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Kitajiang
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test2_UniMERNet

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\[\frac{{\Delta}^{2}{u}_{k-1}\left(x\right)}{{\tau}^{2}}-\frac{1}{2}{q}_{k}\left(\frac{\mathrm{d}^{2}{u}_{k+1}\left(x\right)}{\mathrm{d}{x}^{2}}+\frac{\mathrm{d}^{2}{u}_{k-1}\left(x\right)}{\mathrm{d}{x}^{2}}\right)={f}_{k}\left(x\right)\,,\quad k=1,2,\ldots,{n-1}\,,\]
\[\begin{array}{r l}{\left[\begin{array}{l}{\dot{\theta}_{i}}\\ {\dot{\omega}_{i}}\end{array}\right]=}&{\left[\begin{array}{l l}{0}&{1}\\ {0}&{0}\end{array}\right]\left[\begin{array}{l}{{\theta}_{i}}\\ {{\omega}_{i}}\end{array}\right],}\\ {\left[\begin{array}{l}{\dot{p}_{i}}\\ {\dot{v}_{i}}\end{array}\right]=}&{\left[\begin{array}{l l}{0_{2\times2}}&{I_{2}}\\ {0_{2\times2}}&{\omega_{i}a}\end{array}\right]\left[\begin{array}{l}{{p}_{i}}\\ {{v}_{i}}\end{array}\right],}\end{array}\]
\[\begin{array}{r l}&{\mathrm{Pick~a~coordinate~from~X^{(j)}~,~i.e.,~}(k^{(j)},\ell),}\\ &{\alpha^{(j)}=\mathop{\arg\operatorname*{min}}_{\alpha\in\mathbb{R}}f_{\mu,W}\left(X^{(j)}+\alpha E_{k^{(j)}\ell}\right),}\\ &{X^{(j+1)}=X^{(j)}+\alpha^{(j)}E_{k^{(j)}\ell},}\end{array}\]
\[\begin{array}{r l}{A_{1}\cap B_{2}}&{=\left\{\begin{array}{l l}{0\leq x\leq\frac{\sigma r}{2}\qquad\textrm{i f}-\frac{r}{2}\leq z\leq\frac{\sigma}{2}-\frac{r}{2}}\\ {0\leq x\leq-\sigma z\left(1-\frac{\sigma}{r}\right)^{-1}\qquad\textrm{i f}\frac{\sigma}{2}-\frac{r}{2}\leq z\leq0}\\ {-\left(\frac{\sigma r}{2}+\sigma z\right)\left(1-\frac{\sigma}{r}\right)^{-1}\leq x\leq\frac{\sigma r}{2}\qquad\textrm{i f}\frac{\sigma}{2}-r\leq z\leq-\frac{r}{2}}\end{array}\right.,}\end{array}\]
\[\begin{array}{r}{\operatorname*{inf}_{\pi}{\mathbb E}[g(X_{t_{n}}^{\eta})]=\int g_{n}\prod_{i=1}^{n}p(x_{i},t_{i}|x_{i-1},\eta_{i-1})\pi_{i}(\eta_{i-1}|x_{i-1})\prod_{i}d\eta_{i}\prod_{i}d x_{i},\quad}\\ {g_{n}:=g(x_{1},\cdots,x_{n},\eta_{1},\cdots,\eta_{n})}\end{array}\]
\[\left.\begin{array}{r c l l}{\partial_{t}\left(\epsilon\partial_{t}\vec{A}\right)+\operatorname{curl}_{x}\left(\mu^{-1}\operatorname{curl}_{x}\underline{{A}}\right)}&{=}&{\underline{{j}}}&{\mathrm{~in~}Q,}\\ {\underline{{A}}(0,\cdot)}&{=}&{0}&{\mathrm{~in~}\Omega,}\\ {\partial_{t}\underline{{A}}(0,\cdot)}&{=}&{0}&{\mathrm{~in~}\Omega,}\\ {\gamma_{\mathrm{t}}\vec{A}}&{=}&{0}&{\mathrm{~on~}\Sigma,}\end{array}\right\}\]
\[\begin{array}{r l}&{d(\|u^{k}\|_{\widetilde{s+1}}^{2}\|U\|_{s-1,j}^{2})}\\ {=}&{\|u^{k}\|_{\widetilde{s+1}}^{2}d\|U\|_{s-1,j}^{2}+\|U\|_{s-1,j}^{2}d\|u^{k}\|_{\widetilde{s+1}}^{2}+d\|u^{k}\|_{\widetilde{s+1}}^{2}d\|U\|_{s-1,j}^{2}}\\ {=}&{\Big(\|u^{k}\|_{\widetilde{s+1}}^{2}(I_{1}^{\prime}+I_{2}^{\prime}+I_{4}^{\prime}+I_{5}^{\prime})+\|U\|_{s-1,j}^{2}(J_{1}+J_{2}+J_{4}+J_{5})+(I_{3}^{\prime}+I_{6}^{\prime})(J_{3}+J_{6})\Big)d t}\\ &{+\Big(\|u^{k}\|_{\widetilde{s+1}}^{2}(I_{3}^{\prime}+I_{6}^{\prime})+\|U\|_{s-1,j}^{2}(J_{3}+J_{6})\Big)d W,}\end{array}\]
\[\begin{array}{r l}{\left|\frac{1}{N}\sum_{i=1}^{N}\phi_{i}\left(\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}[\mathbf{Z}_{N}]_{i}\right)-\frac{1}{N}\sum_{i=1}^{n}\phi_{i}\left(\sigma[\mathbf{Z}_{N}]_{i}\right)\right|}&{\leq\frac{1}{N}\sum_{i=1}^{N}\left|\phi_{i}\left(\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}[\mathbf{Z}_{N}]_{i}\right)-\phi_{i}\left(\sigma[\mathbf{Z}_{N}]_{i}\right)\right|}\\ &{\leq\frac{L}{N}\sum_{i=1}^{N}\left|1+\left(\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}+\sigma\right)[\mathbf{Z}_{N}]_{i}\right|\left|\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}-\sigma\right||[\mathbf{Z}_{N}]_{i}|}\\ &{\leq L\left|\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}-\sigma\right|\left(\frac{\|\mathbf{Z}_{N}\|}{\sqrt{N}}+\left(\frac{\|\mathbf{v}_{N}\|}{\|\mathbf{Z}_{N}\|}+\sigma\right)\frac{{\|\mathbf{Z}_{N}\|^{2}}}N\right).}\end{array}\]
\[\begin{array}{r l}{d_{k}}&{=t_{k}-c_{k}=\frac{\mathrm{i}}{k}\mathrm{e}^{-\mathrm{i}k\xi}-\frac{\mathrm{i}\pi}{n\sin{\left(\frac{k\pi}{n}\right)}}\mathrm{e}^{-\mathrm{i}k\left(m+\frac{1}{2}\right)2\pi/n}}\\ &{=\frac{\mathrm{i}}{k}\left(\mathrm{e}^{-\mathrm{i}k\xi}-\mathrm{e}^{-\mathrm{i}k\left(m+\frac{1}{2}\right)2\pi/n}\right)+\mathrm{i}\left(\frac{1}{k}-\frac{\pi}{n\sin{\left(\frac{k\pi}{n}\right)}}\right)\mathrm{e}^{-\mathrm{i}k\left(m+\frac{1}{2}\right)2\pi/n}.}\end{array}\]
\[\begin{array}{r l}{\theta_{0}}&{=\operatorname{atan2}(x\cos\bar{\varphi}+y\sin\bar{\varphi},z)\pm i\ln\left(\lambda+\sqrt{\lambda^{2}-1}\right),}\\ {\lambda}&{=\frac{1}{2a}\frac{a^{2}+x^{2}+y^{2}+z^{2}}{\sqrt{(x\cos\bar{\varphi}+y\sin\bar{\varphi})^{2}+z^{2}}}.}\end{array}\]
\[\begin{array}{r l}{\mathcal{E}_{2}}&{=\{z\in\mathbb{C}\setminus\mathcal{E}_{1}:\mathrm{~f_{n}(z,w)=f_{n'}(z,w)=0~}}\\ &{\qquad\qquad\mathrm{for~some~n,n'\in\{1,2,...,n_\phi\}~s u c h~t h a t~n\ne~n'~a n d~f o r~s o m e~w\in\mathbb{C}~}\}.}\end{array}\]
\[\Psi_{w}\left(\begin{array}{l l}{\alpha_{++}}&{\alpha_{+-}}\\ {\alpha_{-+}}&{\alpha_{--}}\end{array}\right)=(-1)^{w(\alpha)}\left(\begin{array}{l l}{-X_{-+}^{\alpha}}&{-X_{--}^{\alpha}}\\ {X_{++}^{\alpha}}&{X_{+-}^{\alpha}}\end{array}\right),\quad\mathrm{~for~all~}\alpha\in\operatorname{Arc}(\mathbf{\Sigma}).\]
\[\begin{array}{r}{\mathbf{\Theta}_{(1:l_{1})}^{\mathrm{(I)}\mathrm{T}}=\underset{^B}{\angle}(\mathrm{e}^{j\frac{2\pi v}{V}}\mathbf{g}_{(1:l_{1})}^{\mathrm{(I)}\mathrm{H}})=\mathrm{e}^{j\frac{2\pi v}{V}}\underset{^B}{\angle}(\mathbf{g}_{(1:l_{1})}^{\mathrm{(I)}\mathrm{H}}),}\end{array}\]
\[\sum_{i=1}^{N}\langle\mathrm{\boldmath~v~}_{i},\dot{\mathrm{\boldmath~w~}}_{i}\rangle=-\frac{\kappa_{0}}{2N}\sum_{i,j=1}^{N}\phi(r_{j i})|\mathrm{\boldmath~v~}_{j i}|^{2}-\frac{\kappa_{1}}{4N}\sum_{\substack{i,j=1\,j\neq i}}^{N}\Big\langle\mathrm{\boldmath~v~}_{j i},\frac{\mathrm{\boldmath~x~}_{j i}}{r_{j i}}\Big\rangle^{2}-\frac{\kappa_{2}}{4N}\sum_{\substack{i,j=1\,j\neq i}}^{N}(r_{j i}-R_{i j}^{\infty})\Big\langle\frac{\mathrm{\boldmath~x~}_{j i}}{r_{j i}},\mathrm{\boldmath~v~}_{j i}\Big\rangle.\]
\[\begin{array}{r l}{\boldsymbol{a}_{\mathit{\Pi}_{\overline{{I}}}}+\boldsymbol{a}_{\mathcal{T}_{\overline{{I}}}}}&{=\delta\boldsymbol{a}_{p},}\\ {\delta\boldsymbol{a}_{l}+\boldsymbol{a}_{\mathit{\Pi}_{\overline{{S}}}}+\boldsymbol{a}_{\mathcal{T}_{\overline{{S}}}}}&{=\delta\boldsymbol{a}_{\nu}+\delta\boldsymbol{f},}\end{array}\]
\[\mathbf{q}(t_{n},\mathbf{z}_{n};s\mathbf{,\xi})=\mathbf{f}(s,\mathbf{z}_{n}+\mathbf{\phi}\left(t_{n},\mathbf{z}_{n};s-t_{n}\right)+\mathbf{\xi})-\mathbf{f}_{\mathbf{x}}(t_{n},\mathbf{z}_{n})\mathbf{\phi}(t_{n},\mathbf{z}_{n};s-t_{n})-\mathbf{f}_{t}(t_{n},\mathbf{z}_{n})(s-t_{n})-\mathbf{f}(t_{n},\mathbf{z}_{n}).\]
\[P_{e}=\left\{\begin{array}{l l}{\begin{array}{r l r}&{\frac{2^{3/2}}{3\pi^{2}\hbar^{3}}m_{e}^{4}c^{5}\beta_{e}^{5/2}\left[\Bigl\{\frac{2}{5}(\xi_{e}+\phi)^{5/2}+\frac{\pi^{2}}{4}(\xi_{e}+\phi)^{1/2}-\frac{7\pi^{4}}{960}(\xi_{e}+\phi)^{-3/2}\Bigr\}+\right.}\\ &{\left.\frac{\beta_{e}}{2}\Bigl\{\frac{2}{7}(\xi_{e}+\phi)^{7/2}+\frac{5\pi^{2}}{12}(\xi_{e}+\phi)^{3/2}+\frac{7\pi^{4}}{192}(\xi_{e}+\phi)^{-1/2}\Bigr\}\right]}&{\mathrm{for~\beta_{e}<1~},}\\ &{\frac{1}{9\pi^{2}\hbar^{3}}m_{e}^{4}c^{5}\beta_{e}^{7/2}\left[2\Bigl\{(\xi_{e}+\phi)^{3}+\pi^{2}(\xi_{e}+\phi)\Bigr\}+\right.}\\ &{\left.3\beta_{e}\Bigl\{\frac{1}{4}(\xi_{e}+\phi)^{4}+\frac{\pi^{2}}{2}(\xi_{e}+\phi)^{2}+\frac{7\pi^{4}}{60}\Bigr\}\right]}&{\mathrm{for~\beta_{e}>1~},}\end{array}}\end{array}\right.\]
\[\begin{array}{r l}{a_{s}^{*}\left(z^{\mathcal{N}},p^{\mathcal{N}};\boldsymbol{\mu}\right)}&{:=a^{*}\left(z^{\mathcal{N}},p^{\mathcal{N}};\boldsymbol{\mu}\right)+\sum_{K\in\mathcal{T}_{h}}\delta_{K}^{a}\left((T_{S}-T_{S S})p^{\mathcal{N}},\frac{h_{K}}{|\mathbf{\eta}|}\left(-T_{S S}\right)z^{\mathcal{N}}\right)_{K},}\\ {\big(y^{\mathcal{N}}-y_{d},z^{\mathcal{N}};\boldsymbol{\mu}\big)_{s}}&{:=\int_{\Omega_{o b s}}(y^{\mathcal{N}}-y_{d})z^{\mathcal{N}}\ \mathrm{dx}+\sum_{K\in{\mathcal{T}_{h}}_{\vert_{\Omega_{o b s}}}}\delta_{K}^{a}\left(y^{\mathcal{N}}-y_{d},\frac{h_{K}}{|\mathbf{\eta}|}\left(-T_{S S}\right)z^{\mathcal{N}}\right)_{K},}\end{array}\]
\[P_{e}=\left\{\begin{array}{l l}{\begin{array}{r l r}&{\frac{2^{3/2}}{3\pi^{2}\hbar^{3}}m_{e}^{4}c^{5}\beta_{e}^{5/2}\left[\Bigl\{\frac{2}{5}(\xi_{e}+\phi)^{5/2}+\frac{\pi^{2}}{4}(\xi_{e}+\phi)^{1/2}-\frac{7\pi^{4}}{960}(\xi_{e}+\phi)^{-3/2}\Bigr\}+\right.}\\ &{\left.\frac{\beta_{e}}{2}\Bigl\{\frac{2}{7}(\xi_{e}+\phi)^{7/2}+\frac{5\pi^{2}}{12}(\xi_{e}+\phi)^{3/2}+\frac{7\pi^{4}}{192}(\xi_{e}+\phi)^{-1/2}\Bigr\}\right]}&{\mathrm{for~\beta_{e}<1~},}\\ &{\frac{1}{9\pi^{2}\hbar^{3}}m_{e}^{4}c^{5}\beta_{e}^{7/2}\left[2\Bigl\{(\xi_{e}+\phi)^{3}+\pi^{2}(\xi_{e}+\phi)\Bigr\}+\right.}\\ &{\left.3\beta_{e}\Bigl\{\frac{1}{4}(\xi_{e}+\phi)^{4}+\frac{\pi^{2}}{2}(\xi_{e}+\phi)^{2}+\frac{7\pi^{4}}{60}\Bigr\}\right]}&{\mathrm{for~\beta_{e}>1~},}\end{array}}\end{array}\right.\]
\[\phi\beta_{\varepsilon}^{\prime}(\phi)-\beta_{\varepsilon}(\phi)=\left\{\begin{array}{r l r l}&{0}&&{\mathrm{if~|\phi|~\geq~\varepsilon~},}\\ &{\frac{1}{4\varepsilon}(\phi^{2}-\varepsilon^{2})}&&{\mathrm{if~|\phi|~\leq~\varepsilon~}.}\end{array}\right.\]
\[\oint_{\partial\Sigma}{\mathbf{F}\cdot\,\mathrm{d}\mathbf{\gamma}}=\oint_{\partial\Sigma}{\omega_{\mathbf{F}}}=\int_{\Sigma}{\mathrm{d}\omega_{\mathbf{F}}}=\int_{\Sigma}{\star\omega_{\nabla\times\mathbf{F}}}=\iint_{\Sigma}{\nabla\times\mathbf{F}\cdot\,\mathrm{d}\mathbf{\Sigma}}\]