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Kitajiang

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test2_ICPR2024

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\[\begin{aligned}&x-\alpha=-4m/3\alpha\\ &y-b=-4m/3\alpha\mathrm{~by~}(73)\\ &z=-4m/9\alpha\mathrm{~by~}(70)\end{aligned}\]

\[\begin{aligned}&V_{e n}(\mathbf{r},\mathbf{x})=\sum_{l a}Z_{e}U_{a}\left(\mathbf{r}-\mathbf{x}_{l a}\right)\\ &V_{e e}(\mathbf{r},\mathbf{x})=Z_{e}\int_{\Omega}U_{e}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\rho\left(\mathbf{r}^{\prime},\mathbf{x}\right)\mathrm{d}^{3}\mathbf{r}^{\prime}\\ &\rho(\mathbf{r},\mathbf{x})=\sum_{k}\eta_{k}\psi_{k}^{*}(\mathbf{r},\mathbf{x})\psi_{k}(\mathbf{r},\mathbf{x})\\ &V_{x}(\mathbf{r},\mathbf{x})\psi_{k}(\mathbf{r},\mathbf{x})=Z_{e}\int_{\Omega}U_{x}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\rho\left(\mathbf{r}^{\prime},\mathbf{r},\mathbf{x}\right)\psi_{k}\left(\mathbf{r}^{\prime},\mathbf{x}\right)\mathrm{d}^{3}\mathbf{r}^{\prime}\\ &\rho\left(\mathbf{r}^{\prime},\mathbf{r},\mathbf{x}\right)=\sum_{k}\eta_{k}\psi_{k}^{*}\left(\mathbf{r}^{\prime},\mathbf{x}\right)\psi_{k}(\mathbf{r},\mathbf{x})\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}\alpha_{0l}=\left(k_{0}^{2}-\beta_{l}\right)^{1/2},\quad\operatorname{Im}\alpha_{0l}\leq0\\ \alpha_{1l}=\left(\varepsilon_{r}k_{0}^{2}-\beta_{l}\right)^{1/2},\quad\operatorname{Im}\alpha_{1l}\leq0.\end{array}\right\}\end{aligned}\]

\[\begin{array}{l}C_{\mathrm{Fe}(\mathrm{II})}=\frac{\mathrm{C}_{\mathrm{Fe}(\mathrm{II})\mathrm{~solid~}}}{P_{\mathrm{s}/\mathrm{d}}}+\mathrm{C}_{\mathrm{Fe}(\mathrm{II})\mathrm{~dissolved~}}\\ \delta=\frac{\mathrm{C}_{\mathrm{Fe}(\mathrm{II})\mathrm{~dissolved~}}}{\mathrm{C}_{\mathrm{Fe}(\mathrm{II})}}\end{array}\]

\[\begin{aligned}\Delta P&=\Delta P^{(2)}+\Delta P^{(4)}+\Delta P^{(6)}+\cdots\approx\Delta P^{(2)}+J\Delta P^{(2)}+J^{2}\Delta P^{(2)}+\cdots\\ &=\frac{\Delta P^{(2)}}{1-J}\end{aligned}\]

\[\begin{aligned}\mathbf{E}_{t^{-}}^{\mathcal{N}}\left[\omega_{t^{-}}^{y}\tilde{X}_{t^{-}}\tilde{X}_{t^{-}}^{\top}\right]&=\mathrm{E}_{t^{-}}^{\mathcal{N}}\left[\omega_{t^{-}}^{y}X_{t^{-}}X_{t^{-}}^{\top}\right]\\ &-\mathrm{E}_{t^{-}}^{\mathcal{N}}\left[\omega_{t^{-}}^{y}X_{t^{-}}\right]\mu_{t^{-}}^{\top}-\mu_{t^{-}}\mathbf{E}_{t^{-}}^{\mathcal{N}}\left[\omega_{t^{-}}^{y}X_{t^{-}}\right]\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{c}W_{\mathrm{D}}=\left(\mu b^{2}/(4\pi K)\right)\ln\left(R/R_{C}\right)+\hat{W}\approx\kappa_{1}\mu b^{2}/2,\\ W_{\mathrm{T}}\approx W_{\mathrm{SF}}/2=\kappa_{2}\mu b/2\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}\alpha_{i}=i(i+m)-i S\\ \beta_{i}=-C+2p(S-2i-m-1)+S(2i+m+1)-2i^{2}-2i m-2i-m-1,\\ \gamma_{i}=(i+1)(i+m+1)-(i+m+1)S\end{array}\right\}\end{aligned}\]

\[\begin{aligned}-u_{\cdot t}^{1}+f x_{\cdot t}^{2}-w_{1\cdot1}^{1}-w_{1\cdot2}^{2}&=4\\ &=-u_{\cdot t}^{2}-f x_{\cdot t}^{1}-w_{2\cdot1}^{1}-w_{2\cdot2}^{2}\\ &=u^{i}\cdot i\cdot1\cdot2\\ &=x_{\cdot\alpha}^{i}\\ &=\frac{\partial e(\tau)}{\partial x_{\alpha}^{i}}+w_{i}^{\alpha}\cdot i\cdot\alpha\cdot1\cdot2\\ &=i\cdot\alpha\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\dot{x}(t)=-F_{\mathrm{d}}x(t)-f_{v}\left(x_{\mathrm{d}},v_{\mathrm{d}}\right)v(t)\\ \dot{y}(t)=\mathrm{e}^{-a_{1}\tau_{1}}\left(f_{x}\left(x_{\mathrm{d}},v_{\mathrm{d}}\right)x\left(t-\tau_{1}\right)+f_{v}\left(x_{\mathrm{d}},v_{\mathrm{d}}\right)v\left(t-\tau_{1}\right)\right)-A_{w}y(t)-\alpha y_{\mathrm{d}}w(t)\\ \dot{v}(t)=k\mathrm{e}^{-a_{2}\tau_{2}}y\left(t-\tau_{2}\right)-p v(t)\\ \dot{z}(t)=\alpha w_{\mathrm{d}}y(t)-b z(t)+\alpha y_{\mathrm{d}}w(t)\\ \dot{w}(t)=c z(t)-q w(t)\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\mathbf{S}=\left[\begin{array}{l l l l l}0&0&0&0&0\\ 4&0&0&0&0\\ 0&3&0&0&0\\ 0&0&2&0&0\\ 0&0&0&1&0\end{array}\right]\end{aligned}\]

\[\begin{array}{l}\mathrm{P}+\mathrm{S}_{c}-\mathrm{S}=R\cdot\\ =\frac{\epsilon\left(r-\frac{d_{1}}{d}\right)}{r}\\ =\frac{1-r}{r-\frac{d_{1}}{d}}\cdot\cdot\end{array}\]

\[\begin{array}{l}\delta\hat{U}(p)=\frac{-\left(2s^{2}-3\kappa^{2}\right)}{8R_{\alpha}s^{6}}\\ =\frac{-\left(s^{2}-3\kappa^{2}\right)}{16R_{\alpha}s^{5}}\end{array}\]

\[\begin{aligned}\left.\begin{array}{r}(1-\eta)\frac{\partial w_{1}^{(1)}}{\partial\theta}=(1-\eta)f^{(1)}-\eta f^{33}\\ (1-\eta)\frac{\partial w_{3}}{\partial\rho}=(1-\eta)f^{(1)}-\eta f^{11}\\ \frac{1-\eta}{2}\left(\frac{\partial w_{1}}{\partial\rho}+\frac{\partial w_{3}}{\partial\theta}\right)=f^{(1)}\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\phi_{h}=\left(\begin{array}{c}\phi^{+}\\ \phi^{0}\end{array}\right)\end{aligned}\]

\[\begin{array}{l}\frac{\partial w}{\partial t}+\frac{1}{V_{F}}\left\{u A_{x}\frac{\partial w}{\partial x}+v A_{y}R\frac{\partial w}{\partial y}+w A_{z}\frac{\partial w}{\partial z}\right\}\\ =-\frac{1}{\rho}\frac{\partial p}{\partial z}+G_{z}+f_{z}-b_{z}-\frac{R_{s o r}}{\rho V_{F}}\left(w-w_{w}-\delta w_{s}\right)\end{array}\]

\[\begin{aligned}\hat{\gamma}^{1}=\left(\begin{array}{c c}\widehat{E}_{2}&0\\ 0&-\widehat{E}_{2}\end{array}\right),\hat{\gamma}^{2}=\left(\begin{array}{c c}0&\sigma_{x}\\ -\sigma_{x}&0\end{array}\right)\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\epsilon_{x}=\frac{N_{x}\left(A_{26}^{2}-A_{22}A_{66}\right)}{A_{66}A_{12}^{2}-2A_{12}A_{16}A_{26}+A_{22}A_{16}^{2}+A_{11}A_{26}^{2}-A_{11}A_{22}A_{66}}\\ \epsilon_{y}=-\frac{N_{x}\left(A_{16}A_{26}-A_{12}A_{66}\right)}{A_{66}A_{12}^{2}-2A_{12}A_{16}A_{26}+A_{22}A_{16}^{2}+A_{11}A_{26}^{2}-A_{11}A_{22}A_{66}}\\ \epsilon_{x y}=-\frac{N_{x}\left(A_{12}A_{26}-A_{16}A_{22}\right)}{A_{66}A_{12}^{2}-2A_{12}A_{16}A_{26}+A_{22}A_{16}^{2}+A_{11}A_{26}^{2}-A_{11}A_{22}A_{66}}\end{array}\right\}\end{aligned}\]

\[\begin{aligned}P_{\mathrm{QCTB}}&=P^{\sigma_{1}}+P^{\sigma_{2}}\\ &=I+A\left(A^{2}+\delta^{2}I\right)^{-1/2}\end{aligned}\]

\[\begin{array}{l}\hat{\mathbf{\Sigma}}_{\hat{\theta}_{2}}=\mathcal{A}_{2}^{-1}\left(\frac{1}{n_{2}^{2}}\sum_{x\in s_{2}}\hat{\mathcal{R}}^{2}(x)\mathcal{Z}(x)\mathcal{Z}^{t}(x)\right)\mathcal{A}_{2}^{-1}\\ \hat{\mathbf{\Sigma}}_{\hat{\gamma}_{2}}=\left(\mathcal{A}_{1}^{(1)}\right)^{-1}\left(\frac{1}{n_{2}^{2}}\sum_{x\in s_{2}}\hat{\mathcal{R}}_{1}^{2}(x)\mathcal{Z}^{(1)}(x)\mathcal{Z}^{(1)t}(x)\right)\left(\mathcal{A}_{1}^{(1)}\right)^{-1}\end{array}\]

\[\begin{aligned}\Omega_{M}&=\frac{\rho_{M}}{\rho_{\mathrm{cr}}}\cdot\frac{4\omega\rho_{M}}{3\phi^{2}H^{2}}\\ &=\frac{\rho_{k}}{\rho_{\mathrm{cr}}}\cdot\frac{k}{H^{2}a^{2}}\\ &=\frac{\rho_{D}}{\rho_{\mathrm{cr}}}\cdot\frac{4\omega\rho_{D}}{3\phi^{2}H^{2}}\end{aligned}\]

\[\begin{array}{c}x\equiv v_{\mathrm{c}}/v\cdot\Pi\equiv p/p_{\mathrm{c}}\cdot\eta\equiv\frac{1}{3}\left(\Pi+8t/t_{\mathrm{c}}\right)\\ =\Pi-\eta x+3x^{2}-x^{3}\end{array}\]

\[\begin{array}{c}\sum_{q}p_{q}=1\\ \sum_{q}p_{q}x^{q}=\mu\end{array}\]

\[\begin{array}{l}c_{i}=\left(1-\left(\frac{r_{i}}{R}\right)^{2}\right)\quad\mathrm{~if~}r_{i}<R\\ c_{i}=0\quad\mathrm{~if~}r_{i}>R\end{array}\]

\[\begin{aligned}\left.\begin{array}{r l}\varepsilon\dot{x}=f(x,y,\varepsilon)\\ \dot{y}=g(x,y,\varepsilon)\end{array}\right\}\end{aligned}\]

\[\begin{aligned}&\gamma_{t1}=\theta_{\mathrm{st}}-\theta_{\mathrm{tt}}+\mathrm{~local~buckling~terms~}\\ &\gamma_{t1}=\frac{\partial u_{\mathrm{st}}}{\partial z}+\frac{\partial u_{\mathrm{tt}}}{\partial x}+\frac{\partial u}{\partial x}+\frac{\partial w}{\partial x}\frac{\partial w}{\partial z}\\ &\gamma_{t1}=\left[\left(q_{\mathrm{s}}-q_{\mathrm{t}}\right)+\left(q_{\phi}-q_{\tau}\right)\lambda\right]\pi\cos\frac{\pi z}{L}-\frac{2}{b}u+\frac{4x}{b^{2}}w w^{\prime}\end{aligned}\]

\[\begin{aligned}d w_{y^{\prime}}&=-\frac{d\kappa}{d y}d y\frac{y}{4\pi\mathrm{R}^{2}}\left(\frac{1}{1-y y^{\prime}/\mathrm{R}^{2}}+\frac{1}{1+y y^{\prime}/\mathrm{R}^{2}}\right)\\ &=\frac{\kappa_{F}}{2\pi\mathrm{R}^{2}s^{2}}\frac{y^{2}d y}{\sqrt{1-\left(y^{2}/s^{2}\right)\left[1-\left(y y^{\prime}/\mathrm{R}^{2}\right)^{2}\right]}}\end{aligned}\]

\[\begin{aligned}&p_{t+\mathrm{d}t}(y\cdot i)\\ &=\frac{\alpha i\{H-(y+1)\}}{H}p_{t}(y-1\cdot i)\mathrm{d}t+\xi(y+1)p_{t}(y+1\cdot i)\mathrm{d}t\\ &+\frac{\beta y\{V-(i-1)\}}{H}p_{t}(y\cdot i-1)\mathrm{d}t+\delta(i+1)p_{t}(y\cdot i+1)\mathrm{d}t\\ &+\left(1-\frac{\alpha i(H-y)}{H}\mathrm{~d}t-\xi y\mathrm{~d}t-\frac{\beta(V-i)y}{H}\mathrm{~d}t-\delta i\mathrm{~d}t\right)p_{t}(y\cdot i)\end{aligned}\]

\[\begin{array}{l}H=-\nabla_{1}^{2}-\nabla_{2}^{2}-\frac{2z}{r_{1}}-\frac{2z}{r_{2}}+\frac{2}{r_{12}}\\ =k^{2}-z^{2}\end{array}\]

\[\begin{aligned}X_{1}=\left[\begin{array}{c c c c}x_{1}^{1}(1)&x_{1}^{1}(2)&\ldots&x_{1}^{1}(24)\\ x_{1}^{2}(1)&x_{1}^{2}(2)&\ldots&x_{1}^{2}(24)\\ \mathrm{M}&\mathrm{M}&\ldots&\mathrm{M}\\ x_{1}^{N}(1)&x_{1}^{N}(2)&\ldots&x_{1}^{N}(24)\end{array}\right]\end{aligned}\]

\[\begin{aligned}\dot{x}=F(x):=\left(\begin{array}{c}\dot{r}\\ \dot{\theta}\\ b_{0}\cos(\theta)+r\left(a_{1}+a_{2}\dot{\theta}+\dot{\theta}^{2}\right)-\hat{a}_{2}\dot{r}\\ -\frac{1}{r}\left(b_{0}\sin(\theta)+r\left(\hat{a}_{1}+\hat{a}_{2}\dot{\theta}\right)+\dot{r}\left(a_{2}+2\dot{\theta}\right)\right)\end{array}\right)\end{aligned}\]

\[\begin{aligned}(1-C)\frac{d p}{d z}&=(1-C)\frac{\mu_{s}(C)}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right)u_{f}+C S\left(u_{p}-u_{f}\right)\cdot O\leq r\leq R_{1}\\ &=C S\left(u_{f}-u_{p}\right)\cdot b\leq r\leq R_{1}\end{aligned}\]

\[\begin{aligned}\frac{\partial\mathcal{A}_{\mathfrak{D}_{h}(\alpha)}^{\partial\alpha}}{\partial\alpha}&=\frac{\partial}{\partial\alpha}\iint_{\mathfrak{D}_{h}(\alpha)}G^{-1}(x\cdot y)\mathrm{d}x\mathrm{~d}y\\ &=-\int_{f}^{\tau(h\cdot\alpha)}(x(t)-\ln x(t))\mathrm{d}t.\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{c}p_{i i}^{(0)}\left(\zeta_{p}\right)=-\zeta_{p}\left(4\left(1-\kappa^{2}\right)\frac{\partial^{2}w_{3}^{(0)}}{\partial\xi_{i}^{2}}+2\left(1-2\kappa^{2}\right)\frac{\partial^{2}w_{3}^{(0)}}{\partial\xi_{j}^{2}}\right)\\ q_{i i}^{(0)}\left(\zeta_{q}\right)=\frac{1}{2}Q_{1}\left(4\left(1-\kappa^{2}\right)\frac{\partial^{2}w_{3}^{(0)}}{\partial\xi_{i}^{2}}+2\left(1-2\kappa^{2}\right)\frac{\partial^{2}w_{3}^{(0)}}{\partial\xi_{j}^{2}}\right)\end{array}\right\}\end{aligned}\]

\[\begin{array}{l}V_{\mathrm{rigid~}}(t)=V_{0}+\frac{g a_{0}}{L\left(\omega^{2}+\lambda^{2}\right)}\left(\omega\left(e^{-\lambda t}-\cos\omega t\right)+\lambda\sin\omega t\right)\\ H_{r i g i d}(x,t)=\frac{L-x}{g}\left(\frac{d V_{\mathrm{rigid~}}}{d t}+\lambda V_{\mathrm{rigid~}}\right).\end{array}\]

\[\begin{aligned}\frac{d}{d y}A-\star d_{A}\phi&=r\\ &=\frac{d}{d y}\phi-\star\left(F_{A}-\phi\wedge\phi\right)\\ &=d_{A}^{\star}\phi\end{aligned}\]

\[\begin{aligned}&\mathrm{M}_{1}^{\prime}=-\frac{4\pi a^{\prime}p\sigma^{\prime}}{\mathrm{D}_{1}^{\prime}}\frac{a^{\prime}b^{\prime}}{\left(b^{\prime2}+c^{2}\right)}\mathrm{I}+\mathrm{~cubes.~}\\ &\mathrm{N}_{1}^{\prime}=+\frac{4\pi a^{\prime}p\sigma^{\prime}}{\mathrm{D}_{1}^{\prime}}\frac{a^{\prime}c}{b^{2}+c^{2}}\mathrm{I}+\mathrm{~cubes.~}\\ &\mathrm{Q}_{1}^{\prime}=+\frac{8\pi^{2}a^{\prime2}p^{2}}{\mathrm{D}_{1}^{\prime}}\frac{a^{\prime}b^{\prime}}{b^{2}+c^{2}}\mathrm{I}+\ldots\\ &\mathrm{R}_{1}^{\prime}=-\frac{8\pi^{2}a^{\prime2}p^{2}}{\mathrm{D}_{1}^{\prime}}\frac{a^{\prime}c}{b^{\prime2}+c^{2}}\mathrm{I}+\ldots\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}\mathbf{u}^{(\mathrm{C})}(\mathbf{x},0)=\mathbf{u}^{(\mathrm{L})}(\mathbf{x},0)=\mathbf{u}^{(\mathrm{H})}(\mathbf{x},0)=0\\ \sigma^{(\mathrm{C})}(\mathbf{x},0)=\sigma^{(\mathrm{L})}(\mathbf{x},0)=\sigma^{(\mathrm{H})}(\mathbf{x},0)=0\\ p^{(\mathrm{C})}(\mathbf{x},0)=p^{(\mathrm{L})}(\mathbf{x},0)=p^{(\mathrm{H})}(\mathbf{x},0)=0\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{c}\mathrm{T}(x)\mathrm{T}\left(x-\frac{\omega}{n}\right)\ldots\mathrm{T}\left(x-\frac{n-1}{n}\omega\right)\\ \times\mathrm{T}\left(x+\frac{\omega}{n}\right)\mathrm{T}(x)\ldots\mathrm{T}\left(x-\frac{n-2}{n}\omega\right)\\ \cdots\\ \times\mathrm{T}\left(x+\frac{n-1}{n}\omega\right)\ldots\mathrm{T}(x)\\ \mathrm{T}_{p_{n}^{2}}(n x,\omega)=\mathrm{C}^{2x}\left[\frac{1}{n}\right]-\frac{n^{2}x^{2}n-1}{\omega^{2}}\Pi_{r=0}^{-(n-1)}\Pi_{s=0}^{-1}\mathrm{~T}_{p}\left(x+\frac{r+s}{n}\omega,\omega\right)\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\exp(\mathrm{i}\tilde{\omega}\Delta t)\left(\begin{array}{l}U\\ V\end{array}\right)=\left(\begin{array}{c c}1-\frac{1}{2}\theta^{2}+\frac{1}{24}\theta^{4}-\frac{3-\sqrt{3}}{1728}\theta^{6}&1-\frac{1}{6}\theta^{2}+\frac{1}{72}\theta^{4}-\frac{1}{1728}\theta^{6}\\ -\theta^{2}+\frac{1}{6}\theta^{4}-\frac{1}{288}\theta^{6}&1-\frac{1}{2}\theta^{2}+\frac{1}{24}\theta^{4}-\frac{3+\sqrt{3}}{1728}\theta^{6}\end{array}\right)\left(\begin{array}{l}U\\ V\end{array}\right)\end{aligned}\]

\[\begin{aligned}\mathrm{QE}_{\max}&=\eta_{\mathrm{op}}\times\phi_{\mathrm{fl}}\times\eta_{\mathrm{r}}\times\gamma\approx u.2\times1\times O.25\\ &=\times1\end{aligned}\]

\[\begin{aligned}\bar{Q}=-W^{\prime}(b)\left(\begin{array}{c c}0&s1\\ 1&0\end{array}\right)\end{aligned}\]

\[\begin{aligned}\int_{\tau}\boldsymbol{G}\times\boldsymbol{B}\cdot\boldsymbol{J}\mathrm{d}\tau&=-\int_{\tau}\left(J_{\phi}/r\right)\boldsymbol{G}\cdot\nabla\left(r A_{\phi}\right)\mathrm{d}\tau\\ &=\int_{\tau}J_{\phi}A_{\phi}\nabla\cdot\boldsymbol{G}\mathrm{d}\tau-\int_{\tau}\left(J_{\phi}/r\right)\nabla\cdot\left(\boldsymbol{r}A_{\phi}\boldsymbol{G}\right)\mathrm{d}\tau.\end{aligned}\]

\[\begin{array}{l}\sigma_{k}^{t}=f\left(\mathbf{W}_{k},\mathbf{I}^{t},\mathbf{S}^{t}\right)\quad\mathrm{~for~}1\leq k\leq N\\ y_{k}^{t}=g\left(\sigma_{k}^{t}\right)\quad\mathrm{~for~}1\leq k\leq N\end{array}\]

\[\begin{aligned}&\frac{\mathrm{PP}^{\prime}}{p p^{\prime}}=\frac{\mathrm{QQ}^{\prime}}{q q^{\prime}}\cdot\frac{\mathrm{RR}^{\prime}}{r r^{\prime}}\\ &=\frac{\mathrm{QR}^{\prime}}{q r^{\prime}}\cdot\frac{\mathrm{Q}^{\prime}\mathbf{R}}{q^{\prime}r}\end{aligned}\]

\[\begin{aligned}y^{2}&=\left(1-V^{2}/c^{2}\right)\left[x^{\prime}+\left(x^{\prime2}+y^{\prime2}\right)^{\frac{1}{2}}\right]\left[-x+\left(x^{\prime2}+y^{\prime2}\right)^{\frac{1}{2}}\right]\\ &=\left(1-V^{2}/c^{2}\right)y^{\prime2}\end{aligned}\]

\[\begin{array}{l}E^{(1)}\int_{\Omega_{o}^{(1)}}\frac{\partial^{2}u_{i-1}^{(1)}}{\partial x^{2}}+\frac{\partial^{2}u_{i}^{(1)}}{\partial x\partial y}\mathrm{~d}y+E^{(2)}\int_{\Omega_{u}^{(2)}}\frac{\partial^{2}u_{i-1}^{(2)}}{\partial x^{2}}+\frac{\partial^{2}u_{i}^{(2)}}{\partial x\partial y}\mathrm{~d}y\\ =\rho^{(1)}\int_{\Omega_{3}^{(1)}}\frac{\partial^{2}u_{i-1}^{(1)}}{\partial t^{2}}\mathrm{~d}y+\rho^{(2)}\int_{\Omega_{O}^{(2)}}\frac{\partial^{2}u_{i-1}^{(2)}}{\partial t^{2}}\mathrm{~d}y\end{array}\]

\[\begin{aligned}\left.\begin{array}{l}B_{m+1,i}=\frac{1}{2}\left[U_{i-n}\cos\phi_{2k-2}-U_{i}\cos\phi_{2k}+U_{i-n-1}\cos\phi_{2k-2}-U_{i-1}\cos\phi_{2k}\right]_{m+1},\\ C_{m+1,i}=\frac{1}{2}\left[V_{i}-V_{i-1}+V_{i-n}-V_{i-n-1}\right]_{m+1}\end{array}\right\}\end{aligned}\]

\[\begin{array}{l}\frac{\partial\mathcal{F}}{\partial\mathcal{W}_{c d}}+\frac{\partial}{\partial\mathcal{W}_{c d}}\sum_{c^{\prime}}\lambda_{c^{\prime}}\left(\sum_{d^{\prime}}\mathcal{W}_{c^{\prime}d^{\prime}}-A\right)\stackrel{!}{=}0\\ \frac{\partial\mathcal{F}}{\partial\mathcal{R}_{k c}}+\frac{\partial}{\partial\mathcal{R}_{k c}}\sum_{k^{\prime}}\lambda_{k^{\prime}}\left(\sum_{c^{\prime}}\mathcal{R}_{k^{\prime}c^{\prime}}-1\right)\stackrel{!}{=}0\end{array}\]

\[\begin{aligned}\left.\begin{array}{l}u_{1}(\theta,\varphi)=\mathrm{e}^{\mathrm{i}(\sigma t+m\varphi)}f(\theta)\\ u_{2}(\theta,\varphi)=\mathrm{e}^{\mathrm{i}(\sigma t+m\varphi)}g(\theta),\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\left(\begin{array}{l}x_{2}+\omega\delta\\ y_{2}-k\delta\end{array}\right)=\exp(\gamma\theta)\left(\begin{array}{c c}\cos\theta+\gamma\sin\theta&\sin\theta\\ -\left(\gamma^{2}+1\right)\sin\theta&\cos\theta-\gamma\sin\theta\end{array}\right)\left(\begin{array}{l}x_{1}+\omega\delta\\ y_{1}-k\delta\end{array}\right)\end{aligned}\]

\[\begin{aligned}V_{P S P}(t)&=E_{L}+\left(V\left(t_{q}\right)-E_{L}\right)\mathrm{e}^{-\left(t-t_{3}\right)}\left(\frac{1}{\tau_{m}\left(t_{M}\right)}+\frac{1}{\Delta\tau_{m}^{s}}\right)\\ &+Q E_{s}\left(\frac{1}{\tau_{m}^{s}\left(t_{i}\right)}+\frac{1}{\Delta\tau_{m}^{s}}\right)\left(\frac{1}{\tau_{s}}-\frac{1}{\tau_{m}\left(t_{N}\right)}-\frac{1}{\Delta\tau_{m}^{s}}\right)^{-1}\\ &\times\left\{\mathrm{e}^{-\left(t-t_{s}\right)\left(\frac{1}{\tau_{m}\left(t_{D}\right)}+\frac{1}{\Delta\tau_{m}^{s}}\right)}-\mathrm{e}^{-\frac{t-t_{P}}{\tau s}}\right\}\end{aligned}\]

\[\begin{aligned}g=I_{2}I_{1}=\left(\begin{array}{c c c}1&-\sqrt{2}\bar{\xi}&-|\xi|^{2}+i v\\ 0&1&\sqrt{2}\xi\\ 0&0&1\end{array}\right)\end{aligned}\]

\[\begin{array}{c}\frac{\partial E}{\partial t}+\frac{\partial\omega}{\partial k_{j}}\frac{\partial E}{\partial x_{j}}=\frac{-2\pi E}{|\partial L/\partial\omega|}\int\Phi(\boldsymbol{k}\cdot\boldsymbol{K})\delta\{L(\boldsymbol{K}\cdot\omega)\}\mathrm{d}\boldsymbol{K}\\ =-\frac{\partial L}{\partial k_{j}}/\frac{\partial L}{\partial\omega}\end{array}\]

\[\begin{aligned}\mathbf{l}_{1}=l_{1}\left[\begin{array}{c}-c_{\lambda_{1}}\\ 0\\ s_{\lambda_{1}}\end{array}\right];\mathbf{l}_{2}=l_{2}\left[\begin{array}{c}-c_{\lambda_{2}}c_{\alpha}\\ -c_{\lambda_{2}}s_{\alpha}\\ s_{\lambda_{2}}\end{array}\right];\mathbf{l}_{3}=l_{3}\left[\begin{array}{c}-c_{\lambda_{3}}c_{\beta}\\ -c_{\lambda_{3}}s_{\beta}\\ s_{\lambda_{3}}\end{array}\right]\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}t_{y}=-\frac{E d w^{\prime}}{R_{y}}\\ m_{y}=D\left(\frac{1}{R_{y}}+\sigma\frac{\mathrm{d}^{2}w^{\prime}}{\mathrm{d}x^{2}}-\frac{\sigma}{R_{x0}}\right)\end{array}\right\}\end{aligned}\]

\[\begin{aligned}f_{1}(t)&=q\\ &=\mathrm{e}^{-\delta\cdot t}\\ &=2\mathrm{e}^{-2\delta\cdot t}\cdot[1+2\delta\cdot t]\\ &=3\mathrm{e}^{-3\delta\cdot t}\cdot\left[1-8\left(1-\mathrm{e}^{-\delta\cdot t}\right)+12\delta\cdot t\right]\\ &=4\mathrm{e}^{-4\delta\cdot t}\cdot\left[1-24\left(1-\mathrm{e}^{-2\delta\cdot t}\right)+18\delta\cdot t\mathrm{e}^{-2\delta}\right.\\ &+36\delta\cdot t]\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}N_{\mathrm{d}}=1+\varepsilon N_{\mathrm{d}}^{(1)}+\varepsilon^{2}N_{\mathrm{d}}^{(2)}+\varepsilon^{3}N_{\mathrm{d}}^{(3)}+\ldots\\ Z_{\mathrm{d}}=1+\varepsilon Z_{\mathrm{d}}^{(1)}+\varepsilon^{2}Z_{\mathrm{d}}^{(2)}+\varepsilon^{3}Z_{\mathrm{d}}^{(3)}+\ldots\\ \phi=\varepsilon\phi^{(1)}+\varepsilon^{2}\phi^{(2)}+\varepsilon^{3}\phi^{(3)}+\ldots\\ V_{\mathrm{d}x}=\varepsilon V_{\mathrm{dx}}^{(1)}+\varepsilon^{2}V_{\mathrm{dx}}^{(2)}+\varepsilon^{3}V_{\mathrm{d}x}^{(3)}+\ldots\\ V_{\mathrm{d}y,z}=\varepsilon^{2}V_{\mathrm{d}y,z}^{(1)}+\varepsilon^{3}V_{\mathrm{d}y,z}^{(2)}+\varepsilon^{4}V_{\mathrm{d}y,z}^{(3)}+\ldots\end{array}\right\}\end{aligned}\]

\[\begin{aligned}E_{\mathrm{h}}+U_{\mathrm{rel}}&=\frac{(\boldsymbol{m}\times\boldsymbol{E})\cdot\boldsymbol{v}}{c}-\frac{\left(\boldsymbol{v}\times\boldsymbol{m}_{C}\right)\cdot\boldsymbol{E}}{c}\cdot\frac{(\boldsymbol{v}\times\boldsymbol{m})\cdot\boldsymbol{E}}{c}-\frac{\left(\boldsymbol{v}\times\boldsymbol{m}_{H}\right)\cdot\mathbf{E}}{c}\\ &=\frac{\left[\boldsymbol{v}\times\left(\boldsymbol{m}-\boldsymbol{m}_{B}\right)\right]\cdot\mathbf{E}}{c}\cdot\frac{\gamma-1}{\gamma v^{2}}\left(\boldsymbol{m}_{5}\cdot\boldsymbol{v}\right)\frac{(\boldsymbol{v}\times\boldsymbol{v})\cdot\mathbf{E}}{c}\cdot b\end{aligned}\]

\[\begin{aligned}\breve{\boldsymbol{F}}(\sigma)\equiv\left\{\breve{f}_{i j}(\sigma)\right\}=\left[\begin{array}{l l}\int_{0}^{\sigma}\breve{\zeta}_{1}(\tau)^{\mathrm{T}}\boldsymbol{T}_{1}(\tau)\mathrm{d}\tau&\int_{0}^{\sigma}\breve{\zeta}_{2}(\tau)^{\mathrm{T}}\boldsymbol{T}_{1}(\tau)\mathrm{d}\tau\\ \int_{0}^{\sigma}\breve{\zeta}_{1}(\tau)^{\mathrm{T}}\boldsymbol{T}_{2}(\tau)\mathrm{d}\tau&\int_{0}^{\sigma}\breve{\zeta}_{2}(\tau)^{\mathrm{T}}\boldsymbol{T}_{2}(\tau)\mathrm{d}\tau\end{array}\right]\in\mathbb{R}^{2\times2}\end{aligned}\]

\[\begin{array}{l}k_{0}=d\\ k_{i+1}=k_{i}+\Phi_{*,\beta}^{-1}\left(\frac{\Phi_{*,\beta}(d)}{2^{i m}}\right)\quad\mathrm{~for~}i\geq1\\ r_{i}=\frac{R}{2}\left(1+2^{-i}\right)\\ u_{i}=U\left(k_{i},r_{i}\right)\end{array}\]

\[\begin{aligned}\mathcal{B}:=\left(\begin{array}{c}{[s+k+1,n]}\\ k-1\end{array}\right)\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\mathcal{R}(t)=R_{4}(t)-\frac{1}{2}R_{5}(t)+2R_{6}(t)\\ \boldsymbol{\Pi}_{\mathrm{T}}^{\mathrm{e}}=\mathbf{F}^{-1}\mathbf{T}_{5}^{\mathrm{e}}\mathbf{F}^{-\mathrm{T}}\quad\boldsymbol{\Pi}_{\mathrm{L}}^{\mathrm{e}}=\tilde{T}^{\mathrm{e}}\mathbf{F}^{-1}\mathbf{m}\otimes\mathbf{m}\mathbf{F}^{-\mathrm{T}}\\ \boldsymbol{\Pi}_{\mathrm{C}}^{\mathrm{e}}=\mathbf{F}^{-1}\left(\frac{\mu_{T}}{E_{L}}\tilde{T}^{\mathrm{e}}\mathbf{I}\right)\mathbf{F}^{-\mathrm{T}},\quad\boldsymbol{\Pi}_{\mathrm{A}}^{\mathrm{e}}=\mathbf{F}^{-1}\mathbf{T}_{6}^{\mathrm{e}}\mathbf{F}^{-\mathrm{T}}\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\sigma_{i j}=-p\delta_{i j}+\hat{\sigma}_{i j}\\ \pi_{i j}=\beta_{i}d_{j}+\hat{\pi}_{i j}\\ g_{i}=\gamma d_{i}-\beta_{j}d_{i,j}+\hat{g}_{i},\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\bar{D}_{3}=c_{1}\left(P_{3}^{\mathrm{ds}}+P_{3}^{*}\right)+P_{(0)3}^{\mathrm{sp}}\\ \bar{\varepsilon}_{3}=s_{33}^{(E)}\bar{\sigma}_{33}+c_{1}\left(\varepsilon_{3}^{\mathrm{ds}}+\varepsilon_{3}^{*}\right)+\varepsilon_{(0)3^{\prime}}^{\mathrm{sp}}\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\frac{\mathrm{R}m}{\mu\mathrm{S}}&=f_{1}\left(\frac{m\mathrm{~S}}{\nu}\cdot\frac{v}{m}\cdot\frac{m\omega}{\mathrm{S}}\right)\\ &=f_{2}\left(\frac{m\mathrm{~S}}{\nu}\cdot\frac{r}{m}\right)\end{aligned}\]

\[\begin{aligned}D\left(\Delta_{k}\right)&=\Delta_{k}^{4}-\Delta_{k}^{2}|\Omega|^{2}+\left|\Omega_{1}\right|^{2}\left|\Omega_{3}\right|^{2}+\left|\Omega_{2}\right|^{2}\left|\Omega_{4}\right|^{2}\\ &+2\left|\Omega_{1}\right|\left|\Omega_{2}\right|\left|\Omega_{3}\right|\left|\Omega_{4}\right|\cos\phi+i\frac{\Gamma}{2}\Delta_{k}\left(\left|\Omega_{3}\right|^{2}+\left|\Omega_{4}\right|^{2}-\Delta_{k}^{2}\right)\end{aligned}\]

\[\begin{aligned}\bar{\Psi}M_{2}\Psi=\left(\begin{array}{c c}\bar{\psi}_{a}&\bar{\psi}_{b}\end{array}\right)=\left(\begin{array}{l l}m_{s}&m_{k}\\ m_{k}&m_{s}\end{array}\right)\left(\begin{array}{c}\psi_{a}\\ \psi_{b}\end{array}\right)\end{aligned}\]

\[\begin{aligned}\mathcal{O}\left(\underline{\boldsymbol{x}}\cdot\underline{\boldsymbol{y}}^{<i}\cdot\boldsymbol{y}^{i}\right)&=\lambda_{1}\log q\left(\boldsymbol{y}^{i}\mid\underline{\boldsymbol{x}}\right)+\\ &\log p_{\mathrm{LM}}\left(\boldsymbol{y}^{i}\mid\underline{\boldsymbol{y}}^{<i}\right)+\\ &\lambda_{2}\log p_{\mathrm{TM}}\left(\boldsymbol{x}^{i}\mid\boldsymbol{y}^{i}\right)+\lambda_{3}\left|\boldsymbol{y}^{i}\right|+\\ &\mathcal{O}\left(\underline{\boldsymbol{x}}\cdot\underline{\boldsymbol{y}}^{<i-1}\cdot\boldsymbol{y}^{i-1}\right)\end{aligned}\]

\[\begin{array}{l}H_{g o}=\frac{18}{28}\frac{P_{v o}}{P_{a t m}-P_{v o}}\\ =\frac{18}{28}\frac{P_{v i}}{P_{a t m}-P_{v i}}\end{array}\]

\[\begin{aligned}\mathrm{C}=\left[\begin{array}{l l}0&A\\ A^{\dagger}&0\end{array}\right]\end{aligned}\]

\[\begin{aligned}\left[\begin{array}{l}d N_{g}/d t\\ d N_{e}/d t\end{array}\right]=\left[\begin{array}{c c}-\rho\left(\omega_{0},t\right)B_{g\rightarrow e}&A_{e\rightarrow g}+\rho\left(\omega_{0},t\right)B_{e\rightarrow g}\\ \rho\left(\omega_{0},t\right)B_{g\rightarrow e}&-\left(A_{e\rightarrow g}+\rho\left(\omega_{0},t\right)B_{e\rightarrow g}\right)\end{array}\right]\left[\begin{array}{c}N_{g}\\ N_{e}\end{array}\right]\end{aligned}\]

\[\begin{aligned}\left\{\begin{array}{l}R_{\mathrm{h}}=x_{2}R+Q_{\mathrm{l}}\\ R_{\mathrm{l}}=x_{1}R=\frac{1}{\alpha_{l}}\frac{\mathrm{d}Q_{\mathrm{l}}}{\mathrm{d}t}+Q_{\mathrm{l}}\end{array}\right.\end{aligned}\]

\[\begin{aligned}\left[\begin{array}{l}S_{707}\\ S_{710}\end{array}\right]=\left[\begin{array}{l l}\mathrm{LFP}_{707}&\mathrm{FP}_{707}\\ \mathrm{LFP}_{710}&\mathrm{FP}_{710}\end{array}\right]\left[\begin{array}{l}a\\ b\end{array}\right]\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\dot{\mathbf{r}}_{1}=\mathbf{x}_{1}\\ \dot{\mathbf{r}}_{2}=\mathbf{x}_{2}\\ \dot{\mathbf{x}}_{1}=-(1-\lambda)\mu\left(\bar{x}_{1}-a\right)\left(\frac{\mathbf{r}}{|\mathbf{r}|}-\bar{x}_{1}\mathbf{x}_{1}\right)-\lambda\mu\left(\bar{x}_{1b1}-a_{0}\right)\left(\frac{\mathbf{r}_{1b1}}{\left|\mathbf{r}_{1b1}\right|}-\bar{x}_{1b1}\mathbf{x}_{1}\right)\\ \dot{\mathbf{x}}_{2}=-(1-\lambda)\mu\left(\bar{x}_{2}-a\right)\left(-\frac{\mathbf{r}}{|\mathbf{r}|}-\bar{x}_{2}\mathbf{x}_{2}\right)-\lambda\mu\left(\bar{x}_{2b2}-a_{0}\right)\left(\frac{\mathbf{r}_{2b2}}{\left|\mathbf{r}_{2b2}\right|}-\bar{x}_{2b2}\mathbf{x}_{2}\right)\end{array}\right\}\end{aligned}\]

\[\begin{aligned}\mathbb{P}\left(\boldsymbol{X}_{1}\in\mathrm{d}^{3}\boldsymbol{x}_{1}\cdot T_{1}\in\mathrm{d}t_{1}\cdot I_{1}\right.&\left.=i_{1}\cdot\ldots\cdot\boldsymbol{X}_{n}\in\mathrm{d}^{3}\boldsymbol{x}_{n}\cdot T_{n}\in\mathrm{d}t_{n}\cdot I_{n}\cdot i_{n}\right)\\ &=\left\|K_{n}\left(o\cdot\boldsymbol{x}_{1}\cdot t_{1}\cdot i_{1}\cdot\ldots\cdot\boldsymbol{x}_{n}\cdot t_{n}\cdot i_{n}\right)\psi\right\|^{2}\mathrm{~d}^{3}\boldsymbol{x}_{1}\mathrm{~d}t_{1}\cdots\mathrm{d}^{3}\boldsymbol{x}_{n}\mathrm{~d}t_{n}\end{aligned}\]

\[\begin{aligned}X=\left[\begin{array}{c c c c}1&x_{11}&\ldots&x_{1p}\\ 1&x_{21}&\ldots&x_{2p}\\ 1&x_{n1}&\ldots&x_{n p}\end{array}\right]\in\mathbb{R}^{n\times(p+1)};\quad n\geqslant p+1\end{aligned}\]

\[\begin{array}{l}\tau\frac{d V}{d t}=-\left(V-V_{T}\right)-g\times\left(V-E_{s}\right)\\ =\tau_{s}\frac{d g}{d t}\end{array}\]

\[\begin{aligned}C=\left[\begin{array}{c c c}c_{11}&\ldots&c_{1n}\\ \ldots&\ldots&\ldots\\ c_{n1}&\ldots&c_{n n}\end{array}\right]\end{aligned}\]

\[\begin{aligned}f((x a)(y z))&=((f\circ S)\circ(f\circ f))((x a)(y z))\\ &=((f\circ S)\circ f)((x a)(y z))\\ &=\bigvee_{((x a)\cdot(y z))\in A_{(x a)(y z)}}\{(f\circ S)(x a)\wedge f(y z)\}\\ &=\bigvee_{(x\cdot a)\in A_{(x a)}}\{f(x)\wedge S(a)\}\wedge f(y z)\\ &=f(x)\wedge f(y)\wedge f(z)\end{aligned}\]

\[\begin{aligned}Z\left(H_{1}^{+}*H_{04}\right)=\left\{\begin{array}{l}\left\{F^{1}\backslash T_{1}\cap F^{04},\Theta\left(S^{1}\backslash P_{1}\cap S^{04}\backslash P_{04}\right),\left[\tilde{\mathbf{B}}^{1}\right]\right\},\left\{\varnothing,\circ\left(p_{1}\cap p_{04}\right),\left[\tilde{\nabla}\times\tilde{\mathbf{B}}^{1}\right]\right\}\\ \left\{\left\{T_{1},P_{1},Q_{1}\right\},\left\{f^{1}\cap d t,\odot\left(s^{1}\cap p_{04}\right),\left[\partial\tilde{\mathbf{E}}_{1}/\partial\tilde{t}\propto\tilde{\nabla}\times\widetilde{\mathbf{E}}_{1}\right]\right\}\right\}\end{array}\right\},\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}\Delta\mathrm{T}=\cos\mathrm{I}\Delta\mathrm{H}+\sin\mathrm{I}\Delta\mathrm{V}\\ \Delta\mathrm{I}=\frac{1}{2}\sin2\mathrm{I}(\Delta\mathrm{V}/\mathrm{V}-\Delta\mathrm{H}/\mathrm{H})\\ \Delta\mathrm{N}=\cos\mathrm{D}\Delta\mathrm{H}-\mathrm{H}\sin\mathrm{D}\Delta\mathrm{D}\\ \Delta\mathrm{W}=\sin\mathrm{D}\Delta\mathrm{H}+\mathrm{H}\cos\mathrm{D}\Delta\mathrm{D}\end{array}\right\}\end{aligned}\]

\[\begin{aligned}&\theta=1-\frac{L_{\mathrm{free~}}}{L_{i n h}}\\ &\mathrm{IE}\%=1-\left(\frac{L_{\mathrm{free~}}}{L_{i n h}}\right)X100\end{aligned}\]

\[\begin{aligned}\frac{\partial\mathscr{A}_{1}^{l}}{\partial t}+\nabla\cdot c_{1}\mathscr{A}_{1}^{l}&=\frac{\partial\mathscr{A}_{2}^{l}}{\partial y}+\nabla\cdot\boldsymbol{c}_{2}\mathscr{A}_{2}^{l}\\ &=-\frac{\partial\mathscr{A}_{12}^{l}}{\partial t}-\nabla\cdot\boldsymbol{c}_{12}\mathscr{A}_{12}^{l}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}\beta_{1}=\pi-\alpha_{\mathrm{B}4},\beta_{2}=\pi-\alpha_{\mathrm{B}3},\beta_{3}=\pi-\alpha_{\mathrm{A}2},\beta_{4}=\pi-\alpha_{\mathrm{A}1}\\ \gamma_{1}=\pi-\alpha_{\mathrm{A}4},\gamma_{2}=\pi-\alpha_{\mathrm{A}3},\gamma_{3}=\pi-\alpha_{\mathrm{B}2},\gamma_{4}=\pi-\alpha_{\mathrm{B}1}.\end{array}\right\}\end{aligned}\]

\[\begin{array}{l}I_{\mathrm{c}}=\int\dot{m}_{\mathrm{v}}\mathrm{d}V\\ =\int\boldsymbol{j}_{\mathrm{s}}\cdot\mathrm{d}\boldsymbol{S}\\ =\int\rho_{\mathrm{a}}\boldsymbol{v}_{\mathrm{a}}\cdot\mathrm{d}\boldsymbol{S}\end{array}\]

\[\begin{array}{c}T_{i j}=\mu v_{i}v_{j}-S_{i j}\\ =S_{i j}v_{j}\end{array}\]

\[\begin{aligned}U_{\mathrm{m}}(r)=\left\{\begin{array}{l l}\frac{3n_{\mathrm{k}}U_{\mathrm{w}_{0}}^{2}r}{\nu_{\mathrm{m}}}\mathrm{e}^{-\left(n_{\mathrm{k}}w_{0}/\nu_{\mathrm{m}}\right)z}a<r\leq r_{\mathrm{c}}\\ n_{b}\Omega r\mathrm{e}^{-n_{\mathrm{b}}\sqrt{\Omega/\nu_{\mathrm{m}}}z}r_{\mathrm{c}}<r\leq b\end{array}\right.\end{aligned}\]

\[\begin{aligned}&79V=19\cdot3-18\\ &V=\frac{1\cdot3}{79},\mathrm{~which~is~negligible~}\end{aligned}\]

\[\begin{array}{c}d x_{1}=V x_{1}d y_{1}-x_{2}\eta+\frac{e^{4m y_{1}}}{2R}\left(x_{4}d y_{2}-x_{3}d y_{3}\right)\\ =V x_{2}d y_{1}+x_{1}\eta+\frac{e^{4m y_{1}}}{2R}\left(x_{3}d y_{2}+x_{4}d y_{3}\right)\\ =-V x_{3}d y_{1}+x_{4}\eta+\frac{e^{-4m y_{1}}}{2R}\left(x_{2}d y_{2}-x_{1}d y_{3}\right)\\ =-V x_{4}d y_{1}-x_{3}\eta+\frac{e^{-4m y_{1}}}{2R}\left(x_{1}d y_{2}+x_{2}d y_{3}\right)\end{array}\]

\[\begin{aligned}K^{(t+1)}=\left(\begin{array}{l l}K_{B A,p}&K_{B A,v}\\ K_{T A,p}&K_{T A,v}\\ K_{B S,p}&K_{B S,v}\\ K_{T S,p}&K_{T S,v}\end{array}\right)\end{aligned}\]

\[\begin{aligned}f&=\sigma_{2}F^{T}\\ &=\left(\sigma_{2}+\beta_{2}\right)/\left(\sigma_{1}+\beta_{1}\right)\\ &=\sigma_{1}/\left(\sigma_{1}+\beta_{1}\right)\\ &=\sigma_{2}/\left(\sigma_{2}+\beta_{2}\right)\\ &=T\tau_{2}^{-1}\end{aligned}\]

\[\begin{aligned}\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}\left(\rho\varepsilon u_{i}\right)&=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon}{\partial x_{j}}\right]+C_{1\varepsilon}\frac{\varepsilon}{k}\left(P_{k}\right.\\ &\left.+C_{3\varepsilon}P_{b}\right)-C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon}\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{r l}d_{1}=3\pi^{2}\left\{2(1+p r)+3\pi^{2}\Lambda\right\}^{-1}\\ k_{1}=d_{1}\left\{-54p r^{2}+\pi^{4}(\lambda-\Lambda)(93\lambda-148\mu)\right\}/162\pi^{2}\\ \bar{a}=16\gamma k_{\mathrm{c}}^{2},\quad h=16\gamma k_{\mathrm{c}}l_{\mathrm{c}},\quad\bar{b}=16\gamma l_{\mathrm{c}}^{2}\\ \gamma=\pi^{-2}\left\{2(1+p r)+3\pi^{2}(\Lambda-\lambda)\right\}^{-1},\quad v_{1}=v_{2}=0\end{array}\right\}\end{aligned}\]

\[\begin{aligned}&v_{s t e m,i}=2l_{s(1),i}g_{s(1),i}-\frac{1}{3}\left(g_{s(1),i}+g_{s(1),i}g_{s(2m),i}+g_{s(2m),i}\right)\\ &+\sum_{j=2}^{S_{i}}l_{s(j),i}\frac{g_{s(j-1),i}+g_{s(j),i}}{2}\end{aligned}\]

\[\begin{aligned}q_{i}&=Q_{\max}\frac{\left(\tilde{K}_{i}c_{i}\right)^{n_{i}}}{\left(\tilde{K}_{i}c_{i}\right)^{n_{i}}+\left(\tilde{K}_{\mathrm{H}}[\mathrm{H}]\right)^{n_{\mathrm{H}}}}\\ &\times\frac{\left(\left(\tilde{K}_{i}c_{i}\right)^{n_{i}}+\left(\tilde{K}_{\mathrm{H}}[\mathrm{H}]\right)^{n_{\mathrm{H}}}\right)^{p}}{1+\left(\left(\tilde{K}_{i}c_{i}\right)^{n_{i}}+\left(\tilde{K}_{\mathrm{H}}[\mathrm{H}]\right)^{n_{\mathrm{H}}}\right)^{p}}\end{aligned}\]

\[\begin{aligned}&v_{12}^{\mathrm{b}}(r,t)=\sum_{\alpha}e_{\alpha}\iint_{\Sigma_{\alpha}^{\mathrm{b}}}\mathrm{d}s^{\prime}\delta\left(r-r^{\prime}\right)\\ &s_{12}^{\mathrm{b}}(r,t)=\sum_{\alpha}\frac{e_{\alpha}}{c}\iint_{\Sigma_{\alpha}^{\mathrm{b}}}\mathrm{d}s^{\prime}\cdot\dot{r}^{\prime}\delta\left(r-\boldsymbol{r}^{\prime}\right)\end{aligned}\]

\[\begin{aligned}\varrho_{F^{\wedge}}\left(\gamma_{1}\right)=\left[\bar{\varrho}\left(\gamma_{1}\right)\right]\left(\begin{array}{c c}\phi\left(\gamma_{1}\right)\eta^{-1}\left(\gamma_{1}\gamma_{2}^{2}\right)\left(1+a_{2}\right)^{-2}&b_{1}\\ &\eta\left(\gamma_{1}\gamma_{2}^{2}\right)\left(1+a_{2}\right)^{2}\end{array}\right)\end{aligned}\]

\[\begin{aligned}\hat{p}&=-\left(\frac{P}{X}\right)r+\eta\left(\hat{f}_{r r r}+3\hat{f}_{r r}r^{-1}-2\hat{f}_{r}r^{-2}+2\hat{f}r^{-3}\right)r\\ &-\eta\ell^{-2}\left[\hat{f}_{r}+\hat{f}r^{-1}+\frac{r(U-V)}{X}\right]-\rho_{m}^{r}\left(\hat{\psi}-\frac{r E}{X}\right).\end{aligned}\]

\[\begin{aligned}\left.\begin{array}{l}L_{n}^{\prime}=\frac{p-n^{2}(1+a)-\frac{1}{2}n}{a(n+1)\left(n+\frac{1}{2}\right)}\\ y_{n}^{\prime}=\frac{(n-\sigma)(n+\sigma-1)}{a(n+1)\left(n+\frac{1}{2}\right)}\\ N_{n+1}^{\prime}+L_{n}^{\prime}+y_{n}^{\prime}/N_{n}^{\prime}=0\\ N_{n}^{\prime}=-\frac{y_{n}^{\prime}}{L_{n}^{\prime}+N_{n+1}^{\prime}}\end{array}\right\}\end{aligned}\]