}\right ) \left ( \frac{5}{331}\right ) \left ( \frac{7}{331}\right ) \left ( \frac{23}{331}\right ) \\
&= (-1)\left (\frac{331}{3}\right) \left(\frac{331}{5}\right) (-1) \left(\frac{331}{7}\right) (-1)\left (\frac{331}{23}\right ) \\
&= -\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{9}{23}\right )\\
&= -\left ( \frac{1}{3}\right ) \left ( \frac{1}{5}\right ) \left ( \frac{2}{7}\right ) \left ( \frac{3^2}{23}\right )\\
&= -(1) (1) (1) (1) \\
&= -1.
\end{align}
},
{
"math_input": \left\{\begin{array}{l}
(x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+
f(x) \left(\alpha^2 \mu-\beta (\beta\mu-2 \lambda+2)+
x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt]
f(0)=\frac{\sqrt{\alpha} \left(-\frac{1}{2}\right)^{\lambda-\frac{1}{2}}
e^{-\beta\mu} \mu^{\lambda-\frac{1}{2}}
\left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda}
K_{\lambda-\frac{1}{2}}(-\alpha\mu)}{\sqrt{\pi}
\Gamma(\lambda)}, \\[12pt]
f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda} \mu e^{-\beta\mu}
(-\mu )^{\lambda-
\frac{5}{2}} \left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda}
\left((\beta\mu-2 \lambda+1) K_{\lambda -\frac{1}{2}}(-\alpha\mu)-
\alpha\mu K_{\lambda+\frac{1}{2}}(-\alpha\mu)\right)}{\sqrt{\pi}
\Gamma(\lambda)}
\end{array}\right\}
},
{
"math_input": \begin{align}
H_1 &= -\vec{d}\cdot\vec{E} \\
&=-\left(\vec{d}_\text{eg}|\text{e}\rangle\langle\text{g}|+\vec{d}_\text{eg}^*|\text{g}\rangle\langle\text{e}|\right)
\cdot\left(\vec{E}_0e^{-i\omega_Lt}+\vec{E}_0^*e^{i\omega_Lt}\right) \\
&=-\left(\vec{d}_\text{eg}\cdot\vec{E}_0e^{-i\omega_Lt}
+\vec{d}_\text{eg}\cdot\vec{E}_0^*e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|
-\left(\vec{d}_\text{eg}^*\cdot\vec{E}_0e^{-i\omega_Lt}
+\vec{d}_\text{eg}^*\cdot\vec{E}_0^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}| \\
&=-\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|
-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|,
\end{align}
},
{
"math_input": \begin{array}{ll} \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \,&= \,: \hat{f}_1 \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^\dagger \, : \\ & - \,: \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, : + \,: \hat{f}_1^\bullet \,\hat{f}_2 \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, : +\,: \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^\dagger \, : - : \hat{f}_1 \,\hat{f}_2^\bullet \, \hat{f}_1^\dagger \,\hat{f}_2^{\dagger\bullet} \, : \\ & -: \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet\bullet} \,\hat{f}_2^{\dagger\bullet} \, :+: \hat{f}_1^{\bullet\bullet} \,\hat{f}_2^\bullet \, \hat{f}_1^{\dagger\bullet} \,\hat{f}_2^{\dagger\bullet\bullet}: \end{array}
},
{
"math_input": B=\begin{pmatrix}
\lambda & 1 & 0 & 0 & \cdots & 0 \\
0 & \lambda & 1 & 0 & \cdots & 0 \\
0 & 0 & \lambda & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \lambda & 1 \\
0 & 0 & 0 & 0 & 0 & \lambda \\\end{pmatrix}
=
\lambda \begin{pmatrix}
1 & \lambda^{-1} & 0 & 0 & \cdots & 0 \\
0 & 1 & \lambda^{-1} & 0 & \cdots & 0 \\
0 & 0 & 1 & \lambda^{-1} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & 0 & 0 & 1 & \lambda^{-1} \\
0 & 0 & 0 & 0 & 0 & 1 \\\end{pmatrix}=\lambda(I+K)
},
{
"math_input":
\begin{matrix}
\beta_0 & = \beta_0^{(0)} & & & \\
& & \beta_0^{(1)} & & \\
\beta_1 & = \beta_1^{(0)} & & & \\
& & & \ddots & \\
\vdots & & \vdots & & \beta_0^{(n)} \\
& & & & \\
\beta_{n-1} & = \beta_{n-1}^{(0)} & & & \\
& & \beta_{n-1}^{(1)} & & \\
\beta_n & = \beta_n^{(0)} & & & \\
\end{matrix}
},
{
"math_input": \begin{align}
\int_{\mathbb{R}^n} f(x)g(x) \, dx &= \displaystyle\int_{\mathbb{R}^n}\int_0^\infty \int_0^\infty \chi_{f(x)>r}\chi_{g(x)>s} \, dr \, ds \, dx \\[8pt]
&= \int_{\mathbb{R}^n} \int_0^\infty \int_{\mathbb{R}^n}\chi_{f(x)>r\cap g(x)>s} \, dx \, dr \, ds \\[8pt]
&= \int_0^\infty \int_0^\infty \mu\left(\left\{\chi_{f(x)>r\cap g(x)>s}\right\}\right) \, dr \, ds\\[8pt]
&\leq \int_0^\infty \int_0^\infty \min\left(\mu\left(f(x)>r\right);\mu\left(g(x)>s\right)\right) \, dr \, ds\\[8pt]
&= \int_0^\infty \int_0^\infty \min\left(\mu\left(f^*(x)>r\right);\mu\left(g^*(x)>s\right)\right) \, dr \, ds\\[8pt]
&= \int_0^\infty \int_0^\infty \mu\left(\left\{\chi_{f^*(x)>r\cap g^*(x)>s}\right\}\right) \, dr \, ds\\[8pt]
&= \int_{\mathbb{R}^n} f^*(x)g^*(x) \, dx
\end{align}
},
{
"math_input":
\begin{align}
12 \arctan \frac{1}{49} &+ 32 \arctan \frac{1}{57} - 5 \arctan \frac{1}{239}
+ 12 \arctan \frac{1}{110443} \\
&= 12 \arctan \frac{46}{2253} + 32 \arctan \frac{1}{57} - 5 \arctan \frac{1}{239} \\
&= 12 \arctan\frac{3}{79} + 20 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} \\
&= 12 \arctan\frac{1}{18} + 8 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} \ \ \ \ \text{(Gauss)} \\
&= 4 \arctan\frac{1}{18} + 8 \arctan\frac{3}{41} - 5 \arctan\frac{1}{239} \\
&= 4 \arctan\frac{17}{331} + 4 \arctan\frac{123}{836} - \arctan\frac{1}{239} \\
&= 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \ \ \ \ \text{(Machin)} \\
&= 2 \arctan\frac{5}{12} - \arctan\frac{1}{239} \\
&= \arctan\frac{120}{119} - \arctan\frac{1}{239} \\
&= \arctan 1 = \frac{\pi}{4}.
\end{align}
},
{
"math_input":
\begin{align}
r &= \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2 + {x_1}^2} \\
\phi_1 &= \arccot \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} = \arccos \frac{x_{1}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_1}^2}} \\
\phi_2 &= \arccot \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_3}^2}} = \arccos \frac{x_{2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}} \\
&\vdots\\
\phi_{n-2} &= \arccot \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} = \arccos \frac{x_{n-2}}{\sqrt{{x_n}^2+{x_{n-1}}^2+{x_{n-2}}^2}} \\
\phi_{n-1} &= 2\arccot \frac{x_{n-1}+\sqrt{x_n^2+x_{n-1}^2}}{x_n} = \begin{cases}
\arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n\geq 0 \\
2 \pi - \arccos \frac{x_{n-1}}{\sqrt{{x_n}^2+{x_{n-1}}^2}} & x_n < 0
\end{cases} \,.
\end{align}
},
{
"math_input":
\begin{align}
D_\text{fitted}- D_\text{null} &= \left(-2\ln \frac{\text{likelihood of fitted model}} {\text{likelihood of the saturated model}} \right)-\left(-2\ln \frac{\text{likelihood of null model}} {\text{likelihood of the saturated model}}\right) \\
&= -2 \left(\ln \frac{\text{likelihood of fitted model}} {\text{likelihood of the saturated model}}-\ln \frac{\text{likelihood of null model}} {\text{likelihood of the saturated model}}\right)\\
=& -2 \ln \frac{ \left( \frac{\text{likelihood of fitted model}}{\text{likelihood of the saturated model}}\right)}{ \left( \frac{\text{likelihood of null model}}{\text{likelihood of the saturated model}}\right)}\\
=& -2 \ln \frac{\text{likelihood of the fitted model}}{\text{likelihood of null model}}.
\end{align}
},
{
"math_input":
M = \begin{bmatrix}
m_1 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & m_1 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
0 & \cdots & 0 & m_2 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & m_2 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & m_N & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & m_N\\
\end{bmatrix}
},
{
"math_input":
\begin{align}
& \frac{\partial \sigma_{rr}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{r\theta}}{\partial \theta} + \frac{\partial \sigma_{rz}}{\partial z} + \cfrac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta}) + F_r = \rho~\frac{\partial^2 u_r}{\partial t^2} \\
& \frac{\partial \sigma_{r\theta}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta\theta}}{\partial \theta} + \frac{\partial \sigma_{\theta z}}{\partial z} + \cfrac{2}{r}\sigma_{r\theta} + F_\theta = \rho~\frac{\partial^2 u_\theta}{\partial t^2} \\
& \frac{\partial \sigma_{rz}}{\partial r} + \cfrac{1}{r}\frac{\partial \sigma_{\theta z}}{\partial \theta} + \frac{\partial \sigma_{zz}}{\partial z} + \cfrac{1}{r}\sigma_{rz} + F_z = \rho~\frac{\partial^2 u_z}{\partial t^2}
\end{align}
},
{
"math_input":
M = \begin{bmatrix}
m_1 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & m_1 & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
0 & \cdots & 0 & m_2 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots &\vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & m_2 & \cdots & 0 & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & m_n & \cdots & 0 \\
\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
0 & \cdots & 0 & 0 & \cdots & 0 & \cdots & 0 & \cdots & m_n\\
\end{bmatrix}
},
{
"math_input":
\mathbf{A} = \begin{bmatrix}
\mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} & & & \cdots & \mathbf{A}_{(1,n-1)} & \mathbf{A}_{(1,n)} \\
\mathbf{A}_{(2,1)} & \mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} & & & & \mathbf{A}_{(1,n-1)} \\
& \ddots & \ddots & \ddots & & & \vdots \\
& & \mathbf{A}_{(2,1)} & \mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} & & \\
\vdots & & & \ddots & \ddots & \ddots & \\
\mathbf{A}_{(n-1,1)} & & & & \mathbf{A}_{(2,1)} & \mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} \\
\mathbf{A}_{(n,1)} & \mathbf{A}_{(n-1,1)} & \cdots & & & \mathbf{A}_{(2,1)} & \mathbf{A}_{(1,1)}
\end{bmatrix}.
},
{
"math_input":
\begin{align}
\frac{(\mu_2-\mu_1)-(\bar X_2 - \bar X_1)}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} & = \frac{\mu_2-\bar{X}_2}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} - \frac{\mu_1-\bar{X}_1}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \\[10pt]
& = \underbrace{\frac{\mu_2-\bar{X}_2}{S_2/\sqrt{n_2}}}_{\text{This is }T_2} \cdot \underbrace{\left( \frac{S_2/\sqrt{n_2}}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \right)}_{\text{This is }\cos\theta} - \underbrace{\frac{\mu_1-\bar{X}_1}{S_1/\sqrt{n_1}}}_{\text{This is }T_1}\cdot\underbrace{\left( \frac{S_1/\sqrt{n_1}}{\displaystyle\sqrt{\frac{S^2_1}{n_1} + \frac{S^2_2}{n_2} }} \right)}_{\text{This is }\sin\theta}.\qquad\qquad\qquad (1)
\end{align}
},
{
"math_input":
\begin{bmatrix} \hat{\varepsilon}_{11} & \hat{\varepsilon}_{12} & \hat{\varepsilon}_{13} \\
\hat{\varepsilon}_{21} & \hat{\varepsilon}_{22} & \hat{\varepsilon}_{23} \\
\hat{\varepsilon}_{31} & \hat{\varepsilon}_{32} & \hat{\varepsilon}_{33} \end{bmatrix}
= \begin{bmatrix} \ell_{11} & \ell_{12} & \ell_{13} \\ \ell_{21} & \ell_{22} & \ell_{23} \\ \ell_{31} & \ell_{32} & \ell_{33} \end{bmatrix}
\begin{bmatrix} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \end{bmatrix}
\begin{bmatrix} \ell_{11} & \ell_{12} & \ell_{13} \\ \ell_{21} & \ell_{22} & \ell_{23} \\ \ell_{31} & \ell_{32} & \ell_{33} \end{bmatrix}^T
},
{
"math_input": \begin{matrix}\begin{align}
\hat{U}(t) &= e^{-i\hat{H}_{\text{JC}}t/\hbar}\\
&=
\begin{pmatrix}
e^{- i \nu t (\hat{a}^{\dagger} \hat{a} + \frac{1}{2})}\left( \cos t \sqrt{\hat{\varphi} + g^2} - i \delta/2 \frac{\sin t \sqrt{\hat{\varphi} +
g^2}}{\sqrt{\hat{\varphi} + g^2}}\right)
& - i g e^{- i \nu t (\hat{a}^{\dagger} \hat{a} + \frac{1}{2})} \frac{\sin t \sqrt{\hat{\varphi} + g^2}}{\sqrt{\hat{\varphi} + g^2}} \,\hat{a} \\
-i g e^{- i \nu t (\hat{a}^{\dagger} \hat{a} - \frac{1}{2})}\frac{\sin t \sqrt{\hat{\varphi}}} {\sqrt{\hat{\varphi}}}\hat{a}^{\dagger}
& e^{- i \nu t (\hat{a}^{\dagger} \hat{a} - \frac{1}{2})} \left( \cos t \sqrt{\hat{\varphi}} + i \delta/2 \frac{\sin t \sqrt{\hat{\varphi}}}{\sqrt{\hat{\varphi} }}\right)
\end{pmatrix}
\end{align}\end{matrix}
},
{
"math_input": {}_{
x_{N-1}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N},\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\[8pt]
\frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix}
-\sqrt{\frac{c}{b}}{\rm{i}}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{1}{2N},\frac{3}{2N},\cdots,\frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}{2N},\frac{2N-1}{2N};\\[8pt]
\frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix}
}
},
{
"math_input": \begin{align}
\frac{\partial^2 u_\pm}{\partial x^2}&=\frac{\partial}{\partial x}\frac{\partial u_\pm}{\partial x}
\\&=\frac{\partial}{\partial x}(\mp u_{\pm}\pm\pi^{-\frac 1 2}e^{-\frac{x^2}{4t}}t^{-\frac 1 2})
\\&=\mp \frac{\partial{u_{\pm}}}{\partial x}\pm\pi^{-\frac 1 2}t^{-\frac 1 2}\frac{\partial{e^{-\frac{x^2}{4t}}}}{\partial x}
\\&=\mp (\mp u_{\pm}\pm\pi^{-\frac 1 2}e^{-\frac{x^2}{4t}}t^{-\frac 1 2})\pm\pi^{-\frac 1 2}t^{-\frac 1 2}e^{-\frac{x^2}{4t}}\frac{\partial{{-\frac{x^2}{4t}}}}{\partial x}
\\&=u_{\pm}-\pi^{-\frac 1 2}e^{-\frac{x^2}{4t}}t^{-\frac 1 2}\mp\frac 1 2\pi^{-\frac 1 2}e^{-\frac{x^2}{4t}}t^{-\frac 3 2}x
\\&=u_{\pm}+\pi^{-\frac 1 2}e^{-\tfrac {x^2}{4t}}t^{-\frac 3 2}\cdot(\mp\frac x 2-t)
\\&=\frac{\partial u_{\pm}}{\partial t}
\end{align}
},
{
"math_input": {}_{
x_{N}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N},\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\[8pt]
\frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix}
+\sqrt{\frac{c}{b}}{\rm{i}}{}_{N-1}F_{N-2}
\begin{bmatrix}
\frac{1}{2N},\frac{3}{2N},\cdots,\frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}{2N},\frac{2N-1}{2N};\\[8pt]
\frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\[8pt]
-\frac{a^2c^{N-2}}{4b^N\left(N-2\right)^{N-2}}
\end{bmatrix}
}
},
{
"math_input":
y = h \ast x =
\begin{bmatrix}
h_1 & 0 & \ldots & 0 & 0 \\
h_2 & h_1 & \ldots & \vdots & \vdots \\
h_3 & h_2 & \ldots & 0 & 0 \\
\vdots & h_3 & \ldots & h_1 & 0 \\
h_{m-1} & \vdots & \ldots & h_2 & h_1 \\
h_m & h_{m-1} & \vdots & \vdots & h_2 \\
0 & h_m & \ldots & h_{m-2} & \vdots \\
0 & 0 & \ldots & h_{m-1} & h_{m-2} \\
\vdots & \vdots & \vdots & h_m & h_{m-1} \\
0 & 0 & 0 & \ldots & h_m
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\vdots \\
x_n
\end{bmatrix}
},
{
"math_input":
\mathbf{R} \mathbf{U}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
= \mathbf{U}
\underbrace{
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
\frac{-i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
}_{=\;\mathbf{E}}
\begin{pmatrix}
e^{i\phi} & 0 & 0 \\
0 & e^{-i\phi} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
},
{
"math_input":
\mathbf{R} \mathbf{U}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
= \mathbf{U}
\underbrace{
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
\frac{-i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
}_{=\;\mathbf{I}}
\begin{pmatrix}
e^{i\phi} & 0 & 0 \\
0 & e^{-i\phi} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
},
{
"math_input":
\begin{pmatrix}
\frac{\cos^2\gamma}{I_1}+\frac{\sin^2\gamma}{I_2} &
\left(\frac{1}{I_2}-\frac{1}{I_1}\right){\scriptstyle \sin\beta\sin\gamma\cos\gamma}&
-\frac{\cos\beta\cos^2\gamma}{I_1}-\frac{\cos\beta\sin^2\gamma}{I_2} \\
\left(\frac{1}{I_2}-\frac{1}{I_1}\right){\scriptstyle \sin\beta\sin\gamma\cos\gamma}&
\frac{\sin^2\beta\sin^2\gamma}{I_1}+\frac{\sin^2\beta\cos^2\gamma}{I_2} &
\left(\frac{1}{I_1}-\frac{1}{I_2}\right){\scriptstyle \sin\beta\cos\beta\sin\gamma\cos\gamma}\\
-\frac{\cos\beta\cos^2\gamma}{I_1}-\frac{\cos\beta\sin^2\gamma}{I_2} &
\left(\frac{1}{I_1}-\frac{1}{I_2}\right){\scriptstyle \sin\beta\cos\beta\sin\gamma\cos\gamma} &
\frac{\cos^2\beta\cos^2\gamma}{I_1}+ \frac{\cos^2\beta\sin^2\gamma}{I_2}+\frac{\sin^2\beta}{I_3} \\
\end{pmatrix}.
},
{
"math_input": \begin{matrix}
(Df)(\mathbf{a},\mathbf{b}) & = &
\left[\begin{matrix}
\frac{\partial f_1}{\partial x_1}(\mathbf{a},\mathbf{b}) &
\cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a},\mathbf{b})\\
\vdots & \ddots & \vdots\\
\frac{\partial f_m}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a},\mathbf{b})
\end{matrix}\right|\left.
\begin{matrix}
\frac{\partial f_1}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial y_m}(\mathbf{a},\mathbf{b})\\
\vdots & \ddots & \vdots\\
\frac{\partial f_m}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial y_m}(\mathbf{a},\mathbf{b})\\
\end{matrix}\right]\\
& = & \begin{bmatrix} X & | & Y \end{bmatrix}\\
\end{matrix}
},
{
"math_input": \begin{align}
\Rightarrow \eta'(0) &= -\zeta'(0) - \ln 2 = -\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\left[\ln n-\ln (n+1)\right] \\
&= -\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\ln \frac{n}{n+1} \\
&= -\frac{1}{2} \left(\ln \frac{1}{2} - \ln \frac{2}{3} + \ln \frac{3}{4} - \ln \frac{4}{5} + \ln \frac{5}{6} - \cdots\right) \\
&= \frac{1}{2} \left(\ln \frac{2}{1} + \ln \frac{2}{3} + \ln \frac{4}{3} + \ln \frac{4}{5} + \ln \frac{6}{5} + \cdots\right) \\
&= \frac{1}{2} \ln\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\cdots\right) = \frac{1}{2} \ln\frac{\pi}{2} \\
\Rightarrow \zeta'(0) &= -\frac{1}{2} \ln\left(2 \pi\right)
\end{align}
},
{
"math_input": \begin{align}
\Delta K = W &= \int_{\mathbf{r}_0}^{\mathbf{r}_1} \mathbf{F} \cdot d\mathbf{r} \\
&= \int_{t_0}^{t_1} \frac{d}{dt}(\gamma m\mathbf{v})\cdot\mathbf{v}dt \\
&= \left. \gamma m \mathbf{v} \cdot \mathbf{v} \right|^{t_1}_{t_0} - \int_{t_0}^{t_1} \gamma m\mathbf{v} \cdot \frac{d\mathbf{v}}{dt} dt \\
&= \left. \gamma m v^2 \right|^{t_1}_{t_0} - m\int_{v_0}^{v_1} \gamma v\,dv \\
&= m \left( \left. \gamma v^2 \right|^{t_1}_{t_0} - c^2\int_{v_0}^{v_1} \frac{2v/c^2}{2\sqrt{1-v^2/c^2}}\,dv \right) \\
&= \left. m\left(\frac {v^2}{\sqrt{1-v^2/c^2}} + c^2 \sqrt{1-v^2/c^2} \right) \right|^{t_1}_{t_0} \\
&= \left. \frac {mc^2}{\sqrt{1-v^2/c^2}} \right|^{t_1}_{t_0} \\
&= \left. {\gamma mc^2}\right|^{t_1}_{t_0} \\
&= \gamma_1 mc^2 - \gamma_0 mc^2.\end{align}
},
{
"math_input":
\begin{align}
\varepsilon_{11} & = \cfrac{\mathrm{d}u_0}{dx_1} - x_3\cfrac{\mathrm{d}^2w_0}{\mathrm{d}x_1^2} +
\frac{1}{2}\left[
\left(\cfrac{\mathrm{d}u_0}{\mathrm{d}x_1}-x_3\cfrac{\mathrm{d}^2w_0}{\mathrm{d}x_1^2}\right)^2 +
\left(\cfrac{\mathrm{d}w_0}{\mathrm{d}x_1}\right)^2\right] \\
\varepsilon_{22} & = 0 \\
\varepsilon_{33} & = \frac{1}{2}\left(\cfrac{\mathrm{d}w_0}{\mathrm{d}x_1}\right)^2 \\
\varepsilon_{23} & = 0 \\
\varepsilon_{31} & =
\frac{1}{2}\left(\cfrac{\mathrm{d}w_0}{\mathrm{d}x_1}-\cfrac{\mathrm{d}w_0}{\mathrm{d}x_1}\right) -
\frac{1}{2}\left[\left(\cfrac{\mathrm{d}u_0}{\mathrm{d}x_1}-x_3\cfrac{\mathrm{d}^2w_0}{\mathrm{d}x_1^2}\right)
\left(\cfrac{\mathrm{d}w_0}{\mathrm{d}x_1}\right)\right] \\
\varepsilon_{12} & = 0
\end{align}
},
{
"math_input": [\mathbf{a,\ b, \ d}] = (\mathbf{a \times b}) \mathbf{\cdot d } = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{b \cdot} \hat {\mathbf i} & \mathbf{d\cdot} \hat {\mathbf i}\\
\mathbf{a\cdot }\hat {\mathbf j} & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{d\cdot}\hat {\mathbf j}\\ \mathbf{a\cdot} \hat {\mathbf k} & \mathbf{b\cdot} \hat {\mathbf k} & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix} = \begin{vmatrix} \mathbf{a\cdot }\hat {\mathbf i} & \mathbf{a \cdot} \hat {\mathbf j} & \mathbf{a\cdot} \hat {\mathbf k}\\
\mathbf{b\cdot }\hat {\mathbf i} & \mathbf{b\cdot} \hat {\mathbf j} & \mathbf{b\cdot}\hat {\mathbf k}\\ \mathbf{d\cdot} \hat {\mathbf i} & \mathbf{d\cdot} \hat {\mathbf j} & \mathbf{d\cdot }\hat {\mathbf k} \end{vmatrix}
\ ,
},
{
"math_input": \begin{align}
& z\left( {x_1 \,\,x_2 } \right)\,\,\,\, \approx \,\,\,z\left( {\bar x_1 \,\,\bar x_2 } \right)\,\,\, + \,\,\,\,{{\partial z} \over {\partial x_1 }}\left( {x_1 - \,\,\bar x_1 } \right)\,\,\, + \,\,\,{{\partial z} \over {\partial x_2 }}\left( {x_2 - \,\,\bar x_2 } \right)\,\,\, + \,\,\,{{\partial ^2 z} \over {\partial x_1 \partial x_2 }}\left( {x_1 - \,\,\bar x_1 } \right)\left( {x_2 - \,\,\bar x_2 } \right) \\
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,{1 \over 2}\,\,{{\partial ^2 z} \over {\partial x_1^2 }}\left( {x_1 - \,\,\bar x_1 } \right)^2 \,\,\, + \,\,\,\,{1 \over 2}\,\,{{\partial ^2 z} \over {\partial x_2^2 }}\left( {x_2 - \,\,\bar x_2 } \right)^2\end{align}
},
{
"math_input":
\begin{align}
D_\text{fitted}- D_\text{null} &=-2\ln \frac{\text{likelihood of null model}} {\text{likelihood of the saturated model}} -\left(-2\ln \frac{\text{likelihood of fitted model}} {\text{likelihood of the saturated model}} \right)\\
&=-2 \left(\ln \frac{\text{likelihood of null model}} {\text{likelihood of the saturated model}}-\ln \frac{\text{likelihood of fitted model}} {\text{likelihood of the saturated model}}\right)\\
=& -2 \ln \frac{ \left( \frac{\text{likelihood of null model}}{\text{likelihood of the saturated model}}\right)}{ \left( \frac{\text{likelihood of fitted model}}{\text{likelihood of the saturated model}}\right)}\\
=& -2 \ln \frac{\text{likelihood of null model}}{\text{likelihood of the fitted model}}.
\end{align}
},
{
"math_input": \mathbf{H} = \left( \begin{array}{cccccccc}
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
P_z^{\dagger} & P+\Delta & \sqrt{2}Q^{\dagger} & -S^{\dagger}/\sqrt{2} & -\sqrt{2}P_{+}^{\dagger} & 0 & -\sqrt{3/2}S & -\sqrt{2}R \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
E_{el} & P_z & \sqrt{2}P_z & -\sqrt{3}P_{+} & 0 & \sqrt{2}P_{-} & P_{-} & 0 \\
\end{array} \right)
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\cdot \boldsymbol{S} & = \frac{\partial S_{rr}}{\partial r}~\mathbf{e}_r
+ \frac{\partial S_{r\theta}}{\partial r}~\mathbf{e}_\theta
+ \frac{\partial S_{rz}}{\partial r}~\mathbf{e}_z \\[8pt]
& + \cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial \theta} + (S_{rr}-S_{\theta\theta})\right]~\mathbf{e}_r +
\cfrac{1}{r}\left[\frac{\partial S_{\theta\theta}}{\partial \theta} + (S_{r\theta}+S_{\theta r})\right]~\mathbf{e}_\theta +\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial \theta} + S_{rz}\right]~\mathbf{e}_z \\[8pt]
& +
\frac{\partial S_{zr}}{\partial z}~\mathbf{e}_r +
\frac{\partial S_{z\theta}}{\partial z}~\mathbf{e}_\theta +
\frac{\partial S_{zz}}{\partial z}~\mathbf{e}_z
\end{align}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of states} \\
T &=& \text{number of observations} \\
\theta_{i=1 \dots N} &=& \text{emission parameter for an observation associated with state } i \\
\phi_{i=1 \dots N, j=1 \dots N} &=& \text{probability of transition from state } i \text{ to state } j \\
\boldsymbol\phi_{i=1 \dots N} &=& N\text{-dimensional vector, composed of } \phi_{i,1 \dots N} \text{; must sum to 1} \\
x_{t=1 \dots T} &=& \text{state of observation at time } t \\
y_{t=1 \dots T} &=& \text{observation at time } t \\
F(y|\theta) &=& \text{probability distribution of an observation, parametrized on } \theta \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{t-1}}) \\
y_{t=1 \dots T} &\sim& F(\theta_{x_t})
\end{array}
},
{
"math_input":
\begin{align}
& \mathbf{x}_0 := \text{Some initial guess} \\
& \mathbf{r}_0 := \mathbf{b} - \mathbf{A x}_0 \\
& \mathbf{p}_0 := \mathbf{r}_0 \\
& \text{Iterate, with } k \text{ starting at } 0:\\
& \qquad \alpha_k := \frac{\mathbf{r}_k^\mathrm{T} \mathbf{A r}_k}{(\mathbf{A p}_k)^\mathrm{T} \mathbf{A p}_k} \\
& \qquad \mathbf{x}_{k+1} := \mathbf{x}_k + \alpha_k \mathbf{p}_k \\
& \qquad \mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k \\
& \qquad \beta_k := \frac{\mathbf{r}_{k+1}^\mathrm{T} \mathbf{A r}_{k+1}}{\mathbf{r}_k^\mathrm{T} \mathbf{A r}_k} \\
& \qquad \mathbf{p}_{k+1} := \mathbf{r}_{k+1} + \beta_k \mathbf{p}_k \\
& \qquad \mathbf{A p}_{k + 1} := \mathbf{A r}_{k+1} + \beta_k \mathbf{A p}_k \\
& \qquad k := k + 1
\end{align}
},
{
"math_input":
p_c \gets \underbrace{(1-c_c)}_{\!\!\!\!\!\text{discount factor}\!\!\!\!\!}\,
p_c +
\underbrace{\mathbf{1}_{[0,\alpha\sqrt{n}]}(\|p_\sigma\|)}_{\text{indicator function}}
\overbrace{\sqrt{1 - (1-c_c)^2}}^{
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\text{complements for discounted variance}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
\underbrace{\sqrt{\mu_w}
\, \frac{m_{k+1} - m_k}{\sigma_k}}_{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\text{distributed as}\; \mathcal{N}(0,C_k)\;\text{under neutral selection}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of states} \\
T &=& \text{number of observations} \\
\phi_{i=1 \dots N, j=1 \dots N} &=& \text{probability of transition from state } i \text{ to state } j \\
\boldsymbol\phi_{i=1 \dots N} &=& N\text{-dimensional vector, composed of } \phi_{i,1 \dots N} \text{; must sum to 1} \\
\mu_{i=1 \dots N} &=& \text{mean of observations associated with state } i \\
\sigma^2_{i=1 \dots N} &=& \text{variance of observations associated with state } i \\
x_{t=1 \dots T} &=& \text{state of observation at time } t \\
y_{t=1 \dots T} &=& \text{observation at time } t \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{t-1}}) \\
y_{t=1 \dots T} &\sim& \mathcal{N}(\mu_{x_t}, \sigma_{x_t}^2)
\end{array}
},
{
"math_input": \begin{align}
j \left( \frac{4 \left( 5 \, i \pm 1 \right)}{13} \right) = \Bigg( \frac{\left( 1 - \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 - 858585699 \, \sqrt{2} - 540309076 \, \sqrt{5} + 374537880 \, \sqrt{10} \, \pm \, \textit{i} \, \sqrt[4]{5} \left( 693172512 - 595746414 \, \sqrt{2} - 407357424 \, \sqrt{5} + 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3,\\
j \left( \frac{5 \left( 4 \, i \pm 1 \right)}{17} \right) = \Bigg( \frac{\left( 1 - \sqrt{5} \, \right)^{37}}{2^{39}} \Bigg( 1190448488 + 858585699 \, \sqrt{2} - 540309076 \, \sqrt{5} - 374537880 \, \sqrt{10} \, \pm \, \textit{i} \, \sqrt[4]{5} \left( 693172512 + 595746414 \, \sqrt{2} - 407357424 \, \sqrt{5} - 240819696 \, \sqrt{10} \, \right) \Bigg) \Bigg)^3
\end{align}
},
{
"math_input":
\begin{align}
\operatorname{Var} \left( h_r \right) = & \sum_i
\left( \frac{ \partial h_r }{ \partial B_i } \right)^2
\operatorname{Var}\left( B_i \right) + \\
& \sum_i \sum_{j \neq i}
\left( \frac{ \partial h_r }{ \partial B_i } \right)
\left( \frac{ \partial h_r }{ \partial B_j } \right)
\operatorname{Cov}\left( B_i, B_j \right) \\
\operatorname{Cov}\left( h_r, h_s \right) = & \sum_i
\left( \frac{ \partial h_r }{ \partial B_i } \right)
\left( \frac{ \partial h_s }{ \partial B_i } \right)
\operatorname{Var}\left( B_i \right) + \\
& \sum_i \sum_{j \neq i}
\left( \frac{ \partial h_r }{ \partial B_i } \right)
\left( \frac{ \partial h_s }{ \partial B_j } \right)
\operatorname{Cov}\left( B_i, B_j \right)
\end{align}
},
{
"math_input":
\begin{array}{lcl}
K &=& \text{number of mixture components} \\
N &=& \text{number of observations} \\
\theta_{i=1 \dots K} &=& \text{parameter of distribution of observation associated with component } i \\
\phi_{i=1 \dots K} &=& \text{mixture weight, i.e., prior probability of a particular component } i \\
\boldsymbol\phi &=& K\text{-dimensional vector composed of all the individual } \phi_{1 \dots K} \text{; must sum to 1} \\
z_{i=1 \dots N} &=& \text{component of observation } i \\
x_{i=1 \dots N} &=& \text{observation } i \\
F(x|\theta) &=& \text{probability distribution of an observation, parametrized on } \theta \\
z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
x_{i=1 \dots N} &\sim& F(\theta_{z_i})
\end{array}
},
{
"math_input":
\begin{bmatrix}
\mathbf A^T \mathbf A & \mathbf A^T \mathbf B \\
\mathbf B^T \mathbf A & \mathbf B^T \mathbf B
\end{bmatrix}^{-1}
=
\begin{bmatrix}
(\mathbf A^T \mathbf A-\mathbf A^T \mathbf B(\mathbf B^T \mathbf B)^{-1}\mathbf B^T \mathbf A)^{-1}
& -(\mathbf A^T \mathbf A)^{-1}\mathbf A^T \mathbf B(\mathbf B^T \mathbf B-\mathbf B^T \mathbf A(\mathbf A^T \mathbf A)^{-1}\mathbf A^T \mathbf B)^{-1}
\\
-(\mathbf B^T \mathbf B)^{-1}\mathbf B^T \mathbf A(\mathbf A^T \mathbf A-\mathbf A^T \mathbf B(\mathbf B^T \mathbf B)^{-1}\mathbf B^T \mathbf A)^{-1}
& (\mathbf B^T \mathbf B-\mathbf B^T \mathbf A(\mathbf A^T \mathbf A)^{-1}\mathbf A^T \mathbf B)^{-1}
\end{bmatrix}
},
{
"math_input":
\begin{array}{lcl}
N &=& \text{number of mixture components} \\
T &=& \text{number of observations} \\
\theta_{i=1 \dots N} &=& \text{parameter of distribution of observation associated with component } i \\
\phi_{i=1 \dots N} &=& \text{mixture weight, i.e. prior probability of a particular component } i \\
\boldsymbol\phi &=& N\text{-dimensional vector composed of all the individual } \phi_{1 \dots N} \text{; must sum to 1} \\
x_{i=1 \dots T} &=& \text{component of observation } i \\
y_{i=1 \dots T} &=& \text{observation } i \\
F(y|\theta) &=& \text{probability distribution of an observation, parametrized on } \theta \\
x_{i=1 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
y_{i=1 \dots T} &\sim& F(\theta_{x_i})
\end{array}
},
{
"math_input":
\begin{align}
\nabla\cdot\boldsymbol{s} \equiv \cfrac{\partial s_{ij}}{\partial x_i} & =
\mu \left[\cfrac{\partial}{\partial x_i}\left(\cfrac{\partial v_i}{\partial x_j}+\cfrac{\partial v_j}{\partial x_i}\right)\right] + \lambda~\left[\cfrac{\partial}{\partial x_i}\left(\cfrac{\partial v_k}{\partial x_k}\right)\right]\delta_{ij} \\
& = \mu~\cfrac{\partial^2 v_i}{\partial x_i \partial x_j} + \mu~\cfrac{\partial^2 v_j}{\partial x_i\partial x_i} + \lambda~\cfrac{\partial^2 v_k}{\partial x_k\partial x_j} \\
& = (\mu + \lambda)~\cfrac{\partial^2 v_i}{\partial x_i \partial x_j} + \mu~\cfrac{\partial^2 v_j}{\partial x_i^2} \\
& \equiv (\mu + \lambda)~\nabla(\nabla\cdot\mathbf{v}) + \mu~\nabla^2\mathbf{v} ~.
\end{align}
},
{
"math_input":
\begin{align}
dF & = \left(\frac{\partial F}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial u} +\frac{\partial F}{\partial u} \right) \, du + \left(\frac{\partial F}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial v} +\frac{\partial F}{\partial v} \right) \, dv = 0 \\[6pt]
dG & = \left(\frac{\partial G}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial u} +\frac{\partial G}{\partial u} \right) \, du + \left(\frac{\partial G}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial v} +\frac{\partial G}{\partial v} \right) \, dv = 0.
\end{align}
},
{
"math_input":
\begin{align}
\left[\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^3 \right.&
\left.+ \frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda^2 +
\frac{\partial I_3}{\partial \boldsymbol{A}}~\lambda + \frac{\partial I_4}{\partial \boldsymbol{A}}\right]\boldsymbol{\mathit{1}} +
\boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}}~\lambda^3 +
\boldsymbol{A}^T\cdot\frac{\partial I_1}{\partial \boldsymbol{A}}~\lambda^2 +
\boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}}~\lambda +
\boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} \\
&=
\left[I_0~\lambda^3 + I_1~\lambda^2 + I_2~\lambda + I_3\right]
\boldsymbol{\mathit{1}} ~.
\end{align}
},
{
"math_input":
\underline{\underline{\mathsf{A}_\varepsilon}} = \begin{bmatrix}
A_{11}^2 & A_{12}^2 & A_{13}^2 & A_{12}A_{13} & A_{11}A_{13} & A_{11}A_{12} \\
A_{21}^2 & A_{22}^2 & A_{23}^2 & A_{22}A_{23} & A_{21}A_{23} & A_{21}A_{22} \\
A_{31}^2 & A_{32}^2 & A_{33}^2 & A_{32}A_{33} & A_{31}A_{33} & A_{31}A_{32} \\
2A_{21}A_{31} & 2A_{22}A_{32} & 2A_{23}A_{33} & A_{22}A_{33}+A_{23}A_{32} & A_{21}A_{33}+A_{23}A_{31} & A_{21}A_{32}+A_{22}A_{31} \\
2A_{11}A_{31} & 2A_{12}A_{32} & 2A_{13}A_{33} & A_{12}A_{33}+A_{13}A_{32} & A_{11}A_{33}+A_{13}A_{31} & A_{11}A_{32}+A_{12}A_{31} \\
2A_{11}A_{21} & 2A_{12}A_{22} & 2A_{13}A_{23} & A_{12}A_{23}+A_{13}A_{22} & A_{11}A_{23}+A_{13}A_{21} & A_{11}A_{22}+A_{12}A_{21} \end{bmatrix}
},
{
"math_input": \left\{\begin{array}{l}
(x-\mu ) f''(x)-2 f'(x) (-\beta\mu+\lambda+\beta x-1)+
f(x) \left(\alpha^2 \mu-\beta(\beta\mu-2 \lambda+2)+
x \left(\beta^2-\alpha^2\right)\right)=0, \\[12pt]
f(0)=\frac{2^{\frac{1}{2}-\lambda}\sqrt{\frac{\alpha }{\mu}}
e^{-\beta\mu} \left(\mu \left(\alpha-
\frac{\beta^2}{\alpha}\right)\right)^{\lambda}
K_{\lambda-\frac{1}{2}}(\alpha\mu)}{\sqrt{\pi}
\Gamma(\lambda)}, \\[12pt]
f'(0)=\frac{\sqrt{\alpha} 2^{\frac{1}{2}-\lambda}
e^{-\beta\mu} \mu^{\lambda-\frac{3}{2}}
\left(\alpha-\frac{\beta^2}{\alpha}\right)^{\lambda}
\left((\beta\mu-2 \lambda+1)
K_{\lambda-\frac{1}{2}}(\alpha\mu)+\alpha\mu
K_{\lambda+\frac{1}{2}}(\alpha\mu)\right)}{\sqrt{\pi}
\Gamma (\lambda)}
\end{array}\right\}
},
{
"math_input": \begin{align}
F_n(x) =
& \Phi(x) \\
& - \frac{1}{n^{1/2}}\bigg( \tfrac{1}{6}\lambda_3\,\Phi^{(3)}(x) \bigg) \\
& + \frac{1}{n}\bigg( \tfrac{1}{24}\lambda_4\,\Phi^{(4)}(x) + \tfrac{1}{72}\lambda_3^2\,\Phi^{(6)}(x) \bigg) \\
& - \frac{1}{n^{3/2}}\bigg( \tfrac{1}{120}\lambda_5\,\Phi^{(5)}(x) + \tfrac{1}{144}\lambda_3\lambda_4\,\Phi^{(7)}(x) + \tfrac{1}{1296}\lambda_3^3\,\Phi^{(9)}(x)\bigg) \\
& + \frac{1}{n^2}\bigg( \tfrac{1}{720}\lambda_6\,\Phi^{(6)}(x) + \big(\tfrac{1}{1152}\lambda_4^2 + \tfrac{1}{720}\lambda_3\lambda_5\big)\Phi^{(8)}(x) \\
&\qquad\quad + \tfrac{1}{1728}\lambda_3^2\lambda_4\,\Phi^{(10)}(x) + \tfrac{1}{31104}\lambda_3^4\,\Phi^{(12)}(x) \bigg) \\
& + O(n^{-5/2})\,.
\end{align}
},
{
"math_input":
\begin{align}
\frac{\mathrm d F_\varepsilon}{\mathrm d\varepsilon} & =\frac{\mathrm d x}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial x} + \frac{\mathrm d g_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon} + \frac{\mathrm d g_\varepsilon'}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon'} \\
& = \frac{\mathrm d g_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g_\varepsilon}+\frac{\mathrm d g'_\varepsilon}{\mathrm d\varepsilon}\frac{\partial F_\varepsilon}{\partial g'_\varepsilon} \\
& = \eta(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon} + \eta'(x) \frac{\partial F_\varepsilon}{\partial g_\varepsilon'} \ . \\
\end{align}
},
{
"math_input":
\begin{align}
\boldsymbol{\nabla}\phi & = \cfrac{\partial\phi}{\partial \xi^i}~\mathbf{g}^i \\
\boldsymbol{\nabla}\mathbf{v} & = \cfrac{\partial (v^j \mathbf{g}_j)}{\partial \xi^i}\otimes\mathbf{g}^i
= \left(\cfrac{\partial v^j}{\partial \xi^i} + v^k~\Gamma_{ik}^j\right)~\mathbf{g}_j\otimes\mathbf{g}^i
= \left(\cfrac{\partial v_j}{\partial \xi^i} - v_k~\Gamma_{ij}^k\right)~\mathbf{g}^j\otimes\mathbf{g}^i\\
\boldsymbol{\nabla}\boldsymbol{S} & = \cfrac{\partial (S_{jk}~\mathbf{g}^j\otimes\mathbf{g}^k)}{\partial \xi^i}\otimes\mathbf{g}^i
= \left(\cfrac{\partial S_{jk}}{\partial \xi_i}- S_{lk}~\Gamma_{ij}^l - S_{jl}~\Gamma_{ik}^l\right)~\mathbf{g}^j\otimes\mathbf{g}^k\otimes\mathbf{g}^i
\end{align}
},
{
"math_input": V_i(r)=\left\{\begin{array}{ll}
\frac{1}{\epsilon_1}
\frac{\epsilon_1-\epsilon_2}{\epsilon_1+\epsilon_2}
\frac{e^2}{2z}
=\frac{1}{\epsilon_1}
\frac{\Delta\epsilon}{\epsilon_{ave}}
\frac{e^2}{4z}
& z \ge 0\\
\ & \ \\
\frac{1}{\epsilon_2}
\frac{\epsilon_1-\epsilon_2}{\epsilon_1+\epsilon_2}
\frac{e^2}{2z}
=\frac{1}{\epsilon_2}
\frac{\Delta\epsilon}{\epsilon_{ave}}
\frac{e^2}{4z}
& z < 0,\\ \end{array}\right.
},
{
"math_input":
\begin{align}
&\int\limits_{0}^{2\pi} \hat{r}\ {\left(\frac{p}{r}\right)}^3\ \sin^2 u\ \cos u\ du\ =
\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin^2 u \cos^2 u \ du\
+\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin^3 u\ \cos u \ du\ = \\
&\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^2 u \cos^2 u \ du\ +\
3\ {e_g}^2\ \int\limits_{0}^{2\pi}\ \cos^4 u\ \sin^2 u \ du\ \ +\
3\ {e_h}^2\ \int\limits_{0}^{2\pi}\ \sin^4 u\ \cos^2 u \ du\ \right) \\
&+\hat{h}\ 6\ e_g\ e_h\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^4 u \ du = \\
&\hat{g}\ \left(2\pi \left(\frac{1}{8}\ +\ \frac{3}{16}\ {e_g}^2\ +\ \frac{3}{16}\ {e_h}^2\right)\right)
+\hat{h}\ \left(2\pi \left(\frac{3}{8}\ e_g\ e_h\right)\right)
\end{align}
},
{
"math_input": = \, \frac{d}{d t} \left( \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, p_{\delta} \right) \, - \,
\left(
\frac{\partial \bar{x}^{\beta}}{\partial x^{\theta}} \,
\frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \,
\frac{\partial x^{\iota}}{\partial \bar{x}^{\gamma}} \,
\Gamma^{\theta}_{\delta \iota} \,
+ \, \frac{\partial \bar{x}^{\beta}}{\partial x^{\eta}}\,
\frac{\partial^2 x^{\eta}}{\partial \bar{x}^{\alpha} \partial \bar{x}^{\gamma}}
\right) \, \frac{\partial x^{\epsilon}}{\partial \bar{x}^{\beta}} \, p_{\epsilon} \, \frac{\partial \bar{x}^{\gamma}}{\partial x^{\zeta}} \, \frac{d x^{\zeta}}{d t} \, - \, q \, \frac{\partial x^{\delta}}{\partial \bar{x}^{\alpha}} \, F_{\delta \zeta} \, \frac{d x^{\zeta}}{d t} \,
},
{
"math_input":
\begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix} =
\begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \gamma_{23} \\ \gamma_{31} \\ \gamma_{12} \end{bmatrix} =
\cfrac{1}{E}
\begin{bmatrix} 1 & -\nu & -\nu & 0 & 0 & 0 \\
-\nu & 1 & -\nu & 0 & 0 & 0 \\
-\nu & -\nu & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 2(1+\nu) & 0 & 0 \\
0 & 0 & 0 & 0 & 2(1+\nu) & 0 \\
0 & 0 & 0 & 0 & 0 & 2(1+\nu) \end{bmatrix}
\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}
},
{
"math_input": \begin{align}R_{J}(x,y,z,p) & = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\frac{1}{\sqrt{(t + 1)^{5} - (t + 1)^{4} E_{1} + (t + 1)^{3} E_{2} - (t + 1)^{2} E_{3} + (t + 1) E_{4} - E_{5}}} dt \\
& = \frac{3}{2 A^{\frac{3}{2}}} \int _{0}^{\infty}\left( \frac{1}{(t + 1)^{\frac{5}{2}}} - \frac{E_{2}}{2 (t + 1)^{\frac{9}{2}}} + \frac{E_{3}}{2 (t + 1)^{\frac{11}{2}}} + \frac{3 E_{2}^{2} - 4 E_{4}}{8 (t + 1)^{\frac{13}{2}}} + \frac{2 E_{5} - 3 E_{2} E_{3}}{4 (t + 1)^{\frac{15}{2}}} + O(E_{1}) + O(\Delta^{6})\right) dt \\
& = \frac{1}{A^{\frac{3}{2}}} \left( 1 - \frac{3}{14} E_{2} + \frac{1}{6} E_{3} + \frac{9}{88} E_{2}^{2} - \frac{3}{22} E_{4} - \frac{9}{52} E_{2} E_{3} + \frac{3}{26} E_{5} + O(E_{1}) + O(\Delta^{6})\right) \end{align}
},
{
"math_input":
\underline{\underline{\mathsf{A}_\sigma}} = \begin{bmatrix}
A_{11}^2 & A_{12}^2 & A_{13}^2 & 2A_{12}A_{13} & 2A_{11}A_{13} & 2A_{11}A_{12} \\
A_{21}^2 & A_{22}^2 & A_{23}^2 & 2A_{22}A_{23} & 2A_{21}A_{23} & 2A_{21}A_{22} \\
A_{31}^2 & A_{32}^2 & A_{33}^2 & 2A_{32}A_{33} & 2A_{31}A_{33} & 2A_{31}A_{32} \\
A_{21}A_{31} & A_{22}A_{32} & A_{23}A_{33} & A_{22}A_{33}+A_{23}A_{32} & A_{21}A_{33}+A_{23}A_{31} & A_{21}A_{32}+A_{22}A_{31} \\
A_{11}A_{31} & A_{12}A_{32} & A_{13}A_{33} & A_{12}A_{33}+A_{13}A_{32} & A_{11}A_{33}+A_{13}A_{31} & A_{11}A_{32}+A_{12}A_{31} \\
A_{11}A_{21} & A_{12}A_{22} & A_{13}A_{23} & A_{12}A_{23}+A_{13}A_{22} & A_{11}A_{23}+A_{13}A_{21} & A_{11}A_{22}+A_{12}A_{21} \end{bmatrix}
},
{
"math_input": \begin{align}
F_n(x) &= \Phi(x) \\
&\quad -\frac{1}{n^{\frac{1}{2}}}\left(\tfrac{1}{6}\lambda_3\,\Phi^{(3)}(x) \right) \\
&\quad +\frac{1}{n}\left(\tfrac{1}{24}\lambda_4\,\Phi^{(4)}(x) + \tfrac{1}{72}\lambda_3^2\,\Phi^{(6)}(x) \right) \\
&\quad -\frac{1}{n^{\frac{3}{2}}}\left(\tfrac{1}{120}\lambda_5\,\Phi^{(5)}(x) + \tfrac{1}{144}\lambda_3\lambda_4\,\Phi^{(7)}(x) + \tfrac{1}{1296}\lambda_3^3\,\Phi^{(9)}(x)\right) \\
&\quad + \frac{1}{n^2}\left(\tfrac{1}{720}\lambda_6\,\Phi^{(6)}(x) + \left(\tfrac{1}{1152}\lambda_4^2 + \tfrac{1}{720}\lambda_3\lambda_5\right)\Phi^{(8)}(x) + \tfrac{1}{1728}\lambda_3^2\lambda_4\,\Phi^{(10)}(x) + \tfrac{1}{31104}\lambda_3^4\,\Phi^{(12)}(x) \right)\\
&\quad + O \left (n^{-\frac{5}{2}} \right ).
\end{align}
},
{
"math_input": \begin{align}
I_1 &= \sigma_{11}+\sigma_{22}+\sigma_{33} \\
&= \sigma_{kk} \\
I_2 &= \begin{vmatrix}
\sigma_{22} & \sigma_{23} \\
\sigma_{32} & \sigma_{33} \\
\end{vmatrix}
+ \begin{vmatrix}
\sigma_{11} & \sigma_{13} \\
\sigma_{31} & \sigma_{33} \\
\end{vmatrix}
+
\begin{vmatrix}
\sigma_{11} & \sigma_{12} \\
\sigma_{21} & \sigma_{22} \\
\end{vmatrix} \\
&= \sigma_{11}\sigma_{22}+\sigma_{22}\sigma_{33}+\sigma_{11}\sigma_{33}-\sigma_{12}^2-\sigma_{23}^2-\sigma_{31}^2 \\
&= \frac{1}{2}\left(\sigma_{ii}\sigma_{jj}-\sigma_{ij}\sigma_{ji}\right) \\
I_3 &= \det(\sigma_{ij}) \\
&= \sigma_{11}\sigma_{22}\sigma_{33}+2\sigma_{12}\sigma_{23}\sigma_{31}-\sigma_{12}^2\sigma_{33}-\sigma_{23}^2\sigma_{11}-\sigma_{31}^2\sigma_{22} \\
\end{align}
\,\!
},
{
"math_input": \begin{align}
\frac{\partial y}{\partial c}
&= a_0 x^c \ln(x) \sum_{r = 0}^\infty \frac{(c + \alpha)_r (c + \beta)_r}{(c + 1)_r^2} x^r \\
&\quad + a_0 x^c \sum_{r = 0}^\infty \left(\frac{(c + \alpha)_r (c + \beta)_r}{(c + 1)_r^2}
\left\{\sum_{k = 0}^{r - 1} \left(\frac{1}{c + \alpha + k} + \frac{1}{c + \beta + k}
- \frac{2}{c + 1 + k} \right) \right\} \right) x^r \\
&= a_0 x^c \sum_{r = 0}^\infty \frac{(c + \alpha)_r (c + \beta)_r}{(c + 1)_r)^2}
\left(\ln x + \sum_{k = 0}^{r - 1} \left(\frac{1}{c + \alpha + k}
+\frac{1}{c + \beta + k} - \frac{2}{c + 1 + k} \right) \right) x^{r}.
\end{align}
},
{
"math_input": \begin{align}
c_1(t)&=c_1(0)\left[\frac{\Omega_c ^2}{\Omega^2}+\frac{\Omega_p ^2}{\Omega^2}\cos\frac{\Omega t}{2}\right]+c_2(0)\left[-\frac{\Omega_p \Omega_c }{\Omega^2}+\frac{\Omega_p \Omega_c }{\Omega^2}\cos\frac{\Omega t}{2}\right]\\
&\quad-ic_3(0)\frac{\Omega_p }{\Omega}\sin\frac{\Omega t}{2}\\
c_2(t)&=c_1(0)\left[-\frac{\Omega_p \Omega_c }{\Omega^2}+\frac{\Omega_p \Omega_c }{\Omega^2}\cos\frac{\Omega t}{2}\right]+c_2(0)\left[\frac{\Omega_p ^2}{\Omega^2}+\frac{\Omega_c ^2}{\Omega^2}\cos\frac{\Omega t}{2}\right]\\
&\quad-ic_3(0)\frac{\Omega_c }{\Omega}\sin\frac{\Omega t}{2}\\
c_3(t)&=-ic_1(0)\frac{\Omega_p }{\Omega}\sin\frac{\Omega t}{2}-ic_2(0)\frac{\Omega_c }{\Omega}\sin\frac{\Omega t}{2}+c_3(0)\cos\frac{\Omega t}{2}\end{align}
},
{
"math_input": \begin{align}
& \underset{s\to a}{\mathop{\lim }}\,\frac{\left( s-a \right)P_{1}(s)}{P_{2}(s)}=\underset{s\to 0}{\mathop{\lim }}\,\frac{\left( s-0 \right)((2-\gamma )s^{2}+(\alpha +\beta -1)s)}{(s^{3}-s^{2})}=\underset{s\to 0}{\mathop{\lim }}\,\frac{((2-\gamma )s^{2}+(\alpha +\beta -1)s)}{s^{2}-s} \\
& =\underset{s\to 0}{\mathop{\lim }}\,\frac{((2-\gamma )s+(\alpha +\beta -1))}{s-1}=1-\alpha -\beta \text{ }\text{. } \\
& \underset{s\to a}{\mathop{\lim }}\,\frac{\left( s-a \right)^{2}P_{0}(s)}{P_{2}(s)}=\underset{s\to 0}{\mathop{\lim }}\,\frac{\left( s-0 \right)^{2}\left( -\alpha \beta \right)}{(s^{3}-s^{2})}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\left( -\alpha \beta \right)}{s-1}=\alpha \beta \text{ }\text{.}
\end{align}
},
{
"math_input": \begin{align}
y = \frac{E}{(\alpha + \beta - \gamma + 1)_{\gamma - \alpha - \beta - 1}}
&\sum_{r = 1 - \gamma}^\infty \frac{(\alpha)_r (\beta)_r}{(1)_r (1)_{r + \alpha + \beta - \gamma}} (1 - x)^r + {}\\
+ F(1 - x)^{\gamma - \alpha - \beta}
& \sum_{r = 0}^\infty \frac{(\gamma - \alpha - \beta)(\gamma - \beta)_r (\gamma - \alpha)_r}
{(1 + \gamma - \alpha - \beta)_r (1)_r}
\Biggl(\ln(1 - x) + \frac{1}{\gamma - \alpha - \beta} + {} \\
&\quad+ \sum_{k = 0}^{r - 1} \left(\frac{1}{k + \gamma - \beta} + \frac{1}{k + \gamma - \alpha}
- \frac{1}{1 + k + \gamma - \alpha - \beta} - \frac{1}{1 + k} \right) \Biggr) (1 - x)^r
\end{align}
},
{
"math_input":
\begin{align}
ds^2 &=
\begin{bmatrix}
du&dv
\end{bmatrix}
\begin{bmatrix}
E&F\\
F&G
\end{bmatrix}
\begin{bmatrix}
du\\dv
\end{bmatrix}\\
&=\begin{bmatrix}
du'&dv'
\end{bmatrix}
\begin{bmatrix}
\frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'}\\
\frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
\end{bmatrix}^\mathrm{T}
\begin{bmatrix}
E&F\\
F&G
\end{bmatrix}
\begin{bmatrix}
\frac{\partial u}{\partial u'} & \frac{\partial u}{\partial v'}\\
\frac{\partial v}{\partial u'} & \frac{\partial v}{\partial v'}
\end{bmatrix}
\begin{bmatrix}
du'\\dv'
\end{bmatrix}\\
&=
\begin{bmatrix}
du'&dv'
\end{bmatrix}
\begin{bmatrix}
E'&F'\\
F'&G'
\end{bmatrix}
\begin{bmatrix}
du'\\dv'
\end{bmatrix}\\
&=(ds')^2.
\end{align}
},
{
"math_input": \begin{bmatrix}y_{1,t} \\ y_{2,t}\\ \vdots \\ y_{k,t}\end{bmatrix}=\begin{bmatrix}c_{1} \\ c_{2}\\ \vdots \\ c_{k}\end{bmatrix}+
\begin{bmatrix}
a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1\\
a_{2,1}^1&a_{2,2}^1 & \cdots & a_{2,k}^1\\
\vdots& \vdots& \ddots& \vdots\\
a_{k,1}^1&a_{k,2}^1 & \cdots & a_{k,k}^1
\end{bmatrix}
\begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\\ \vdots \\ y_{k,t-1}\end{bmatrix}
+ \cdots +
\begin{bmatrix}
a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\
a_{2,1}^p&a_{2,2}^p & \cdots & a_{2,k}^p\\
\vdots& \vdots& \ddots& \vdots\\
a_{k,1}^p&a_{k,2}^p & \cdots & a_{k,k}^p
\end{bmatrix}
\begin{bmatrix}y_{1,t-p} \\ y_{2,t-p}\\ \vdots \\ y_{k,t-p}\end{bmatrix}
+ \begin{bmatrix}e_{1,t} \\ e_{2,t}\\ \vdots \\ e_{k,t}\end{bmatrix}
},
{
"math_input":
\begin{bmatrix}
C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix} =
\begin{bmatrix}
C_{11} & C_{12} & C_{13} & -C_{14} & -C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & -C_{24} & -C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & -C_{34} & -C_{35} & C_{36} \\
-C_{14} & -C_{24} & -C_{34} & C_{44} & C_{45} & -C_{46} \\
-C_{15} & -C_{25} & -C_{35} & C_{45} & C_{55} & -C_{56} \\
C_{16} & C_{26} & C_{36} & -C_{46} & -C_{56} & C_{66} \end{bmatrix}
},
{
"math_input":
\begin{align}
\operatorname{cn}(x+y) & =
{\operatorname{cn}(x)\;\operatorname{cn}(y)
- \operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y)
\over {1 - k^2 \;\operatorname{sn}^2 (x) \;\operatorname{sn}^2 (y)}}, \\[8pt]
\operatorname{sn}(x+y) & =
{\operatorname{sn}(x)\;\operatorname{cn}(y)\;\operatorname{dn}(y) +
\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{dn}(x)
\over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}, \\[8pt]
\operatorname{dn}(x+y) & =
{\operatorname{dn}(x)\;\operatorname{dn}(y)
- k^2 \;\operatorname{sn}(x)\;\operatorname{sn}(y)\;\operatorname{cn}(x)\;\operatorname{cn}(y)
\over {1 - k^2 \;\operatorname{sn}^2 (x)\; \operatorname{sn}^2 (y)}}.
\end{align}
},
{
"math_input":
\begin{align}
\lambda_0(\mu-\mu_0)^2 + n(\bar{x} -\mu)^2&=\lambda_0 \mu^2 + 2 \lambda_0 \mu \mu_0 + \lambda_0 \mu_0^2 + n \mu^2 - 2 n \bar{x} \mu + n \bar{x}^2 \\
&= (\lambda_0 + n) \mu^2 - 2(\lambda_0 \mu_0 + n \bar{x}) \mu + \lambda_0 \mu_0^2 +n \bar{x}^2 \\
&= (\lambda_0 + n)( \mu^2 - 2 \frac{\lambda_0 \mu_0 + n \bar{x}}{\lambda_0 + n} \mu ) + \lambda_0 \mu_0^2 +n \bar{x}^2 \\
&= (\lambda_0 + n)\left(\mu - \frac{\lambda_0 \mu_0 + n \bar{x}}{\lambda_0 + n} \right) ^2 + \lambda_0 \mu_0^2 +n \bar{x}^2 - \left( \frac{\lambda_0 \mu_0 +n \bar{x}}{\lambda_0 + n} \right)^2 \\
&= (\lambda_0 + n)\left(\mu - \frac{\lambda_0 \mu_0 + n \bar{x}}{\lambda_0 + n} \right) ^2 + \frac{\lambda_0 n (\bar{x} - \mu_0 )^2}{\lambda_0 +n}
\end{align}
},
{
"math_input": \begin{align}
P_3(\boldsymbol{x}) = f ( \boldsymbol{a} ) + &\frac{\partial f}{\partial x_1}( \boldsymbol{a} ) v_1 + \frac{\partial f}{\partial x_2}( \boldsymbol{a} ) v_2 + \frac{\partial^2 f}{\partial^2 x_1}( \boldsymbol{a} ) \frac {v_1^2}{2!} + \frac{\partial^2 f}{\partial x_1 \partial x_2}( \boldsymbol{a} ) v_1 v_2 + \frac{\partial^2 f}{\partial^2 x_2}( \boldsymbol{a} ) \frac{v_2^2}{2!} \\
& + \frac{\partial^3 f}{\partial x_1^3}( \boldsymbol{a} ) \frac{v_1^3}{3!} + \frac{\partial^3 f}{\partial^2 x_1 \partial x_2}( \boldsymbol{a} ) \frac{v_1^2 v_2}{2!} + \frac{\partial^3 f}{\partial x_1 \partial^2 x_2}( \boldsymbol{a} ) \frac{v_1 v_2^2}{2!} + \frac{\partial^3 f}{\partial^3 x_2}( \boldsymbol{a} ) \frac{v_2^3}{3!}
\end{align}
},
{
"math_input": \begin{align}
& x=a\cos \left( \alpha \right)+b\sin \left( \alpha \right) \\
& y=-a\sin \left( \alpha \right)+b\cos \left( \alpha \right) \\
& \cos \left( \alpha \right)={{a}^{T}}x\quad \sin \left( \alpha \right)={{b}^{T}}x \\
& y=-a{{b}^{T}}x+b{{a}^{T}}x=\left( b{{a}^{T}}-a{{b}^{T}} \right)x \\
& \\
& {x}'=x\cos \left( \beta \right)+y\sin \left( \beta \right) \\
& \ \ \ =\left[ I\cos \left( \beta \right)+\left( b{{a}^{T}}-a{{b}^{T}} \right)\sin \left( \beta \right) \right]x \\
& \\
& R=I\cos \left( \beta \right)+\left( b{{a}^{T}}-a{{b}^{T}} \right)\sin \left( \beta \right) \\
& \quad =I\cos \left( \beta \right)+G\sin \left( \beta \right) \\
& \\
& G=b{{a}^{T}}-a{{b}^{T}} \\
\end{align}
},
{
"math_input":
\begin{align}
p(\boldsymbol\eta|\mathbf{X},\boldsymbol\chi,\nu)& \propto p(\mathbf{X}|\boldsymbol\eta) p_\pi(\boldsymbol\eta|\boldsymbol\chi,\nu) \\
& = \left( \prod_{i=1}^n h(x_i) \right) g(\boldsymbol\eta)^n
\exp\left(\ \boldsymbol\eta^{\rm T} \Big(\sum_{i=1}^n \mathbf{T}(x_i)\Big) \ \right)
f(\boldsymbol\chi,\nu) g(\boldsymbol\eta)^\nu \exp(\boldsymbol\eta^{\rm T} \boldsymbol\chi) \\
& \propto g(\boldsymbol\eta)^n
\exp\left(\ \boldsymbol\eta^{\rm T} \Big(\sum_{i=1}^n \mathbf{T}(x_i)\Big) \ \right)
g(\boldsymbol\eta)^\nu \exp(\boldsymbol\eta^{\rm T} \boldsymbol\chi) \\
& \propto g(\boldsymbol\eta)^{\nu + n} \exp\left(\ \boldsymbol\eta^{\rm T} \Big(\boldsymbol\chi + \sum_{i=1}^n \mathbf{T}(x_i)\Big) \ \right)
\end{align}
},
{
"math_input": \begin{align}\langle f|g\rangle-\langle g|f\rangle &= \int_{-\infty}^{\infty} \psi^*(x) \, x \cdot \left(-i \hbar \frac{d}{dx}\right) \, \psi(x) \, dx \\
&{} \, \, \, \, \, - \int_{-\infty}^{\infty} \psi^*(x) \, \left(-i \hbar \frac{d}{dx}\right) \cdot x \, \psi(x) dx \\
&= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \frac{d(x \psi(x))}{dx} \right] \, dx \\
&= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \left[ \left(-x \cdot \frac{d\psi(x)}{dx}\right) + \psi(x) + \left(x \cdot \frac{d\psi(x)}{dx}\right)\right] \, dx \\
&= i \hbar \cdot \int_{-\infty}^{\infty} \psi^*(x) \psi(x) \, dx \\
&= i \hbar \cdot \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx \\
&= i \hbar\end{align}
},
{
"math_input": \begin{align}
\big|\operatorname{E}f(Y_n) - \operatorname{E}f(X_n)\big|
&\leq \operatorname{E}|f(Y_n) - f(X_n)| \\
&= \operatorname{E}\Big[|f(Y_n) - f(X_n)|\mathbf{1}_{\{|Y_n-X_n|<\varepsilon\}}\Big] +
\operatorname{E}\Big[|f(Y_n) - f(X_n)|\mathbf{1}_{\{|Y_n-X_n|\geq\varepsilon\}}\Big] \\
&\leq \operatorname{E}\Big[K|Y_n - X_n|\mathbf{1}_{\{|Y_n-X_n|<\varepsilon\}}\Big] +
\operatorname{E}\Big[2M\mathbf{1}_{\{|Y_n-X_n|\geq\varepsilon\}}\Big] \\
&\leq K\varepsilon\, \operatorname{Pr}\big[|Y_n-X_n|<\varepsilon\big] +
2M\, \operatorname{Pr}\big[|Y_n-X_n|\geq\varepsilon\big] \\
&\leq K\varepsilon + 2M\, \operatorname{Pr}\big[|Y_n-X_n|\geq\varepsilon\big]
\end{align}
},
{
"math_input":
\begin{array}{lll}
& GS_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
1 & . & . & . & . & . & . \\
. & 3 & . & . & . & . & . \\
. & . & 6 & . & . & . & . \\
. & . & . & 10 & . & . & . \\
. & . & . & . & 15 & . & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & . & . & . & . & . & . \\
. & 1 & . & . & . & . & . \\
1 & . & 1 & . & . & . & . \\
. & 3 & . & 1 & . & . & . \\
3 & . & 6 & . & 1 & . & . \\
. & 15 & . & 10 & . & 1 & . \\
15 & . & 45 & . & 15 & . & 1
\end{smallmatrix}
\right ]
;\quad
\end{array}
},
{
"math_input":
\begin{align}
\frac{\pi}{6} \rho_p d_p^3 \frac{\text{d} \boldsymbol{U}_p}{\text{d} t}
&= \underbrace{3 \pi \mu d_p \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 1}}
- \underbrace{\frac{\pi}{6} d_p^3 \boldsymbol{\nabla} p}_{\text{term 2}}
+ \underbrace{\frac{\pi}{12} \rho_f d_p^3\,
\frac{\text{d}}{\text{d} t} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)}_{\text{term 3}}
\\ &
+ \underbrace{\frac{3}{2} d_p^2 \sqrt{\pi \rho_f \mu}
\int_{t_{_0}}^t \frac{1}{\sqrt{t-\tau}}\, \frac{\text{d}}{\text{d} \tau} \left( \boldsymbol{U}_f - \boldsymbol{U}_p \right)\,
\text{d} \tau}_{\text{term 4}}
+ \underbrace{\sum_k \boldsymbol{F}_k}_{\text{term 5}} .
\end{align}
},
{
"math_input":
\begin{align}
(x_N * y)[n]\ &=\ \int_{0}^{1} \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\
&=\ \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot \int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\
&=\ \frac{1}{N} \sum_{k=0}^{N-1} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot e^{i 2 \pi \frac{n}{N} k}\\
&=\ \scriptstyle{DFT}^{-1} \displaystyle \big[\ \scriptstyle{DFT}\displaystyle \{x_N\}\cdot \ \scriptstyle{DFT}\displaystyle \{y_N\}\ \big],
\end{align}
},
{
"math_input":
\begin{align}
(x_N * y)[n] & = \int_{0}^{1} \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df \\
& = \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot \int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df \\
& = \frac{1}{N} \sum_{k=0}^{N-1} \scriptstyle{DFT}\displaystyle\{x_N\}[k]\cdot \scriptstyle{DFT}\displaystyle\{y_N\}[k]\cdot e^{i 2 \pi \frac{n}{N} k} \\
& = \scriptstyle{DFT}^{-1} \displaystyle \big[ \scriptstyle{DFT}\displaystyle \{x_N\}\cdot \scriptstyle{DFT}\displaystyle \{y_N\} \big],
\end{align}
},
{
"math_input": p_n=\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1 & 0 & \cdots & 2e_2 \\ e_2 & e_1 & 1& & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
\\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{\begin{vmatrix}1 & 0 & \cdots & \\ e_1 & 1 & 0 & \cdots \\ e_2 & e_1 & 1& \\ \vdots&&\ddots&\ddots
\\ e_{n-1} & \cdots & e_2 & e_1 & (-1)^{n-1} \end{vmatrix}}
=\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1 & 0 & \cdots & 2e_2 \\ e_2 & e_1 & 1& & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
\\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{(-1)^{n-1}}
=\begin{vmatrix}e_1 & 1 & 0 & \cdots\\ 2e_2 & e_1 & 1 & 0 & \cdots\\ 3e_3 & e_2 & e_1 & 1 \\ \vdots &&& \ddots & \ddots
\\ ne_n & e_{n-1} & \cdots & & e_1 \end{vmatrix}.
},
{
"math_input": \begin{matrix}
\left(\Delta A_r - {2 A_r \over r^2}
- {2 \over r^2\sin\theta}{\partial \left(A_\theta \sin\theta\right) \over \partial\theta}
- {2 \over r^2\sin\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat r} & + \\
\left(\Delta A_\theta - {A_\theta \over r^2\sin^2\theta}
+ {2 \over r^2}{\partial A_r \over \partial \theta}
- {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\phi \over \partial \phi}\right) \boldsymbol{\hat\theta} & + \\
\left(\Delta A_\phi - {A_\phi \over r^2\sin^2\theta}
+ {2 \over r^2\sin\theta}{\partial A_r \over \partial \phi}
+ {2 \cos\theta \over r^2\sin^2\theta}{\partial A_\theta \over \partial \phi}\right) \boldsymbol{\hat\phi} & \end{matrix}
},
{
"math_input": \begin{align}
\boldsymbol{a} &=\frac{\operatorname{d}^2\boldsymbol{r}}{\operatorname{d}t^2} = \frac{\operatorname{d}}{\operatorname{d}t}\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t} = \frac{\operatorname{d}}{\operatorname{d}t} \left( \left[\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t}\right] + \boldsymbol{\omega} \times \boldsymbol{r}\ \right) \\
&= \left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] + \frac{\operatorname{d} \boldsymbol{\omega}}{\operatorname{d}t}\times\boldsymbol{r} + 2 \boldsymbol{\omega}\times \left[ \frac{\operatorname{d} \boldsymbol{r}}{\operatorname{d}t} \right] + \boldsymbol{\omega}\times ( \boldsymbol{\omega} \times \boldsymbol{r}) \ .
\end{align}
},
{
"math_input":
\begin{align}
\vec{a}_A & = \sum_{B \not = A} \frac{G m_B \vec{n}_{BA}}{r_{AB}^2} \\
& {} \quad{} + \frac{1}{c^2} \sum_{B \not = A}
\frac{G m_B \vec{n}_{BA}}{r_{AB}^2}
\left[ v_A^2+2v_B^2 - 4( \vec{v}_A \cdot \vec{v}_B) - \frac{3}{2} ( \vec{n}_{AB} \cdot \vec{v}_B)^2 \right. \\
& {} \qquad {} \left. {} -
4 \sum_{C \not = A} \frac{G m_C}{r_{AC}} -
\sum_{C \not = B} \frac{G m_C}{r_{BC}} + \frac{1}{2}( (\vec{x}_B-\vec{x}_A) \cdot \vec{a}_B ) \right] \\
& {}\quad{} + \frac{1}{c^2} \sum_{B \not = A} \frac{G m_B}{r_{AB}^2}\left[\vec{n}_{AB}\cdot(4\vec{v}_A-3\vec{v}_B)\right](\vec{v}_A-\vec{v}_B) \\
& {} \quad {} + \frac{7}{2c^2} \sum_{B \not = A}{ \frac{G m_B \vec{a}_B }{r_{AB}}}
\end{align}
},
{
"math_input":
\begin{array}{lcl}
N,T &=& \text{as above} \\
\theta_{i=1 \dots N}, \phi_{i=1 \dots N, j=1 \dots N}, \boldsymbol\phi_{i=1 \dots N} &=& \text{as above} \\
x_{t=1 \dots T}, y_{t=1 \dots T}, F(y|\theta) &=& \text{as above} \\
\alpha &=& \text{shared hyperparameter for emission parameters} \\
\beta &=& \text{shared hyperparameter for transition parameters} \\
H(\theta|\alpha) &=& \text{prior probability distribution of emission parameters, parametrized on } \alpha \\
\theta_{i=1 \dots N} &\sim& H(\alpha) \\
\boldsymbol\phi_{i=1 \dots N} &\sim& \operatorname{Symmetric-Dirichlet}_N(\beta) \\
x_{t=2 \dots T} &\sim& \operatorname{Categorical}(\boldsymbol\phi_{x_{i-t}}) \\
y_{t=1 \dots T} &\sim& F(\theta_{x_t})
\end{array}
},
{
"math_input": \Sigma=\begin{bmatrix}
{Var \left (X_{1(1)} \right)} & {Cov \left (X_{1(1)},X_{1(2)} \right)} & Cov \left (X_{1(1)},X_{1(3)} \right) & \cdots & Cov \left (X_{1(1)},X_{1(k)} \right) \\
{Cov \left (X_{1(2)},X_{1(1)} \right)} & {Var \left (X_{1(2)} \right)} & {Cov \left(X_{1(2)},X_{1(3)} \right)} & \cdots & Cov \left(X_{1(2)},X_{1(k)} \right) \\
Cov \left (X_{1(3)},X_{1(1)} \right) & {Cov \left (X_{1(3)},X_{1(2)} \right)} & Var \left (X_{1(3)} \right) & \cdots & Cov \left (X_{1(3)},X_{1(k)} \right) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
Cov \left (X_{1(k)},X_{1(1)} \right) & Cov \left (X_{1(k)},X_{1(2)} \right) & Cov \left (X_{1(k)},X_{1(3)} \right) & \cdots & Var \left (X_{1(k)} \right) \\
\end{bmatrix}.
},
{
"math_input": \begin{align}
\frac{\partial u}{\partial t} &= \frac{\partial^2 u}{\partial x^2} &&\text{in } \{(t,x): 0 < x < s(t), t>0\}, && \text{the heat equation},\\
-\frac{\partial u}{\partial x}(0, t) &= f(t), && t>0, &&\text{the Neumann condition at the left end of the ice block describing the given heat flux}, &&\\
u\big(s(t),t\big) &= 0, && t>0, &&\text{the Dirichlet condition at the right end of the block setting the temperature to that of melting/freezing},\\
\frac{\mathrm{d}s}{\mathrm{d}t} &= -\frac{\partial u}{\partial x}\big(s(t), t\big), && t>0, &&\text{Stefan condition},\\
u(x,0) &= 0, && x\geq 0, &&\text{initial temperature distribution},\\
s(0) &= 0, && &&\text{initial melt depth of the ice block}.
\end{align}
},
{
"math_input": \forall (\alpha,\beta)\ \mathbf T \left( \alpha
\begin{bmatrix}
\vec f \\
\ \\
0 \\
\end{bmatrix} + \beta
\begin{bmatrix}
0 \\
\ \\
\vec b \\
\end{bmatrix} \right ) = \alpha
\begin{bmatrix} 1 \\
0 \\
\vdots \\
0 \\
\epsilon_f \\
\end{bmatrix} + \beta
\begin{bmatrix} \epsilon_b \\
0 \\
\vdots \\
0 \\
1 \\
\end{bmatrix}.
},
{
"math_input": \begin{align}
\sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n A_{i,\sigma_i}
&=\sgn([1,2,3]) \prod_{i=1}^n A_{i,[1,2,3]_i} + \sgn([1,3,2]) \prod_{i=1}^n A_{i,[1,3,2]_i} + \sgn([2,1,3]) \prod_{i=1}^n A_{i,[2,1,3]_i} \\ &+ \sgn([2,3,1]) \prod_{i=1}^n A_{i,[2,3,1]_i} + \sgn([3,1,2]) \prod_{i=1}^n A_{i,[3,1,2]_i} + \sgn([3,2,1]) \prod_{i=1}^n A_{i,[3,2,1]_i}
\\
&=\prod_{i=1}^n A_{i,[1,2,3]_i} - \prod_{i=1}^n A_{i,[1,3,2]_i} - \prod_{i=1}^n A_{i,[2,1,3]_i} + \prod_{i=1}^n A_{i,[2,3,1]_i} + \prod_{i=1}^n A_{i,[3,1,2]_i} - \prod_{i=1}^n A_{i,[3,2,1]_i}
\\
&=A_{1,1}A_{2,2}A_{3,3}-A_{1,1}A_{2,3}A_{3,2}-A_{1,2}A_{2,1}A_{3,3}+A_{1,2}A_{2,3}A_{3,1} \\
& \qquad +A_{1,3}A_{2,1}A_{3,2}-A_{1,3}A_{2,2}A_{3,1}.
\end{align}
},
{
"math_input": \begin{align}
\sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n a_{i,\sigma_i}
&=\sgn([1,2,3]) \prod_{i=1}^n a_{i,[1,2,3]_i} + \sgn([1,3,2]) \prod_{i=1}^n a_{i,[1,3,2]_i} + \sgn([2,1,3]) \prod_{i=1}^n a_{i,[2,1,3]_i} \\ &+ \sgn([2,3,1]) \prod_{i=1}^n a_{i,[2,3,1]_i} + \sgn([3,1,2]) \prod_{i=1}^n a_{i,[3,1,2]_i} + \sgn([3,2,1]) \prod_{i=1}^n a_{i,[3,2,1]_i}
\\
&=\prod_{i=1}^n a_{i,[1,2,3]_i} - \prod_{i=1}^n a_{i,[1,3,2]_i} - \prod_{i=1}^n a_{i,[2,1,3]_i} + \prod_{i=1}^n a_{i,[2,3,1]_i} + \prod_{i=1}^n a_{i,[3,1,2]_i} - \prod_{i=1}^n a_{i,[3,2,1]_i}
\\
&=a_{1,1}a_{2,2}a_{3,3}-a_{1,1}a_{2,3}a_{3,2}-a_{1,2}a_{2,1}a_{3,3}+a_{1,2}a_{2,3}a_{3,1} \\
& \qquad +a_{1,3}a_{2,1}a_{3,2}-a_{1,3}a_{2,2}a_{3,1}.
\end{align}
},
{
"math_input": \Sigma=\begin{bmatrix}
{Var \left (X_{1(1)} \right)} & {Cov \left (X_{1(1)},X_{1(2)} \right)} & Cov \left (X_{1(1)},X_{1(3)} \right) & \cdots & Cov \left (X_{1(1)},X_{1(k)} \right) \\
{Cov \left (X_{1(2)},X_{1(1)} \right)} & {Var \left (X_{1(2)} \right)} & {Cov \left(X_{1(2)},X_{1(3)} \right)} & \cdots & Cov \left(X_{1(2)},X_{1(k)} \right) \\
Cov \left (X_{1(3)},X_{1(1)} \right) & {Cov \left (X_{1(3)},X_{1(2)} \right)} & Var \left (X_{1(3)} \right) & \cdots & Cov \left (X_{1(3)},X_{1(k)} \right) \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
Cov \left (X_{1(k)},X_{1(1)} \right) & Cov \left (X_{1(k)},X_{1(2)} \right) & Cov \left (X_{1(k)},X_{1(3)} \right) & \cdots & Var \left (X_{1(k)} \right) \\
\end{bmatrix}.
},
{
"math_input": \begin{align}
\boldsymbol{r}_i&=\boldsymbol{b}-\boldsymbol{Ax}_i\\
&=\boldsymbol{b}-\boldsymbol{A}(\boldsymbol{x}_0+\boldsymbol{V}_i\boldsymbol{y}_i)\\
&=\boldsymbol{r}_0-\boldsymbol{AV}_i\boldsymbol{y}_i\\
&=\boldsymbol{r}_0-\boldsymbol{V}_{i+1}\boldsymbol{\tilde{H}}_i\boldsymbol{y}_i\\
&=\boldsymbol{r}_0-\boldsymbol{V}_i\boldsymbol{H}_i\boldsymbol{y}_i-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\\
&=\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{v}_1-\boldsymbol{V}_i(\lVert\boldsymbol{r}_0\rVert_2\boldsymbol{e}_1)-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\\
&=-h_{i+1,i}(\boldsymbol{e}_i^\mathrm{T}\boldsymbol{y}_i)\boldsymbol{v}_{i+1}\text{.}\end{align}
},
{
"math_input": \left\{\begin{array}{l}
(x-\mu ) f''(x) \left(\delta^2+(x-\mu)^2\right)+f'(x) \left(-\delta ^2-2 \beta
(x-\mu) \left(\delta^2+(x-\mu)^2\right)\right)+f(x) \left(\alpha ^2 \mu ^3-\beta^2 \mu
\left(\delta^2+\mu^2\right)+\beta \delta^2+x^3 \left(\beta^2-\alpha^2\right)+
3 \mu x^2 (\alpha-\beta) (\alpha+\beta)+3 \mu^2 x \left(\beta^2-\alpha^2\right)+
\beta^2 \delta^2 x\right)=0, \\
f(0)=\frac{\gamma e^{\alpha \left(-\sqrt{\delta ^2+\mu^2}\right)-\beta \mu}}
{2 \alpha \delta K_1(\gamma \delta)}, \\
f'(0)=\frac{\gamma e^{\alpha \left(-\sqrt{\delta^2+\mu^2}\right)-\beta \mu}
\left(\alpha \mu+\beta \sqrt{\delta^2+\mu^2}\right)}{2 \alpha \delta
\sqrt{\delta^2+\mu^2} K_1(\gamma \delta)}
\end{array}\right\}
},
{
"math_input":
\begin{align}
\cfrac{\partial W}{\partial\boldsymbol{C}} & =
\cfrac{\partial W}{\partial \lambda_1}~\cfrac{\partial \lambda_1}{\partial\boldsymbol{C}} +
\cfrac{\partial W}{\partial \lambda_2}~\cfrac{\partial \lambda_2}{\partial\boldsymbol{C}} +
\cfrac{\partial W}{\partial \lambda_3}~\cfrac{\partial \lambda_3}{\partial\boldsymbol{C}} \\
& = \boldsymbol{R}^T\cdot\left[\cfrac{1}{2\lambda_1}~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
\cfrac{1}{2\lambda_2}~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
\cfrac{1}{2\lambda_3}~\cfrac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]\cdot\boldsymbol{R}
\end{align}
},
{
"math_input":
\begin{align}
\operatorname{var}\left(\frac{k}{n}\right) & =
\operatorname{E}\left[\operatorname{var}\left(\left.\frac{k}{n}\right|\theta\right)\right] +
\operatorname{var}\left[\operatorname{E}\left(\left.\frac{k}{n}\right|\theta\right)\right] \\
& =
\operatorname{E}\left[\left(\left.\frac{1}{n}\right)\theta(1-\theta)\right|\mu,M\right] +
\operatorname{var}\left(\theta|\mu,M\right) \\
& =
\frac{1}{n}\left(\mu(1-\mu)\right) + \frac{n_{i}-1}{n_{i}}\frac{(\mu(1-\mu))}{M+1} \\
& =
\frac{\mu(1-\mu)}{n}\left(1+\frac{n-1}{M+1}\right).
\end{align}
},
{
"math_input":
\begin{align}
\cfrac{\partial W}{\partial\boldsymbol{C}} & =
\cfrac{\partial W}{\partial \lambda_1}~\cfrac{\partial \lambda_1}{\partial\boldsymbol{C}} +
\cfrac{\partial W}{\partial \lambda_2}~\cfrac{\partial \lambda_2}{\partial\boldsymbol{C}} +
\cfrac{\partial W}{\partial \lambda_3}~\cfrac{\partial \lambda_3}{\partial\boldsymbol{C}} \\
& = \boldsymbol{R}^T\cdot\left[\cfrac{1}{2\lambda_1}~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
\cfrac{1}{2\lambda_2}~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
\cfrac{1}{2\lambda_3}~\cfrac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]\cdot\boldsymbol{R}
\end{align}
},
{
"math_input": \begin{align}
\left(\Delta A_r - \frac{2 A_r}{r^2}
- \frac{2}{r^2\sin\theta} \frac{\partial \left(A_\theta \sin\theta\right)}{\partial\theta}
- \frac{2}{r^2\sin\theta}{\frac{\partial A_\phi}{\partial \phi}}\right) &\boldsymbol{\hat{r}} \\
+ \left(\Delta A_\theta - \frac{A_\theta}{r^2\sin^2\theta}
+ \frac{2}{r^2} \frac{\partial A_r}{\partial \theta}
- \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\phi}{\partial \phi}\right) &\boldsymbol{\hat{\theta}} \\
+ \left(\Delta A_\phi - \frac{A_\phi}{r^2\sin^2\theta}
+ \frac{2}{r^2\sin\theta} \frac{\partial A_r}{\partial \phi}
+ \frac{2 \cos\theta}{r^2\sin^2\theta} \frac{\partial A_\theta}{\partial \phi}\right) &\boldsymbol{\hat{\phi}}
\end{align}
},
{
"math_input":
\begin{array}{lll}
& LAH_7=\exp
\left (
\left [
\begin{smallmatrix}
. & . & . & . & . & . & . \\
2 & . & . & . & . & . & . \\
. & 6 & . & . & . & . & . \\
. & . &12 & . & . & . & . \\
. & . & . & 20 & . & . & . \\
. & . & . & . & 30 & . & . \\
. & . & . & . & . & 42 & .
\end{smallmatrix}
\right ]
\right )
=
\left [
\begin{smallmatrix}
1 & . & . & . & . & . & . & . \\
2 & 1 & . & . & . & . & . & . \\
6 & 6 & 1 & . & . & . & . & . \\
24 & 36 & 12 & 1 & . & . & . & . \\
120 & 240 & 120 & 20 & 1 & . & . & . \\
720 & 1800 & 1200 & 300 & 30 & 1 & . & . \\
5040 & 15120 & 12600 & 4200 & 630 & 42 & 1 & . \\
40320 & 141120 & 141120 & 58800 & 11760 & 1176 & 56 & 1
\end{smallmatrix}
\right ]
;\quad
\end{array}
},
{
"math_input": \begin{array}{rccrcrcrcr}
{\color{Brown}{P}} {\color{RoyalBlue}{Q}} & {{=}}&&({\color{Brown}{2x}}\cdot{\color{RoyalBlue}{2x}})
&+&({\color{Brown}{2x}}\cdot{\color{RoyalBlue}{5y}})&+&({\color{Brown}{2x}}\cdot {\color{RoyalBlue}{xy}})&+&({\color{Brown}{2x}}\cdot{\color{RoyalBlue}{1}})
\\&&+&({\color{Brown}{3y}}\cdot{\color{RoyalBlue}{2x}})&+&({\color{Brown}{3y}}\cdot{\color{RoyalBlue}{5y}})&+&({\color{Brown}{3y}}\cdot {\color{RoyalBlue}{xy}})&+&
({\color{Brown}{3y}}\cdot{\color{RoyalBlue}{1}})
\\&&+&({\color{Brown}{5}}\cdot{\color{RoyalBlue}{2x}})&+&({\color{Brown}{5}}\cdot{\color{RoyalBlue}{5y}})&+&
({\color{Brown}{5}}\cdot {\color{RoyalBlue}{xy}})&+&({\color{Brown}{5}}\cdot{\color{RoyalBlue}{1}})
\end{array}
},
{
"math_input":
\begin{align}
& T(x_1,\dots,x_d)\\
= {} & \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty
\frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d) \\
= {} & f(a_1, \dots,a_d) + \sum_{j=1}^d \frac{\partial f(a_1, \dots,a_d)}{\partial x_j} (x_j - a_j) \\
& {} + \frac{1}{2!} \sum_{j=1}^d \sum_{k=1}^d \frac{\partial^2 f(a_1, \dots,a_d)}{\partial x_j \partial x_k} (x_j - a_j)(x_k - a_k) \\
& {} + \frac{1}{3!} \sum_{j=1}^d\sum_{k=1}^d\sum_{l=1}^d \frac{\partial^3 f(a_1, \dots,a_d)}{\partial x_j \partial x_k \partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \dots
\end{align}
},
{
"math_input": \begin{bmatrix}
\sin^2 (\theta+\alpha)K_n & -K_n \sin(\theta+\alpha)\cos(\theta+\alpha) & \cos(\theta+\alpha)K_s L\sin(\alpha) \\
+\cos^2(\theta+\alpha)K_s & +K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & -\sin(\theta+\alpha)K_n L\cos(\alpha) \\
\\
-K_n\sin(\theta+\alpha)\cos(\theta+\alpha) & \sin^2(\theta+\alpha)K_s & \cos(\theta+\alpha)K_n L\cos(\alpha) \\
+K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & +\cos^2(\theta+\alpha)K_n & +\sin(\theta+\alpha)K_s L\sin(\alpha) \\
\\
\cos(\theta+\alpha)K_s L\sin(\alpha) & \cos(\theta+\alpha)K_n L\cos(\alpha) & L^2\cos^2(\alpha)K_n \\
-\sin(\theta+\alpha)K_n L\cos(\alpha) & +\sin(\theta+\alpha)K_s L\sin(\alpha) & +L^2\sin^2(\alpha)K_s
\end{bmatrix}
},
{
"math_input":
\begin{align}
&\int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^3\ \sin^3 u\ du\ =
-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin^4 u \ du\
+\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin^3 u\ \cos u \ du\ = \\
&-\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^4 u \ du\ +\
3\ {e_g}^2\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^4 u \ du\ \ +\
3\ {e_h}^2\ \int\limits_{0}^{2\pi}\ \sin^6 u \ du\ \right) \\
&+\hat{h}\ 6\ e_g\ e_h\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^4 u \ du = \\
&-\hat{g}\ \left(2\pi \left(\frac{3}{8}\ +\ \frac{3}{16}\ {e_g}^2\ +\ \frac{15}{16}\ {e_h}^2\right)\right)
+\hat{h}\ \left(2\pi \left(\frac{3}{8}\ e_g\ e_h\right)\right)
\end{align}
},
{
"math_input":
\begin{align}
T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty
\frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d) \\
&= f(a_1, \dots,a_d) + \sum_{j=1}^d \frac{\partial f(a_1, \dots,a_d)}{\partial x_j} (x_j - a_j) \\
&\quad {} + \frac{1}{2!} \sum_{j=1}^d \sum_{k=1}^d \frac{\partial^2 f(a_1, \dots,a_d)}{\partial x_j \partial x_k} (x_j - a_j)(x_k - a_k) \\
&\quad {} + \frac{1}{3!} \sum_{j=1}^d\sum_{k=1}^d\sum_{l=1}^d \frac{\partial^3 f(a_1, \dots,a_d)}{\partial x_j \partial x_k \partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \dots
\end{align}
},
{
"math_input":
\begin{array}{ l l}
f(x,y) \approx & \, \frac{f(Q_{11})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y_2-y) \, + \\
& \, \frac{f(Q_{21})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y_2-y) \, + \\
& \, \frac{f(Q_{12})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y-y_1) \, + \\
& \, \frac{f(Q_{22})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y-y_1) \\
\qquad = & \, \frac{1}{(x_2-x_1)(y_2-y_1)} \Big( f(Q_{11})(x_2-x)(y_2-y) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{21})(x-x_1)(y_2-y) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{12})(x_2-x)(y-y_1) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{22})(x-x_1)(y-y_1) \quad \Big)
\end{array}
},
{
"math_input":
\begin{align}
b_N &= b_0 + \frac{1}{2} \operatorname{E}_\mu \left[\sum_{n=1}^N (x_n-\mu)^2 + \lambda_0(\mu - \mu_0)^2\right] \\
&= b_0 + \frac{1}{2} \operatorname{E}_\mu \left[ (\lambda_0+N)\mu^2 -2(\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n)\mu + (\textstyle\sum_{n=1}^N x_n^2) + \lambda_0\mu_0^2 \right] \\
&= b_0 + \frac{1}{2} \left[ (\lambda_0+N)\operatorname{E}_\mu[\mu^2] -2(\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n)\operatorname{E}_\mu[\mu] + (\textstyle\sum_{n=1}^N x_n^2) + \lambda_0\mu_0^2 \right] \\
&= b_0 + \frac{1}{2} \left[ (\lambda_0+N)(\lambda_N^{-1} + \mu_N^2) -2(\lambda_0\mu_0 + \textstyle\sum_{n=1}^N x_n)\mu_N + (\textstyle\sum_{n=1}^N x_n^2) + \lambda_0\mu_0^2 \right] \\
\end{align}
},
{
"math_input": \begin{bmatrix}
\sin^2 (\theta+\alpha)K_n & -K_n \sin(\theta+\alpha)\cos(\theta+\alpha) & \cos(\theta+\alpha)K_s L\sin(\alpha) \\
+\cos^2(\theta+\alpha)K_s & +K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & -\sin(\theta+\alpha)K_n L\cos(\alpha) \\
\\
-K_n\sin(\theta+\alpha)\cos(\theta+\alpha) & \sin^2(\theta+\alpha)K_s & \cos(\theta+\alpha)K_n L\cos(\alpha) \\
+K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & +\cos^2(\theta+\alpha)K_n & +\sin(\theta+\alpha)K_s L\sin(\alpha) \\
\\
\cos(\theta+\alpha)K_s L\sin(\alpha) & \cos(\theta+\alpha)K_n L\cos(\alpha) & L^2\cos^2(\alpha)K_n \\
-\sin(\theta+\alpha)K_n L\cos(\alpha) & +\sin(\theta+\alpha)K_s L\sin(\alpha) & +L^2\sin^2(\alpha)K_s
\end{bmatrix}
},
{
"math_input": \begin{align}
H_2 = \frac{1}{2}
&\begin{pmatrix}\begin{array}{rrrr}
1 & 1 & 1 & 1\\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1\\
1 & -1 & -1 & 1
\end{array}\end{pmatrix}\\
H_3 = \frac{1}{2^{\frac{3}{2}}}
&\begin{pmatrix}\begin{array}{rrrrrrrr}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1
\end{array}\end{pmatrix}\\
(H_n)_{i,j} = \frac{1}{2^{\frac{n}{2}}} &(-1)^{i \cdot j}
\end{align}
},
{
"math_input":
\begin{bmatrix}\sigma_{11}\\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} =
\begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\
c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\
c_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\
c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\
c_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\
c_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212}
\end{bmatrix}
\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix}
},
{
"math_input": \begin{align}
y_1 &= b_0 x^{1 - \gamma} \left(\frac{(\alpha + 1 - \gamma)_{\gamma - 1} (\beta + 1 - \gamma)_{\gamma - 1}}
{(2 - \gamma)_{\gamma - 2} (1)_{\gamma - 1}} x^{\gamma - 1}
+ \frac{(\alpha + 1 - \gamma)_{\gamma} (\beta + 1 - \gamma)_{\gamma}}
{(2 - \gamma)_{\gamma - 2} (1) (1)_{\gamma}} x^{\gamma}
+ \cdots \right) \\
&= \frac{b_0}{(2 - \gamma)_{\gamma - 2}} x^{1 - \gamma}
\sum_{r = \gamma - 1}^\infty \frac{(\alpha + 1 - \gamma)_r (\beta + 1 - \gamma)_r}
{(1)_r (1)_{r + 1 - \gamma}} x^r.
\end{align}
},
{
"math_input":
\begin{align}
\frac{\partial \mathcal{L}}{\partial f}
& - \frac{\partial}{\partial x_1}\left(\frac{\partial \mathcal{L}}{\partial f_{,1}}\right)
- \frac{\partial}{\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{,2}}\right)
+ \frac{\partial^2}{\partial x_1^2}\left(\frac{\partial \mathcal{L}}{\partial f_{,11}}\right)
+ \frac{\partial^2}{\partial x_1\partial x_2}\left(\frac{\partial \mathcal{L}}{\partial f_{,12}}\right)
+ \frac{\partial^2}{\partial x_2^2}\left(\frac{\partial \mathcal{L}}{\partial f_{,22}}\right) \\
& - \dots
+ (-1)^n \frac{\partial^n}{\partial x_2^n}\left(\frac{\partial \mathcal{L}}{\partial f_{,22\dots 2}}\right) = 0
\end{align}
},
{
"math_input": \begin{align}
H \star W &= \left(\frac{1}{2}m \omega^2 x^2 + \frac{p^2}{2m}\right) \star W \\
&= \left(\frac{1}{2}m \omega^2 \left( x+\frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{p} \right)^2 + \frac{1}{2m}\left(p - \frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{x}\right)^2\right) ~ W\\
&= \left( \frac{1}{2}m \omega^2 \left(x^2 - \frac{\hbar^2}{4} \stackrel{\rightarrow }{\partial }_{p}^2 \right) + \frac{1}{2m}\left( p^2 - \frac{\hbar^2}{4} \stackrel{\rightarrow }{\partial }_{x}^2 \right) \right) ~ W\\
&\, \, \, \, \, + \frac{i \hbar}{2} \left(m \omega^2 x \stackrel{\rightarrow }{\partial }_{p} - \frac{p}{m} \stackrel{\rightarrow }{\partial }_{x}\right) ~ W \\
&= E \cdot W.
\end{align}
},
{
"math_input": \begin{align}
\Rightarrow I(n) &= \int_0^\pi \sin^nxdx=\int_0^\pi u dv = uv |_{x=0}^{x=\pi}-\int_0^\pi v du \\
{} &= -\sin^{n-1}x\cos x |_{x=0}^{x=\pi} - \int_0^\pi - \cos x(n-1) \sin^{n-2}x \cos x dx \\
{} &= 0 - (n-1) \int_0^\pi -\cos^2x \sin^{n-2}x dx, n > 1 \\
{} &= (n - 1) \int_0^\pi (1-\sin^2 x) \sin^{n-2}x dx \\
{} &= (n - 1) \int_0^\pi \sin^{n-2}x dx - (n - 1) \int_0^\pi \sin^{n}x dx \\
{} &= (n - 1) I(n-2)-(n-1) I(n) \\
{} &= \frac{n-1}{n} I(n-2) \\
\Rightarrow \frac{I(n)}{I(n-2)}
&= \frac{n-1}{n} \\
\Rightarrow \frac{I(2n-1)}{I(2n+1)}
&=\frac{2n+1}{2n}
\end{align}
},
{
"math_input":
\begin{align}
{{\partial w }\over{\partial t }} &\sim \frac{W}{T} \\[1.2ex]
u {\frac{\partial w}{\partial x}} &\sim U\frac{W}{L} &\qquad
v {\frac{\partial w}{\partial y}} &\sim U\frac{W}{L} &\qquad
w {\frac{\partial w}{\partial z}} &\sim W\frac{W}{H} \\[1.2ex]
{\frac{u^2}{R}} &\sim \frac{U^2}{R} &\qquad
{\frac{v^2}{R}} &\sim \frac{U^2}{R} \\[1.2ex]
\frac{1}{\varrho}\frac{\partial p}{\partial z} &\sim \frac{1}{\varrho}\frac{\Delta P}{H} &\qquad
\Omega u \cos \varphi &\sim \Omega U \\[1.2ex]
\nu \frac{\partial^2 w}{\partial x^2} &\sim \nu \frac{W}{L^2} &\qquad
\nu \frac{\partial^2 w}{\partial y^2} &\sim \nu \frac{W}{L^2} &\qquad
\nu \frac{\partial^2 w}{\partial z^2} &\sim \nu \frac{W}{H^2}
\end{align}
},
{
"math_input":
\begin{array}{|rlr|rlr|rlr|}
\hline
\alpha & \mathrm{alpha} & 1 & \iota & \mathrm{iota} & 10 & \varrho & \mathrm{rho} & 100 \\ \beta & \mathrm{beta} & 2 & \kappa & \mathrm{kappa} & 20 & & & \\ \gamma & \mathrm{gamma} & 3 & \lambda & \mathrm{lambda} & 30 & & & \\ \delta & \mathrm{delta} & 4 & \mu & \mathrm{mu} & 40 & & & \\ \varepsilon & \mathrm{epsilon} & 5 & \nu & \mathrm{nu} & 50 & & & \\ \stigma & \mathrm{stigma\ (archaic)} & 6 & \xi & \mathrm{xi} & 60 & & & \\ \zeta & \mathrm{zeta} & 7 & \omicron & \mathrm{omicron} & 70 & & & \\ \eta & \mathrm{eta} & 8 & \pi & \mathrm{pi} & 80 & & & \\ \vartheta & \mathrm{theta} & 9 & \koppa & \mathrm{koppa\ (archaic)} & 90 & & & \\ \hline
\end{array}
},
{
"math_input": \boldsymbol{\sigma}= \sigma_{ij} = \left[{\begin{matrix} \mathbf{T}^{(\mathbf{e}_1)} \\
\mathbf{T}^{(\mathbf{e}_2)} \\
\mathbf{T}^{(\mathbf{e}_3)} \\
\end{matrix}}\right] =
\left[{\begin{matrix}
\sigma _{11} & \sigma _{12} & \sigma _{13} \\
\sigma _{21} & \sigma _{22} & \sigma _{23} \\
\sigma _{31} & \sigma _{32} & \sigma _{33} \\
\end{matrix}}\right] \equiv \left[{\begin{matrix}
\sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\
\sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\
\sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\
\end{matrix}}\right] \equiv \left[{\begin{matrix}
\sigma _x & \tau _{xy} & \tau _{xz} \\
\tau _{yx} & \sigma _y & \tau _{yz} \\
\tau _{zx} & \tau _{zy} & \sigma _z \\
\end{matrix}}\right],
},
{
"math_input":
\begin{bmatrix}
\mathbf{e}_1'(t) \\
\mathbf{e}_2'(t) \\
\vdots \\
\mathbf{e}_{n-1}'(t) \\
\mathbf{e}_n'(t) \\
\end{bmatrix}
=
\left\Vert \gamma'\left(t\right) \right\Vert
\begin{bmatrix}
0 & \chi_1(t) & \cdots & 0 & 0 \\
-\chi_1(t) & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 0 & \chi_{n-1}(t) \\
0 & 0 & \cdots & -\chi_{n-1}(t) & 0 \\
\end{bmatrix}
\begin{bmatrix}
\mathbf{e}_1(t) \\
\mathbf{e}_2(t) \\
\vdots \\
\mathbf{e}_{n-1}(t) \\
\mathbf{e}_n(t) \\
\end{bmatrix}
},
{
"math_input":
\begin{array}{ l l}
f(x,y) \approx & \, \frac{f(Q_{11})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y_2-y) \, + \\
& \, \frac{f(Q_{21})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y_2-y) \, + \\
& \, \frac{f(Q_{12})}{(x_2-x_1)(y_2-y_1)} (x_2-x)(y-y_1) \, + \\
& \, \frac{f(Q_{22})}{(x_2-x_1)(y_2-y_1)} (x-x_1)(y-y_1) \\
= & \, \frac{1}{(x_2-x_1)(y_2-y_1)} \Big( f(Q_{11})(x_2-x)(y_2-y) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{21})(x-x_1)(y_2-y) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{12})(x_2-x)(y-y_1) \, + \\
& \, \qquad \qquad \qquad \qquad \; \; f(Q_{22})(x-x_1)(y-y_1) \quad \Big)
\end{array}
},
{
"math_input": \begin{align}
\sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n A_{i,\sigma_i}
&=\sgn([1,2,3]) \prod_{i=1}^n A_{i,[1,2,3]_i} + \sgn([1,3,2]) \prod_{i=1}^n A_{i,[1,3,2]_i} + \sgn([2,1,3]) \prod_{i=1}^n A_{i,[2,1,3]_i} \\ &+ \sgn([2,3,1]) \prod_{i=1}^n A_{i,[2,3,1]_i} + \sgn([3,1,2]) \prod_{i=1}^n A_{i,[3,1,2]_i} + \sgn([3,2,1]) \prod_{i=1}^n A_{i,[3,2,1]_i}
\\
&=\prod_{i=1}^n A_{i,[1,2,3]_i} - \prod_{i=1}^n A_{i,[1,3,2]_i} - \prod_{i=1}^n A_{i,[2,1,3]_i} + \prod_{i=1}^n A_{i,[2,3,1]_i} + \prod_{i=1}^n A_{i,[3,1,2]_i} - \prod_{i=1}^n A_{i,[3,2,1]_i}
\\
&=A_{1,1}A_{2,2}A_{3,3}-A_{1,1}A_{2,3}A_{3,2}-A_{1,2}A_{2,1}A_{3,3}+A_{1,2}A_{2,3}A_{3,1}+A_{1,3}A_{2,1}A_{3,2}-A_{1,3}A_{2,2}A_{3,1}.
\end{align}
},
{
"math_input": \begin{align}
(\mathcal{L}_X \Lambda)^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} = X^\gamma \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s , \gamma} \, & - X^{\alpha_1}{}_{, \gamma} \Lambda^{\gamma \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \cdots - X^{\alpha_r}{}_{, \gamma} \Lambda^{\alpha_1 \cdots \alpha_{r-1} \gamma}{}_{\beta_1 \cdots \beta_s} \\
& + X^{\gamma}{}_{, \beta_1} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\gamma \beta_2 \cdots \beta_s} + \cdots + X^{\gamma}{}_{, \beta_s} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \gamma} \\
& + wX^{\gamma}{}_{, \gamma} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s}}\,.
\end{align}
},
{
"math_input": \mathbf J_{\mathbf F}(r, \theta, \varphi) = \begin{bmatrix}
\dfrac{\partial x}{\partial r} &
\dfrac{\partial x}{\partial \theta} &
\dfrac{\partial x}{\partial \varphi} \\[1em]
\dfrac{\partial y}{\partial r} &
\dfrac{\partial y}{\partial \theta} &
\dfrac{\partial y}{\partial \varphi} \\[1em]
\dfrac{\partial z}{\partial r} &
\dfrac{\partial z}{\partial \theta} &
\dfrac{\partial z}{\partial \varphi}
\end{bmatrix} = \begin{bmatrix}
\sin \theta \cos \varphi &
r \cos \theta \cos \varphi &
- r \sin \theta \sin \varphi \\
\sin \theta \sin \varphi &
r \cos \theta \sin \varphi &
r \sin \theta \cos \varphi \\
\cos \theta &
- r \sin \theta &
0
\end{bmatrix}.
},
{
"math_input":
\begin{align}
&\int\limits_{0}^{2\pi} \hat{t}\ {\left(\frac{p}{r}\right)}^3\ \sin u\ du\ =
-\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin^2 u \ du\
+\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin u\ \cos u \ du\ = \\
&-\hat{g}\ \left(\int\limits_{0}^{2\pi}\ \sin^2 u \ du\ +\
3\ {e_g}^2\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^2 u \ du\ \ +\
3\ {e_h}^2\ \int\limits_{0}^{2\pi}\ \sin^6 u \ du\ \right) \\
&+\hat{h}\ 6\ e_g\ e_h\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^2 u \ du = \\
&-\hat{g}\ \left(2\pi \left(\frac{1}{2}\ +\ \frac{3}{8}\ {e_g}^2\ +\ \frac{9}{8}\ {e_h}^2\right)\right)
+\hat{h}\ \left(2\pi \left(\frac{3}{4}\ e_g\ e_h\right)\right)
\end{align}
},
{
"math_input": \begin{align}
Var( X(t+1) | X(t)=i) &= Var(X(t)) + Var(\Delta(t+1)| X(t)=i) \\
&= 0 + E\left [\Delta(t+1)^2| X(t)=i \right ] - E[\Delta(t+1)| X(t)=i]^2\\
&= (i-1-i)^2 \cdot P_{i,i-1} + (i-i)^2 \cdot P_{i,i} + (i+1-i)^2 \cdot P_{i,i+1} - E[\Delta(t+1)| X(t)=i]^2\\
&= P_{i,i-1} + P_{i,i+1} - E[\Delta(t+1)| X(t)=i]^2\\
&= \frac{(N-i)i}{(r i + N -i)N} + \frac{(N-i)i(1+s)}{(r i + N -i)N} - E[\Delta(t+1)| X(t)=i]^2\\
&= i (N-i)\frac{2+s}{(r i + N -i)N} - E[\Delta(t+1)| X(t)=i]^2\\
&= i (N-i)\frac{2+s}{(r i + N -i)N} - \left (p s \dfrac{1-p}{ps + 1} \right )^2\\
&= p(1-p)\frac{2+s (ps + 1)}{(ps + 1)^2} - p(1-p) \frac{p s^2(1-p)}{(p s + 1)^2}\\
&= p(1-p)\dfrac{2+2ps + s + p^2 s^2}{(ps +1)^2}
\end{align}
},
{
"math_input": \begin{align}
& \left| + \right\rangle =\left( \begin{matrix}
\cos \frac{\theta }{2}{{e}^{-i\frac{\phi }{2}}} \\
{} \\
\sin \frac{\theta }{2}{{e}^{i\frac{\phi }{2}}} \\
\end{matrix} \right)\equiv \cos \frac{\theta }{2}{{e}^{-i\frac{\phi }{2}}}\left| 1 \right\rangle +\sin \frac{\theta }{2}{{e}^{i\frac{\phi }{2}}}\left| 2 \right\rangle \\
& \left| - \right\rangle =\left( \begin{matrix}
-\sin \frac{\theta }{2}{{e}^{i\frac{\phi }{2}}} \\
{} \\
\cos \frac{\theta }{2}{{e}^{-i\frac{\phi }{2}}} \\
\end{matrix} \right)\equiv -\sin \frac{\theta }{2}{{e}^{-i\frac{\phi }{2}}}\left| 1 \right\rangle +\cos \frac{\theta }{2}{{e}^{i\frac{\phi }{2}}}\left| 2 \right\rangle \\
\end{align}
},
{
"math_input":
\begin{align}
\uparrow_{\alpha} t &= \mathbf{SYN}\ t \\
\uparrow_{\tau_1 \to \tau_2} v &=
\mathbf{LAM} (\lambda S.\ \uparrow_{\tau_2} (\mathbf{app}\ (v, \downarrow^{\tau_1} S))) \\
\uparrow_{\tau_1 \times \tau_2} v &=
\mathbf{PAIR} (\uparrow_{\tau_1} (\mathbf{fst}\ v), \uparrow_{\tau_2} (\mathbf{snd}\ v)) \\[1ex]
\downarrow^{\alpha} (\mathbf{SYN}\ t) &= t \\
\downarrow^{\tau_1 \to \tau_2} (\mathbf{LAM}\ S) &=
\mathbf{lam}\ (x, \downarrow^{\tau_2} (S\ (\uparrow_{\tau_1} (\mathbf{var}\ x))))
\text{ where } x \text{ is fresh} \\
\downarrow^{\tau_1 \times \tau_2} (\mathbf{PAIR}\ (S, T)) &=
\mathbf{pair}\ (\downarrow^{\tau_1} S, \downarrow^{\tau_2} T)
\end{align}
},
{
"math_input":
\begin{align}
\delta J(y)(h)&=\left.\frac{d}{d\varepsilon} J(y + \varepsilon h)\right|_{\varepsilon = 0}\\
&= \left.\frac{d}{d\varepsilon} \int_a^b (y + \varepsilon h)(y^\prime + \varepsilon h^\prime) \ dx\right|_{\varepsilon = 0}\\
&= \left.\frac{d}{d\varepsilon} \int_a^b (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\right|_{\varepsilon = 0}\\
&= \left.\int_a^b \frac{d}{d\varepsilon} (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\right|_{\varepsilon = 0}\\
&= \left.\int_a^b (yh^\prime + y^\prime h + 2\varepsilon hh^\prime) \ dx\right|_{\varepsilon = 0}\\
&= \int_a^b (yh^\prime + y^\prime h) \ dx
\end{align}
},
{
"math_input":
\begin{align}
\varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} ~;~~
\varepsilon_{\theta\theta} = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) ~;~~
\varepsilon_{zz} = \cfrac{\partial u_z}{\partial z} \\
\varepsilon_{r\theta} & = \cfrac{1}{2}\left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) ~;~~
\varepsilon_{\theta z} = \cfrac{1}{2}\left(\cfrac{\partial u_\theta}{\partial z} + \cfrac{1}{r}\cfrac{\partial u_z}{\partial \theta}\right) ~;~~
\varepsilon_{zr} = \cfrac{1}{2}\left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right)
\end{align}
},
{
"math_input":
\begin{align}
\frac{D^3F(P_0)}{DP^3} & =\frac{D^2F'(P_0)}{DP^2}=\frac{DF''(P_0)}{DP}=\frac{F''(P_1 < P < P_3)-F''(P_0 < P < P_2)}{P_1-P_0}, \\[10pt]
& {\color{white}.}\qquad\qquad\qquad\qquad\qquad\ \ \ne\frac{F''(P_1)-F''(P_0)}{P_1-P_0}, \\[10pt]
& =\frac{\frac{F'(P_2 < P < P_3)-F'(P_1 < P < P_2)}{P_1-P_0}-\frac{F'(P_1 < P < P_2)-F'(P_0 < P < P_1)}{P_1-P_0}}{P_1-P_0}, \\[10pt]
& =\frac{F'(P_2 < P < P_3)-2F'(P_1 < P < P_2)+F'(P_0 < P < P_1)}{(P_1-P_0)^2}, \\[10pt]
& =F[P_0,P_1,P_2,P_3]=\frac{F(P_3)-3F(P_2)+3F(P_1)-F(P_0)}{(P_1-P_0)^3}, \\[10pt]
& =F'''(P_0 < P < P_3)=\sum_{TN=1}^{UT=\infty}\frac{F'''(P_{(tn)})}{UT}, \\[10pt]
& =G''(P_0 < P < P_3)\ =H'(P_0 < P < P_3)=I(P_0 < P < P_3).
\end{align}
},
{
"math_input": \begin{bmatrix}
\sin^2 (\theta+\alpha)K_n & -K_n \sin(\theta+\alpha)\cos(\theta+\alpha) & \cos(\theta+\alpha)K_s L\sin(\alpha) \\
+\cos^2(\theta+\alpha)K_s & +K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & -\sin(\theta+\alpha)K_n L\cos(\alpha) \\
-K_n\sin(\theta+\alpha)\cos(\theta+\alpha) & \sin^2(\theta+\alpha)K_s & \cos(\theta+\alpha)K_n L\cos(\alpha) \\
-K_s\sin(\theta+\alpha)\cos(\theta+\alpha) & +\cos^2(\theta+\alpha)K_n & +\sin(\theta+\alpha)K_s L\sin(\alpha) \\
\cos(\theta+\alpha)K_s L\sin(\alpha) & \cos(\theta+\alpha)K_n L\cos(\alpha) & L^2\cos^2(\alpha)K_n \\
-\sin(\theta+\alpha)K_n L\cos(\alpha) & +\sin(\theta+\alpha)K_s L\sin(\alpha) & +L^2\sin^2(\alpha)K_s
\end{bmatrix}
},
{
"math_input": J_F(r,\theta,\phi) =\begin{bmatrix}
\dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} \\[3pt]
\dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} \\[3pt]
\dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} \\
\end{bmatrix}=\begin{bmatrix}
\sin\theta\, \cos\phi & r\, \cos\theta\, \cos\phi & -r\, \sin\theta\, \sin\phi \\
\sin\theta\, \sin\phi & r\, \cos\theta\, \sin\phi & r\, \sin\theta\, \cos\phi \\
\cos\theta & -r\, \sin\theta & 0
\end{bmatrix}.
},
{
"math_input": \begin{align}
\frac{\partial u}{\partial t} &= \frac{\partial^2 u}{\partial x^2} &&\text{in } \{(x,t): 0 < x < s(t), t>0\}, && \text{the heat equation},\\
-\frac{\partial u}{\partial x}(0, t) &= f(t), && t>0, &&\text{the Neumann condition at the left end of the domain describing the inlet heat flux}, &&\\
u\big(s(t),t\big) &= 0, && t>0, &&\text{the Dirichlet condition at the water-ice interface: setting melting/freezing temperature},\\
\frac{\mathrm{d}s}{\mathrm{d}t} &= -\frac{\partial u}{\partial x}\big(s(t), t\big), && t>0, &&\text{Stefan condition},\\
u(x,0) &= 0, && x\geq 0, &&\text{initial temperature distribution},\\
s(0) &= 0, && &&\text{initial depth of the melted ice block}.
\end{align}
},
{
"math_input":
\begin{align}
p_i &\propto \Omega_B\left(E_B\right) \\
&= \Omega_B \left(E - E_i\right) \\
\Rightarrow k \ln \Omega_B \left( E - E_i \right) & \approx k \ln \Omega_B \left(E\right) - \frac{\partial}{\partial E} \left(k \ln \Omega_B\left(E\right)\right) E_i \\
& \approx k \ln \Omega_B \left(E\right) - \frac{\partial S_B}{\partial E} E_i \\
& \approx k \ln \Omega_B \left(E\right) - \frac{E_i}{T} \\
\Rightarrow k \ln p_i & \approx k \ln \Omega_B \left(E\right) - \frac{E_i}{T} \\
\Rightarrow p_i & \propto e^{\ln \Omega_B \left(E\right) - \frac{E_i}{kT}} \\
\Rightarrow p_i & \propto \Omega_B \left(E\right) e^{ - \frac{E_i}{kT}} \\
\Rightarrow p_i & \propto e^{- \frac{E_i}{kT}}.
\end{align}
},
{
"math_input":
\begin{array}{lcl}
K,N &=& \text{as above} \\
\phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
z_{i=1 \dots N}, x_{i=1 \dots N} &=& \text{as above} \\
\mu_{i=1 \dots K} &=& \text{mean of component } i \\
\sigma^2_{i=1 \dots K} &=& \text{variance of component } i \\
\mu_0, \lambda, \nu, \sigma_0^2 &=& \text{shared hyperparameters} \\
\mu_{i=1 \dots K} &\sim& \mathcal{N}(\mu_0, \lambda\sigma_i^2) \\
\sigma_{i=1 \dots K}^2 &\sim& \operatorname{Inverse-Gamma}(\nu, \sigma_0^2) \\
\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
x_{i=1 \dots N} &\sim& \mathcal{N}(\mu_{z_i}, \sigma^2_{z_i})
\end{array}
},
{
"math_input": \forall (\alpha,\beta)\ \mathbf T \left( \alpha
\begin{bmatrix}
\vec f \\
\ \\
0 \\
\end{bmatrix} + \beta
\begin{bmatrix}
0 \\
\ \\
\vec b
\end{bmatrix} \right ) = \alpha
\begin{bmatrix} 1 \\
0 \\
\vdots \\
0 \\
\epsilon_f \\
\end{bmatrix} + \beta
\begin{bmatrix} \epsilon_b \\
0 \\
\vdots \\
0 \\
1
\end{bmatrix}.
},
{
"math_input":
\begin{align}
&\int\limits_{0}^{2\pi} \hat{r}\ {\left(\frac{p}{r}\right)}^3\ \cos u\ du\ =
\hat{g}\ \int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \cos^2 u \ du\
+\hat{h}\int\limits_{0}^{2\pi}\ {\left(\frac{p}{r}\right)}^3\ \sin u\ \cos u \ du\ = \\
&\hat{g}\ \left(\int\limits_{0}^{2\pi} \cos^2 u \ du\ +\
3\ {e_g}^2\ \int\limits_{0}^{2\pi}\ \cos^4 u \ du\ \ +\
3\ {e_h}^2\ \int\limits_{0}^{2\pi}\ \sin^2 u\ \cos^2 u \ du\ \right) \\
&+\hat{h}\ 6\ e_g\ e_h\ \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^2 u \ du = \\
&\hat{g}\ \left(2\pi \left(\frac{1}{2}\ +\ \frac{9}{8}\ {e_g}^2\ +\ \frac{3}{8}\ {e_h}^2\right)\right)
+\hat{h}\ \left(2\pi \left(\frac{3}{4}\ e_g\ e_h\right)\right)
\end{align}
},
{
"math_input":
\begin{bmatrix}
6.1917 & -0.3411 & 1.2418 & 0.1492 & 0.1583 & 0.2742 & -0.0724 & 0.0561 \\
0.2205 & 0.0214 & 0.4503 & 0.3947 & -0.7846 & -0.4391 & 0.1001 & -0.2554 \\
1.0423 & 0.2214 & -1.0017 & -0.2720 & 0.0789 & -0.1952 & 0.2801 & 0.4713 \\
-0.2340 & -0.0392 & -0.2617 & -0.2866 & 0.6351 & 0.3501 & -0.1433 & 0.3550 \\
0.2750 & 0.0226 & 0.1229 & 0.2183 & -0.2583 & -0.0742 & -0.2042 & -0.5906 \\
0.0653 & 0.0428 & -0.4721 & -0.2905 & 0.4745 & 0.2875 & -0.0284 & -0.1311 \\
0.3169 & 0.0541 & -0.1033 & -0.0225 & -0.0056 & 0.1017 & -0.1650 & -0.1500 \\
-0.2970 & -0.0627 & 0.1960 & 0.0644 & -0.1136 & -0.1031 & 0.1887 & 0.1444 \\
\end{bmatrix}
},
{
"math_input":
\begin{align}
\varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
\varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
\varepsilon_{zz} & = \cfrac{\partial u_z}{\partial z} \\
\varepsilon_{r\theta} & = \cfrac{1}{2}\left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) \\
\varepsilon_{\theta z} & = \cfrac{1}{2}\left(\cfrac{\partial u_\theta}{\partial z} + \cfrac{1}{r}\cfrac{\partial u_z}{\partial \theta}\right) \\
\varepsilon_{zr} & = \cfrac{1}{2}\left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right)
\end{align}
},
{
"math_input":
\begin{matrix}
P & = & \begin{bmatrix}
0.9 & 0.1 \\
0.5 & 0.5
\end{bmatrix}
\\
\mathbf{q} P & = & \mathbf{q}
& \mbox{(} \mathbf{q} \mbox{ is unchanged by } P \mbox{.)}
\\
& = & \mathbf{q}I
\\
\mathbf{q} (P - I) & = & \mathbf{0} \\
& = & \mathbf{q} \left( \begin{bmatrix}
0.9 & 0.1 \\
0.5 & 0.5
\end{bmatrix}
-
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
\right)
\\
& = & \mathbf{q} \begin{bmatrix}
-0.1 & 0.1 \\
0.5 & -0.5
\end{bmatrix}
\end{matrix}
},
{
"math_input": \begin{array}{rccrcrcrcr}
{\color{BrickRed}P}{\color{RoyalBlue}Q}&{{=}}&&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}2x})
&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}2x}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}2x}\cdot{\color{RoyalBlue}1})
\\&&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}3y}\cdot{\color{RoyalBlue}5y})&+&({\color{BrickRed}3y}\cdot {\color{RoyalBlue}xy})&+&
({\color{BrickRed}3y}\cdot{\color{RoyalBlue}1})
\\&&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}2x})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}5y})&+&
({\color{BrickRed}5}\cdot {\color{RoyalBlue}xy})&+&({\color{BrickRed}5}\cdot{\color{RoyalBlue}1})
\end{array}
},
{
"math_input":
\begin{align}
(h_\text{eff})_{AB}&=E\begin{pmatrix} \delta_{ab}&0\\\\0&\delta_{\bar a\bar b}\end{pmatrix}
+\frac{1}{2E}\begin{pmatrix} (\tilde m^2)_{ab}&0\\\\0&(\tilde m^2)_{\bar a\bar b}^*\end{pmatrix} \\\\
&\quad+\frac{1}{E}\begin{pmatrix}[(a_L)^\alpha p_\alpha-(c_L)^{\alpha\beta} p_\alpha p_\beta]_{ab}&
-i\sqrt2p_\alpha(\epsilon_+)_\beta[(g^{\alpha\beta\gamma}p_\gamma-H^{\alpha\beta})]_{a\bar b}\\\\
i\sqrt2p_\alpha(\epsilon_+)_\beta^*[(g^{\alpha\beta\gamma}p_\gamma-H^{\alpha\beta})]_{\bar ab}^*&
[(a_R)^\alpha p_\alpha-(c_R)^{\alpha\beta} p_\alpha p_\beta]_{\bar a\bar b}\end{pmatrix} .
\end{align}
},
{
"math_input": \qquad =\begin{cases}
\displaystyle \frac{\,\pi\,}{2n}\tan\frac{\,\pi\,}{2n}\ln2\pi + \frac{\pi}{n}\sum_{l=1}^{n-1} (-1)^{l-1}
\sin\frac{\,\pi l\,}{n}\cdot
\ln\left\{\!\frac{\Gamma\!\left(\!\displaystyle\frac{1}{\,2\,}+\displaystyle\frac{l}{\,2n}\!\right) }{\Gamma\!\left(\!\displaystyle\frac{l}{\,2n}\!\right)}\right\}
,\quad n=2,4,6,\ldots \\[10mm]
\displaystyle \frac{\,\pi\,}{2n}\tan\frac{\,\pi\,}{2n}\ln\pi + \frac{\pi}{n}\!\!\!\!\!
\sum_{l=1}^{\;\;\;\frac{1}{2}(n-1)} \!\!\!\! (-1)^{l-1} \sin\frac{\,\pi l\,}{n}\cdot
\ln\left\{\!\frac{\Gamma\!\left(1-\displaystyle\frac{\,l}{n}\!\right) }{\Gamma\!\left(\!\displaystyle\frac{\,l}{n}\!\right)}\right\} ,\qquad n=3,5,7,\ldots
\end{cases}
},
{
"math_input": \mathbf T^n \begin{bmatrix} \vec f^{n-1} \\ 0 \\ \end{bmatrix} =
\begin{bmatrix}
\ & \ & \ & t_{-n+1} \\
\ & \mathbf T^{n-1} & \ & t_{-n+2} \\
\ & \ & \ & \vdots \\
t_{n-1} & t_{n-2} & \dots & t_0 \\
\end{bmatrix}
\begin{bmatrix} \ \\
\vec f^{n-1} \\
\ \\
0 \\
\ \\
\end{bmatrix} =
\begin{bmatrix} 1 \\
0 \\
\vdots \\
0 \\
\epsilon_f^n \\
\end{bmatrix}.
},
{
"math_input":
\begin{align}
\mathbf{P}(\tau, \mu | \mathbf{X}) & \propto \tau^{\frac{n}{2} + \alpha_0 - \frac{1}{2}} \exp \left[-\tau \left( \frac{1}{2} n s + \beta_0 \right) \right] \exp \left[- \frac{\tau}{2} \left( \left(\lambda_0 + n \right) \left(\mu- \frac{\lambda_0 \mu_0 + n \bar{x}}{\lambda_0 + n} \right)^2 + \frac{\lambda_0 n (\bar{x} - \mu_0 )^2}{\lambda_0 +n} \right) \right]\\
& \propto \tau^{\frac{n}{2} + \alpha_0 - \frac{1}{2}} \exp \left[-\tau \left( \frac{1}{2} n s + \beta_0 + \frac{\lambda_0 n (x - \mu_0 )^2}{2(\lambda_0 +n)} \right) \right] \exp \left[- \frac{\tau}{2} \left(\lambda_0 + n \right) \left(\mu- \frac{\lambda_0 \mu_0 + n \bar{x}}{\lambda_0 + n} \right)^2 \right]
\end{align}
},
{
"math_input": \begin{align}
a_1 &= \frac{(c + \alpha)(c + \beta)}{(c + 1)(c + \gamma)} a_0 \\
a_2 &= \frac{(c + \alpha + 1)(c + \beta + 1)}{(c + 2)(c + \gamma + 1)} a_1
= \frac{(c + \alpha + 1)(c + \alpha)(c + \beta)(c + \beta + 1)}
{(c + 2)(c + 1)(c + \gamma)(c + \gamma + 1)} a_0
= \frac{(c + \alpha)_2 (c + \beta)_2}{(c + 1)_2 (c + \gamma)_2} a_0 \\
a_3 &= \frac{(c + \alpha + 2)(c + \beta + 2)}{(c + 3)(c + \gamma + 2)} a_2
= \frac{(c + \alpha)_2 (c + \alpha + 2)(c + \beta )_2 (c + \beta + 2)}
{(c + 1)_2 (c + 3)(c + \gamma)_2 (c + \gamma + 2)} a_0 \\
&= \frac{(c + \alpha)_3 (c + \beta)_3}
{(c + 1)_3 (c + \gamma)_3} a_0
\end{align}
},
{
"math_input": \begin{array}{ll}
{\rm (BL1)}\colon & (A \rightarrow B) \rightarrow ((B \rightarrow C) \rightarrow (A \rightarrow C)) \\
{\rm (BL2)}\colon & A \otimes B \rightarrow A\\
{\rm (BL3)}\colon & A \otimes B \rightarrow B \otimes A\\
{\rm (BL4)}\colon & A \otimes (A \rightarrow B) \rightarrow B \otimes (B \rightarrow A)\\
{\rm (BL5a)}\colon & (A \rightarrow (B \rightarrow C)) \rightarrow (A \otimes B \rightarrow C)\\
{\rm (BL5b)}\colon & (A \otimes B \rightarrow C) \rightarrow (A \rightarrow (B \rightarrow C))\\
{\rm (BL6)}\colon & ((A \rightarrow B) \rightarrow C) \rightarrow (((B \rightarrow A) \rightarrow C) \rightarrow C)\\
{\rm (BL7)}\colon & \bot \rightarrow A
\end{array}
},
{
"math_input":
\begin{align}
\sin^2 x & = \sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{(-1)^i}{(2i + 1)!} \frac{(-1)^j}{(2j + 1)!} x^{(2i + 1) + (2j + 1)} \\
& = \sum_{n = 1}^\infty \left(\sum_{i = 0}^{n - 1} \frac{(-1)^{n - 1}}{(2i + 1)!(2(n - i - 1) + 1)!}\right) x^{2n} \\
& = \sum_{n = 1}^\infty \left( \sum_{i = 0}^{n - 1} {2n \choose 2i + 1} \right) \frac{(-1)^{n - 1}}{(2n)!} x^{2n},\\
\cos^2 x & = \sum_{i = 0}^\infty \sum_{j = 0}^\infty \frac{(-1)^i}{(2i)!} \frac{(-1)^j}{(2j)!} x^{(2i) + (2j)} \\
& = \sum_{n = 0}^\infty \left(\sum_{i = 0}^n \frac{(-1)^n}{(2i)!(2(n - i))!}\right) x^{2n} \\
& = \sum_{n = 0}^\infty \left( \sum_{i = 0}^n {2n \choose 2i} \right) \frac{(-1)^n}{(2n)!} x^{2n}.
\end{align}
},
{
"math_input": \begin{align}
\mathbf{r} & =\mathbf{r}\left ( t \right ) = r \mathbf{\hat{e}}_r\\
\mathbf{v} & = v \mathbf{\hat{e}}_r + r\,\frac{d\theta}{dt}\mathbf{\hat{e}}_\theta + r\,\frac{d\phi}{dt}\,\sin\theta \mathbf{\hat{e}}_\phi \\
\mathbf{a} & = \left( a - r\left(\frac{d\theta}{dt}\right)^2 - r\left(\frac{d\phi}{dt}\right)^2\sin^2\theta \right)\mathbf{\hat{e}}_r \\
& + \left( r \frac{d^2 \theta}{dt^2 } + 2v\frac{d\theta}{dt} - r\left(\frac{d\phi}{dt}\right)^2\sin\theta\cos\theta \right) \mathbf{\hat{e}}_\theta \\
& + \left( r\frac{d^2 \phi}{dt^2 }\,\sin\theta + 2v\,\frac{d\phi}{dt}\,\sin\theta + 2 r\,\frac{d\theta}{dt}\,\frac{d\phi}{dt}\,\cos\theta \right) \mathbf{\hat{e}}_\phi
\end{align} \,\!
},
{
"math_input": \begin{align}
A &= 63365028312971999585426220 \\
&\quad + 28337702140800842046825600\sqrt{5} \\
&\quad + 384\sqrt{5} (10891728551171178200467436212395209160385656017 \\
&\qquad + 4870929086578810225077338534541688721351255040\sqrt{5})^{1/2} \\
B &= 7849910453496627210289749000 \\
&\quad + 3510586678260932028965606400\sqrt{5} \\
&\quad + 2515968\sqrt{3110}(6260208323789001636993322654444020882161 \\
&\qquad + 2799650273060444296577206890718825190235\sqrt{5})^{1/2} \\
C &= -214772995063512240 \\
&\quad - 96049403338648032\sqrt{5} \\
&\quad - 1296\sqrt{5}(10985234579463550323713318473 \\
&\qquad + 4912746253692362754607395912\sqrt{5})^{1/2}
\end{align}
},
{
"math_input":
\begin{array}{cl}
\displaystyle\frac{x:\sigma \in \Gamma \quad \tau = \mathit{inst}(\sigma)}{\Gamma \vdash x:\tau}&[\texttt{Var}]\\ \\
\displaystyle\frac{\Gamma \vdash e_0:\tau_0 \quad \Gamma \vdash e_1 : \tau_1 \quad \tau'=\mathit{newvar} \quad \mathit{unify}(\tau_0,\ \tau_1 \rightarrow \tau') }{\Gamma \vdash e_0\ e_1 : \tau'}&[\texttt{App}]\\ \\
\displaystyle\frac{\tau = \mathit{newvar} \quad \Gamma,\;x:\tau\vdash e:\tau'}{\Gamma \vdash \lambda\ x\ .\ e : \tau \rightarrow \tau'}&[\texttt{Abs}]\\ \\
\displaystyle\frac{\Gamma \vdash e_0:\tau \quad\quad \Gamma,\,x:\bar{\Gamma}(\tau) \vdash e_1:\tau'}{\Gamma \vdash \texttt{let}\ x = e_0\ \texttt{in}\ e_1 : \tau'}&[\texttt{Let}]
\end{array}
},
{
"math_input": \mathbf T^n \begin{bmatrix} 0 \\ \vec b^{n-1} \\ \end{bmatrix} =
\begin{bmatrix}
t_0 & \dots & t_{-n+2} & t_{-n+1} \\
\vdots & \ & \ & \ \\
t_{n-2} & \ & \mathbf T^{n-1} & \ \\
t_{n-1} & \ & \ & \ \\
\end{bmatrix}
\begin{bmatrix} \ \\
0 \\
\ \\
\vec b^{n-1} \\
\ \\
\end{bmatrix} =
\begin{bmatrix} \epsilon_b^n \\
0 \\
\vdots \\
0 \\
1 \\
\end{bmatrix}.
},
{
"math_input": \begin{align}
V^{(n)}[R] & = \int_0^R \int_0^{2 \pi} V^{(n-2)}\left[\sqrt{R^2-r^2}\right] \, r \, d\theta \, dr \\
& = 2 \pi \int_0^R V^{(n-2)}\left[\sqrt{R^2-r^2}\right] \, r \, dr \\
& = 2 \pi \int_0^R \frac{\pi^{\frac{n-2}{2}}}{\Gamma(\frac{n}{2})} \, \left({R^2-r^2}\right)^{\frac{n-2}{2}} \, r \, dr \\
& = \frac{\pi^{\frac{n}{2}}}{\frac{1}{2}\Gamma(\frac{n}{2})} \int_0^R \, \left({R^2-r^2}\right)^{\frac{n-2}{2}} \, r \, dr \\
& = \frac{\pi^{\frac{n}{2}}}{ \frac{n}{2}\Gamma(\frac{n}{2})} \left[- \left(R^2-r^2\right)^{\frac{n}{2}} \right]_{r=0}^{r=R} \\
& = \frac{\pi^{\frac{n}{2}} R^n}{\Gamma(\frac{n}{2} + 1)}
\end{align}
},
{
"math_input": \mathbf T^n \begin{bmatrix} \vec f^{n-1} \\ 0 \\ \end{bmatrix} =
\begin{bmatrix}
\ & \ & \ & t_{-n+1} \\
\ & \mathbf T^{n-1} & \ & t_{-n+2} \\
\ & \ & \ & \vdots \\
t_{n-1} & t_{n-2} & \dots & t_0 \\
\end{bmatrix}
\begin{bmatrix} \ \\
\vec f^{n-1} \\
\ \\
0 \\
\ \\
\end{bmatrix} =
\begin{bmatrix} 1 \\
0 \\
\vdots \\
0 \\
\epsilon_f^n
\end{bmatrix}.
},
{
"math_input":
\begin{align}
\cfrac{d}{dt}\left( \int_{\Omega(t)} \mathbf{f}(\mathbf{x},t)~\text{dV}\right) & =
\int_{\Omega_0} \left[\lim_{\Delta t \rightarrow 0} \cfrac{
\hat{\mathbf{f}}(\mathbf{X},t+\Delta t)~J(\mathbf{X},t+\Delta t) -
\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)}{\Delta t} \right]~\text{dV}_0 \\
& = \int_{\Omega_0} \frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)~J(\mathbf{X},t)]~\text{dV}_0 \\
& = \int_{\Omega_0} \left(
\frac{\partial }{\partial t}[\hat{\mathbf{f}}(\mathbf{X},t)]~J(\mathbf{X},t)+
\hat{\mathbf{f}}(\mathbf{X},t)~\frac{\partial }{\partial t}[J(\mathbf{X},t)]\right) ~\text{dV}_0
\end{align}
},
{
"math_input": \begin{align}h_{jk} & =
h\left(\tfrac{\partial}{\partial\theta_j}, \tfrac{\partial}{\partial\theta_k}\right) \\
& = \frac{1}{4} \mathrm{E}\left[
\frac{\partial\log p}{\partial\theta_j}
\frac{\partial\log p}{\partial\theta_k}
\right]
+ \mathrm{E}\left[
\frac{\partial\alpha}{\partial\theta_j}
\frac{\partial\alpha}{\partial\theta_k}
\right]
- \mathrm{E}\left[ \frac{\partial\alpha}{\partial\theta_j} \right]
\mathrm{E}\left[ \frac{\partial\alpha}{\partial\theta_k} \right] \\
& - \frac{i}{2}\mathrm{E}\left[
\frac{\partial\log p}{\partial\theta_j}
\frac{\partial\alpha}{\partial\theta_k}
-
\frac{\partial\alpha}{\partial\theta_j}
\frac{\partial\log p}{\partial\theta_k}
\right]
\end{align}
},
{
"math_input":
\begin{align}
\mathbb{S}\, &=\, \begin{pmatrix} S_{xx} & S_{xy} \\ S_{yx} & S_{yy} \end{pmatrix}\,
=\, \mathbb{I}\, \left( \frac{c_g}{c_p} - \frac12 \right)\, E\,
+\, \frac{1}{k^2}\, \begin{pmatrix} k_x\, k_x & k_x\, k_y \\[2ex] k_y\, k_x & k_y\, k_y \end{pmatrix}\, \frac{c_g}{c_p}\, E,
\\
\mathbb{I}\, &=\, \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\quad \text{and}
\\
\nabla \boldsymbol{U}\, &=\,
\begin{pmatrix}
\displaystyle \frac{\partial U_x}{\partial x} & \displaystyle \frac{\partial U_y}{\partial x}
\\[2ex]
\displaystyle \frac{\partial U_x}{\partial y} & \displaystyle \frac{\partial U_y}{\partial y}
\end{pmatrix},
\end{align}
},
{
"math_input": \begin{align}
\bold{A}+\bold{B} & = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
\begin{bmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\
\end{align}\,\!
},
{
"math_input":
\begin{align}
& \left(\frac{2D^{\mathrm{face}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} - \left(1+\frac{2D^{\mathrm{face}}}{D^{\mathrm{beam}}}\right)\cfrac{\mathrm{d}^2 w}{\mathrm{d} x^2} + \left(\cfrac{1}{S^{\mathrm{core}}}\right)~\cfrac{\mathrm{d} Q}{\mathrm{d} x} = \frac{M}{D^{\mathrm{beam}}} \\
& \left(\frac{D^{\mathrm{beam}}}{S^{\mathrm{core}}}\right)\cfrac{\mathrm{d}^3 w_s}{\mathrm{d} x^3} - \left(1+\frac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\cfrac{\mathrm{d} w_s}{\mathrm{d} x} - \cfrac{1}{S^{\mathrm{core}}}~\cfrac{\mathrm{d} M}{\mathrm{d} x} = -\left(1+\cfrac{D^{\mathrm{beam}}}{2D^{\mathrm{face}}}\right)\frac{Q}{S^{\mathrm{core}}}\,
\end{align}
},
{
"math_input": \begin{align}
{\scriptstyle{\Pi}}X&=\cos(\phi_s)\cos(\phi_f)\cos(\lambda_s)\cos(\lambda_f);\\
{\scriptstyle{\Pi}}Y&=\cos(\phi_s)\cos(\phi_f)\sin(\lambda_s)\sin(\lambda_f);\\
{\scriptstyle{\Pi}}Z&=\sin(\phi_s)\sin(\phi_f);\\
\cos(\Delta\lambda)&=\frac{{\scriptstyle{\Pi}}X\!\!+\!{\scriptstyle{\Pi}}Y}{\cos(\phi_s)\cos(\phi_f)}=\cos(\lambda_s)\cos(\lambda_f)+\sin(\lambda_s)\sin(\lambda_f);\\
\Delta\widehat{\sigma}&=\arccos\Big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y+{\scriptstyle{\Pi}}Z\Big)
=\arccos\Big({\scriptstyle{\Pi}}Z+\big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y\big)\Big),\\
&=\arccos\Big(\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)\Big)
.\end{align}\,\!
},
{
"math_input":
\begin{align}
& \Pr(z_n=k\mid\mathbb{Z}^{(-n)},\boldsymbol{\alpha}) \\
\propto\ & \Pr(z_n=k,\mathbb{Z}^{(-n)}\mid\boldsymbol{\alpha}) \\
=\ &\ \frac{\Gamma\left(A\right)}{\Gamma\left(N+A\right)}\prod_{j=1}^K\frac{\Gamma(n_{j}+\alpha_{j})}{\Gamma(\alpha_{j})} \\
\propto\ & \prod_{j=1}^K\Gamma(n_{j}+\alpha_{j}) \\
=\ & \Gamma(n_{k}+\alpha_{k})\prod_{j\not=k}\Gamma(n_{j}+\alpha_{j}) \\
=\ & \Gamma(n_k^{(-n)}+1+\alpha_{k})\prod_{j\not=k}\Gamma(n_j^{(-n)}+\alpha_{j}) \\
=\ & (n_k^{(-n)}+\alpha_{k}) \Gamma(n_k^{(-n)}+\alpha_{k})\prod_{j\not=k}\Gamma(n_j^{(-n)}+\alpha_{j}) \\
=\ & (n_k^{(-n)}+\alpha_{k}) \prod_{j}\Gamma(n_j^{(-n)}+\alpha_{j}) \\
\propto\ & n_k^{(-n)}+\alpha_{k}
\end{align}
},
{
"math_input": \begin{align}
f(z) &= \frac{1}{\pi^k\sqrt{\det(\Gamma)\det(P)}}\,
\exp\!\left\{-\frac12 \begin{pmatrix}(\overline{z}-\overline\mu)' & (z-\mu)'\end{pmatrix}
\begin{pmatrix}\Gamma&C\\\overline{C}'&\overline\Gamma\end{pmatrix}^{\!\!-1}\!
\begin{pmatrix}z-\mu \\ \overline{z}-\overline{\mu}\end{pmatrix}
\right\} \\[8pt]
&= \tfrac{\sqrt{\det\left(\overline{P^{-1}}-\overline{R}'P^{-1}R\right)\det(P^{-1})}}{\pi^k}\,
e^{ -(\overline{z}-\overline\mu)'\overline{P^{-1}}(z-\mu) +
\operatorname{Re}\left((z-\mu)'R'\overline{P^{-1}}(z-\mu)\right)},
\end{align}
},
{
"math_input":
\begin{align}
\operatorname{E}[S^2]
&= \operatorname{E}\left[ \frac{1}{n}\sum_{i=1}^n \left(X_i-\overline{X}\right)^2 \right]
= \operatorname{E}\bigg[ \frac{1}{n}\sum_{i=1}^n \big((X_i-\mu)-(\overline{X}-\mu)\big)^2 \bigg] \\[8pt]
&= \operatorname{E}\bigg[ \frac{1}{n}\sum_{i=1}^n (X_i-\mu)^2 -
2(\overline{X}-\mu)\frac{1}{n}\sum_{i=1}^n (X_i-\mu) +
(\overline{X}-\mu)^2 \bigg] \\[8pt]
&= \operatorname{E}\bigg[ \frac{1}{n}\sum_{i=1}^n (X_i-\mu)^2 - (\overline{X}-\mu)^2 \bigg]
= \sigma^2 - \operatorname{E}\left[ (\overline{X}-\mu)^2 \right] < \sigma^2.
\end{align}
},
{
"math_input": \begin{align}
{\scriptstyle{\Pi}}X&=\cos(\phi_s)\cos(\phi_f)\cos(\lambda_s)\cos(\lambda_f);\\
{\scriptstyle{\Pi}}Y&=\cos(\phi_s)\cos(\phi_f)\sin(\lambda_s)\sin(\lambda_f);\\
{\scriptstyle{\Pi}}Z&=\sin(\phi_s)\sin(\phi_f);\\
\frac{{\scriptstyle{\Pi}}X\!\!+\!{\scriptstyle{\Pi}}Y}{\cos(\phi_s)\cos(\phi_f)}&=\cos(\lambda_s)\cos(\lambda_f)+\sin(\lambda_s)\sin(\lambda_f)=\cos(\Delta\lambda);\\
\Delta\widehat{\sigma}&=\arccos\Big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y+{\scriptstyle{\Pi}}Z\Big)
=\arccos\Big({\scriptstyle{\Pi}}Z+\big({\scriptstyle{\Pi}}X+{\scriptstyle{\Pi}}Y\big)\Big),\\
&=\arccos\Big(\sin(\phi_s)\sin(\phi_f)+\cos(\phi_s)\cos(\phi_f)\cos(\Delta\lambda)\Big).\end{align}\,\!
},
{
"math_input": \begin{align}
\bold{A}+\bold{B} & = \begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn} \\
\end{bmatrix} +
\begin{bmatrix}
b_{11} & b_{12} & \cdots & b_{1n} \\
b_{21} & b_{22} & \cdots & b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
b_{m1} & b_{m2} & \cdots & b_{mn} \\
\end{bmatrix} \\
& = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12} & \cdots & a_{1n} + b_{1n} \\
a_{21} + b_{21} & a_{22} + b_{22} & \cdots & a_{2n} + b_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} + b_{m1} & a_{m2} + b_{m2} & \cdots & a_{mn} + b_{mn} \\
\end{bmatrix} \\
\end{align}\,\!
},
{
"math_input": \begin{align}
d\mathbf x_1 \cdot d\mathbf x_2&=dx_1dx_2\cos\theta_{12} \\
\mathbf F \cdot d\mathbf X_1\cdot \mathbf F\cdot d\mathbf X_2&= \sqrt {d\mathbf X_1 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_1}\cdot \sqrt {d\mathbf X_2 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_2} \cos\theta_{12} \\
\frac{d\mathbf X_1\cdot \mathbf F^T\cdot\mathbf F\cdot d\mathbf X_2}{dX_1dX_2}&=\frac{\sqrt {d\mathbf X_1 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_1}\cdot \sqrt {d\mathbf X_2 \cdot \mathbf F^T\cdot\mathbf F \cdot d\mathbf X_2}}{dX_1dX_2}\cos\theta_{12}\\
\mathbf I_1 \cdot \mathbf C \cdot \mathbf I_2&= \Lambda_{\mathbf I_1}\Lambda_{\mathbf I_2}\cos\theta_{12}\\
\end{align}\,\!
},
{
"math_input":
\begin{align}
\eta(x,t) =& a \left\{
\cos \theta
+ \tfrac12 (k a)\, \cos 2\theta
+ \tfrac38 (k a)^2\, \cos 3\theta
\right\}
\\ &
+ \mathcal{O}\left( (ka)^4 \right),
\\
\Phi(x,z,t) =& a\frac{\omega}{k}\, \text{e}^{kz}\, \sin \theta
+ \mathcal{O}\left( (ka)^4 \right),
\\
c =& \frac{\omega}{k} = \left( 1 + \tfrac12 (ka)^2 \right)\, \sqrt{\frac{g}{k}}
+ \mathcal{O}\left( (ka)^4 \right), \text{ and}
\\
\theta(x,t) =& kx - \omega t,
\end{align}
},
{
"math_input": (Df)(\mathbf{a},\mathbf{b}) = \left[\begin{matrix}
\frac{\partial f_1}{\partial x_1}(\mathbf{a},\mathbf{b}) &
\cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a},\mathbf{b})\\
\vdots & \ddots & \vdots\\
\frac{\partial f_m}{\partial x_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a},\mathbf{b})
\end{matrix}\right|\left.
\begin{matrix}
\frac{\partial f_1}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_1}{\partial y_m}(\mathbf{a},\mathbf{b})\\
\vdots & \ddots & \vdots\\
\frac{\partial f_m}{\partial y_1}(\mathbf{a},\mathbf{b}) & \cdots & \frac{\partial f_m}{\partial y_m}(\mathbf{a},\mathbf{b})\\
\end{matrix}\right] = [X|Y]
},
{
"math_input": \begin{align}
Var( X(t+1) | X(t)=i) &= Var(X(t)) + Var(\Delta(t+1)| X(t)=i)\\
&= 0 + E[\Delta(t+1)^2| X(t)=i] - E[\Delta(t+1)| X(t)=i]^2\\
&= (i-1-i)^2 \cdot P_{i,i-1} + (i-i)^2 \cdot P_{i,i} + (i+1-i)^2 \cdot P_{i,i+1} - E[\Delta(t+1)| X(t)=i]^2\\
&= P_{i,i-1} + P_{i,i+1} - E[\Delta(t+1)| X(t)=i]^2\\
&= \frac{(N-i)i}{(r i + N -i)N} + \frac{(N-i)i(1+s)}{(r i + N -i)N} - E[\Delta(t+1)| X(t)=i]^2\\
&= i (N-i)\frac{2+s }{(r i + N -i)N} - E[\Delta(t+1)| X(t)=i]^2\\
&= i (N-i)\frac{2+s }{(r i + N -i)N} - (p s \dfrac{1-p}{p s + 1})^2\\
&= p(1-p)\frac{2+s (p s + 1)}{(p s + 1)^2} - p(1-p) \frac{p s^2(1-p)}{(p s + 1)^2}\\
&= p(1-p)\dfrac{2+2 p s + s + p^2 s^2 }{(p s +1)^2}
\end{align}
},
{
"math_input":
\begin{align}
\eta(x,t) =& a \left\{
\cos \theta
+ \tfrac12 (k a)\, \cos 2\theta
+ \tfrac38 (k a)^2\, \cos 3\theta
\right\}
\\ &
+ \mathcal{O}\left( (ka)^4 \right),
\\
\Phi(x,z,t) =& \frac{\omega}{k}\, \text{e}^{kz}\, \sin \theta
+ \mathcal{O}\left( (ka)^4 \right),
\\
c =& \frac{\omega}{k} = \left( 1 + \tfrac12 (ka)^2 \right)\, \sqrt{\frac{g}{k}}
+ \mathcal{O}\left( (ka)^4 \right), \text{ and}
\\
\theta(x,t) =& kx - \omega t,
\end{align}
},
{
"math_input": \begin{align}
f(x,p) \star g(x,p) &= f\left(x+\frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{p} , p - \frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot g(x,p) \\
&= f(x,p) \cdot g\left(x -\frac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{p} , p + \frac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{x}\right) \\
&= f\left(x +\frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{p} , p\right) \cdot g\left(x -\frac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{p} , p\right) \\
&= f\left(x , p - \frac{i \hbar}{2} \stackrel{\rightarrow }{\partial }_{x}\right) \cdot g\left(x , p + \frac{i \hbar}{2} \stackrel{\leftarrow }{\partial }_{x}\right).
\end{align}
},
{
"math_input":
\begin{align}
&(1 - t)^2 \mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^2 \mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0 + (1 - t)^{2}t\mathbf{P}_0 + 2(1 - t)^2 t\mathbf{P}_1 \\
&+ 2(1 - t)t^2 \mathbf{P}_1 + (1 - t)t^2 \mathbf{P}_2 + t^3 \mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0
+ (1 - t)^2 t \left( \mathbf{P}_0 + 2\mathbf{P}_1\right)
+ (1 - t) t^2 \left(2\mathbf{P}_1 + \mathbf{P}_2\right)
+ t^{3}\mathbf{P}_2 \\
= {} &(1 - t)^3 \mathbf{P}_0
+ 3(1 - t)^2 t \left( \frac{\mathbf{P}_0 + 2\mathbf{P}_1}{3} \right)
+ 3(1 - t) t^2 \left( \frac{2\mathbf{P}_1 + \mathbf{P}_2}{3} \right)
+ t^{3}\mathbf{P}_2
\end{align}
},
{
"math_input": \begin{align} \mathcal{G}(\Pi_\beta) &= \displaystyle \sum_\alpha \Pi_\alpha \left| \langle \psi_\alpha | \psi_\beta \rangle \right|^2 \\
&= \displaystyle \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{1}{d+1} \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{d}{d+1}\sum_\alpha \frac{1}{d}\Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \left( \Pi_\beta + I \right)
\end{align}
},
{
"math_input": \begin{align}
\nabla^2 U &= -\left[\begin{align}
&M_x\left({\partial^2 B_x \over {\partial x}^2} +
{\partial^2 B_x \over {\partial y}^2} +
{\partial^2 B_x \over {\partial z}^2}\right) +\\
&M_y\left({\partial^2 B_y \over {\partial x}^2} +
{\partial^2 B_y \over {\partial y}^2} +
{\partial^2 B_y \over {\partial z}^2}\right) +\\
&M_z\left({\partial^2 B_z \over {\partial x}^2} +
{\partial^2 B_z \over {\partial y}^2} +
{\partial^2 B_z \over {\partial z}^2}\right)\end{align}\right]\\
&= -(M_x \nabla^2 B_x + M_y \nabla^2 B_y + M_z \nabla^2 B_z)
\end{align}
},
{
"math_input":
\begin{align}
\mathbb{P}(S_{X_t+1}>x) & {} = \int_0^\infty \mathbb{P}(S_{X_t+1}>x \mid J_{X_t} = s) f_S(s) \, ds \\[12pt]
& {} = \int_0^\infty \mathbb{P}(S_{X_t+1}>x | S_{X_t+1}>t-s) f_S(s)\, ds \\[12pt]
& {} = \int_0^\infty \frac{\mathbb{P}(S_{X_t+1}>x \, , \, S_{X_t+1}>t-s)}{\mathbb{P}(S_{X_t+1}>t-s)} f_S(s) \, ds \\[12pt]
& {} = \int_0^\infty \frac{ 1-F(\max \{ x,t-s \}) }{1-F(t-s)} f_S(s) \, ds \\[12pt]
& {} = \int_0^\infty \min \left\{\frac{ 1-F(x) }{1-F(t-s)},\frac{ 1-F(t-s) }{1-F(t-s)}\right\} f_S(s) \, ds \\[12pt]
& {} = \int_0^\infty \min \left\{\frac{ 1-F(x) }{1-F(t-s)},1\right\} f_S(s) \, ds \\[12pt]
& {} \geq 1-F(x) \\[12pt]
& {} = \mathbb{P}(S_1>x)
\end{align}
},
{
"math_input": \begin{align} \langle \psi | \psi \rangle & = \left ( c_1| \uparrow_z \rangle + c_2| \downarrow_z \rangle \right ) \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) \\
& = c_1| \uparrow_z \rangle \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) + c_2| \downarrow_z \rangle \left ( c_1^{*} \langle \uparrow_z | + c_2^{*} \langle \downarrow_z | \right ) \\
& = c_1 c_1^{*} \langle \uparrow_z | \uparrow_z \rangle + c_1 c_2^{*} \langle \downarrow_z | \uparrow_z \rangle + c_2 c_1^{*} \langle \uparrow_z | \downarrow_z \rangle + c_2 c_2^{*} \langle \downarrow_z | \downarrow_z \rangle \\
& = |c_1|^2+|c_2|^2 \\
& = 1
\end{align}
},
{
"math_input": \begin{align} \mathcal{G}(\Pi_\beta) &= \displaystyle \sum_\alpha \Pi_\alpha \left| \langle \psi_\alpha | \psi_\beta \rangle \right|^2 \\
&= \displaystyle \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{1}{d+1} \Pi_\beta + \frac{1}{d+1} \sum_{\alpha \neq \beta} \Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \Pi_\beta + \frac{d}{d+1}\sum_\alpha \frac{1}{d}\Pi_\alpha \\
&= \displaystyle \frac{d}{d+1} \left( \Pi_\beta + I \right) \end{align}
},
{
"math_input":
\begin{matrix}
\tau_I(\tau) &=& \tau\\
\tau_I(a) &=& a \mbox{ if } a \notin I\\
\tau_I(a) &=& \tau \mbox{ if } a \in I\\
\tau_I(x + y) &=& \tau_I(x) + \tau_I(y)\\
\tau_I(x \cdot y) &=& \tau_I(x) \cdot \tau_I(y)\\
\partial_H(\tau) &=& \tau\\
x \cdot \tau &=& x\\
\tau \cdot x &=& \tau \cdot x + x\\
a\cdot(\tau\cdot x + y) &=& a\cdot(\tau\cdot x + y) + a\cdot x \\
\tau \cdot x \vert\lfloor y &=& \tau\cdot ( x \vert\vert y)\\
\tau \vert\lfloor x &=& \tau \cdot x \\
\tau \vert x &=& \delta\\
x \vert \tau &=& \delta\\
\tau\cdot x \vert y &=& x \vert y\\
x \vert \tau\cdot y &=& x \vert y\\
(x + y)\vert z &=& x\vert z + y\vert z\\
x \vert (y + z) &=& x\vert y + x\vert z
\end{matrix}
},
{
"math_input": {\color{OliveGreen}\text{M}
\begin{cases}\text{u}\\\text{o}\\\text{ou}\end{cases}
\begin{cases}\varnothing\\\text{'}\end{cases}
\begin{cases}\varnothing\\\text{a}\end{cases}
\begin{cases}\text{mm}\\\text{m}\end{cases}
\text{a}
\text{r}}
~~~~
{\color{MidnightBlue}\begin{cases}\text{Al}\\\text{al}\\\text{El}\\\text{el}\\\varnothing\end{cases}
\begin{cases}\text{-}\\\textvisiblespace\\\varnothing\end{cases}}
{\color{RedViolet}\begin{cases}\text{Q}\\\text{G}\\\text{K}\\\text{Kh}\end{cases}
\text{a}
\begin{cases}\text{d}\\\text{dh}\\\text{dd}\\\text{dhdh}\\\text{th}\\\text{zz}\end{cases}
\text{a}
\begin{cases}\text{f}\\\text{ff}\end{cases}
\begin{cases}\text{i}\\\text{y}\end{cases}}
},
{
"math_input":
\begin{align}
\ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln \left( \prod_p \frac{1}{1-p^{-1}}\right)
= \sum_p \ln \left( \frac{1}{1-p^{-1}}\right) = \sum_p - \ln(1-p^{-1}) \\
& {} = \sum_p \left( \frac{1}{p} + \frac{1}{2p^2} + \frac{1}{3p^3} + \cdots \right) \\
& {} = \left( \sum_{p}\frac{1}{p} \right) + \sum_p \frac{1}{p^2} \left( \frac{1}{2} + \frac{1}{3p} + \frac{1}{4p^2} + \cdots \right) \\
& {} < \left( \sum_p \frac{1}{p} \right) + \sum_p \frac{1}{p^2} \left( 1 + \frac{1}{p} + \frac{1}{p^2} + \cdots \right) \\
& {} = \left( \sum_p \frac{1}{p} \right) + \left( \sum_p \frac{1}{p(p-1)} \right) \\
& {} = \left( \sum_p \frac{1}{p} \right) + C
\end{align}
},
{
"math_input":
\begin{array}{llll}
1: & x:\alpha \vdash x : \alpha & [\texttt{Var}] & (x:\alpha \in \left\{x:\alpha\right\})\\
2: & \vdash \lambda x.x : \alpha\rightarrow\alpha & [\texttt{Abs}] & (1)\\
3: & \vdash \lambda x.x : \forall \alpha.\alpha\rightarrow\alpha & [\texttt{Gen}] & (2),\ (\alpha \not\in free(\epsilon))\\
4: & id:\lambda\alpha.\alpha\rightarrow\alpha \vdash id : \lambda\alpha.\alpha\rightarrow\alpha & [\texttt{Var}] & (id:\lambda\alpha.\alpha\rightarrow\alpha \in \left\{id : \lambda\alpha.\alpha\rightarrow\alpha\right\})\\
5: & \vdash \textbf{let}\, id = \lambda x . x\ \textbf{in}\ id\, :\,\forall\alpha.\alpha\rightarrow\alpha & [\texttt{Let}] & (3),\ (4)\\
\end{array}
},
{
"math_input": \begin{align}
&\quad A(a, \lambda) B(b, \lambda) + A(a, \lambda) B(b', \lambda) + A(a', \lambda) B(b, \lambda) - A(a', \lambda) B(b', \lambda)\\
&= A(a, \lambda) \left[B(b, \lambda) + B(b', \lambda)\right] + A(a', \lambda) \left[B(b, \lambda) - B(b', \lambda)\right]\\
&\leq \big| A(a, \lambda) \left[B(b, \lambda) + B(b', \lambda)\right] + A(a', \lambda) \left[B(b, \lambda) - B(b', \lambda)\right] \big|\\
&\leq \big| A(a, \lambda) \left[B(b, \lambda) + B(b', \lambda)\right] \big| + \big| A(a', \lambda) \left[B(b, \lambda) - B(b', \lambda)\right] \big|\\
&\leq \big| B(b, \lambda) + B(b', \lambda) \big| + \big| B(b, \lambda) - B(b', \lambda) \big| \leq 2
\end{align}
},
{
"math_input":
\begin{bmatrix}
\hat{\textbf{x}}_{t\mid t} \\
\hat{\textbf{x}}_{t-1\mid t} \\
\vdots \\
\hat{\textbf{x}}_{t-N+1\mid t} \\
\end{bmatrix}
=
\begin{bmatrix}
\textbf{I} \\
0 \\
\vdots \\
0 \\
\end{bmatrix}
\hat{\textbf{x}}_{t\mid t-1}
+
\begin{bmatrix}
0 & \ldots & 0 \\
\textbf{I} & 0 & \vdots \\
\vdots & \ddots & \vdots \\
0 & \ldots & I \\
\end{bmatrix}
\begin{bmatrix}
\hat{\textbf{x}}_{t-1\mid t-1} \\
\hat{\textbf{x}}_{t-2\mid t-1} \\
\vdots \\
\hat{\textbf{x}}_{t-N+1\mid t-1} \\
\end{bmatrix}
+
\begin{bmatrix}
\textbf{K}^{(0)} \\
\textbf{K}^{(1)} \\
\vdots \\
\textbf{K}^{(N-1)} \\
\end{bmatrix}
\textbf{y}_{t\mid t-1}
},
{
"math_input": \begin{align}P(\alpha)&=|c_0|^2\delta^2(\alpha-\alpha_0)+|c_1|^2\delta^2(\alpha-\alpha_1) \\
&\, \, \, \, \, +2c_0^*c_1
e^{|\alpha|^2-\frac{1}{2}|\alpha_0|^2-\frac{1}{2}|\alpha_1|^2}
e^{(\alpha_1^*-\alpha_0^*)\cdot\partial/\partial(2\alpha^*-\alpha_0^*-\alpha_1^*)}
e^{(\alpha_0-\alpha_1)\cdot\partial/\partial(2\alpha-\alpha_0-\alpha_1)}
\cdot \delta^2(2\alpha-\alpha_0-\alpha_1) \\
&\, \, \, \, \, +2c_0c_1^*
e^{|\alpha|^2-\frac{1}{2}|\alpha_0|^2-\frac{1}{2}|\alpha_1|^2}
e^{(\alpha_0^*-\alpha_1^*)\cdot\partial/\partial(2\alpha^*-\alpha_0^*-\alpha_1^*)}
e^{(\alpha_1-\alpha_0)\cdot\partial/\partial(2\alpha-\alpha_0-\alpha_1)}
\cdot \delta^2(2\alpha-\alpha_0-\alpha_1).
\end{align}
},
{
"math_input":
\begin{align}
\sigma_{ij} &=
-\begin{pmatrix}
p&0&0\\
0&p&0\\
0&0&p
\end{pmatrix} +
\mu \begin{pmatrix}
2 \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} & \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} \\
\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} & 2 \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} \\
\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} & \frac{\partial w}{\partial y} + \frac{\partial v}{\partial z} & 2\frac{\partial w}{\partial z}
\end{pmatrix} \\
&= -p I + \mu (\nabla \mathbf{v} + (\nabla \mathbf{v})^T)
\end{align}
},
{
"math_input":
\mathbf{Q}_3=
\begin{pmatrix}
1 \; 0 \; 0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\
1 \; 1 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \\
0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \\
1 \; 0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \\
0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \\
0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1
\end{pmatrix},
},
{
"math_input": \begin{align}
y = \frac{E}{(\beta + 1 - \alpha)_{\alpha - \beta - 1}}
&\sum_{r = \alpha - \beta}^\infty \frac{(\beta)_r (\beta + 1 - \gamma)_r}
{(1)_r (1)_{r + \beta - \alpha}} x^{-r} \\
+ F x^{-\alpha} &\sum_{r = 0}^\infty \frac{(\alpha - \beta) (\alpha)_r (\alpha + 1 - \gamma)_r}
{(1)_r (\alpha + 1 - \beta)_r} \Biggl(\ln x^{-1} + \frac{1}{\alpha -\beta } \\
&\quad+ \sum_{k = 0}^{r - 1} \left(\frac{1}{\alpha + k} + \frac{1}{\alpha + 1 + k - \gamma} -\frac{1}{1 + k}
-\frac{1}{\alpha + 1 + k - \beta} \right) \Biggr) x^{-r}
\end{align}
},
{
"math_input": = -
\left.
\left( \begin{array}{cccc}
\tfrac{\partial^2}{\partial \theta_1^2}
& \tfrac{\partial^2}{\partial \theta_1 \partial \theta_2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_1 \partial \theta_n} \\
\tfrac{\partial^2}{\partial \theta_2 \partial \theta_1}
& \tfrac{\partial^2}{\partial \theta_2^2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_2 \partial \theta_n} \\
\vdots &
\vdots &
\ddots &
\vdots \\
\tfrac{\partial^2}{\partial \theta_n \partial \theta_1}
& \tfrac{\partial^2}{\partial \theta_n \partial \theta_2}
& \cdots
& \tfrac{\partial^2}{\partial \theta_n^2} \\
\end{array} \right)
\ell(\theta)
\right|_{\theta = \theta^*}
},
{
"math_input": \
\left\{
\begin{array}{l}
\frac{1}{3} \rho l_1^{2} \left(l_1 + 3 l_2\right)\ddot{\alpha}_1 + \frac{1}{2} \rho l_1 l_2^{2} \cos(\alpha_1 - \alpha_2)\ddot{\alpha}_2 + \frac{1}{2} \rho l_1 l_2^{2} \sin(\alpha_1 - \alpha_2)\dot{\alpha}_2^{2} + (k_1 + k_2)\alpha_1 - k_2\alpha_2\,+ \\ [5mm]
+ (\beta_1 + \beta_2)\dot{\alpha}_1 - \beta_2 \dot{\alpha}_2 - l_1 P \sin(\alpha_1 - \alpha_2) = 0 , \\ [5mm]
\frac{1}{2} \rho l_1 l_2^{2} \cos(\alpha_1 - \alpha_2)\ddot{\alpha}_1 + \frac{1}{3} \rho l_2^{3}\ddot{\alpha}_2 - \frac{1}{2} \rho l_1 l_2^{2} \sin(\alpha_1 - \alpha_2)\dot{\alpha}_1^{2} - k_2(\alpha_1 - \alpha_2) - \beta_2(\dot{\alpha}_1 - \dot{\alpha}_2) = 0 ,
\end{array}
\right.
},
{
"math_input": B_n(x_1,\dots,x_n) = \det\begin{bmatrix}x_1 & {n-1 \choose 1} x_2 & {n-1 \choose 2}x_3 & {n-1 \choose 3} x_4 & {n-1 \choose 4} x_5 & \cdots & \cdots & x_n \\ \\
-1 & x_1 & {n-2 \choose 1} x_2 & {n-2 \choose 2} x_3 & {n-2 \choose 3} x_4 & \cdots & \cdots & x_{n-1} \\ \\
0 & -1 & x_1 & {n-3 \choose 1} x_2 & {n-3 \choose 2} x_3 & \cdots & \cdots & x_{n-2} \\ \\
0 & 0 & -1 & x_1 & {n-4 \choose 1} x_2 & \cdots & \cdots & x_{n-3} \\ \\
0 & 0 & 0 & -1 & x_1 & \cdots & \cdots & x_{n-4} \\ \\
0 & 0 & 0 & 0 & -1 & \cdots & \cdots & x_{n-5} \\ \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \vdots \\ \\
0 & 0 & 0 & 0 & 0 & \cdots & -1 & x_1 \end{bmatrix}.
},
{
"math_input":
\begin{align}
I_0~\boldsymbol{\mathit{1}} - \frac{\partial I_1}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_0}{\partial \boldsymbol{A}} & = 0 \\
I_1~\boldsymbol{\mathit{1}} - \frac{\partial I_2}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} -
I_2~\boldsymbol{\mathit{1}} - \frac{\partial I_3}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_2}{\partial \boldsymbol{A}} & = 0 \\
I_3~\boldsymbol{\mathit{1}} - \frac{\partial I_4}{\partial \boldsymbol{A}}~\boldsymbol{\mathit{1}} - \boldsymbol{A}^T\cdot\frac{\partial I_3}{\partial \boldsymbol{A}} & = 0 ~.
\end{align}
},
{
"math_input": \begin{align}
y = \frac{G}{(\alpha + 1 - \beta)_{\beta - \alpha - 1}}
\sum_{r = \beta - \alpha}^\infty &\frac{(\alpha)_r (\alpha + 1 - \gamma)_r}
{(1)_r (1)_{r + \alpha - \beta}} x^{-r} \\
{} + H x^{-\beta} \sum_{r = 0}^\infty &\frac{(\beta - \alpha) (\beta)_r (\beta + 1 - \gamma)_r}
{(1)_r (\beta + 1 - \alpha)_r} \Biggl(\ln x^{-1} + \frac{1}{\beta - \alpha } \\
&\;+ \sum_{k = 0}^{r - 1} \left(\frac{1}{\beta + k} + \frac{1}{\beta + 1 + k - \gamma} - \frac{1}{1 + k}
- \frac{1}{\beta + 1 + k - \alpha} \right) \Biggr) x^{-r}
\end{align}
},
{
"math_input":
\begin{align}
\frac {df}{d\phi}
&= \int_0^{2\pi}\;\frac{\partial}{\partial\phi}\left(e^{\phi\cos\theta}\;\cos(\phi\sin\theta)\right)\;d\theta\,
\\
&= \int_0^{2\pi}\;e^{\phi\cos\theta}\;\left(\cos\theta\cos(\phi\sin\theta)\; - \;\sin\theta\sin(\phi\sin\theta)\right)\;d\theta\,
\\
&= \int_0^{2\pi}\;\frac {1}{\phi}\;\frac {\partial}{\partial\theta}\left(e^{\phi\cos\theta}\;\sin(\phi\sin\theta)\right)\;d\theta\,
\\
&= \frac {1}{\phi}\;\int_0^{2\pi}\;d\left(e^{\phi\cos\theta}\;\sin(\phi\sin\theta)\right)\,
\\
&= \frac {1}{\phi}\;\left(e^{\phi\cos\theta}\;\sin(\phi\sin\theta)\right)\;\bigg|_0^{2\pi}\,
\\
&= 0.\,
\end{align}
},
{
"math_input":
Pr\begin{cases}
Ds\begin{cases}
Sp(\pi)\begin{cases}
Va:\\
S^{0},\cdots,S^{T},O^{0},\cdots,O^{T}\\
Dc:\\
\begin{cases}
& P\left(S^{0}\wedge\cdots\wedge O^{T}|\pi\right)\\
= & \left[\begin{array}{c}
P\left(S^{0}\wedge O^{0}|\pi\right)\\
\prod_{t=1}^{T}\left[P\left(S^{t}|S^{t-1}\wedge\pi\right)\times P\left(O^{t}|S^{t}\wedge\pi\right)\right]\end{array}\right]\end{cases}\\
Fo:\\
\begin{cases}
P\left(S^t \mid S^{t-1}\wedge\pi\right)\equiv G\left(S^{t},A\bullet S^{t-1},Q\right)\\
P\left(O^t \mid S^t \wedge\pi\right)\equiv G\left(O^{t},H\bullet S^{t},R\right)\end{cases}\end{cases}\\
Id\end{cases}\\
Qu:\\
P\left(S^T \mid O^0 \wedge\cdots\wedge O^{T}\wedge\pi\right)\end{cases}
},
{
"math_input": {}_{-\frac{1}{5} +e^{\frac{6}{5}} {}_4F_3\left(-\frac{1}{5},\frac{1}{20},\frac{3}{10},\frac{11}{20};\frac{1}{5},\frac{2}{5},\frac{3}{5};\frac{256}{3125e^6}\right)+\frac{2}{25e^{\frac{6}{5}}}{}_4F_3\left(\frac{1}{5},\frac{9}{20},\frac{7}{10},\frac{19}{20};\frac{3}{5},\frac{4}{5},\frac{7}{5};\frac{256}{3125e^6}\right)-\frac{4}{125e^{\frac{12}{5}}}{}_4F_3\left(\frac{2}{5},\frac{13}{20},\frac{9}{10},\frac{23}{20};\frac{4}{5},\frac{6}{5},\frac{8}{5};\frac{256}{3125e^6}\right)+\frac{7}{625e^{\frac{18}{5}}}{}_4F_3\left(\frac{3}{5},\frac{17}{20},\frac{11}{10},\frac{27}{20};\frac{6}{5},\frac{7}{5},\frac{9}{5};\frac{256}{3125e^6}\right)-\pi\approx 2.89221114964408683\times10^{-8}}
},
{
"math_input":
\mathbf{Q}_2=
\begin{pmatrix}
0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \\
1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \\
0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \\
1 \; 0 \; 1 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \\
0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \\
0 \; 0 \; 1 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 1 \; 0
\end{pmatrix},
},
{
"math_input":
\begin{array}{lcl}
K,N &=& \text{as above} \\
\theta_{i=1 \dots K}, \phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
z_{i=1 \dots N}, x_{i=1 \dots N}, F(x|\theta) &=& \text{as above} \\
\alpha &=& \text{shared hyperparameter for component parameters} \\
\beta &=& \text{shared hyperparameter for mixture weights} \\
H(\theta|\alpha) &=& \text{prior probability distribution of component parameters, parametrized on } \alpha \\
\theta_{i=1 \dots K} &\sim& H(\theta|\alpha) \\
\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
x_{i=1 \dots N} &\sim& F(\theta_{z_i})
\end{array}
},
{
"math_input": {} = \begin{vmatrix} 1 & 2 \\ 5 & 6 \end{vmatrix} \cdot \begin{vmatrix} 11 & 12 \\ 15 & 16 \end{vmatrix}
- \begin{vmatrix} 1 & 3 \\ 5 & 7 \end{vmatrix} \cdot \begin{vmatrix} 10 & 12 \\ 14 & 16 \end{vmatrix}
+ \begin{vmatrix} 1 & 4 \\ 5 & 8 \end{vmatrix} \cdot \begin{vmatrix} 10 & 11 \\ 14 & 15 \end{vmatrix}
+ \begin{vmatrix} 2 & 3 \\ 6 & 7 \end{vmatrix} \cdot \begin{vmatrix} 9 & 12 \\ 13 & 16 \end{vmatrix}
- \begin{vmatrix} 2 & 4 \\ 6 & 8 \end{vmatrix} \cdot \begin{vmatrix} 9 & 11 \\ 13 & 15 \end{vmatrix}
+ \begin{vmatrix} 3 & 4 \\ 7 & 8 \end{vmatrix} \cdot \begin{vmatrix} 9 & 10 \\ 13 & 14 \end{vmatrix}
},
{
"math_input":
\begin{align}
\frac{\partial F}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial u} & = -\frac{\partial F}{\partial u} \\[6pt]
\frac{\partial G}{\partial x} \frac{\partial x}{\partial u} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial u} & = -\frac{\partial G}{\partial u} \\[6pt]
\frac{\partial F}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial F}{\partial y} \frac{\partial y}{\partial v} & = -\frac{\partial F}{\partial v} \\[6pt]
\frac{\partial G}{\partial x} \frac{\partial x}{\partial v} +\frac{\partial G}{\partial y} \frac{\partial y}{\partial v} & = -\frac{\partial G}{\partial v}.
\end{align}
},
{
"math_input":
\begin{align}
\sigma(X,Y)
&= \operatorname{E}\left[\left(X - \operatorname{E}\left[X\right]\right) \left(Y - \operatorname{E}\left[Y\right]\right)\right] \\
&= \operatorname{E}\left[X Y - X \operatorname{E}\left[Y\right] - \operatorname{E}\left[X\right] Y + \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right]\right] \\
&= \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] + \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right] \\
&= \operatorname{E}\left[X Y\right] - \operatorname{E}\left[X\right] \operatorname{E}\left[Y\right].
\end{align}
},
{
"math_input":
\mathbf{Q}_1 =
\begin{pmatrix}
1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \\
0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \\
0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \\
0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 1 \; 1 \; 0 \\
0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1 \; 0 \; 1 \; 1 \; 0 \; 1 \; 1 \; 1 \; 1 \\
0 \; 0 \; 0 \; 0 \; 0 \; 1 \; 0 \; 0 \; 0 \; 0 \; 1 \; 1 \; 0 \; 0 \; 1 \; 0 \; 0 \; 1 \; 1 \; 0 \; 1
\end{pmatrix},
},
{
"math_input":
\begin{matrix}
\sin 0 & = & \sin 0^\circ & = & \sqrt{0}/2 & = & \cos 90^\circ & = & \cos \left( \frac {\pi} {2} \right) \\ \\
\sin \left( \frac {\pi} {6} \right) & = & \sin 30^\circ & = & \sqrt{1}/2 & = & \cos 60^\circ & = & \cos \left( \frac {\pi} {3} \right) \\ \\
\sin \left( \frac {\pi} {4} \right) & = & \sin 45^\circ & = & \sqrt{2}/2 & = & \cos 45^\circ & = & \cos \left( \frac {\pi} {4} \right) \\ \\
\sin \left( \frac {\pi} {3} \right) & = & \sin 60^\circ & = & \sqrt{3}/2 & = & \cos 30^\circ & = & \cos \left( \frac {\pi} {6} \right)\\ \\
\sin \left( \frac {\pi} {2} \right) & = & \sin 90^\circ & = & \sqrt{4}/2 & = & \cos 0^\circ & = & \cos 0
\end{matrix}
},
{
"math_input": \begin{align}
& \mu \in \left[\, \hat\mu + t_{n-1,\alpha/2}\, \frac{1}{\sqrt{n}}s,\ \
\hat\mu + t_{n-1,1-\alpha/2}\,\frac{1}{\sqrt{n}}s \,\right] \approx
\left[\, \hat\mu - |z_{\alpha/2}|\frac{1}{\sqrt n}s,\ \
\hat\mu + |z_{\alpha/2}|\frac{1}{\sqrt n}s \,\right], \\
& \sigma^2 \in \left[\, \frac{(n-1)s^2}{\chi^2_{n-1,1-\alpha/2}},\ \
\frac{(n-1)s^2}{\chi^2_{n-1,\alpha/2}} \,\right] \approx
\left[\, s^2 - |z_{\alpha/2}|\frac{\sqrt{2}}{\sqrt{n}}s^2,\ \
s^2 + |z_{\alpha/2}|\frac{\sqrt{2}}{\sqrt{n}}s^2 \,\right],
\end{align}
},
{
"math_input":
\begin{matrix}
S&S\\
S 10_H \overline{10}_H&S 10_H^{\alpha\beta} \overline{10}_{H\alpha\beta}\\
10_H 10_H H_d&\epsilon_{\alpha\beta\gamma\delta\epsilon}10_H^{\alpha\beta}10_H^{\gamma\delta} H_d^{\epsilon}\\
\overline{10}_H\overline{10}_H H_u&\epsilon^{\alpha\beta\gamma\delta\epsilon}\overline{10}_{H\alpha\beta}\overline{10}_{H\gamma\delta}H_{u\epsilon}\\
H_d 10 10&\epsilon_{\alpha\beta\gamma\delta\epsilon}H_d^{\alpha}10_i^{\beta\gamma}10_j^{\delta\epsilon}\\
H_d \bar{5} 1 &H_d^\alpha \bar{5}_{i\alpha} 1_j\\
H_u 10 \bar{5}&H_{u\alpha} 10_i^{\alpha\beta} \bar{5}_{j\beta}\\
\overline{10}_H 10 \phi&\overline{10}_{H\alpha\beta} 10_i^{\alpha\beta} \phi_j\\
\end{matrix}
},
{
"math_input": \mathcal{M}\{\xi\in B\}=\begin{cases} \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 1-\underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\}, & \text{if } \underset{f(B_1,B_2,\cdots,B_n)\subset B^c}{\operatorname{sup} }\;\underset{1\le k\le n}{\operatorname{min} }\mathcal{M}_k\{\xi_k\in B_k\} > 0.5 \\ 0.5, & \text{otherwise} \end{cases}
},
{
"math_input":
\begin{align}
\varphi_1 = - \frac{\partial w^K}{\partial x_1}
- \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_1^K
+ \frac{\partial }{\partial x_1}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
+ \frac{1}{c^2}\frac{\partial \Omega}{\partial x_2} \\
\varphi_2 = - \frac{\partial w^K}{\partial x_2}
- \frac{1}{\kappa G h}\left(1 - \frac{1}{\mathcal{A}} - \frac{\mathcal{B} c^2}{2}\right)Q_2^K
+ \frac{\partial }{\partial x_2}\left(\frac{D}{\kappa G h \mathcal{A}}\nabla^2 \Phi + \Phi - \Psi\right)
+ \frac{1}{c^2}\frac{\partial \Omega}{\partial x_1}
\end{align}
},
{
"math_input":
K=\left[
\begin{array}{ccccc}
\frac{E^{(1)}A^{(1)}}{L^{(1)}} & -\frac{E^{(1)}A^{(1)}}{L^{(1)}} & 0 & ... & 0 \\
-\frac{E^{(1)}A^{(1)}}{L^{(1)}} & \frac{E^{(1)}A^{(1)}}{L^{(1)}} + \frac{E^{(2)}A^{(2)}}{L^{(2)}} & -\frac{E^{(2)}A^{(2)}}{L^{(2)}} & ... & 0 \\
0 & -\frac{E^{(2)}A^{(2)}}{L^{(2)}} & \frac{E^{(2)}A^{(2)}}{L^{(2)}}+ \frac{E^{(3)}A^{(3)}}{L^{(3)}} & ... & 0 \\
... & ... & ... & ... & ... \\
0 & 0 & ... & \frac{E^{(Ne-1)}A^{(Ne-1)}}{L^{(Ne-1)}} + \frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} & -\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} \\
0 & 0 & ... & -\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}} & \frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}}
\end{array}
\right]
},
{
"math_input":
\begin{align}
\delta K & = \int_0^T \int_{\Omega^0} \int_{-h}^h \rho \left[
\left(\dot{u}^0_\alpha - x_3~\dot{w}^0_{,\alpha}\right)~
\left(\delta\dot{u}^0_\alpha - x_3~\delta\dot{w}^0_{,\alpha}\right)
+ \dot{w}^0~\delta\dot{w}^0\right] ~\mathrm{d}x_3~\mathrm{d}A~\mathrm{d}t \\
& = \int_0^T \int_{\Omega^0} \int_{-h}^h \rho
\left(\dot{u}^0_\alpha~\delta\dot{u}^0_\alpha
- x_3~\dot{w}^0_{,\alpha}~ \delta\dot{u}^0_\alpha
- x_3~\dot{u}^0_\alpha~\delta\dot{w}^0_{,\alpha}
+ x_3^2~\dot{w}^0_{,\alpha}~\delta\dot{w}^0_{,\alpha}
+ \dot{w}^0~\delta\dot{w}^0\right) ~\mathrm{d}x_3~\mathrm{d}A~\mathrm{d}t
\end{align}
},
{
"math_input": \begin{bmatrix}
c_{1}\\
c_{2}\\
\vdots\\
c_{n}\end{bmatrix}
=
\begin{bmatrix}
\left\langle p_{1},p_{1}\right\rangle & \left\langle p_{2},p_{1}\right\rangle & \cdots & \left\langle p_{n},p_{1}\right\rangle \\
\left\langle p_{1},p_{2}\right\rangle & \left\langle p_{2},p_{2}\right\rangle & \cdots & \left\langle p_{n},p_{2}\right\rangle \\
\vdots & \vdots & \ddots & \vdots\\
\left\langle p_{1},p_{n}\right\rangle & \left\langle p_{2},p_{n}\right\rangle & \cdots & \left\langle p_{n},p_{n}\right\rangle \end{bmatrix}^{-1}
\begin{bmatrix}
\left\langle x,p_{1}\right\rangle \\
\left\langle x,p_{2}\right\rangle \\
\vdots\\
\left\langle x,p_{n}\right\rangle \end{bmatrix},
},
{
"math_input":
C_{4,1/2}=
\begin{bmatrix}
c_1 & c_2 & c_3&c_4\\
-c_2 &c_1&-c_4&c_3\\
-c_3&c_4&c_1&-c_2\\
-c_4&-c_3&c_2&c_1\\
c_1^* & c_2^*&c_3^*&c_4^*\\
-c_2^* &c_1^*&-c_4^*&c_3^*\\
-c_3^*&c_4^*&c_1^*&-c_2^*\\
-c_4^*&-c_3^*&c_2^*&c_1^*
\end{bmatrix}
\quad\mbox{and}\quad{}
C_{4,3/4}=
\begin{bmatrix}
c_1&c_2&\frac{c_3}{\sqrt 2}&\frac{c_3}{\sqrt 2}\\
-c_2^*&c_1^*&\frac{c_3}{\sqrt 2}&-\frac{c_3}{\sqrt 2}\\
\frac{c_3^*}{\sqrt 2}&\frac{c_3^*}{\sqrt 2}&\frac{\left(-c_1-c_1^*+c_2-c_2^*\right)}{2}&\frac{\left(-c_2-c_2^*+c_1-c_1^*\right)}{2}\\
\frac{c_3^*}{\sqrt 2}&-\frac{c_3^*}{\sqrt 2}&\frac{\left(c_2+c_2^*+c_1-c_1^*\right)}{2}&-\frac{\left(c_1+c_1^*+c_2-c_2^*\right)}{2}
\end{bmatrix}
},
{
"math_input": \begin{align}
\tan(\theta_1 + \theta_2) &
= \frac{ e_1 }{ e_0 - e_2 }
= \frac{ x_1 + x_2 }{ 1 \ - \ x_1 x_2 }
= \frac{ \tan\theta_1 + \tan\theta_2 }{ 1 \ - \ \tan\theta_1 \tan\theta_2 }
,
\\[10pt]
\tan(\theta_1 + \theta_2 + \theta_3) &
= \frac{ e_1 - e_3 }{ e_0 - e_2 }
= \frac{ (x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3) }{ 1 \ - \ (x_1x_2 + x_1 x_3 + x_2 x_3) },
\\[10pt]
\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &
= \frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\ \\ &
= \frac{ (x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4) },
\end{align}
},
{
"math_input": \begin{align}
\tan(\theta_1 + \theta_2) &
= \frac{ e_1 }{ e_0 - e_2 }
= \frac{ x_1 + x_2 }{ 1 \ - \ x_1 x_2 }
= \frac{ \tan\theta_1 + \tan\theta_2 }{ 1 \ - \ \tan\theta_1 \tan\theta_2 }
,
\\[8pt]
\tan(\theta_1 + \theta_2 + \theta_3) &
= \frac{ e_1 - e_3 }{ e_0 - e_2 }
= \frac{ (x_1 + x_2 + x_3) \ - \ (x_1 x_2 x_3) }{ 1 \ - \ (x_1x_2 + x_1 x_3 + x_2 x_3) },
\\[8pt]
\tan(\theta_1 + \theta_2 + \theta_3 + \theta_4) &
= \frac{ e_1 - e_3 }{ e_0 - e_2 + e_4 } \\[8pt] &
= \frac{ (x_1 + x_2 + x_3 + x_4) \ - \ (x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4) }{ 1 \ - \ (x_1 x_2 + x_1 x_3 + x_1 x_4 + x_2 x_3 + x_2 x_4 + x_3 x_4) \ + \ (x_1 x_2 x_3 x_4) },
\end{align}
},
{
"math_input": \mathbf T^n \begin{bmatrix} 0 \\ \vec b^{n-1} \\ \end{bmatrix} =
\begin{bmatrix}
t_0 & \dots & t_{-n+2} & t_{-n+1} \\
\vdots & \ & \ & \ \\
t_{n-2} & \ & \mathbf T^{n-1} & \ \\
t_{n-1} & \ & \ &
\end{bmatrix}
\begin{bmatrix} \ \\
0 \\
\ \\
\vec b^{n-1} \\
\ \\
\end{bmatrix} =
\begin{bmatrix} \epsilon_b^n \\
0 \\
\vdots \\
0 \\
1
\end{bmatrix}.
},
{
"math_input":
\begin{alignat}{2}
\epsilon(0,\omega) & \simeq 1 + V_q \sum_{k,i}{ \frac{q_i \frac{\partial f_k}{\partial k_i}}{\hbar \omega_0 - \frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}} }\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}(1+\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0})\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0}\\
& = 1 - V_q \frac{q^2}{m \omega_0^2} \sum_k{f_k}\\
& = 1 - V_q \frac{q^2 N}{m \omega_0^2} \\
& = 1 - \frac{4 \pi e^2}{\epsilon q^2 L^3} \frac{q^2 N}{m \omega_0^2} \\
& = 1 - \frac{\omega_{pl}^2}{\omega_0^2}
\end{alignat}
},
{
"math_input":
\begin{array}{lcl}
K,N &=& \text{as above} \\
\theta_{i=1 \dots K}, \phi_{i=1 \dots K}, \boldsymbol\phi &=& \text{as above} \\
z_{i=1 \dots N}, x_{i=1 \dots N}, F(x|\theta) &=& \text{as above} \\
\alpha &=& \text{shared hyperparameter for component parameters} \\
\beta &=& \text{shared hyperparameter for mixture weights} \\
H(\theta|\alpha) &=& \text{prior probability distribution of component parameters, parametrized on } \alpha \\
\theta_{i=1 \dots K} &\sim& H(\alpha) \\
\boldsymbol\phi &\sim& \operatorname{Symmetric-Dirichlet}_K(\beta) \\
z_{i=1 \dots N} &\sim& \operatorname{Categorical}(\boldsymbol\phi) \\
x_{i=1 \dots N} &\sim& F(\theta_{z_i})
\end{array}
},
{
"math_input": \begin{align}
\frac{v_1 (1- z)}{J_1+ (1 - z)} &= \frac{v_2 z}{J_2+ z} \\
J_2 v_1+ z v_1 - J_2 v_1 z - z^2 v_1 &= z v_2 J_1+ v_2 z - z^2 v_2\\
z^2 (v_2 - v_1) - z \underbrace{(v_2 - v_1 + J_1 v_2 + J_2 v_1)}_{B} + v_1 J_2 &= 0\\
z = \frac{B - \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} &= \frac{B - \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{2 (v_2 - v_1)} \cdot \frac{B + \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}{B + \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\
z &= \frac{ 4 (v_2 - v_1) v_1 J_2}{2 (v_2 - v_1)} \cdot \frac{1}{B + \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}\\
z &= \frac{ 2 v_1 J_2}{B + \sqrt{B^2 - 4 (v_2 - v_1) v_1 J_2}}. \qquad \qquad (3)
\end{align}
},
{
"math_input":
\begin{align}
\int_V \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i dV
&= \int_V \frac12 \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)
+ \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i dV \\
&= \int_V \left[ \epsilon_{ij}
+ \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i dV \\
&= \int_V \epsilon_{ij} \sigma_{ij} - u_i f_i dV\\
&= \int_V \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f dV
\end{align}
}
]
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