harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0\rightarrow{\calO}_{\pi^{*}{\calS}}(\sigma|_{\pi^{*}{\calS}})\rightarrow{\calO}_{\pi^{*}{\calS}}(2\sigma|_{\pi^{*}{\calS}})\rightarrow{\calO}_{\pi^{*}{\calS}}(2\sigma|_{\pi^{*}{\calS}})|_{\sigma|_{\pi^{*}{\calS}}}\rightarrow0
\delta_{\perp}\kappa_{1}=\kappa_{3}\kappa_{2}\Psi_{3}+2\kappa_{2}\Psi_{2}{}^{\prime}+\kappa_{2}^{\prime}\Psi_{2}+\Psi_{1}{}^{\prime\prime}-\left(\kappa_{1}^{2}+\kappa_{2}{}^{2}\right)\Psi_{1}\,.
\chi^{\theta}(p^{\mu}):=\left(\begin{array}{c}{\left(\xi\,\Theta_{[1/2]}\right)\,\chi_{_L}^{\ast}(p^{\mu})}\\{\left(\xi\,\Theta_{[1/2]}\right)^{\ast}\,\chi_{_R}^{\ast}(p^{\mu})}\\\end{array}\right)\quad,
f(T)\tilde{\mathcal{G}}_{S}^{\mu\nu}\partial_{\mu}\theta_{R}^{T}\Gamma^{0}\partial_{\nu}\theta_{L}=\frac{1}{2}f(T)\tilde{\mathcal{G}}_{S}^{\mu\nu}\partial_{\mu}\overline{{\theta}}(1-\Gamma_{11})\partial_{\nu}\theta
\psi(z,t)=d_{0}(t)\psi_{0}(z)+d_{1}(t)\psi_{1}+d_{1}^{\ast}(t)\psi_{1}^{\ast}(z)+\underset{-M_{-},\neq0}{\overset{M_{+}}{\sum}}\left(a_{n}(t)\psi_{+}(k_{n}^{+},z)+a_{n}^{\ast}(t)\psi_{+}^{\ast}(k_{n}^{+},z)\right)
I=\intd^{4}x\left[\frac{1}{2}\pi_{i}^{\alpha}\dot{A_{i}}^{\alpha}-\sqrt{1+B_{i}^{\alpha}B_{i}^{\alpha}+\frac{1}{4}(\epsilon^{ijk}B_{j}^{\alpha}{\cal{L}}_{\alpha\beta}B_{k}^{\beta})^{2}}+1\right],
\mathrm{~e}^{ie\int_{0}^{1/T}d\tauA_{0}(\tau,\vec{x})}\rightarrow\mathrm{~e}^{ie^{2}/\kappa}\cdot\mathrm{~e}^{ie\int_{0}^{1/T}d\tauA_{0}(\tau,\vec{x})}
\Psi^{0}=\frac{R_{vac}^{2}}{\sqrt{\sum_{s=1}^{N}|\Pi^{s}|^{2}+R_{vac}^{2}}},...,\quad\Psi^{i}=\Pi^{i}\frac{R_{vac}}{\sqrt{\sum_{s=1}^{N}|\Pi^{s}|^{2}+R_{vac}^{2}}}.
Q=+{\frac{\pi}{2}}+\rho\left[\int_{1}^{n^{2}}{\frac{du^{\prime}}{1-u^{\prime}}}\Im\left\{{\frac{i\sqrt{u^{\prime}-1}}{\sqrt{n^{2}-u^{\prime}}+i\rho\sqrt{u^{\prime}-1}}}\right\}\right].
{\epsilonQ}{\left|\phi_{0}\right>}={\frac{1}{\sqrt{p^{+}}}}\left({\sqrt2}{p^{+}}{\zeta^{a}}+{p^{i}}{\eta^{\dot{a}}}{\gamma_{{\dot{a}}a}^{i}}\right){S_{0}^{a}}{\left|\phi_{0}\right>}.
\delta_{\Gamma}\langleI\rangle_{cl.}=0\,\rightarrow\Gamma^{\lambda}{}_{\mu\nu}=\{_{\mu}{}^{\lambda}{}_{\nu}\}+{\frac{1}{2}}\left(\phi_{\mu}\delta_{\nu}^{\lambda}+\phi_{\nu}\delta_{\mu}^{\lambda}-g_{\mu\nu}g^{\lambda\rho}\phi_{\rho}\right)\
\int{\frac{dl^{-}}{2\pi}}\,e^{\alpha_{1}l^{-}T_{1}}e^{\alpha_{2}(k^{--}l^{-})T_{2}}=e^{\alpha_{2}k^{-}T_{2}}\delta(\alpha_{1}T_{1}-\alpha_{2}T_{2})=e^{k^{-}\tau_{2}}\delta(\tau_{1}-\tau_{2}).
\begin{array}{rcl}{G_{\sigma}^{(n)}(\zeta^{\prime\prime},\zeta_{2n-1},\zeta_{2n})|_{\zeta_{2n}=sx^{-1}\zeta_{2n-1}}}&{=}&{G_{s\sigma}^{(n-1)}(\zeta^{\prime\prime})\otimesu_{s}~~~~(s=\pm),}\\\end{array}
R_{nm}^{\prime\prime}(r)+\frac{1}{r}R_{nm}^{\prime}(r)-\left(\frac{m^{2}}{r^{2}}-\frac{4m}{1+r^{2}}-\frac{4(1-r^{2})}{(1+r^{2})^{2}}\right)R_{nm}(r)=-\varepsilon_{nm}^{2}R_{nm}(r).
H=\sum_{k}\omega_{k}A_{1,1}\left(k\right)+\sum_{n,\left\{m\right\},\left\{m^{\prime}\right\}}\sum_{k_{1},\ldotsk_{n}}^{\prime}V_{\left\{m\right\},\left\{m^{\prime}\right\}}\left(k_{1},\ldotsk_{n}\right)\prod_{i=1}^{n}A_{m_{i},m_{i}^{\prime}}\left(k_{i}\right),
n\:C\;\alpha_{1}^{1-p}\:\beta_{1}^{1-q-\gamma}\left(\prod_{j=2}^{p}\;\:\alpha_{1}^{a_{j}-\alpha_{j}}\:\alpha_{j}^{-a_{j}}\right)\beta_{i_{0}}^{\gamma}\left(\prod_{k=2}^{q}\beta_{1}^{b_{k}-\beta_{k}}\:\beta_{k}^{-b_{k}}\right),
\begin{array}{c}{\displaystyle{E(R=x\,c/\Omega)=}}\\{\displaystyle{=\frac{\hbar\,\Omega}{x}\,\sum_{\nu=1/2}^{\nu_{0}}\,\nu\,\left[\zeta_{\nu}(-1,x)-\zeta_{\nu}(-1,0^{+})\right]=}}\\{\displaystyle{=\hbar\,\Omega\,\sum_{\nu=1/2}^{\nu_{0}}\,F(x,\nu),}}\\\end{array}
V_{ij}^{a}\=\\frac{1}{2\sqrt2}\,\Sigma^{2}\,f_{xij}^{a}\,K^{x}~,\qquadT_{ij}^{a}\=\\frac12\,\Sigma^{-1}\,L_{i}^{I}{}^{k}\,f_{xkj}^{a}\,K_{I}^{x},
\begin{array}{l}{2n^{\prime}-2(n_{2}+\cdots+n_{j})+\beta_{i-1j-1}^{(k-1)}\geqa_{1}^{(j)}\geq\cdots\geqa_{n_{j}-n_{j+1}}^{(j)}\geq0,~~~~2\leqj\leqk-2,}\\{n^{\prime}-(n_{2}+\cdots+n_{k-1})\geqb_{1}\geq\cdots\geqb_{n_{k-1}}\geq0.}\\\end{array}
{\frac{\pi^{2}}{8a}}\int_{0}^{\lambda}{du{\frac{1}{1-u^{2}}}\left[{\frac{(A^{3}-3\delta^{2}A)}{(A^{2}+\delta^{2})^{3}}}+{\frac{1+u^{2}}{1-u^{2}}}{\frac{A}{(A^{2}+\delta^{2})}}\right]}.
\psi_{i}=\psi_{i}^{T}+\left(\frac{1}{2}\gamma_{i}-\frac{1}{2}{\hat{k}_{i}}({\bf{\hat{k}}\cdot\gamma})\right)\gamma^{j}\psi_{j}+\left(\frac{3}{2}{\hat{k}_{i}}-\frac{1}{2}\gamma_{i}({\bf{\hat{k}}\cdot\gamma})\right){\bf{\hat{k}}\cdot\psi}\,
{\frac{135}{256}}v_{4}^{4}(\vec{v}_{3}\cdot\vec{\nabla}_{4})^{2}{\frac{1}{\vert\vec{x}_{4}\vert^{7}}}(\vec{v}_{12}\cdot\vec{\nabla}_{3})^{2}{\frac{1}{\vert\vec{x}_{3}\vert^{5}}}{\frac{1}{\vert\vec{x}_{12}\vert^{5}}}.
\begin{array}{r|r|r|r}{n}&{m}&{N}&{\bar{N}}\\\hline{1}&{1}&{1}&{0}\\{1}&{-1}&{0}&{1}\\{-1}&{1}&{0}&{1}\\{-1}&{-1}&{1}&{0}\\{2}&{0}&{0}&{0}\\{-2}&{0}&{0}&{0}\\{0}&{2}&{0}&{0}\\{0}&{-2}&{0}&{0}\\\end{array}
S[\vec{r}]\=\{\frac{Z}{2}}\,\intd^{D}\!x\(\nabla\vec{r})^{2}\+\b_{R}\,\mu^{\epsilon}\Z_{b}\\int\intd^{D}\!x\,d^{D}\!y\\delta\big(\vec{r}(x)-\vec{r}(y)\big)\.
{\calL}_{E_{AB}}w_{{\calM}_{0}}^{(a)}=\frac{g^{2}\lambda}{8\pi}d\langleE_{AB},w_{\mathrm{flat}}^{(a)}\rangle=\frac{g^{2}\lambda}{8\pi}{\calL}_{E_{AB}}w_{\mathrm{flat}}^{(a)}=0.
{\frac{1}{\sqrt{2\pi\gamma_{1}}}}\tilde{b}_{1}^{y=\pm1}=\pmb_{-1}^{y=\pm1}\equiv{\frac{b}{\sqrt{2}}},\qquad\pmb_{1}^{y=\pm1}(E)=b_{-1}^{y=\pm1}(E)\equiv{\frac{b(E)}{\sqrt{2}}}.
\hat{H}_{I}^{(q)}(\hat{x},\hat{p})=\frac{f^{2}}{6}\sum_{k=1}^{\infty}\frac{V^{(k)}(0)}{k!}\hat{x}^{k}\biggl(k\hat{x}^{2}\partial_{\hat{x}}^{2}+k(k+2)\hat{x}\partial_{\hat{x}}+\frac{1}{6}k(k-1)(2k+5)\biggr).
D_{a}=\partial_{a}+{\frac{1}{4}}\partial_{a}x^{m}\big[\(\omega_{\mu\num}-{\frac{1}{2}}H_{\mu\num}\rho_{3})\Gamma^{\mu\nu}+({\frac{1}{3!}}F_{\mu\nu\lambda}\Gamma^{\mu\nu\lambda}\rho_{1}+{\frac{1}{2\cdot5!}}F_{\mu\nu\lambda\rho\kappa}\Gamma^{\mu\nu\lambda\rho\kappa}\rho_{0})\Gamma_{m}\\big]
{\calL}=\frac{\kappa}{2}\epsilon^{\mu\nu\rho}\sum_{n=0}^{\infty}\left(\sum_{m=0}^{n}A_{\mu}^{(n)}\partial_{\nu}A_{\rho}^{(m)}+\sum_{m=n+1}^{\infty}A_{\mu}^{(m)}\partial_{\nu}A_{\rho}^{(n)}\right)+(\tilde{D}^{\mu}\phi)^{*}(\tilde{D}_{\mu}\phi),
\alpha_{1}(\sigma,\bar{\sigma})=\frac{\partial_{\bar{z}}\varphi_{0}}{{\calH}_{0}}=\frac{\partial_{\bar{z}}\varphi_{0}}{\partial_{z}\varphi_{0}\partial_{\bar{z}}\varphi_{0}+V(\varphi_{0})}\;,\quad\bar{\alpha}_{1}(\sigma,\bar{\sigma})=\frac{\partial_{z}\varphi_{0}}{{\calH}_{0}}=\frac{\partial_{z}\varphi_{0}}{\partial_{z}\varphi_{0}\partial_{\bar{z}}\varphi_{0}+V(\varphi_{0})}\,,
L=\frac{1}{2}|\bigtriangledown_{m}^{2}\phi_{0}\,|^{2}+\frac{1}{2}|\bigtriangledown_{(d-m)}\phi_{0}\,|^{2}+\delta_{0}\frac{1}{2}|\bigtriangledown_{m}\phi_{0}\,|^{2}+\frac{1}{2}t_{0}\phi_{0}^{2}+\frac{1}{4!}\lambda_{0}\phi_{0}^{4}.
S_{D}=-\intd^{4}\sigma\sqrt{-\mathrm{det}\,\left(G_{ij}+\frac{1}{\sqrt{1+e^{2\phi}C_{0}^{2}}}(e^{\frac{\phi}{2}}\tilde{F}_{ij}+C_{ij}+e^{\phi}C_{0}b_{ij}\right)}+\int\Omega_{D},
X^{0}=\frac{1}{2}\,,\quadX^{1}=\frac{1}{2}(ut-n^{a}n^{a})\,,\quadX^{2}=-\frac{1}{2}u\,,\quadX^{3}=\frac{1}{2}t\,,\quadX^{a}=\frac{1}{\sqrt2}\,n^{a}\,
SO(4)\niG_{4}=\left(\begin{array}{cc}{e^{i\Theta_{\mu\nu}\sigma_{\mu\nu}}}&{0}\\{0}&{e^{i\bar{\Theta}_{\mu\nu}\bar{\sigma}_{\mu\nu}}}\\\end{array}\right)\equiv\left(\begin{array}{cc}{g}&{0}\\{0}&{\bar{g}}\\\end{array}\right)\;,
Z\sim\sum_{\alpha,\tilde{\beta}}{\mathrm{Tr}}\,(-)^{\tilde{\beta}}C\Big[\begin{array}{c}{\alpha}\\{\tilde{\beta}}\\\end{array}\Big]\Big(1+\delta_{\alpha}(-)^{b_{i}}C\Big[\begin{array}{c}{\alpha}\\{b_{i}}\\\end{array}\Big]\Big)R^{\alpha}NS^{F+\alpha}.
{\tilde{\delta}}\biggl[{\dot{\varphi}}^{a}-\omega^{ab}{\frac{\partialH}{\partial\varphi^{b}}}\biggr]=\int~{\calD}\lambda_{a}~exp~i\int\lambda_{a}\biggl[{\dot{\varphi}}^{a}-\omega^{ab}{\frac{\partialH}{\partial\varphi^{b}}}\biggr]dt
W_{np}\sime^{\frac{24\pi^{2}}{b_{a}}S}\prod_{i}e^{\frac{3\sigma_{1}(T_{i},U_{i})}{2b_{a}}\frac{|G_{i}|}{|G|}}\prod_{i}\Bigl(\eta(T_{i})\bigr)^{\frac{-6(b_{a}^{N=2})_{i}}{b_{a}}\frac{|G_{i}|}{|G|}}
W_{L}=\sum_{p=1}^{k_{d}}\lambda_{p}\int{\frac{1}{2^{2p}p!}}\delta_{[\mu_{1}\mu_{2}...\mu_{2p-1}\mu_{2p}]}^{[\nu_{1}\nu_{2}...\nu_{2p-1}\nu_{2p}]}R_{\\\nu_{1}\nu_{2}}^{\mu_{1}\mu_{2}}...R_{\\\nu_{2p-1}\nu_{2p}}^{\mu_{2p-1}\mu_{2p}}\equiv\sum_{p=1}^{k_{d}}\lambda_{p}W_{p}
S\simT_{p}\int\left(-1-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{8}\left(F_{\mu\nu}F^{\nu\lambda}F_{\lambda\sigma}F^{\sigma\mu}-\frac{1}{4}(F_{\mu\nu}F^{\mu\nu})^{2}\right)+\cdots\right)
V_{\mu\nu}^{i}=\partial_{\mu}V_{\nu}^{i}-\partial_{\nu}V_{\mu}^{i}\,\,\,,\;\;\;F_{\mu\nu}^{i}=V_{\mu}^{k}\partial_{k}V_{\nu}^{i}-V_{\nu}^{k}\partial_{k}V_{\mu}^{i}\,\,.
{\mathcalG}_{l+\nu}(r^{\prime\prime},r^{\prime};E)=\int_{0}^{\infty}dz\,\frac{1}{\left[\eta_{l+\nu}(z;\kappa)\right]^{-1}-\lambda}\;\Psi_{z,l+\nu}(r^{\prime\prime};\kappa)\Psi_{z,l+\nu}(r^{\prime};\kappa)\;,
\langle0_{R}|T_{00}|0_{R}\rangle=\langleE_{vac}\rangle=\sum_{n1,n2=1}^{\infty}\int{\frac{d^{3}k}{(2\pi)^{3}}}\sqrt{{{\vec{k}}^{2}+{\frac{4\pi^{2}(n_{1}^{2}+n_{2}^{2})}{R^{2}}}}}
G(f)_{\beta}^{b(n)}=\Sigma_{m=0}^{n}\{\tilde{\theta}_{\beta}^{b(n-m)},\tilde{f}^{(m)}\}_{(q,p)}+\Sigma_{m=0}^{(n-2)}\{\tilde{\theta}_{\beta}^{b(n-m)},\tilde{f}^{(m+2)}\}_{(\phi)}+\{\tilde{\theta}_{\beta}^{b(n+1)},\tilde{f}^{(1)}\}_{(\phi)}
\sqrt{|\hat{\hat{g}}|}{\textstyle\frac{-1}{2\cdot4!}}\hat{\hat{G}}{}^{2}=\sqrt{|g_{E}|}\left\{{\textstyle\frac{-1}{2\cdot4!}}K^{6/7}F_{(4)}^{2}+{\textstyle\frac{1}{2\cdot3!}}K^{-3/7}\vec{F}_{(3)}^{\T}{\calM}^{-1}\vec{F}_{(3)}-{\textstyle\frac{1}{4}}K^{-12/7}F_{(2)}^{2}\right\}\,.
\ddot{\chi}+\dot{\alpha}_{0}\left(1-\frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right)\dot{\chi}-\frac{1}{3}\left(1+\frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right)e^{-2\alpha_{0}}\nabla^{2}\chi=0.
\left(Q_{2}^{\dagger}\right)^{31}=-\left(Q_{2}^{\dagger}\right)^{42}=-{\frac{1}{\sqrt{m}}}\left[-{\frac{\partial}{\partialr}}+{\frac{1}{r}}\left(i{\frac{\partial}{\partial\theta}}-1\right)-{\frac{e^{2}}{c\kappa}}{\frac{\partialG}{\partialr}}\right]e^{-i\theta}
a_{A}(k)\,a_{A}^{\dagger}(k^{\prime})+q^{-2}\sum_{l,l^{\prime}\in{\calI}}{\hat{\calR}}\left(k^{\prime},l;k,l^{\prime}\right)a_{A}^{\dagger}(l^{\prime})\,a_{A}(l)=\delta_{k\;k^{\prime}}\;,
\Bigl[\vartheta_{1}^{\prime}(\sigma_{1}),\vartheta_{1}^{\prime}(\sigma_{2})\Bigr]=\Bigl[\vartheta_{2}^{\prime}(\sigma_{1}),\vartheta_{2}^{\prime}(\sigma_{2})\Bigr]=2\pii\,\delta^{\prime}(\sigma_{1}-\sigma_{2})
\begin{aligned}{\overline{{u}}(\vec{k},\sigma)\gamma^{0}u(\vec{k},\tau)=u^{\dagger}(\vec{k},\sigma)u(\vec{k},\tau)}&{{}=2E\,\delta_{\sigma\tau}}\\{u^{\dagger}(\vec{k},\sigma)v(-\vec{k},\tau)}&{{}=0,}\\\end{aligned}
\dot{\varrho}_{m}+3{\frac{\dot{a}_{0}}{a_{0}}}\left(\varrho_{m}+{\frac{\wp_{m}}{c^{2}}}\right)=-\dot{\varrho}_{\delta\sigma}-{\frac{\dot{G}_{4}}{G_{4}}}\varrho+{\frac{3kc\dot{c}}{4\piG_{4}a_{0}^{2}}}-{\frac{3Cc\dot{c}}{4\piG_{4}a_{0}^{4}}}.
{\calS}(e_{\mu}^{i},\omega_{\mu})=\int{d^{2}}x\;e\bigg(\frac{1}{16\alpha}R_{\mu\nu}{}^{ij}R^{\mu\nu}{}_{ij}-\frac{1}{8\beta}T_{\mu\nu}{}^{i}T^{\mu\nu}{}_{i}-\gamma\bigg),
\varepsilon_{\beta}^{*}(-\vec{p},s)\Bigl[\Gamma_{d}^{i\alpha\beta}(\vec{p},-\vec{p})+\Gamma_{e}^{i\alpha\beta}(\vec{p},-\vec{p})\Bigr]\varepsilon_{\alpha}(\vec{p},s)=-\frac{2e^{3}}{8\pi\theta}\epsilon^{ij}q_{j}.
{\calL}=\overline{{D_{\mu}\phi^{i}}}D^{\mu}\phi^{i}+\phi^{i\dagger}\phi^{i}+{\frac{\lambda}{4}}(\phi^{i\dagger}\phi^{i})^{2}+{\frac{\eta}{3}}(\phi^{i\dagger}\phi^{i})^{3}+i\theta\epsilon_{\mu\rho\nu}A_{\mu}\partial_{\nu},
ds^{2}=g_{\alpha\beta}^{(2)}(t,r)dx^{\alpha}dx^{\beta}+r^{2}d\Omega^{2}+g_{ab}^{(p-1)}(x^{c})dx^{a}dx^{b}+g_{kl}^{(7-p)}(x^{m})dx^{k}dx^{l}\;,
\langle\Phi_{H.W.}^{(\lambda_{1},\mu_{1}}(z_{1},\bar{z_{1}})\Phi_{H.W.}^{(\lambda_{2},\mu_{2})}(z_{2},\bar{z_{2}})\Phi_{T_{z}^{3},Y^{3}}^{(\lambda_{3},\mu_{3})}(z_{3},\bar{z_{3}})\tilde{\Phi}_{L.W.}^{(\lambda_{4},\mu_{4})}(z_{4},\bar{z_{4}})\rangle.
p_{0\mu}=p_{\mu}-\frac{1}{2}\tilde{S}^{h_{1}h_{2}}\;\omega_{h_{1}h_{2}\mu}-\frac{1}{2}\tilde{S}^{k_{1}k_{2}}\;\omega_{k_{1}k_{2}\mu}-\frac{1}{2}\tilde{S}^{h_{1}k_{1}}\;\omega_{h_{1}k_{1}\mu},
=\frac{1}{2}\frac{\partial^{2}\phi}{\partials^{2}}+\phi-\phi^{3}+\frac{1}{2N}\frac{\partialN}{\partials}\frac{\partial\phi}{\partials}+2l_{0}^{2}\frac{1}{\sqrt{g}N}\frac{\partial}{\partial\sigma^{j}}\left(G^{jk}\sqrt{g}N\frac{\partial\phi}{\partial\sigma^{k}}\right),
\{~\varphi(x)~,~\varphi(y)~\}_{x^{+}=y^{+}}^{\mathrm{DB}}~=~\frac{\mu^{2}}{N}~\frac{1}{2}\left\{-\frac{1}{2}~\epsilon(x^{-}-y^{-})+\frac{(x^{-}-y^{-})}{2L}\right\}~,
[{\calJ}^{\mu\nu},{\calJ}^{\rho\sigma}]=\eta^{\mu\rho}{\calJ}^{\nu\sigma}-\eta^{\nu\sigma}{\calJ}^{\mu\rho}+\eta^{\nu\rho}{\calJ}^{\mu\sigma}-\eta^{\mu\rho}{\calJ}^{\nu\sigma}\,,
\ddot{\phi}({\bfk}_{n},t)+\omega_{n}^{2}\phi({\bfk}_{n},t)+\sum_{n^{\prime}}\Gamma_{nn^{\prime}}^{D}{\dot{\phi}}({\bfk}_{n^{\prime}},t)+\omega_{n}\sum_{n^{\prime}}\Gamma_{nn^{\prime}}^{C}\phi({\bfk}_{n^{\prime}},t)+\frac{\deltaV}{\delta\phi(-{\bfk}_{n},t)}=\eta({\bfk}_{n},t),
\begin{array}{l}{(\mathrm{{Bose}\oplus{Fermi\oscillators})_{L}^{\left(l_{L}\right)}}}\\{\times(\mathrm{{Bose}\oplus{Fermi\oscillators})_{R}^{\left(l_{R}\right)}}}\\{\times\,\,|vac,\,\,p^{\mu};\vec{m},\vec{n}>}\\\end{array}
{\left(\begin{matrix}{1}&{0}&{2g}\\{g}&{1}&{g^{2}}\\{0}&{0}&{1}\\\end{matrix}\right)}{\left(\begin{matrix}{1}&{0}&{2h}\\{h}&{1}&{h^{2}}\\{0}&{0}&{1}\\\end{matrix}\right)}=\left(\begin{matrix}{1}&{0}&{{2(g+h)}}\\{{g+h}}&{1}&{{(g+h)^{2}}}\\{0}&{0}&{1}\\\end{matrix}\right)
P_{\lambda}(\vec{q},\vec{k}):={\frac{|\Sigma_{\lambda}|^{2}}{p_{\perp}^{4}-2p_{\perp}^{2}(q_{\perp}^{2}+k_{\perp}^{2})+(q_{\perp}^{2}-k_{\perp}^{2})^{2}}}\,.
\frac{\partial}{\partialz}\bigg(\frac{R^{\prime}R^{2}}{\sqrt{R^{2}(1+{R^{\prime}}^{2})+B^{2}}}\bigg)-\frac{R(1+{R^{\prime}}^{2})}{\sqrt{R^{2}(1+{R^{\prime}}^{2}+B^{2})}}+hR=0.
\left(\begin{array}{l}{u_{\nu}\left(\eta\right)}\\{v_{\nu}\left(\eta\right)}\\\end{array}\right)\rightarrow\left(\begin{array}{l}{1}\\{0}\\\end{array}\right)e^{i\nu\eta}+S_{F}\left(\nu\right)\left(\begin{array}{l}{0}\\{1}\\\end{array}\right)e^{-i\nu\eta}\\as\\eta\rightarrow-\infty\,
\Big((\mathbf{8}_{s},\mathbf{4})\otimes(\mathbf{8}_{s},\mathbf{4})\Big)_{\mathrm{symm}}=(\mathbf{28},\mathbf{1})\oplus(\mathbf{1},\mathbf{10})\oplus(\mathbf{28},\mathbf{5})\oplus(\mathbf{35}^{-},\mathbf{10})\,.
\begin{array}{l}{Cl(\eta_{1})>0\;\;\LongrightarrowCl(\eta_{1}\oplus\eta_{2})=Cl(\eta_{1})\otimesCl(\eta_{2})}\\{Cl(\eta_{1})<0\;\;\LongrightarrowCl(\eta_{1}\oplus\eta_{2})=Cl(\eta_{1})\otimesCl(-\eta_{2})\,.}\\\end{array}
\hat{H}_{T}=\frac{1}{2m}[{\hat{p}_{u}}^{2}+\frac{1}{u^{2}}{\hat{p}_{\varphi}}^{2}+{\hat{p}_{v}}^{2}+\frac{1}{v^{2}}{\hat{p}_{\varphi}}^{2}]-e^{2}+\frac{m\omega^{2}}{2}(u^{2}+v^{2})
=\;e^{4\pi\sum_{b=1}^{N}2n_{b}\lambda^{(b)}}\left\langle\prod_{b=1}^{N}\left[\prod_{i_{b}}^{q_{b}}:e^{i2\sqrt{\pi}\varphi^{(b)}(x_{i_{b}}^{(b)})}:_{C^{2}}\prod_{j_{b}}^{2n_{b}-q_{b}}:e^{-i2\sqrt{\pi}\varphi^{(b)}(y_{j_{b}}^{(b)})}:_{C^{2}}\right]\right\rangle_{C^{2}}\;.
S_{dual}=\intd^{6}x\left[-{\frac16}G_{{\mu}{\nu}{\sigma}}G^{{\mu}{\nu}{\sigma}}+{\frac12}{\lambda}_{{\mu}{\nu}}{\calG}^{{\mu}{\rho}{\sigma}}{{\calG}^{\nu}}_{{\rho}{\sigma}}+A_{{\nu}{\sigma}}{\partial}_{\mu}G^{{\mu}{\nu}{\sigma}}\right].
\deltav_{\underline{{\mu}}}^{~A}={\frac{1}{2}}v_{\underline{{\mu}}}^{~B}\Gamma_{BA}^{i}~i_{\delta}f^{i}~-{\frac{1}{4}}v_{\underline{{\mu}}}^{~B}\Gamma_{BA}^{ij}~i_{\delta}A^{ij}~,\qquad
\frac{\mu^{2}}{2^{7}(\pi)^{\frac52}(s-1)}\frac{\Gamma(\frac32-s)}{\Gamma(2-s)}\intd^{4}k\\tilde{S}(k)\tilde{S}(-k)\left(\frac{k^{2}}{\mu^{2}}\right)^{-s+1}\,.
B_{a}({\vec{x}},t)^{av}=\frac{\mu_{0}}{2\pi}\intd^{3}x^{\prime}\frac{\epsilon_{abc}\epsilon_{bde}\{\nabla_{d}M_{e}({\vec{x}}^{\prime})^{av}\}(x_{c}-x_{c}^{\prime})}{\mid{\vec{x}}-{\vec{x}}^{\prime}\mid^{3}}\,
{\calL}=\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-\frac{m^{2}}{4}(B_{\mu\nu}+\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(B^{\mu\nu}+\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu})
f_{TTT}=4\pi^{2}\frac{U-{\bar{U}}}{(T-{\bar{T}})^{2}}\intd^{2}\tau{\bar{F}}({\bar{\tau}})\sum_{P_{L},P_{R}}P_{L}{\bar{P}}_{R}^{3}q^{P_{L}^{2}/2}{\bar{q}}^{P_{R}^{2}/2}.
{\frac{\ddot{\bar{a}}_{R}}{\bar{a}_{R}}}=-{\frac{4\pi\bar{G}_{4}}{3}}\left[\varrho+{\frac{3}{\bar{c}_{4}^{2}}}\wp-{\frac{2V_{r}}{c^{2}\sqrt{1+R_{\tau}^{2}/c^{2}}}}\right],
\int_{0}^{2\lambda-\sum_{\beta=1}^{\alpha-1}p^{\beta}}dp^{\alpha}\longrightarrow\int_{-(\lambda-\sum_{\beta=1}^{\alpha-1}2^{\beta-1}{p^{\beta}}^{\prime})/2}^{(\lambda-\sum_{\beta=1}^{\alpha-1}2^{\beta-1}{p^{\beta}}^{\prime})/2}d{p^{\alpha}}^{\prime}\,
\{Q_{\alpha},Q_{\beta}\}=\left(\Gamma^{\mu}{\calC}^{-1}\right)_{\alpha\beta}P_{\mu}+\left(\Gamma^{\mu\nu}{\calC}^{-1}\right)_{\alpha\beta}Z_{\mu\nu}^{2}+\left(\Gamma^{\mu\nu\rho\sigma\tau}{\calC}^{-1}\right)_{\alpha\beta}Z_{\mu\nu\rho\sigma\tau}^{5}\,.
\begin{array}{rl}{Z^{\prime}(s)}&{:=U(a(s))Z(s)}\\{Z^{\prime*}(s)}&{:=Z(s)^{*}U^{-1}(a(s))}\\{A_{j}^{\prime}(s)}&{:=\frac{1}{r(s)}U(a(s))A_{j}(s)U^{-1}(a(s))+U(a(s))\frac{dU^{-1}}{ds}(a(s)),}\\{A_{j}^{\prime*}(s)}&{:=\frac{1}{r(s)^{*}}U(a(s))A_{j}^{*}(s)U^{-1}(a(s))+U(a(s))\frac{dU^{-1}}{ds}(a(s)),j=1,2}\\\end{array}
\left[\frac{\partial^{2}}{\partialp_{0}^{2}}-\frac{\partial^{2}}{\partial\vec{p}^{2}}\right]\Psi(p)=\frac{4\lambda_{F}\Psi(p)}{(p_{0}^{2}-\vec{p}^{2}+W^{2}-m^{2})^{2}-4p_{0}^{2}W^{2})}
\begin{array}{l}{LG_{2(1,Z)}=Z_{R}^{1/3}Z_{L}^{2/3}LG_{2}Z_{L}^{-2/3}Z_{R}^{-1/3},}\\{G_{2(1,Z)}=Z_{R}^{1/3}Z_{L}^{2/3}G_{2}Z_{L}^{-2/3}Z_{R}^{-1/3}.}\\\end{array}
\left\{A,B\right\}=\left(\frac{\partialA}{\partialq^{i}}\frac{\partialB}{\partialp_{i}}+\frac{\partialA}{\partialp_{i}}\frac{\partialB}{\partialq^{i}}\right)+(-1)^{\sigma(A)}\left(\frac{\partialA}{\partial\theta^{i}}\frac{\partialB}{\partial\pi_{i}}+\frac{\partialA}{\partial\pi^{i}}\frac{\partialB}{\partial\theta_{i}}\right)\.
4\pi\rho^{4}\Theta_{4\;\mu\nu}=(1+\chi){\Big\lbrace}f_{\mu}^{\prime}f_{\nu}^{\prime}-(f_{\mu}^{\prime},V_{\nu}^{\prime})-V_{\mu}^{\prime}V_{\nu}^{\prime}{\big\lbrace}\chi+(1+\chi)f^{2}{\big\rbrace}-\frac{g_{\mu\nu}}{2}(1+f^{2}){\Big\rbrace},
(x_{13}^{2})^{-2r}(x_{24}^{2})^{-2}(\frac{v}{u})^{2}\sum_{D\in\{{\bf{84}},{\bf{105}},{\bf{175}}\}}\sum_{\substack{a,b;\,s\inS_{r}\,(s(a)>s(b))}}\frac{\sigma_{D}}{N^{2}}P_{D}^{I_{a}I_{r+1},J_{s(a)}J_{s(b)}}\delta^{I_{b}J_{r+1}}\prod_{\substack{i=1\,i\notin\{a,b\}}}^{r}\delta^{I_{i}J_{s(i)}}
{\calL}=\frac{1}{8\gamma}\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}+\frac{k}{16\pi}\mathrm{Tr}(A\,dA+\frac{2}{3}A^{3})+\frac{1}{4}\mathrm{Tr}(\partial\phi+[A,\phi])^{2}-V(\phi).
|N_{n,n^{\prime}}(\{\sigma\},\{\sigma^{\prime}\})\rangle\star|N_{m,m^{\prime}}(\{\tau\},\{\tau^{\prime}\})\rangle=\delta_{n^{\prime},m}\sum_{P}\prod_{i}\delta(\sigma_{i}^{\prime}-\tau_{P_{i}})|N_{n,n^{\prime}}(\{\sigma\},\{\tau^{\prime}\})\rangle,
{\vec{\rho}}_{a}(\alpha)={\vec{\rho}}_{a}-\sum_{i,j=1}^{N}\sum_{b=1}^{N-1}{\vec{n}}_{+}\cdot{\vec{\rho}}_{b}{\frac{H_{i}H_{j}}{H_{M}}}\gamma_{aj}(\gamma_{bi}-\gamma_{bj}){\vec{J}}_{j}^{(2)}(\alpha).
\!\!\!\!\!\!\!\!\nabla{\calC}_{3}=\{H_{T},{\calC}_{3}\}\approx\{{\calC}_{5},{\calC}_{3}\}=-p\cdot\nabla^{2}X+2\sqrt{-\gamma}\;V_{*}\;\nabla^{2}X\cdot\nabla^{2}X\;\Pi\cdotP
S_{0}=-i\frac{k}{4\pi}\int_{D\timesR}dxdtd\tau\partial_{\tau}[\partial_{x}\phi*(\partial_{x}+\partial_{t})\phi]+\frac{k}{12\pi}\int_{D\timesR}dxdtd\tau\varepsilon^{\mu\nu\rho}\partial_{\mu}\phi*\partial_{\nu}\phi*\partial_{\rho}\phi.
\langleJ_{mn}^{1}\rangle_{0}=\frac{\hbar}{2}\:(C_{n}\:\delta_{m\,n+1}+C_{-n}\:\delta_{m\,n-1})\;,~~~~\langleJ_{mn}^{2}\rangle_{0}=\frac{-i\hbar}{2}\:(C_{n}\:\delta_{m\,n+1}-C_{-n}\:\delta_{m\,n-1})\;.
x^{+}={\frac{\sqrt{3}}{4\mu}}(\tau-{\frac{2}{3}}\,\psi),\;x^{-}=\mu{\frac{L^{2}}{2}}({\frac{3}{2}}\,\tau+\psi),\;\hat{\phi_{1}}=\phi_{1}+{\frac{1}{3}}\,\psi,\;\theta^{2}={\frac{2\,y^{2}}{\sqrt{3}L^{2}}},\;\rho^{2}={\frac{4\,r^{2}}{3\sqrt{3}L^{2}}}
I_{g}=\frac{1}{2\pi\kappa}\intd^{2}x\epsilon^{\mu\nu}\Bigl(\eta_{a}(\partial_{\mu}e_{\nu}^{a}+\omega_{\mu}\epsilon_{~b}^{a}e_{\nu}^{b})+\eta_{2}\partial_{\mu}\omega_{\nu}+\eta_{3}(\partial_{\mu}a_{\nu}+\frac{1}{2}\epsilon_{ab}e_{\mu}^{a}e_{\nu}^{b})\Bigr)\,.
S=\sum_{C}\sum_{x(\niC)}\bigg|\left(\frac{\delta}{\delta\sigma^{t\mu}(x)}-iA_{t\mu}(x)\right)\Psi[C]\bigg|^{2}+\sum_{x}-\frac{\varepsilon}{2\cdot3!}F_{\mu\nu\lambda}F^{\mu\nu\lambda}-V(|\Psi[C]|^{2}),
R_{\mu\nu}-\nabla_{\mu}\Phi\nabla_{\nu}\Phi-{\frac{1}{4}}e^{4\Phi}H_{\mu\rho\sigma}H_{\nu}^{~~\rho\sigma}+{\frac{1}{6}}g_{\mu\nu}e^{4\Phi}H^{2}+{\frac{4}{k}}g_{\mu\nu}e^{-2\Phi}=0,
{\calW}(q,p)=-\frac{1}{(4\pi)^{6}}\frac{\Gamma\left(2\epsilon\right)\Gamma^{5}\left(\frac{1}{2}-\frac{\epsilon}{2}\right)}{\Gamma\left(1-\epsilon\right)\Gamma\left(\frac{3}{2}-\frac{3\epsilon}{2}\right)}\left(-\frac{4\pi\mu^{2}}{p^{2}+i\eta}\right)^{2\epsilon}I_{\epsilon}(q,p)\;,
ds^{2}=\sum_{i,k=0}^{D-s-3}e^{2F_{i}(r)}\eta_{ik}f^{\delta_{ii_{0}}}(r)dX^{i}dX^{k}+e^{2B(r)}\left(r^{2}d\Omega_{s+1}^{2}+f^{-1}(r)dr^{2}\right),
\dot{H}=\frac{{\dot{\varphi}}^{2}}{4F}-\frac{1}{2}\left(H^{2}+\frac{{\rho}_{\varphi}}{6F}\right)-\frac{\ddot{F}}{2F}+\frac{p_{m}}{4F}+\frac{\rho_{m}}{6F}\,,
{\frac{p}{2N}}\\mathrm{BPS}:\\W^{[0,\ldots,0,a_{p},\ldots,a_{N}]}(\theta^{1},\theta^{2},\ldots,\theta^{2N-p})=(W^{1\ldotsp})^{a_{p}}\ldots(W^{1\ldotsN})^{a_{N}}
c_{ab}=\left(\begin{array}{ccccc}{0}&{-\omega_{1}}&{}&{}&{}\\{\omega_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{-\omega_{\frac{p+1}{2}}}\\{}&{}&{}&{\omega_{\frac{p+1}{2}}}&{0}\\\end{array}\right).
I_{\beta}^{(\pm)}(\psi;\,k)=\pm\frac{1}{2\pi}\int_{0}^{a}d\eta\,\frac{\psi(i\eta)}{-k\pmi(\eta-\beta)}\pm\frac{1}{2\pii}\int_{0}^{\infty}d\xi\,\frac{\psi(\xi\pmia)}{\xi-k\pmi(a-\beta)}+\psi(k\pmi\beta).
\frac{1}{r^{2}}\frac{d}{dr}\frac{r^{2}}{(1+{y_{c}^{\prime}}^{2})^{3/2}}\frac{d}{dr}\psi=\frac{1}{r^{2}}\frac{d}{dr}\left\{\stackrel{-}{(+)}\frac{(r^{4}-r_{0}^{4})^{3/2}}{r^{4}}\right\}\frac{d}{dr}\psi=\omega\psi
\left\vertE,p_{1}^{\mathrm{min}},\ldots,p_{N-1}^{\mathrm{min}}\right>=\left\vert\begin{array}{c}{E,\left[0\right]}\\{\left[q\right]}\\\end{array}\right>=\left\vert\begin{array}{c}{E,0,\ldots,0}\\{q_{1},\ldots,q_{k},\ldots,q_{N}}\\\end{array}\right>.
\deltau=\sum_{i=1}^{3}dx^{i}\vee\frac{\partialu}{\partialx^{i}}-dt\vee\frac{\partialu}{\partialt}=du+\sum_{i=1}^{3}e_{i}\frac{\partialu}{\partialx^{i}}-e_{i}\frac{\partialu}{\partialt}.
\frac{1}{z}F(z)+z^{l+1}F(\frac{1}{z})+\frac{\psi(1)(-1)^{l}}{2(l+1)}\frac{1-z^{l+1}}{1-z}=\frac{1}{\mu}(-1)^{l}(1-z)^{l}F^{\prime}(\frac{1}{1-z}).
\gamma^{0}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,,\quad\gamma^{i}=\left(\begin{array}{cc}{0}&{\sigma_{i}}\\{-\sigma_{i}}&{0}\\\end{array}\right)\,,\quad\gamma^{5}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,
G_{\\nu}^{\mu}(f)={\frac{\kappa^{2}}{l\Phi}}T_{\quad\\nu}^{B\mu}+{\frac{\kappa^{2}(1+\Phi)}{l\Phi}}T_{\quad\\nu}^{A\mu}+{\frac{1}{\Phi}}\left(\Phi_{\;\nu}^{;\mu}-\delta_{\nu}^{\mu}\Phi_{\;\alpha}^{;\alpha}\right)+{\frac{\omega(\Phi)}{\Phi^{2}}}\left(\Phi^{;\mu}\Phi_{;\nu}-{\frac{1}{2}}\delta_{\nu}^{\mu}\Phi^{;\alpha}\Phi_{;\alpha}\right)\,
{\bfZ_{\alpha}}=\left(\begin{matrix}{\partial_{\alpha}V-C_{\alpha}}\\{\partial_{\alpha}U+B_{\alpha}}\\\end{matrix}\right)\;,\;\;\;{\calM}=\frac{1}{\lambda_{2}}\left(\begin{matrix}{1}&{\lambda_{1}}\\{\lambda_{1}}&{|\lambda|^{2}}\\\end{matrix}\right)\;,
u=\frac{\bar{y}_{1}y_{1}^{\prime}+\bar{y}_{2}y_{2}^{\prime}}{y_{1}y_{2}^{\prime}-y_{1}^{\prime}y_{2}},\quad\bar{u}=\frac{y_{1}\bar{y}_{1}^{\prime}+y_{2}\bar{y}_{2}^{\prime}}{\bar{y}_{1}\bar{y}_{2}^{\prime}-\bar{y}_{1}^{\prime}\bar{y}_{2}},
\hat{\mathrm{H}}_{\mathrm{phys}}=\frac{1}{2\mathrm{L}}\hat{\pi}_{b}^{2}-\frac{1}{2}(\xi_{+}+\xi_{-})+\sum_{p>0}\frac{1}{p}\mathrm{E}_{p}(\overline{{\rho}}_{\mathrm{tot,+}}(-p)\overline{{\rho}}_{\mathrm{tot,+}}(p)+\overline{{\rho}}_{\mathrm{tot,-}}(p)\overline{{\rho}}_{\mathrm{tot,-}}(-p)).
{\frac{\partial\psi^{\beta}}{\partial_{\mu}\phi_{1}}}\Rightarrow\sqrt2{\frac{\partial\psi^{\beta}}{\partial_{\mu}\omega_{1}}}=\sqrt2{\frac{\partial\psi^{\beta}}{\partial\omega_{1}}}{\frac{\partial\omega_{1}}{\partialM_{\mu}}}=\sqrt2{\frac{\partial\psi^{\beta}}{\partial\omega_{1}}}{\frac{M_{\mu}}{\omega_{1}}}
\varepsilon^{\mu\nu\alpha\beta}\varepsilon_{abcd}[\chiV_{\alpha}^{c}D_{\nu}V_{\beta}^{d}+\frac{1}{2}V_{\alpha}^{c}v_{\beta}^{d}\chi,_{\nu}]+\frac{\kappa}{4}\sqrt{-g}V^{c\mu}\varepsilon_{abcd}\overline{{\Psi}}\gamma^{5}\gamma^{d}\Psi=0,
<u_{0}|g_{2}V^{\prime\prime\prime}+\frac{1}{4}V^{\prime\prime\prime\prime}g_{0}^{2}|u_{0}>-\frac{1}{2}<u_{0}|g_{0}V^{\prime\prime\prime}\hat{h}^{-1}g_{0}V^{\prime\prime\prime}|u_{0}>\,\,<\,0.
f_{3}(T,A)=\frac{T^{3}}{8}\left[\frac{A}{3}\{A,\tau_{3}\}^{2}+c_{1}^{(3)}(T)A\{A,\tau_{3}\}[\tau_{3},A]+c_{2}^{(3)}(T)(A^{3}-A\tau_{3}A\tau_{3}A)\right],