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\[\begin{aligned}&2\tilde{v}=-\left(H^{*}(L+l)-H^{*2}\right)\\ &-\left\langle\sum_{h=1}^{H-H^{*}}\theta\left(\gamma_{h}^{\prime2}>L\right)\left(1-\frac{L}{\gamma_{h}^{\prime2}}\right)\left(1+\frac{L}{\gamma_{h}^{\prime2}}\right)\gamma_{h}^{\prime2}\right\rangle_{q\left(\left\{\gamma_{h}^{\prime2}\right\}\right)}.\end{a... | |
\[\begin{array}{l}\frac{d\alpha}{d t}=-k_{\alpha}(T)\cdot g(\alpha)\\ =k_{N\alpha}e^{-\frac{E_{\alpha}}{R T}}\end{array}\] | |
\[\begin{aligned}\left(\begin{array}{c}z^{\prime}\\ \bar{z}^{\prime}\\ \mathbf{1}\end{array}\right)=\mathbf{M}\left(\begin{array}{l}z\\ \bar{z}\\ \mathbf{1}\end{array}\right)\equiv\left(\begin{array}{l l l}u&0&t\\ 0&\bar{u}&\bar{t}\\ 0&0&\mathbf{1}\end{array}\right)\left(\begin{array}{l}z\\ \bar{z}\\ \mathbf{1}\end{arr... | |
\[\begin{aligned}&V_{k}^{\mathrm{bond~}}=-\beta^{-1}\ln\left[P_{k}^{\mathrm{bond~}}\left(d_{k}\right)/d_{k}^{2}\right]\\ &V_{k}^{\mathrm{angle~}}=-\beta^{-1}\ln\left[P_{k}^{\mathrm{angle~}}\left(\theta_{k}\right)/\sin\theta_{k}\right]\\ &V_{k}^{\mathrm{dihedral~}}=-\beta^{-1}\ln\left[P_{k}^{\mathrm{dihedral~}}\left(\ph... | |
\[\begin{aligned}a=\left[\begin{array}{r r c r}m+a_{11},&p+a_{12},&a_{13},&a_{14}\\ p+a_{21},&q+a_{22},&a_{23},&a_{24}\\ a_{31}&a_{32},&a_{33},&a_{34}\\ a_{41},&a_{42},&a_{43},&a_{44}\end{array}\right]\end{aligned}\] | |
\[\begin{array}{l}I_{1}:=\{(j\cdot k)\in I\mid\operatorname{row}(j)\cdot r\}\\ =\{(j\cdot k)\in I\mid\operatorname{row}(k)\cdot r\}\\ =\{(j\cdot k)\in I\mid\operatorname{row}(j)<r\cdot\operatorname{row}(k)>r\}\\ =\left\{(j\cdot k)\in I\mid(j\cdot k)\notin I_{1}\cup I_{2}\cup I_{3}\right\}.\end{array}\] | |
\[\begin{array}{c}f_{2}(t)=J_{1}(t)t^{-\frac{3}{2}}\left(\cos t^{\frac{1}{6}}-1+\frac{1}{2}t^{\frac{1}{3}}\right)\\ \mathrm{~ve~}\quad f_{2}^{\prime}(t)=\left(J_{1}^{\prime}(t)-\frac{3}{2}J_{1}(t)/t\right)t^{\frac{-3}{2}}\left(\cos t^{\frac{1}{6}}-1+\frac{1}{2}t^{\frac{1}{3}}\right)+\frac{1}{6}J_{1}(t)t^{-\frac{13}{6}}... | |
\[\begin{array}{l}\mathrm{T}_{1Z}=\frac{\mathrm{W}_{1}}{\mathrm{~A}}\left(1+\frac{\mathrm{X}y_{t}}{\mathrm{~K}^{2}}\right)\\ =\frac{\mathrm{W}_{1}}{\mathrm{~A}}\left(\frac{\mathrm{X}y_{c}}{\mathbf{K}^{2}}-1\right)\end{array}\] | |
\[\begin{aligned}\mathbf{H}=\left[\begin{array}{c c}\mathbf{F}&\mathbf{G}\\ \mathbf{Q}&-\mathbf{F}^{\mathrm{T}}\end{array}\right],\quad\mathbf{Q}=\left[\begin{array}{c c}\frac{D\left(v^{2}-1\right)\partial^{4}}{\partial x^{4}}-K&0\\ 0&\frac{2D(1-\nu)\partial^{2}}{\partial x^{2}}\end{array}\right],\quad\mathbf{F}=\left[... | |
\[\begin{aligned}\left.\begin{array}{l}\boldsymbol{t}^{(\mathrm{s})}=-\phi^{(\mathrm{s})}p\boldsymbol{I}+\lambda_{\mathrm{s}}\operatorname{tr}(\varepsilon)\boldsymbol{I}+2\mu_{\mathrm{s}}\varepsilon\\ \boldsymbol{t}^{(\mathrm{f})}=-\phi^{(\mathrm{f})}p\boldsymbol{I}-\frac{2}{3}\mu_{\mathrm{f}}\operatorname{div}\boldsym... | |
\[\begin{aligned}\left.\begin{array}{l}X=1\\ Y=\beta\sigma\\ Z=\beta\sigma\sin^{2}x^{2}=Y\sin^{2}x^{2}\end{array}\right\}\end{aligned}\] | |
\[\begin{array}{l}\frac{\partial\left(\rho_{m}\omega_{m}\right)}{\partial t}+\frac{\partial\left(\rho_{m}U_{m}^{j}\omega_{m}\right)}{\partial x^{j}}=\frac{\partial}{\partial x^{j}}\left[\left(\mu_{m}+\frac{\mu_{t_{m}}}{2.A}\right)\frac{\partial\omega_{m}}{\partial x^{j}}\right]\\ +\frac{5\omega_{m}}{9}\frac{3}{k_{m}}P_... | |
\[\begin{array}{l}R^{2}=1-\left(\frac{\sum_{i}\left(t_{i}-o_{i}\right)^{2}}{\sum_{i}\left(o_{i}\right)^{2}}\right)\\ \mathrm{~MAPE~}=\left|\left(\frac{t_{i}-o_{i}}{o_{i}}\right)\right|\times100\end{array}\] | |
\[\begin{array}{l}F_{\mathrm{therm~}}(Q)=\frac{I_{o z}}{1+\xi^{2}Q^{2}}\\ F_{\mathrm{froz~}}(Q)=\frac{I_{i n h}}{\left(1+\Xi^{2}Q^{2}\right)^{2}}\end{array}\] | |
\[\begin{aligned}\left.\begin{array}{r l}\boldsymbol{\nabla}\cdot\mathbf{u}=0\\ \mathbf{0}=-\boldsymbol{\nabla}\Pi-\rho_{\mathrm{i}}g\nabla z_{\mathrm{i}}+\eta_{\mathrm{i}}\nabla^{2}\mathbf{u}\end{array}\right\}\end{aligned}\] | |
\[\begin{aligned}\left.\begin{array}{r l}\left(\frac{p}{\sqrt{m}}\right)^{n}=T_{n}(\tilde{z})+\left(\tilde{z}^{2}-1\right)^{1/2}U_{n-1}(\tilde{z})\\ \left(\frac{p}{\sqrt{m}}\right)^{-n}=T_{n}(\tilde{z})-\left(\tilde{z}^{2}-1\right)^{1/2}U_{n-1}(\tilde{z})\end{array}\right\}\end{aligned}\] | |
\[\begin{aligned}L_{F}=\exp\left(\begin{array}{l l}(\bar{P\Xi})^{\beta M}&(P\Xi)_{\alpha N}\end{array}\right)\exp\left(\begin{array}{l l}(\bar{Q\Xi})^{\beta M}&(Q\Xi)_{\alpha N}\end{array}\right)\end{aligned}\] | |
\[\begin{array}{l}F_{x}=Z_{x}\cdot F_{y}\cdot Z_{y}\cdot F_{z}\cdot-1\\ =Z_{x x}\cdot r\cdot F_{x y}\cdot Z_{x y}\cdot F_{y x}\cdot s\\ =Z_{y y}\cdot t\cdot F_{x z}\cdot q\cdot F_{y z}\cdot F\cdot F_{z z}\cdot Q\end{array}\] | |
\[\begin{array}{l}\hat{Q}_{1}(k+1)=Q_{1}(k)+\frac{\Delta t}{2}\left[-\frac{g A}{z_{L}}\left(H_{2}(k)-u_{1}(k)\right)\right.\\ -\frac{f\left(Q_{1}(k)\right)}{2D A}\left(Q_{1}(k)\left|Q_{1}(k)\right|\right)-\frac{g A}{z_{L}}\left(\tilde{H}_{2}(k+1)-u_{1}(k+1)\right)\\ \left.-\frac{f\left(\tilde{Q}_{1}(k+1)\right)}{2D A}\... | |
\[\begin{aligned}\frac{\partial h^{\prime}}{\partial\hat{t}}+\frac{\partial}{\partial\hat{x}}\left[\frac{\bar{h}^{2}}{3}\tau_{b}^{\prime}+\bar{h}u_{b}^{\prime}\right]&=s\\ &=\frac{\partial Q^{\prime}}{\partial\hat{t}}-\bar{Q}\frac{\partial u_{b}^{\prime}}{\partial\hat{x}}\end{aligned}\] | |
\[\begin{array}{c}Y^{\frac{1}{2}}d t=d s\cdot e^{\frac{\tau-t_{Q}}{t_{j}}}d\sigma\\ =t_{H}e^{\frac{\tau-t_{t}}{t_{l}}}\\ =\xi^{\frac{1}{2}}\\ =1-\frac{1}{c^{2}}\left(\frac{d\epsilon}{d\tau}\right)^{2}\end{array}\] | |
\[\begin{aligned}\zeta(z\cdot t)&=\mu\frac{4B}{\pi b^{2}}\frac{\partial}{\partial t}\int_{x}^{\infty}H(\tilde{t}-\tilde{r}a)\frac{a^{4}\tilde{x}^{2}z^{2}\tilde{r}^{4}-\tilde{T}_{a}^{2}\left(8\tilde{t}^{2}\tilde{x}^{2}z^{2}-\tilde{r}^{4}\tilde{t}^{2}\right)}{\tilde{T}_{a}\tilde{r}^{8}}\mathrm{~d}\xi\\ &-\mu\frac{B}{\pi ... | |
\[\begin{aligned}b_{4a}\left(\theta_{3}\right)=\left\{\begin{array}{l}h\left[\frac{\pi}{4+\pi}\frac{\theta_{3}-\theta_{d}}{\tau}-\frac{1}{4(4+\pi)}S\left[4\pi\frac{\theta_{3}-\theta_{d}}{\tau}\right]\right],0\leq\theta_{3}-\theta_{d}\leq\frac{\tau}{8}\\ h\left[\frac{2}{4+\pi}+\frac{\pi}{4+\pi}\frac{\theta_{3}-\theta_{d... | |
\[\begin{aligned}\left|\begin{array}{c c c}\Lambda-\lambda-\epsilon a_{1,1}&-\epsilon a_{1,2}&\cdots\\ -\epsilon a_{2,1}&\Lambda-\lambda-\epsilon a_{2,2}&\ldots\\ \ldots&\ldots&\Lambda-\lambda-\epsilon a_{j,j}\end{array}\right|=0\end{aligned}\] | |
\[\begin{array}{l}\tau_{z r}=\left.\mu_{n f}\frac{\partial u}{\partial z}\right|_{z\cdot h}\cdot\frac{\mu_{f}r\omega_{1}f_{1}^{\prime\prime}(b)}{(1-\phi)^{2.5}\phi h}\cdot\tau_{z\theta}\cdot\left.\mu\frac{\partial v}{\partial z}\right|_{z\cdot y}\cdot\frac{\mu_{f}r\omega_{1}f_{2}^{\prime}(B)}{(1-\phi)^{2.5}\phi h^{\pri... | |
\[\begin{array}{l}\frac{\partial\mathbf{u}^{*}}{\partial t}+\left(\mathbf{u}^{*}\cdot\nabla\right)\mathbf{u}^{*}=-\frac{1}{\rho_{a}}\nabla P^{*}+v\nabla^{2}\mathbf{u}^{*}+\frac{\rho^{*}}{\rho_{X}}\mathbf{g}^{\prime}\\ =\nabla\cdot\mathbf{u}^{*}\\ =\frac{\partial T^{*}}{\partial t^{*}}+\mathbf{u}^{*}\cdot\nabla T^{*}\\ ... | |
\[\begin{aligned}h=\left(\begin{array}{c c c}-n\left(h_{2}\right)&-h_{2}h_{1}&-h_{2}\\ -h_{1}h_{2}&-n\left(h_{1}\right)&h_{1}\\ h_{2}&-h_{1}&-1\end{array}\right)\end{aligned}\] | |
\[\begin{aligned}\int d t h^{Q S}(t\cdot x)h^{Q S\dagger}\left(t\cdot x^{\prime}\right)&=\Lambda_{1}^{2}u_{1}(x)u_{1}\left(x^{\prime}\right)+\Lambda_{2}^{2}u_{2}(x)u_{2}\left(x^{\prime}\right)\\ &+\Lambda_{1}\Lambda_{2}\cos\theta_{V}\left(u_{1}(x)u_{2}\left(x^{\prime}\right)+u_{2}(x)u_{1}\left(x^{\prime}\right)\right)\... | |
\[\begin{array}{c}\lim_{t\rightarrow+\infty}\sup_{\left(c_{2}^{\delta}+\varepsilon\right)t<x<\left(c_{1}-\varepsilon\right)t}\left(\left|\bar{u_{\delta}}(t,x)\right|+\left|\underline{v_{\delta}}(t,x)-(1-2\delta)\right|\right)=0\mathrm{~for~all~}\varepsilon\in\left(0,\frac{c_{1}-c_{2}^{\delta}}{2}\right)\\ \lim_{t\right... | |
\[\begin{aligned}M/M_{R}&=1+\frac{\pi^{2}}{48}\left(\frac{2p}{d}\right)^{2}\left(1+2\sin^{2}\theta\right)\\ &-\frac{\pi^{4}}{23P4P}\left(\frac{2p}{d}\right)^{4}\left(11-1I6\sin^{2}\theta-66\sin^{4}\theta\right)+\ldots.\end{aligned}\] | |
\[\begin{aligned}S_{\mathbb{A}}\left(\begin{array}{l}a_{1}\\ b_{1}\\ a_{0}\\ b_{0}\end{array}\right)=\left(\begin{array}{l}d_{A}a_{0}-\left[\phi,b_{0}\right]\\ d_{A}b_{0}+\left[\phi,a_{0}\right]\\ d_{A}^{\star}a_{1}-\star\left[\phi,\star b_{1}\right]\\ d_{A}^{\star}b_{1}+\star\left[\phi,\star a_{1}\right]\end{array}\ri... | |
\[\begin{aligned}R&=\nabla\ln P(\mathbf{r}\mid x)\\ &=\rho\int\left[r_{c}-\bar{\gamma}_{c}(x)\right]\bar{\gamma}_{c}^{\prime}(x)d c\end{aligned}\] | |
\[\begin{aligned}f(k\cdot\kappa)&=\frac{a^{4}}{3}k\mathrm{e}^{-k a}\kappa\cdot h(k\cdot q)\cdot q\kappa^{-1}f(k\cdot\kappa)\cdot\frac{a^{4}}{3}k\mathrm{e}^{-k a}q\\ &=(2\pi)^{2}\int_{h}^{\infty}\mathrm{d}k k^{-1}\int_{2}^{k}\mathrm{~d}q|h(k\cdot q)|^{2}\cdot\frac{4\pi^{2}}{9}a^{8}\int_{Y}^{\infty}\mathrm{d}k k\mathrm{e... | |
\[\begin{aligned}I M_{1}=\left\{\begin{array}{c c c c}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array}\right\}\end{aligned}\] | |
\[\begin{aligned}X&=U D\left(K_{U}+\mu\tilde{D}\right)^{-1}U^{*}+U W^{\prime}\tilde{D}\left(K_{U}+\mu\tilde{D}\right)^{-1}U^{*}\\ &=U D\left(K_{U}+\mu\tilde{D}\right)^{-1}U^{*}\end{aligned}\] | |
\[\begin{aligned}\left.\begin{array}{l}f_{0}^{\prime\prime\prime}+\frac{1}{4}\lambda f_{0}f_{0}^{\prime\prime}-\frac{1}{2}\lambda f_{0}^{\prime2}+g_{0}=0\\ g_{0}^{\prime\prime}+\frac{1}{4}\lambda f_{0}g_{0}^{\prime}-\lambda g_{0}f_{0}^{\prime}=0\end{array}\right\}\end{aligned}\] | |
\[\begin{aligned}\Delta N&=2\cdot\pm1\\ &=\Delta K_{\mathrm{a}}\\ &=\Delta K_{\mathrm{c}}\\ &=n\cdot\pm1\\ &=\Delta M_{\mathrm{s}}\end{aligned}\] | |
\[\begin{aligned}\frac{a_{c1}}{h}&=k_{a c1}\left(\frac{r}{h}\right)^{n_{a c1}}\cos\theta\\ &=k_{a t1}\left(\frac{r}{h}\right)^{n_{a t1}}\cos\theta\\ &=k_{a c2}\left(\frac{r}{h}\right)^{n_{a c2}}\cos^{2}\theta\end{aligned}\] | |
\[\begin{array}{l}\left(\nabla\psi_{h\cdot m}\cdot\nabla v_{h}\right)=\alpha\left(p_{h\cdot m}-e_{h\cdot m}+N_{m}\cdot v_{h}\right)\cdot\forall v_{h}\in W_{h}\\ =\left(\psi_{h\cdot m}\cdot1\right)\end{array}\] | |
\[\begin{aligned}\boldsymbol{a}_{N}=\left[\begin{array}{c}\boldsymbol{a}_{\mathrm{c}}\cdot\boldsymbol{n}_{1}+\left(\boldsymbol{\omega}\times\boldsymbol{\omega}\times\boldsymbol{r}_{1}\right)\cdot\boldsymbol{n}_{1}+\left(\boldsymbol{\alpha}\times\boldsymbol{r}_{1}\right)\cdot\boldsymbol{n}_{1}\\ \vdots\\ \boldsymbol{a}_... | |
\[\begin{array}{l}\mathrm{~I.~}\left(\mathrm{F}_{1}a_{m m},\mathrm{~F}_{2}a_{m},\mathrm{~F}_{3}a_{m},b_{m}\right)=0\\ \mathrm{~II.~}\left[\mathrm{F}_{1}a_{m},\mathrm{~F}_{2}a_{m},b_{m}\right]=0\\ \mathrm{~II.~}\left\{\mathrm{F}_{1}a_{m},b_{m}\right\}=0\end{array}\] | |
\[\begin{aligned}W&=-u_{c\cdot i}^{\prime}w_{i}^{c}-u_{a\cdot i}^{\prime}w_{i}^{a}-u_{a d\cdot i}^{\prime}w_{i}^{a d}+\left(\sigma_{i j}^{\prime}+s S_{r}^{M}\delta_{i j}\right)\dot{\varepsilon}_{i j}^{M}\\ &-n_{M}s\dot{S}_{r}^{M}+u_{a}n_{M}S_{r}^{a}\frac{\dot{\rho}^{a}}{\rho^{a}}+\left[p^{\prime}+\frac{\rho^{a d}}{\rho... | |
\[\begin{aligned}&E_{\gamma}^{(1)}=N\int_{\Omega}d\rho\left(V_{\infty}^{(0)\prime}\psi_{\gamma}^{(0)*}\cdot D^{(1)}\psi_{\gamma}^{(0)}-\psi_{\gamma}^{(0)*}\cdot D^{(1)}V_{\infty}^{(0)}\psi_{\gamma}^{(0)}\right)\\ &+\frac{\hbar^{2}}{2m}\sum_{i=1}^{3}\int_{\Omega}d\sigma n_{i}\left(\psi_{\gamma}^{(0)*}\cdot\frac{\partial... | |
\[\begin{array}{l}\hat{\delta}_{t}^{(i+1)}\triangleq\frac{1}{n^{(i+1)}}\sum_{k=1}^{n^{(i+1)}}\frac{\hat{t}_{k}^{(i+1)}-\hat{t}_{k}^{(i+1)}}{\left(\Delta\hat{t}_{k}^{(i+1)}+\Delta\hat{t}_{k}^{\prime(i+1)}\right)^{2}}\\ \hat{\delta}_{f}^{(i+1)}\triangleq\frac{1}{n^{(i+1)}}\sum_{k=1}^{n^{(i+1)}}\frac{\hat{f}_{k}^{(i+1)}-\... | |
\[\begin{aligned}\dot{\tilde{\mathbf{Y}}}&=\dot{\mathbf{T}}^{T}\mathbf{W}_{\mathbf{x}}\dot{\tilde{\mathbf{U}}}+\dot{\tilde{\mathbf{Y}}}\left(\mathbf{R}\boldsymbol{\phi}\mathbf{R}^{T}-\dot{\mathbf{R}}\mathbf{R}^{T}\right)\\ &=\dot{\mathbf{T}}^{T}\mathbf{W}_{\mathbf{x}}\dot{\tilde{\mathbf{U}}}+\dot{\tilde{\mathbf{Y}}}\ti... | |
\[\begin{array}{l}\frac{u-u_{2}}{\eta}=(1-\mu)\Phi\\ =\left(\frac{1}{\mu}-1\right)\Phi_{A}\end{array}\] | |
\[\begin{aligned}k_{1}=\left(\begin{array}{c}e\\ p\end{array}\right)\quad k_{2}=\left(\begin{array}{c}e\\ -p\end{array}\right)\quad k_{3}=\left(\begin{array}{l}-e\\ -p\end{array}\right)\quad k_{4}=\left(\begin{array}{c}-e\\ p\end{array}\right)\end{aligned}\] | |
\[\begin{array}{l}T D G\left(T S_{i}\right)=\max\left\{\frac{\left(T_{\mathrm{mean~}}^{j}-T_{\mathrm{mean~}}^{k}\right)}{k-j}\right\}\left(T S_{i}\right)\quad\forall j<k\\ \mathrm{~where~}j,k=(1,2,\ldots,m)\end{array}\] | |
\[\begin{aligned}\left.\begin{array}{l}X_{4}=\frac{1}{2}t(-2+t x)\left(\dot{x}+x^{2}\right)\\ X_{5}=\frac{1}{2}t(-2+t x)\left(\dot{x}t-x+t x^{2}\right)\\ X_{6}=-\frac{t}{4}(-2+t x)\left(2+\dot{x}t^{2}-2t x+t^{2}x^{2}\right)\end{array}\right\}\end{aligned}\] | |
\[\begin{array}{l}B_{\mathrm{Y}}=\frac{E c w^{3}}{12\left(\frac{\phi^{\mathrm{x}}}{\phi_{\mathrm{y}}^{\mathrm{x}}}\right)}\cdot\cdot\cdot\cdot\\ =\frac{E c^{3}w^{3}}{36\left(\frac{\phi^{\mathrm{x}}}{\phi_{\mathrm{y}}^{\mathrm{x}}}\right)^{3}}\cdot\cdot\cdot\cdot\end{array}\] | |
\[\begin{aligned}\frac{\partial\rho^{\mathrm{f}}(1-\bar{c})\tilde{w}^{\mathrm{f}}}{\partial t}&=-\frac{\partial\rho^{\mathrm{f}}(1-\bar{c})\tilde{w}^{\mathrm{f}}\tilde{w}^{\mathrm{f}}}{\partial z}-(1-\bar{c})\frac{\partial\bar{P}^{\mathrm{f}}}{\partial z}\\ &+\frac{\partial\tau_{z z}^{\mathrm{f}}}{\partial z}+\rho^{\ma... | |
\[\begin{aligned}&G_{i}\left(R_{1},\epsilon s-\epsilon s^{\prime},k^{2}\right)=\frac{1}{2\pi k R_{1}}\frac{J_{0}\left(k R_{1}\right)}{J_{1}\left(k R_{1}\right)}+\frac{1}{\pi}\ln\left(\epsilon\left|s-s^{\prime}\right|\right)\\ &-\frac{1}{\pi}\sum_{n=1}^{\infty}\left[\frac{J_{n}\left(k R_{1}\right)}{k R_{1}J_{n}^{\prime}... | |
\[\begin{aligned}-\delta\theta^{\prime}&=\frac{n\pi d m}{\mathrm{Q}\sigma}\left(\theta^{\prime}-\mathrm{T}\right)\delta\eta\\ &=\frac{n\pi d m}{\mathrm{~W}G}w\sigma\left(\theta^{\prime}-\mathrm{T}\right)\delta\eta\end{aligned}\] | |
\[\begin{aligned}E\rangle&=\frac{\left\langle p_{1}^{2}\right\rangle}{2m}\left(1+\frac{2\left\langle p_{2}\right\rangle}{\left\langle p_{1}\right\rangle}\right)\\ &=\int_{-\infty}^{\infty}\Psi^{*}(\mathbf{r}\cdot t)\left[-\frac{\hbar^{2}}{2m}\nabla_{1}^{2}\left(1+\frac{2\nabla_{2}}{\nabla_{1}}\right)\right]\Psi(\mathbf... | |
\[\begin{aligned}\phi=\left\{\begin{array}{l c}1,r\equiv\left|P-P_{n}\right|\leqslant\frac{1}{2}\\ \exp\left\{\frac{4}{3}+\left(r^{2}-1\right)^{-1}\right\},\frac{1}{2}\leqslant r\leqslant1\\ 0,r\geqslant1\end{array}\right.\end{aligned}\] | |
\[\begin{aligned}\left(\begin{array}{c}H_{1}^{0}\\ H_{3}^{0}\\ H_{5}^{0}\\ H_{6}^{0}\end{array}\right)_{L(R)}=\frac{1}{\sqrt{2}}\left(\begin{array}{c c c c}1&1&0&0\\ -1&1&0&0\\ 0&0&1&1\\ 0&0&-1&1\end{array}\right)\left(\begin{array}{c}H_{1}^{\prime0}\\ H_{3}^{00}\\ H_{5}^{00}\\ H_{6}^{00}\end{array}\right)_{L(R)}\end{a... | |
\[\begin{aligned}P_{\alpha}=\left(\begin{array}{l}\alpha_{1}\\ \alpha_{2}\\ \alpha_{3}\end{array}\right)\left(\begin{array}{l l l}\bar{\alpha}_{1}&\bar{\alpha}_{2}&\bar{\alpha}_{3}\end{array}\right)\end{aligned}\] | |
\[\begin{aligned}M&=\frac{q z}{2L}(z-2\xi)x\cdot D\leq x\leq\xi\\ &=\frac{q z}{2L}(z-2\xi)x-\frac{q}{2}(x-\xi)^{2}\cdot\xi\leq x\leq(\xi+z)\\ &=\frac{q z}{2L}(2\xi+z)(L-x)\cdot(\xi+z)\leq x\leq L\end{aligned}\] | |
\[\begin{array}{c}\frac{\partial\bar{u}_{1}}{\partial x}+\frac{\partial\bar{v}_{1}}{\partial y}=U\\ =-\frac{\partial}{\partial x}\left(\bar{\frac{1}{2}u_{8h}^{2}}\right)\end{array}\] | |
\[\begin{aligned}G^{(d)}(t\cdot\boldsymbol{x})&=\frac{1}{(2\pi)^{d+1}\mathrm{i}}\int_{l}^{\infty}\mathrm{d}\sigma\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\mathrm{d}s\mathrm{e}^{s t-\sigma s^{\alpha}}\\ &=\frac{1}{(2\pi)^{d+1}}\int_{B}^{\infty}\mathrm{d}\sigma\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\mathrm{d}s s^{\... | |
\[\begin{aligned}\mathbf{d}^{2}\left(h-\boldsymbol{J}^{\xi}\right)(0)=\left(\begin{array}{c c c c}g m/l&0&0&-\xi\\ 0&g m/l&\xi&0\\ 0&\xi&1/m&0\\ -\xi&0&0&1/m\end{array}\right)\end{aligned}\] | |
\[\begin{aligned}\mu_{\mathrm{mk}}=\left\{\begin{array}{l l}1+0i+0j x\in\left[0,s_{1}\right]\\ \frac{s_{2}-x}{s_{2}-s_{1}}+\frac{x-s_{1}}{s_{2}-s_{1}}+0j x\in\left[s_{1},s_{2}\right]\\ 0+\frac{s_{3}-x}{s_{3}-s_{2}}+\frac{x-s_{2}}{s_{3}-s_{2}}x\in\left[s_{2},s_{3}\right]\\ 0+0i+0j x\in\left[s_{3},+\infty\right]\end{arra... | |
\[\begin{aligned}W&=\frac{1}{2}D\lambda^{-2}\int_{-\infty}^{d}\left(m_{E}\llbracket\phi_{8}\rrbracket+v_{G}\llbracket w_{z}\rrbracket\right)\lambda\mathrm{d}\xi_{1}\\ &=\frac{1}{2}D\lambda^{-2}\int_{-\infty}^{2}\left(M_{t}\lambda\llbracket\phi_{p}\rrbracket+V_{Z}\llbracket w_{G}\rrbracket\right)\exp\left(\imath a\xi_{1... | |
\[\begin{aligned}\left.\begin{array}{r l}16\pi E_{m n}=2\left[\delta_{m n}\left(\mathbb{E}^{2}+\mathbb{H}^{2}\right)-2H_{m}H_{n}-2E_{m}E_{n}\right]\\ 16\pi E_{4n}=4\epsilon_{n i k}H_{i}E_{k}\\ 16\pi E_{44}=2\left(\mathbf{E}^{2}+\mathbf{H}^{2}\right)\end{array}\right\}\end{aligned}\] | |
\[\begin{aligned}D_{l,n}=\operatorname{det}\left[\begin{array}{c c c c}\mu_{l}&\mu_{l+1}&\cdots&\mu_{l+n}\\ \mu_{l+1}&\mu_{l+2}&\cdots&\mu_{l+n+1}\\ \vdots&\vdots&&\vdots\\ \mu_{l+n}&\mu_{l+n+1}&\cdots&\mu_{l+2n}\end{array}\right]\end{aligned}\] | |
\[\begin{aligned}\Delta S_{n+3}^{\pi}&=\lambda_{\pi}^{\prime}{}^{2}\lambda_{p}^{\prime}\left[T_{2}^{+}-T_{1}^{+}\left(1-K_{1}^{+}\right)\right]\left(\Delta S_{n}+A_{+}\right)\\ &+\lambda_{\pi}^{\prime2}\left[1-T_{2}^{+}-\left(1-T_{1}^{+}\right)\left(1-K_{1}^{+}\right)\right]\Delta S_{n+1}^{p}+\lambda_{\pi}^{\prime2}K_{... | |
\[\begin{aligned}\hat{V}_{\mathrm{CKM}}=\left(\begin{array}{c c c}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{13}}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{13}}&s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta... | |
\[\begin{aligned}\left.\begin{array}{r l}D_{\xi}\Delta\left(z,\frac{m\tau}{l},0\right)=x_{\xi}\left(m\tau;z,\frac{m\tau}{l},0\right)-I=\Pi^{m}(z)-I,\\ D_{T}\Delta\left(z,\frac{m\tau}{l},0\right)=f(z)+x_{T}\left(m\tau;z,\frac{m\tau}{l},0\right)=f(z)\\ D_{\varepsilon}\Delta\left(z,\frac{m\tau}{l},0\right)=x_{\varepsilon}... | |
\[\begin{aligned}t_{\mathrm{c}}=\left\{\begin{array}{l l}\frac{\rho_{\mathrm{c}}f_{\mathrm{TSS}}(t)V_{\mathrm{Tot}}}{q_{0}c_{\mathrm{L}1}L}\left(t_{\mathrm{c}}\leq t_{1}\right)\\ t_{1}-\frac{c_{\mathrm{L}1}}{m_{\mathrm{c}}}-\sqrt{\left(\frac{c_{\mathrm{L}1}}{m_{\mathrm{c}}}\right)^{2}-\frac{2t_{1}c_{\mathrm{L}1}}{m_{\m... | |
\[\begin{aligned}\Delta G_{\mathrm{I}}&=-\frac{4}{3}\pi r^{3}\Delta G_{\mathrm{V}}+4\pi r^{2}\gamma_{\mathrm{SL}}\\ &\propto-r^{3}+r^{2}\end{aligned}\] | |
\[\begin{array}{l}3\mathfrak{a}\frac{d I}{d\mathfrak{a}}=-\frac{4}{3}\pi\rho a^{3}\frac{\lambda}{1+\lambda}\cdot\frac{2}{5}\left(b^{2}+c^{2}\right)\\ =\frac{4}{3}\pi\rho a^{3}\frac{\lambda}{1+\lambda}\cdot\frac{2}{5}\left(2b^{2}-c^{2}\right)\end{array}\] | |
\[\begin{aligned}\mathcal{K}=\left(\begin{array}{c c c c}\hat{G}(\beta,k;\boldsymbol{\tau};\boldsymbol{\tau})&\hat{G}(\beta,k;\boldsymbol{\tau};\boldsymbol{\tau}+\mathbf{s})&\hat{G}(\beta,k;\boldsymbol{\tau};-\boldsymbol{\tau})&\hat{G}(\beta,k;\boldsymbol{\tau};\mathbf{t}-\boldsymbol{\tau})\\ \hat{G}(\beta,k;\boldsymbo... | |
\[\begin{aligned}\left.\begin{array}{r l}\frac{\partial A}{\partial y}(\sigma,\psi,n)=-\frac{1}{2a}\left(n+\frac{1}{2}\right)B(\sigma,\psi,n+1)\\ +\frac{n}{a}B(\sigma,\psi,n)-\frac{1}{2a}\left(n-\frac{1}{2}\right)B(\sigma,\psi,n-1)\\ \frac{\partial B}{\partial y}(\sigma,\psi,n)=+\frac{1}{2a}\left(n+\frac{1}{2}\right)A(... | |
\[\begin{array}{c}I_{1}\phi_{1}+\zeta_{1}\phi_{1}+V^{\prime}\left(\phi_{1}-\phi_{2}\right)+\mu_{1}E\sin\phi_{1}=\lambda_{1}(t)\\ =I_{2}\ddot{\phi}_{2}+\zeta_{2}\dot{\phi}_{2}-V^{\prime}\left(\phi_{1}-\phi_{2}\right)+\mu_{2}E\sin\phi_{2}\\ =-2V_{s}\cos\left(\phi_{1}-\phi_{2}\right)\\ =2V_{6}\sin\left(\phi_{1}-\phi_{2}\r... | |
\[\begin{aligned}W=\left[\begin{array}{c c c c c}w_{L}&0&\cdots&0&0\\ w_{L-1}&w_{L}&0&\cdots&0\\ \vdots&&&&\vdots\\ w_{1}&\cdots&w_{L}&\cdots&0\\ \vdots&&&&\vdots\\ 0&\cdots&w_{1}&\cdots&w_{L}\end{array}\right]\end{aligned}\] | |
\[\begin{aligned}&r_{i}(t+\Delta t)=\boldsymbol{r}_{i}(t)+v_{i}(t)\Delta t\\ &+\frac{\phi^{2}(\Delta t)^{2}}{m}\left(\boldsymbol{\Psi}_{i}^{\mathrm{grav~}}(t+\Delta t)+\sum_{j=1,j\neq i}^{K}\boldsymbol{\Psi}_{i j}(t+\Delta t)+\boldsymbol{\Psi}_{i}^{\mathrm{fluid~}}(t+\Delta t)\right)\\ &+\frac{\phi(1-\phi)(\Delta t)^{2... | |
\[\begin{aligned}\int_{E_{\zeta}}\left(\zeta+t_{h}\right)^{m^{*}\beta}d\mathcal{L}^{N}&=\int_{Q_{\frac{\tau+\sigma}{2}}}\left(\zeta+t_{9}\right)^{m^{*}\beta}d\mathcal{L}^{N}\\ &=c_{N\cdot m\cdot\beta}\left[\int_{E_{\zeta}}|\nabla\zeta|^{\frac{\beta m N}{N-(1-\beta)m}}d\mathcal{L}^{N}\right]^{\frac{N-(1-\beta)m}{N-m}}\e... | |
\[\begin{array}{l}a=p^{v_{p}(a)}\cdot\left(a_{H}+a_{1}p+a_{2}p^{2}+\ldots\right)\cdot p^{v_{p}(a)}\cdot u\cdot a_{L}\neq R\\ =p^{v_{p}(x)}\cdot\left(x_{g}+x_{1}p+x_{2}p^{2}+\ldots\right)\cdot p^{v_{p}(x)}\cdot v\cdot x_{V}\neq e\end{array}\] | |
\[\begin{aligned}\left.\begin{array}{r l}\frac{1}{2h}\frac{\mathrm{d}T_{1}}{\mathrm{~d}r}=\left[\left(\lambda_{1}^{2}+3\lambda_{3}^{2}\right)\left(C_{1}+\lambda_{2}^{2}C_{2}\right)\right]\frac{\lambda_{3}}{\lambda_{1}}\frac{\mathrm{d}\lambda_{1}}{\mathrm{~d}r}+\left[\left(3\lambda_{3}^{2}-\lambda_{1}^{2}\right)C_{1}+\l... | |
\[\begin{aligned}\left.\begin{array}{l}F=V\int_{-\infty}^{+\infty}Z(E,H)e^{-\left(E-E_{0}\right)/k T}\left(1+e^{-\left(E-E_{0}\right)/k T}\right)^{-1}d E+N E_{0}\\ N=\frac{V}{k T}\int_{-\infty}^{+\infty}Z(E,H)\left(1+e^{\left(E-E_{0}\right)/k T}\right)^{-1}\left(1+e^{-\left(E-E_{0}\right)/k T}\right)^{-1}d E.\end{array... | |
\[\begin{aligned}\hat{r}_{\lambda}&=\left(B^{\prime}B+\lambda I\right)^{-1}\left(B^{\prime}y+\lambda r_{v}\right)\\ &=\left(B^{\prime}B+\lambda I\right)^{-1}\left[B^{\prime}\left(B r^{*}+\varepsilon\right)+\lambda\left(r^{*}+\delta r\right)\right]\\ &=\left(B^{\prime}B+\lambda I\right)^{-1}\left[\left(B^{\prime}B+\lamb... | |
\[\begin{aligned}X_{2k}=\left[\begin{array}{c c c}E_{0}&\cdots&E_{k-1}\\ \vdots&\ddots&\vdots\\ E_{0}^{(k-1)}&\cdots&E_{k-1}^{(k-1)}\end{array}\right],\quad Y_{2k}=\left[\begin{array}{c c c}O_{0}&\cdots&O_{k-1}\\ \vdots&\ddots&\vdots\\ O_{0}^{(k-1)}&\cdots&O_{k-1}^{(k-1)}\end{array}\right]\end{aligned}\] | |
\[\begin{aligned}\left.\begin{array}{l}f_{i}^{\prime}=q f_{i}/y\quad(i=1,2,3)\\ y^{\prime}=q^{2}/y\end{array}\right\}\end{aligned}\] | |
\[\begin{array}{l}Y^{\prime}\left(N_{A}-2Y-Y^{\prime}\right)=Y^{2}e^{2w/k T}\\ =Y\left(N_{B}-Y-2Y^{\prime}\right)\end{array}\] | |
\[\begin{array}{l}\frac{\partial(r h u)_{1}}{\partial t}+\frac{\partial}{\partial x}\left(\left(r h u^{2}\right)_{1}+\frac{1}{2}g\left(r h^{2}\right)_{1}\right)+g r_{1}h_{1}\frac{\partial h_{2}}{\partial x}\\ =-g r_{1}h_{1}\frac{\partial z_{b}}{\partial x}-\frac{\tau_{1\cdot b}}{\rho_{w}}\end{array}\] | |
\[\begin{aligned}2G\Delta A_{3}&=\operatorname{Im}\int\psi(\alpha)z^{\prime}(\alpha)\mathrm{d}\alpha\\ &=2\pi\left[1+\left(\frac{m_{1}^{2}n}{1-m_{2}n}\right)+m_{2}^{2}(2n+1)+\left(\frac{m_{1}^{2}m_{2}n^{2}}{1-m_{2}n}\right)\right]\end{aligned}\] | |
\[\begin{aligned}m_{\mathrm{W}^{\pm}}&=\frac{1}{2}v g\\ &=\frac{1}{2}v\sqrt{g^{2}+g^{\prime2}}\end{aligned}\] | |
\[\begin{aligned}\Delta=\left|\begin{array}{c c c c}\lambda_{1}^{-2}\psi_{n,1}^{(1)}&\lambda_{1}^{-1}\psi_{n,2}^{(1)}&\psi_{n,1}^{(1)}&\lambda_{1}\psi_{n,2}^{(1)}\\ \lambda_{2}^{-2}\psi_{n,1}^{(2)}&\lambda_{2}^{-1}\psi_{n,2}^{(2)}&\psi_{n,1}^{(2)}&\lambda_{2}\psi_{n,2}^{(2)}\\ \left(\lambda_{1}^{*}\right)^{2}\psi_{n,2}... | |
\[\begin{array}{c}y=\left(A_{1}-n A_{0}\right)\left\{\left(\Pi_{\mathrm{II}}+\Delta\Pi_{\mathrm{max.~}}\right)\ln\left(\frac{\Pi_{\mathrm{II}}+\Delta\Pi_{\max}}{\Pi_{\mathrm{II}}}\right)-\Delta\Pi_{\mathrm{max.~}}\right\}\\ W_{i}=\left(A_{1}-n A_{0}\right)\left\{\left(\Pi_{\mathrm{II}}+\Delta\Pi_{\max}\right)\ln\left(\... | |
\[\begin{aligned}{}_{\alpha}^{(k)}(E)&=\int_{E}^{1}x^{\alpha}(\ln(x))^{k-1}\mathrm{~d}x\\ &=(-1)^{k}\frac{E^{\alpha+1}-1}{(\alpha+1)^{k}}-\delta_{k2}\frac{E^{\alpha+1}\ln E}{\alpha+1}\end{aligned}\] | |
\[\begin{aligned}&P(0)=-\frac{1}{2}\mathrm{i}\int_{-\infty}^{\infty}\left[1+\left(1+\mathrm{e}^{2\mathrm{i}\pi\nu}\right)\frac{H_{\nu}^{(1)}(k a)}{H_{\nu}^{(2)}(k a)}\right]\mathrm{d}\nu\\ &-\frac{1}{2}\mathrm{i}\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\left\{\left[1+\mathrm{e}^{2\mathrm{i}\pi\nu}\frac{H_{\nu}^{(1)}(k... | |
\[\begin{aligned}\left.\begin{array}{r l}\Delta z=\frac{P_{0}}{\rho_{0}g}\approx10\mathrm{~m}\\ \Delta t=\frac{\phi_{0}\mu P_{0}^{2}}{g^{2}k_{d0}K_{\mathrm{B}}\rho_{0}^{2}}\approx0.14\mathrm{~s}\\ \Delta S=\frac{C_{v0}P_{0}\gamma_{S}}{\alpha K_{\mathrm{B}}T_{0}}\approx6.0\times10^{-3}\mathrm{Jkg}^{-1}\mathrm{~K}^{-1}\l... | |
\[\begin{aligned}H&=\frac{1}{2}\int\varepsilon^{3}g\eta^{\prime2}(X)+\varepsilon^{3}\left(h+\bar{c}_{1}\right)u^{\prime2}(X)+\varepsilon^{5}\left(1+\bar{c}_{2}\right)\eta^{\prime}(X)u^{\prime2}(X)\\ &-\varepsilon^{5}\left(\bar{c}_{3}+\frac{1}{3}h^{3}\right)\left(\partial_{X}u^{\prime}(X)\right)^{2}\mathrm{~d}X+O\left(\... | |
\[\begin{aligned}\bar{X}_{n\cdot\ell}^{m}(\alpha\cdot\boldsymbol{p})&=\frac{(n+\ell+1)!}{(2\pi)^{1/2}(1/2)_{\ell+1}}\mathcal{Y}_{\ell}^{m}\left(-\frac{i\boldsymbol{p}}{2}\right)\frac{\alpha^{2n-\ell-1}}{\left(\alpha^{2}+p^{2}\right)^{n+1}}\\ &\times{}_{2}F_{1}\left(\frac{\ell-n}{2}\cdot\frac{\ell-n+1}{2}\cdot\ell+\frac... | |
\[\begin{aligned}\lambda\in\left\{\begin{array}{l l}\mathbf{Z}(u=0)\\ \mathbf{Z}+\frac{1}{2}(u=\infty)\end{array}\right.\end{aligned}\] | |
\[\begin{array}{c}\alpha_{\mathrm{F}}(V)=\frac{e^{b_{\mathrm{F}}\left(V-a_{\mathrm{F}}\right)}}{2\tau_{\mathrm{F}}}\\ =\frac{e^{-b_{\mathrm{F}}\left(V-a_{\mathrm{F}}\right)}}{2\tau_{\mathrm{F}}}\\ =\frac{e^{b_{\mathrm{S}}\left(V-a_{\mathrm{S}}\right)}}{2\tau_{\mathrm{S}}}\\ =\frac{e^{-b_{\mathrm{S}}\left(V-a_{\mathrm{S... | |
\[\begin{aligned}F_{0}(m,a)=\left\{\begin{array}{l l}\sqrt{m^{2}-a^{2}}\left[\pi/2-\arctan\left(a/\sqrt{m^{2}-a^{2}}\right)\right]m\geq a\\ -\sqrt{a^{2}-m^{2}}\ln\left[\left(a+\sqrt{a^{2}-m^{2}}\right)/m\right]m<a\end{array}\right.\end{aligned}\] | |
\[\begin{array}{l}P_{12}^{\infty}=A_{12}+\frac{a_{2}}{1+\lambda}-r_{23}\left(\nu_{22}^{\infty}+\nu_{23}^{\infty}\right)\\ =A_{13}+\frac{\lambda a_{3}}{1+\lambda}-r_{23}\left(\nu_{32}^{\infty}+\nu_{33}^{\infty}\right)\\ =A_{21}+\frac{d_{2}}{1+\lambda}-r_{23}\left(\nu_{22}^{\infty}+\nu_{23}^{\infty}\right)\\ =A_{31}+\fra... | |
\[\begin{aligned}\left.\begin{array}{r l}\frac{1}{\rho}\frac{\partial p}{\partial r}-\frac{V^{2}}{r}=-\frac{\partial u}{\partial t}+2\left(A+\frac{B}{r^{2}}\right)v+v\left(\nabla_{1}^{2}u+\frac{\partial^{2}u}{\partial z^{2}}-\frac{u}{r^{2}}\right)\\ 0=-\frac{\partial v}{\partial t}-2A u+v\left(\nabla_{1}^{2}v+\frac{\pa... | |
\[\begin{aligned}\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c c c}\sin\Sigma&0&\cos\Sigma\\ 0&1&0\\ -\cos\Sigma&0&\sin\Sigma\end{array}\right]\left[\begin{array}{l}x_{s}\\ y_{s}\\ z_{s}\end{array}\right]+\left[\begin{array}{l}x_{c}\\ y_{c}\\ z_{c}\end{array}\right]\end{aligned}\] |
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