harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}{\psi^{\prime}(t,x)}&{=}&{e^{-ie\epsilon(t,x)}\psi(t,x)=e^{-i\frac{2\pik}{L}x}\psi(t,x)\,}\\{{A^{1}}^{\prime}(t,x)}&{=}&{A^{1}(t,x)-\partial_{1}\epsilon(t,x)=A^{1}(t,x)-\frac{2\pik}{eL}\,}\\{\pi_{1}^{\prime}(t,x)}&{=}&{\pi_{1}(t,x)\,\}\\{{A^{0}}^{\prime}(t,x)}&{=}&{A^{0}(t,x)+\partial_{0}\epsilon(t,x)=A^{0}(t,x)\.}\\\end{array}
Z=\sum_{n=1}^{\infty}{\frac{f^{2n}\Lambda^{4n}}{(n!)^{2}}}\,\int\!\prod_{i=1}^{n}\bigl(d^{2}x_{i}d^{2}y_{i}\bigr)\,\int\!{\calD}\theta_{0}\,e^{-S_{\mathrm{eff}}[\theta_{0};x_{i},y_{i}]},
O_{+}(z,\bar{z})=\left({\frac{1}{2}}L_{ab}J^{a}(z)\phi^{b}(z)~+~x\phi^{3}(z)J_{c}(z)\right)\overline{{\left({\frac{1}{2}}L_{ab}J^{a}(z)\phi^{b}(z)~+~x\phi^{3}(z)J_{c}(z)\right)}},
i\frac{\partial}{\partialt}\Phi(1,\cdots,N;t)=-\frac{1}{2m}\sum_{p=1}^{N}\left\{\partial_{i}^{(p)}+{\calA}_{i}^{a}({\bfr}_{p};1,\cdots,p-1,p+1,\cdots,N)T_{(p)}^{a}\right\}^{2}\Phi(1,\cdots,N;t).
\tilde{\varphi}_{\mu\nu,\alpha}(k)=k_{\mu}(1-\alpha)(\tilde{A}_{\nu}+\xi_{\nu}^{\lambda}a^{\lambda})-\,k_{\nu}\,(\tilde{A}_{\mu}+\xi_{\mu}^{\lambda}a^{\lambda})+\alpha\epsilon_{\mu\nu\rho\sigma}k^{\rho}\tilde{A}^{\sigma}\;.
T=-\frac{L_{p}^{(3-d)}}{4\pi}\left\{\frac{(d-2)}{A}\left(k+\frac{A^{2}}{\ell^{2}}+\dot{A}^{2}\right)^{1/2}+\frac{\ddot{A}+A/\ell^{2}}{\sqrt{k+\dot{A}^{2}+A^{2}/\ell^{2}}}\right\}.
F_{z_{i}z_{j}z_{p}}:=\sum_{I=1}^{k+2}\Big[A^{I}(z)\partial_{z_{i}}\partial_{z_{j}}\partial_{z_{p}}B_{I}(z)-B_{I}(z)\partial_{z_{i}}\partial_{z_{j}}\partial_{z_{p}}A^{I}(z)\Big].
\widetilde{g}^{-1}(p^{0})=m-\gamma^{0}(p^{0}+\mu)-\frac{ie^{2}}{(2\pi)^{3}}\int_{-\infty}^{\infty}dk^{0}\gamma^{0}\widetilde{g}(k^{0})\gamma^{0}\widetilde{D}(p^{0}-k^{0}),
S_{J}=\intd^{4}x\{i\partial_{\mu}\xi^{\prime}(x)\partial^{\mu}\eta^{\prime}(x)-im^{2}\xi^{\prime}(x)\eta^{\prime}(x)-J_{\eta}\frac{i}{\partial_{\mu}\partial^{\mu}+m^{2}-i\epsilon}J_{\xi}\}
\bar{Z}_{+}(z_{2})Z_{+}(z_{1})=F_{+}(z_{12})=\left(\begin{array}{cc}{ix_{\bar{2}1}{\cdot\tilde{\sigma}}\,\tau_{+}}&{-2\bar{\theta}_{21b}\,p_{+}}\\{-2\theta_{21}^{a}\,\tau_{+}}&{\delta_{~b}^{a}\,p_{+}}\\\end{array}\right)
\begin{array}{lll}{c_{1}=c^{z}\begin{array}{l}{;}\\\end{array}}&{c_{2}=\partialc^{z}\begin{array}{l}{;}\\\end{array}}&{c_{3}=\rho_{zz}c^{z}-\frac{1}{2}\partial^{2}c^{z}\begin{array}{l}{;}\\\end{array}}\\\end{array}
\widetildeP(\rho)=\frac{2^{2k_{1}k_{2}}}{(2\pi)^{(k_{1}-k_{2})^{2}}}\intP(\sigma)w^{*}(\sigma,\rho)d[\sigma]\quad\mathrm{and}\quadP(\sigma)=\frac{2^{2k_{1}k_{2}}}{(2\pi)^{(k_{1}-k_{2})^{2}}}\int\widetildeP(\rho)w(\sigma,\rho)d[\rho]\.
\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{w_{T}^{2}}&{i2^{1/2}e\kappa|\chi|\epsilon^{ij}\frac{k^{j}}{k^{2}}}&{0}\\{0}&{-i2^{1/2}e\kappa|\chi|\epsilon^{ij}\frac{k^{j}}{k^{2}}}&{\frac{w_{A}^{2}}{k^{2}}}&{0}\\{0}&{0}&{0}&{w_{G}^{2}}\\\end{array}\right)
\left<x|F(L)|x^{\prime}\right>=\int_{0}^{\infty}\!\!d\lambda\,\mu(\lambda)F(\lambda^{2}/a^{2})\int\!\!\frac{d^{N-1}k}{(2\pi)^{2\rho_{N}}}(yy^{\prime})^{\rho_{N}}e^{i{\bfk}\cdot{\bfu}}K_{i\lambda}(ky)K_{i\lambda}(ky^{\prime}),
Z_{g}=\int_{M_{g}}\prod_{i=1}^{3g-3}\mathrm{d}^{2}y^{i}\,{\frac{|\mathrm{det}\langle\phi^{i},\mu^{j}\rangle|^{2}}{\mathrm{det}\langle\phi^{i}|\phi^{j}\rangle}}\mathrm{det}^{\prime}(\bigtriangledown_{z}\bigtriangledown_{-1}^{z})\int{\calD}X\mathrm{e}^{-S[X]}.
\sigma^{\alpha\beta}=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{([\alpha]_{q}[\beta]_{q})^{1/2}q^{-(\alpha+\beta)/2}(q-q^{-1})}&{q^{-\beta}}&{0}\\{0}&{q^{-\alpha}}&{0}&{0}\\{0}&{0}&{0}&{-q^{-\alpha-\beta}}\\\end{array}\right)
(\bar{\theta}_{+}^{1}\Gamma^{+}\delta\theta_{+}^{1}+\gamma\bar{\theta}_{-}^{1}\Gamma^{-}\delta\theta_{-}^{1})(\theta_{+}^{1}\Gamma^{4}\Gamma^{-}\theta_{-}^{1}-\theta_{-}^{1}\Gamma^{4}\Gamma^{+}\theta_{+}^{1})~.
\begin{array}{lcr}{P_{-}=\ointdz{\frac{3}{5}}z^{-{2/3}}(\Psi^{+}(z)\Psi^{-}(0))}\\{P_{+}=\ointd{\bar{z}}{{\frac{3}{5}}z^{-{2/3}}(\bar{\Psi}^{+}(\bar{z})\bar{\Psi}^{-}(0))}.}\\\end{array}
\sum_{l=0}^{N-1}\left(\frac{e\Phi}{2\pi}-l-1+\frac{1}{2I}\right)^{-1}=\left(\sum_{m=0}^{CN^{\frac{1}{3}}}+\sum_{CN^{\frac{1}{3}}}^{\gammaN}+\sum_{\gammaN}^{N-1}\right)(m+\epsilon+g(m))^{-1},
{\frac{\ddot{\hat{a}}_{0}}{\hat{a}_{0}}}-{\frac{\ddot{\varepsilon}}{\varepsilon}}=-{\frac{4\pi\hat{G}_{4}}{3}}(\hat{\varrho}+3{\frac{\hat{\wp}}{c_{0}^{2}}})-{\frac{\hat{\calC}c_{0}^{2}}{\hat{a}_{0}^{4}}},
R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}~^{(4)}R=-\ell_{P}^{2}e^{-1}f\frac{\delta\stackrel{\sim}{\calL}_{m}}{\deltag^{\mu\nu}}-\ell_{P}^{2}e^{-1}\frac{\delta{\calL}_{g}}{\deltag^{\mu\nu}}
\deltaK=\delta\left(h^{a}{}_{b}\nabla_{a}n^{b}\right)=\deltah^{a}{}_{b}\nabla_{a}n^{b}+h^{a}{}_{b}(\deltaC)^{b}{}_{ac}n^{c}+h^{a}{}_{b}\nabla_{a}\deltan^{b}\,,
L_{grav}={{\triangle}^{grav}}(T,{\bar{T}})({\calR}_{abcd}^{2}-4{\calR}_{ab}^{2}+{\calR}^{2})+\nonumber\,{\Theta}^{grav}(T,{\bar{T}}){\epsilon}^{abcd}{\calR}_{abef}{\calR}_{cd}^{ef}
\left|{\frac{B}{A}}\right|~=~\frac{1}{\sqrt{2}}(2\omegar_{0})^{n}\left|{\frac{\Gamma(n)}{\Gamma(2n)}}{\frac{\Gamma(\gamma)~\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)~\Gamma(\gamma-\beta)}}\right|~~.
\lambda=\frac{\widehat{B}_{E}^{2}L^{2}}{A^{2}G^{\prime}(\xi_{3})\epsilon_{E}}\chi+\frac{\widehat{B}_{E}^{2}L^{2}G^{\prime\prime}(\xi_{3})}{2A^{2}G^{\prime}(\xi_{3})^{2}}\chi^{2}+1,
\mathcal{M}_{\alpha_{k}}^{i}(\theta)=\left(\begin{array}{cc}{T_{i}^{\alpha_{k}}(\theta)^{-1}}&{-R_{i}^{\alpha_{k}}(\theta)T_{i}^{\alpha_{k}}(\theta)^{-1}}\\{-R_{i}^{\alpha_{k}}(-\theta)T_{i}^{\alpha_{k}}(-\theta)^{-1}}&{T_{i}^{\alpha_{k}}(-\theta)^{-1}}\\\end{array}\right)\,.
\left(\begin{array}{cc}{0}&{v^{4}}\\{v^{2}}&{v^{3}}\\\end{array}\right)=\{p_{1},Z\}^{*}=\left(\begin{array}{cc}{0}&{-p_{2}/p_{3}\,-i\gammap_{1}z^{4}}\\{-p_{4}/p_{3}\,+i\gammap_{1}z^{2}}&{p_{1}/p_{3}}\\\end{array}\right)\quad.
\begin{array}{llll}{\mathrm{~R~symmetry~singlet~(a=0):}}&{d\geqs+2,}&{s\geq1,}&{r=0}\\{}&{d\geq1,}&{s=0,}&{r=0}\\{\mathrm{~R~symmetry~triplet~(a=2):}}&{d=2,}&{s=0,}&{r=0\quad\mathrm{(current~m-t)}}\\{}&{d=s+4,}&{s\geq0,}&{r=0}\\{\mathrm{~R~symmetry~5-plet~(a=4):}}&{d=4,}&{s=0,}&{r=0}\\\end{array}
F(p,m,\varepsilon)=i\int\frac{d^{4}k}{(k_{0}^{2}-|\mathbf{k}|^{2}-m^{2}+i\varepsilon)[(k+p)_{0}^{2}-|\mathbf{k}+\mathbf{p}|^{2}-m^{2}+i\varepsilon]}
\left[J_{ij}(n),J_{i^{\prime}j^{\prime}}(n^{\prime})\right]=nN_{f}\left(\delta_{ij^{\prime}}\delta_{i^{\prime}j}-\frac{1}{N_{c}}\delta_{ij}\delta_{i^{\prime}j^{\prime}}\right)\delta_{-n^{\prime}}^{n}+\delta_{ij^{\prime}}J_{i^{\prime}j}(n+n^{\prime})-\delta_{i^{\prime}j}J_{ij^{\prime}}(n+n^{\prime}).
\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{\omega^{\sigmak_{2}}w(x_{3},x_{13},x_{1}|\sigma)w(x_{4},x_{24},x_{2}|k_{4}+\sigma)}{w(x_{8},x_{58},x_{5}|k_{3}+\sigma)w(x_{7}/\omega,x_{67},x_{6}|k_{1}+\sigma)}\Bigg\}_{0}.
H_{I\sigmaij}^{(2)}={\underbrace{[H_{I\sigma^{\prime}}^{B},H_{I\sigma^{\prime}T}^{B}]_{ij}}}_{R}=\sum_{k}H_{Iik}^{B}H_{Ikj}^{B}\left\{\frac{g_{\sigmajik}}{P_{i}^{-}-P_{k}^{-}}+\frac{g_{\sigmaijk}}{P_{j}^{-}-P_{k}^{-}}\right\},
\langleout\,\omega^{\prime}\omega^{\prime\prime\prime}\vertin\rangle=e^{iW}(\beta\alpha^{-1})_{\omega^{\prime}\omega^{\prime\prime\prime}}^{*}=\sum_{\omega}\beta_{\omega^{\prime}\omega}^{*}\langleout\,\omega^{\prime\prime\prime}\vert\omega\,in\rangle.
\tilde{p}^{2}=(\gamma+\lambda_{1})^{2}(-x^{2})+\frac{(\tilde{p}x^{\prime})^{2}}{x^{2}}\;\Longrightarrow\lambda_{1}+\gamma=\frac{\sqrt{(\tilde{p}x^{\prime})^{2}-\tilde{p}^{2}x^{2}}}{-x^{2}}{.}
\begin{array}{rcl}{\vertv\rangle\rangle}&{=}&{\left([v+b\gamma]_{q}!\right)^{-1/2}\left(a_{q}^{\dagger}\right)^{v}\vert0\rangle}\\{}&{=}&{\left(v!\right)^{-1/2}\left(a^{\dagger}\right)^{v}\vert0\rangle}\\{}&{=}&{\vertv\rangle~.}\\\end{array}
E_{\alpha}={\frac{\beta\pi^{2}\rho_{0}^{2}}{2}}\left(-\sum_{i=1}^{p}v_{i}^{2}+\beta\sum_{i=0}^{q}w_{i}^{2}\right),\\Q_{\alpha}=\pi\rho_{0}\left(\sum_{i=1}^{p}v_{i}-\beta\sum_{i=0}^{q}w_{i}\right)
\left.{\calP}_{\delta\sigma_{i}}\right|_{s}=2\kappa^{2}\left[{\frac{C(\mu_{i})}{r_{i}+(1/2)}}\right]^{2}\left({\frac{\bar{H}_{i}}{2\pi}}\right)_{s}^{2}\left({\frac{k}{k_{s}}}\right)^{3-2\mu_{i}}\,,
+\;\frac1{\left(q-1\right)!}\left[\,\left(-\delta_{M}\delta_{N}+\delta_{N}\delta_{M}\mpC_{MN}^{\;K}\,\cdot\delta_{K}\right)A_{\nu_{2}\ldots\nu_{q}}\right]\Lambda^{\nu_{2}\ldots\nu_{q}}\cdot\delta\left(\sigma-\sigma^{\prime}\right)\quad\cdots\quad,
\tilde{\psi}_{+,m}^{(n)}(x)=\frac{1}{\tilde{{\calN}}_{m,+}^{(n)}}\left(\begin{array}{c}{i\sqrt{m}(d_{-})^{m-1}\phi_{2}^{(n)}(z,\bar{z})}\\{(d_{-})^{m}\phi_{2}^{(n)}(z,\bar{z})}\\\end{array}\right)
S=\left.\intd^{4}x\,\overline{{\psi}}\left(\mathrm{i}\gamma^{\mu}\partial_{\mu}-m\right)\psi\right|_{x^{\perp}=0}+\mathcal{M}\!\int\!d^{4}x\!\int\!dx^{\perp}\,\overline{{\psi}}\left(\mathrm{i}\gamma^{\mu}\partial_{\mu}+\mathrm{i}\gamma^{\perp}\partial_{\perp}-M\right)\psi.
A=-\frac{n_{0}^{2}}{4},\,\,\,B=\frac{{\bfn}^{2}}{4(1-\xi)}-\frac{\xi({\bfn}\cdot{\bfN})^{2}}{4N_{0}^{2}(1-\xi)(1-\xi+\xi{\bfN}^{2}/N_{0}^{2})}.
\left|\mathcal{B}_{phys}\right\rangle=\int_{-\infty}^{\infty}d\rho\sum_{m\inZ}\\sum_{r\inZ_{+}}\,e^{-i\rho\Theta}\,\left|B_{p^{+}=m+\alpha,\,p^{-}=-\frac{r}{m+\alpha},\,p_{2}=\rho}\right\rangle.
f\starg=f\g+\sum_{m=1}^{2j}\frac{(2j-m)!}{m!(2j)!}\J^{a_{1}b_{1}}J^{a_{2}b_{2}}\cdotsJ^{a_{m}b_{m}}\\partial_{a_{1}}\cdots\partial_{a_{m}}f\,partial_{b_{1}}\cdots\partial_{b_{m}}g.
F_{\mu\nu}^{\alpha\beta}=\delta^{\alpha\beta}f_{\mu\nu}^{\alpha}+g_{\mathrm{YM}}D_{\mu}^{\alpha\beta}\bar{A}_{\nu}^{\alpha\beta}-g_{\mathrm{YM}}D_{\nu}^{\alpha\beta}\bar{A}_{\mu}^{\alpha\beta}-ig_{\mathrm{YM}}^{2}\left[\bar{A}_{\mu},\bar{A}_{\nu}\right]^{\alpha\beta}
\Big\langlee^{-ie\sum_{b}(j_{\mu}^{(b)},a_{\mu}^{(b)})}\Big\rangle=e^{-\frac{e^{2}}{2\pi}\sum_{b,c}(\varepsilon_{\mu\nu}\partial_{\mu}a_{\nu}^{(b)},M^{(b,c)}\varepsilon_{\rho\sigma}\partial_{\rho}a_{\sigma}^{(c)})}\;,
e^{-S(q)}={\calN}\int\prod_{n=0}\left\{dX_{n0}\prod_{k=1}d^{2}X_{nk}\right\}e^{-{\calS}_{E}}~\delta(\sum_{n=0}X_{n0}-q_{0})\prod_{k=1}\delta(\sum_{n=0}X_{nk}-q_{k}),
i\frac{\partial}{\partialt}\rho_{\;\;b}^{a}=h_{\;\;c}^{a}\rho_{\;\;b}^{c}-\rho_{\;\;b}^{c}h_{\;\;c}^{a}+i\frac{g^{2}}{2}\left(F_{\;\;c}^{a}\rho_{\;\;b}^{c}+\rho_{\;\;b}^{c}F_{\;\;c}^{a}\right)
\frac{dg_{2}}{d\xi}-(\frac{1}{f}-3f+\frac{2}{f}\frac{df}{d\xi})g_{2}=-\frac{hf_{2}}{2f^{2}}-\frac{3hf_{2}}{2}-\frac{h^{3}}{4f^{3}}-\frac{h^{3}}{4f}.
x^{i}(t,\sigma^{1},\sigma^{2},..,\sigma^{p})=x^{i}(t_{0},\sigma^{1},\sigma^{2},..,\sigma^{p})+\nu^{i}(\sigma^{1},\sigma^{2},..,\sigma^{p})\int_{t_{o}}^{t}~dt\;R^{-1}(t),
\mu^{2}=-\frac{1}{2}\left(\frac{\alpha^{2}-A^{2}}{l^{4}}+\frac{1}{L^{2}}-\sqrt{{\left(\frac{\alpha^{2}+A^{2}}{l^{4}}+\frac{1}{L^{2}}\right)^{2}-\frac{4A^{2}\alpha^{2}}{l^{4}}}}\right)
\begin{array}{rl}{}&{TrW(w,z_{2})W(z_{1},w)W(w,z_{1})W(z_{2},w)}\\{=}&{TrRW(z_{1},w)W(w,z_{2})R^{-1}RW(z_{2},w)W(w,z_{1})R^{-1}}\\{=}&{TrW(z_{1},z_{2})W(z_{2},z_{1})}\\{=}&{TrW(z_{1},z_{1})}\\\end{array}
\frac{\partialS}{\partialt}+H[f,\deltaS/\deltaf,t]-4CP^{-1/2}\sum_{a,b}\intdx\,\left(K_{x}^{ab}\frac{\delta^{2}P^{1/2}}{\deltaf_{x}^{a}\deltaf_{x}^{b}}+\frac{\deltaK_{x}^{ab}}{\deltaf_{x}^{a}}\frac{\deltaP^{1/2}}{\deltaf_{x}^{b}}\right)=0.
\langleQ_{\mu}^{i}\rangle=\frac{\deltaW}{\deltaJ^{\mu,i}},\,\,\,\,\langleC^{i}\rangle=\frac{\deltaW}{\delta\overline{{\xi}}^{i}},\,\,\,\,\langle\overline{{C}}^{i}\rangle=-\frac{\deltaW}{\delta\xi^{i}}
I(l_{IJ})=\left\{\begin{array}{ccc}{l_{01}}&{l_{02}}&{l_{03}}\\{l_{23}}&{l_{13}}&{l_{12}}\\\end{array}\right\}^{2}=\int_{G^{4}}\[dg_{I}]\prod_{I<J}\chi_{l_{IJ}}(g_{J}g_{I}^{-1}),
\frac{\Psi_{\sigma}^{(n)}(w^{\prime},x^{4}w_{2n}|\zeta^{\prime},x^{2}\zeta_{2n})}{\Psi_{\sigma}^{(n)}(w_{2n},w^{\prime}|\zeta)}=(-1)^{n-1}x^{-3}\prod_{a\inA\atopa\neq2n}\frac{w_{a}-xz_{2n}}{x^{4}z_{2n}-xw_{a}}\prod_{j=1}^{2n-1}\frac{\zeta_{2n}}{\zeta_{j}}\frac{z_{j}-xw_{2n}}{x^{3}w_{2n}-z_{j}}.
A_{(b)}=g^{4}\intd^{26}k\int_{0}^{1}dx_{1}dx_{3}dz_{2}dz_{4}~x_{1}^{-\alpha(-(k-p_{1})^{2})-1}\,x_{3}^{-\alpha(-(k-p_{3})^{2})-1}\,(z_{2}z_{4})^{-\alpha(-(p_{1}-p_{3})^{2})-1}.
\{\eta_{MN}\}=\left(\begin{array}{cc}{g_{\alpha\beta}+g_{\rho\sigma}A_{\alpha}^{\rho}A_{\beta}^{\sigma}}&{g_{\mu\rho}A_{\beta}^{\rho}}\\{A_{\alpha}^{\rho}g_{\rho\nu}}&{g_{\mu\nu}}\\\end{array}\right)\,,
c_{n}\left[\phi(x)\right]\equiv{\frac{1}{Z_{0}}}{\frac{\left(-1\right)^{n}}{n!}}{\frac{1}{4^{n}}}\left(\lambda\int\mathrm{d^{d}}x\phi^{4}\right)^{n}e^{-\int\mathrm{d^{d}}x\left[{\frac{1}{2}}\left(\partial_{\mu}\phi\right)^{2}+{\frac{1}{2}}m^{2}\phi^{2}\right]}
{\calS}\equiv\mathrm{ln}({\cal{E}}\rho)=S(\alpha_{0},\beta_{0})+\mathrm{ln}\frac{{\cal{E}}}{\sqrt{\frac{\partial^{2}\mathrm{ln}Z}{\partial\beta^{2}}|_{(\alpha_{0},\beta_{0})}}}+\mathrm{higher~order~terms}.
{\frac{\dot{\varrho}_{m}}{\varrho_{m}}}=-3{\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}(1+w)-2{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}{\frac{\varrho_{c}}{\varrho_{m}}}+2{\frac{\dot{\bar{c}}_{4}}{\bar{c}_{4}}}{\frac{\varrho+\varrho_{\delta\sigma}}{\varrho_{m}}}-{\frac{\varrho}{\varrho_{m}}}{\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}.
{\bfr}(\lambda)=\frac{\lambda^{\frac{3}{2}}+\lambda^{-\frac{3}{2}}}{\lambda^{\frac{3}{2}}-\lambda^{-\frac{3}{2}}}\sum_{i=1}^{3}H_{i}\otimesH_{i}+\frac{2}{\lambda^{\frac{3}{2}}-\lambda^{-\frac{3}{2}}}\sum_{(ij)}\left(\lambda^{-\frac{1}{2}}E_{ij}\otimes{E}_{ji}+\lambda^{\frac{1}{2}}E_{ji}\otimes{E}_{ij}\right),
\int_{-\infty}^{+\infty}\left[k^{2}+A^{2}\right]^{-s}d^{m}k=\frac{\pi^{\frac{m}{2}}\Gamma\left(s-\frac{m}{2}\right)}{\Gamma\left(s\right)}\left(A^{2}\right)^{\frac{m}{2}-s},
(i\mathrm{~}{\bf{q}}_{m})\mathrm{~}X_{{\bf{q}}_{m}r}=-(\frac{1}{N_{r}^{0}})\frac{1}{1+\frac{1}{N_{r}^{0}}\sum_{{\bf{k}}_{n}}\delta\rho({\bf{k}}_{n}r)T_{{\bf{k}}_{n}}({\bf{q}}_{m})}[\mathrm{~}\sum_{{\bf{p}}_{n}}\delta{\bf{j}}({\bf{p}}_{n}r)T_{{\bf{p}}_{n}}({\bf{q}}_{m})\mathrm{~}]\mathrm{~}\delta_{{\bf{q}}_{m},{\bf{0}}}+\mathrm{~}{\bf{F}}([\rho];{\bf{q}}_{m}r)
H_{N}\rightarrow{\frac{1}{2T_{0}}}\int_{0}^{p^{+}/T_{0}}d\sigma\Big[{\calP}(\sigma)^{2}+T_{0}^{2}x^{\prime}(\sigma)^{2}-iT_{0}S(\sigma)S^{\prime}(\sigma)+iT_{0}\tilde{S}(\sigma)\tilde{S}^{\prime}(\sigma)\Big]\;.
\psi_{f}=\sum_{n}(\phi_{f}^{n}e_{n}+\chi_{f}^{n}\gamma_{5}e_{n});\\\\\bar{\psi}_{f}=\sum_{n}(\bar{\phi}_{f}^{n}e_{n}^{\dagger}+\bar{\chi}_{f}^{n}e_{n}^{\dagger}\gamma_{5}).
f(x;u,\gamma,\tau)~\equiv~{\frac{sinh\big({\frac{\pii}{\tau}}(1-\gamma)x\big)~sinh\big({\frac{2\pii}{\tau}}\gammaux\big)~sinh\big({\frac{\pii(N-1)}{\tau}}\gammax\big)}{x~sinh\big({\frac{\pii}{\tau}}x\big)~sinh\big({\frac{\piiN}{\tau}}\gammax\big)}}\.
{\calG}=\frac{1}{2}M_{ij}d{\bfx}_{i}\cdotd{\bfx}_{j}+\frac{g^{4}}{2(4\pi)^{2}}(M^{-1})_{ij}(d\xi_{i}+{\bfW}_{ik}\cdotd{\bfx}_{k})(d\xi_{j}+{\bfW}_{jl}\cdotd{\bfx}_{l}).
G_{\mu\nu}^{\prime}=\frac{1}{2}\left(\begin{array}{cccccc}{-1+\tilde{v}^{2}}&{0}&{0}&{0}&{0}&{-\tilde{v}}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{-\tilde{v}}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right).
D_{\mu\nu}^{-1}(k)=Z_{3}[k^{2}g_{\mu\nu}\!-\!k_{\mu}k_{\nu}(1\!-\!\eta^{-1})]+ie^{2}Z_{2}Tr\int\!\frac{d^{n}p}{(2\pi)^{n}}\gamma_{\nu}S(p)\Gamma_{\mu}(p,p-k)S(p-k)
ds_{B}^{2}=e^{\phi}|z|^{-1}dzd\bar{z}+e^{\phi}\left(\frac{\partial\xi}{\partialz^{\prime}}\bar{z}^{-\frac12}dz^{\prime}d\bar{z}+c.c.\right)+\left(e^{\phi}\left|\frac{\partial\xi}{\partialz^{\prime}}\right|^{2}+e^{\psi}\right)dz^{\prime}d\bar{z^{\prime}}.
\begin{array}{ccccccc}{\nablaZ_{AB}}&{=}&{P_{ABCD}\overline{{Z}}^{CD}}&{\longrightarrow}&{\nabla\widehat{Z}_{AB}}&{=}&{{\calP}_{A}^{A^{\prime}}\,{\calP}_{B}^{B^{\prime}}P_{A^{\prime}B^{\prime}CD}\,\overline{{Z}}^{CD}\,}\\\end{array}
\begin{array}{l}{\left(:X^{-}\,P^{+}:\right)(z)\times\,\left(:X^{-}\,P^{+}:\right)(w)=}\\{=\,:X^{-}(z)\,P^{+}(z)X^{-}(w)\,P^{+}(w):}\\{+\,:X^{-}(z)\,\,P^{+}(w):\,<P^{+}(z)X^{-}(w)>+}\\{+\,\,:P^{+}(z)\,X^{-}(w):\,<X^{-}(z)P^{+}(w)>}\\{<P^{+}(z)X^{-}(w)>\,<X^{-}(z)P^{+}(w)>}\\{\quad\rightarrow\frac1{\left(z-w\right)^{2}}+\cdots}\\\end{array}
H_{1}(\vec{x},\vec{z})=1+\sum_{i}{\frac{P_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(4-r)^{2}}}|\vec{z}-\vec{z}_{0}|^{4-r}]^{{\frac{1}{2}}(p-1+{\frac{r}{4-r}})}}},
\sigma_{V}(p^{2})=\frac{A(p^{2})}{p^{2}A^{2}(p^{2})+B^{2}(p^{2})},\;\;\;\;\;\mathrm{and}\;\;\;\;\sigma_{S}(p^{2})=\frac{B(p^{2})}{p^{2}A^{2}(p^{2})+B^{2}(p^{2})}.
I=\intd\mu_{\tilde{C}}[\varphi]e^{2e\sum_{a=1}^{N}\Big[\sum_{i=1}^{m_{a}}\varphi(y_{i}^{(a)}\!+\hat{\tau})+\sum_{i=1}^{m_{a}^{\prime}}\varphi({y^{\prime}}_{i}^{(a)})-\sum_{i=1}^{n_{a}}\varphi(x_{i}^{(a)}\!+\hat{\tau})-\sum_{i=1}^{n_{a}^{\prime}}\varphi({x^{\prime}}_{i}^{(a)})\Big]}\;.
\betaM_{B}=\frac{1}{(2\pi)^{5}l_{s}^{6}g_{\mathrm{str}}}\times\pi^{3}r_{0}^{5}\times\pir_{0}\sqrt{\rho^{2}-\frac{1}{\rho^{2}}}=\frac{N}{4}\sqrt{\frac{\eta}{2}}\sqrt{\rho^{2}-\frac{1}{\rho^{2}}}.
\frac{1}{2^{n-l}}\frac{(n-1)!}{(2n-1)!}\left|\sum_{k_{1}=0}^{n-l}\sum_{k_{2}=0}^{n-l-k_{1}}\frac{(n-l)!}{k_{1}!k_{2}!k_{3}!}\frac{\partial^{2(n-l)}V^{l}(q)}{\partialq_{1}^{2k_{1}}\partialq_{2}^{2k_{2}}\partialq_{3}^{2k_{3}}}\right|_{k_{3}=n-l-k_{1}-k_{2}}.
V^{\mathrm{LR}}=\{{\frac{\sigmar}{2}}[({\frac{m^{2}+{\bfq}_{\mathrm{L}}^{2}}{m^{2}+{\bfq}^{2}}})^{\frac{1}{2}}+{\frac{\sqrt{m^{2}+{\bfq}^{2}}}{\vert{\bfq}_{T}\vert}}\mathrm{arcsin}{\frac{\vert{\bfq}_{T}\vert}{\sqrt{m^{2}+{\bfq}^{2}}}}\}_{\mathrm{ord}}+O({\frac{\sigma^{2}}{m^{4}}}))
r=y_{j}^{\dagger}[r_{1}^{\psi}y_{i}y_{i}^{\dagger}y_{j}+r_{2}^{\psi}y_{i}y_{j}^{\dagger}y_{i}+r_{3}^{\psi}Y_{ij}~y_{i}+r_{4}^{\psi}h_{ijkk}y_{i}]
m^{rs}[\vec{\eta}(t)]=\sum_{i=1}^{N}m_{i}\eta_{i}^{r}(t)\eta_{i}^{s}(t)-m\eta^{r}(t)\eta^{s}(t)-\Big(\eta^{r}(t)m^{s}[\vec{\eta}(t)]+\eta^{s}(t)m^{r}[\vec{\eta}(t)]\Big),
H_{\mathrm{triangle}}(R)=24\Lambda^{3}L_{3}R-2\sqrt3\Lambda^{2}L_{3}+\sqrt3\left(5\pi-\frac{16}{3}\right)\Lambda^{2}L_{3}e^{-\sqrt3R\Lambda}+{\calO}\left(e^{-2\sqrt3R\Lambda}\right).
\partial^{\mu}(\sqrt{-g}R_{\mu})\,=\,\frac{1}{192\pi^{2}}\left[(N_{c}^{2}-1)-\frac{1}{3}\sum_{i}(-1+c_{i}^{(0)})\right]\epsilon_{\mu\nu\lambda\delta}R^{\mu\nu\sigma\rho}R_{\sigma\rho}^{\lambda\delta}\;,
\timesD_{S+U+T+N+K}{\frac{\partial^{2}f}{\partial\phi_{A}^{(U)}\partial\phi_{C}^{(T)}}}\Biggl(D_{R}E_{B}^{J}(h)D_{J+M-R-S}E_{D}^{M}(g)-(H\leftrightarrowG)\Biggr).
\boxed{\Big(\sum_{n=1}^{\infty}a(n)k(n)z_{2}^{n}\Big)\Big(\sum_{n=0}^{\infty}a(n)z_{2}^{n}\Big)=\Big(\sum_{n=1}^{\infty}a(n)b(n)z_{2}^{n}\Big)\Big(\sum_{n=1}^{\infty}a(n)s(n)z_{2}^{n}\Big)},
\int_{-L}^{v_{n-1}}dv_{n}e^{-ieA_{-}(v_{n-1}-v_{n})}=\int_{-L}^{\infty}dv_{n}\int_{-\infty}^{\infty}{\frac{dk^{+}}{2\pi}}{\frac{ie^{-i(k^{+}+i/L)(v_{n-1}-v_{n})}}{k^{+}-eA_{-}(u_{n})+i/L}}\;,
{\calL}_{class}(\phi_{1},\phi_{2},\lambda,\alpha)=-\varphi_{1}(\phi_{1},\alpha)\Box\phi_{2}+m^{2}\varphi_{1}(\phi_{1},\alpha)\phi_{2}+{\frac{\lambda}{4}}\varphi_{1}^{2}(\phi_{1},\alpha)\phi_{2}^{2}+J_{1}\phi_{1}+J_{2}\phi_{2}.
[{\calT}_{\alpha},{\calT}_{\beta}^{\dagger}]=\bar{U}_{\alpha\beta}^{\gamma}{\calT}_{\gamma}+{\calT}_{\gamma}^{\dagger}\bar{U}_{\beta\alpha}^{\dagger\gamma}+[\Omega,\bar{U}_{\alpha\beta}^{\gamma}]\bar{C}_{\gamma}(-1)^{\varepsilon_{\alpha}+\varepsilon_{\beta}}+\bar{C}_{\gamma}^{\dagger}[\bar{U}_{\beta\alpha}^{\dagger\gamma},\Omega](-1)^{\varepsilon_{\alpha}+\varepsilon_{\beta}},
e^{{\frac{i\hbar}{2}}(\stackrel{\leftarrow}{\partial}_{x}\stackrel{\rightarrow}{\partial}_{p}-\stackrel{\leftarrow}{\partial}_{p}\stackrel{\rightarrow}{\partial}_{x})+\hbar\stackrel{\leftarrow}{\partial}_{\theta}\stackrel{\rightarrow}{\partial}_{\theta}}~.
\mathrm{det}\left(\eta_{\mu\nu}+(2\pi\alpha^{\prime}){\calF}_{\mu\nu}\right)=-\left[1+\frac{1}{2}(2\pi\alpha^{\prime})^{2}{\calF}^{2}-\frac{1}{16}(2\pi\alpha^{\prime})^{4}({\calF}\tilde{\calF})^{2}\right].
\sum_{m=0}^{\infty}\sum_{j=1}^{23}{\frac{(\Gamma({\frac{j}{24}}+m))^{2}(-)^{m}\prod_{j\neqk=1}^{23}\Gamma({\frac{k-j}{24}}-m)}{m!(m+{\frac{j}{24}})\prod_{k=-2}^{8}\Gamma({\frac{k}{8}}-m-{\frac{j}{24}})\prod_{l=10}^{1}\Gamma({\frac{l}{12}}-m-{\frac{j}{24}})}}z^{-m-{\frac{j}{24}}}.
{\bfF}_{2w}=\left(\begin{array}{cc}{0}&{\frac{D_{2w}}{|D_{2w}|}}\\{\frac{D_{2w}}{|D_{2w}|}}&{0}\\\end{array}\right),\quad\epsilon=\left(\begin{array}{cc}{{\Gamma}}&{0}\\{0}&{{\Gamma}}\\\end{array}\right).
\hat{G}_{\alpha\beta}=i\,\left[\partial_{\alpha}-\fraci4B_{\alpha}^{mn}\sigma_{mn},\,\partial_{\beta}-\fraci4B_{\beta}^{kl}\sigma_{kl}\right]=\left(\begin{array}{cc}{\hat{F}_{\alpha\beta}^{-}}&{0}\\{0}&{\hat{F}_{\alpha\beta}^{+}}\\\end{array}\right);
S=\intd\tau\left\{p_{a}\dot{x}^{a}-\frac{\varsigma}{2}q_{\alpha}\dot{q}^{\alpha}-\frac{e(\tau)}{2}(p^{2}+m^{2})+\lambda(\tau)\left(\frac{\varsigma}{4}(p^{a}\sigma_{a})_{\alpha\beta}q^{\alpha}q^{\beta}+ms\right)\right\}
S_{\mu\nu}:=\left(\begin{array}{c}{G_{\mu\nu}}\\{E_{\mu\nu}}\\\end{array}\right)=\left(\begin{array}{c}{g_{\mu\nu}}\\{H_{\mu\nu}}\\\end{array}\right)+i\left(\begin{array}{c}{B_{\mu\nu}}\\{M_{\mu\nu}}\\\end{array}\right)\,,
S_{abc}=\theta_{ja}\left(F_{bc}^{j}+\lambda_{b}A_{c}^{j}-\lambda_{c}A_{b}^{j}\right)+\theta_{jb}\left(F_{ca}^{j}+\lambda_{c}A_{a}^{j}-\lambda_{a}A_{c}^{j}\right)-\theta_{jc}\left(F_{ab}^{j}+\lambda_{a}A_{b}^{j}-\lambda_{b}A_{a}^{j}\right),
=-(2\pi\sqrt{\alpha^{\prime}})^{2}\,\left(\frac{1}{2}+\frac{1}{(2\pi\sqrt{\alpha^{\prime}})^{2}}\tilde{b}\right)+(2\pi\sqrt{\alpha^{\prime}})^{2}=(2\pi\sqrt{\alpha^{\prime}})^{2}\,\left(\frac{1}{2}-\frac{1}{(2\pi\sqrt{\alpha^{\prime}})^{2}}\tilde{b}\right)
{\calL}^{(0)}=\pi_{ij}\dot{A}^{ij}+{\frac{1}{2}}\,\pi_{ij}\pi^{ij}+2\partial_{i}A_{j0}\pi^{ij}-{\frac{1}{2}}\,\partial_{i}A_{jk}\partial^{i}A^{jk}+\partial_{i}A_{jk}\partial^{j}A^{ik}\,,
k^{2}=\frac{\ddot{F}}{{(F_{1}|F|^{2})_{1}+2F|F_{1}|^{2}-\frac{1}{2}[(F_{1}\bar{F}+\bar{F}_{1}F)F]_{1}-(F_{1}\bar{F}+\bar{F}_{1}F)F_{1}}}.
I_{M}=<-\frac{1}{3!}m^{2}a_{mnp}a^{mnp}-\frac{1}{2!}m^{2}\psi^{2}+\frac{1}{4!}m\epsilon^{mnpq}\psiG_{mnpq}+\frac{1}{3!}m\epsilon^{mnpq}(a_{mnp}-C_{mnp})\partial_{q}\phi>.
\langle\Psi|\hat{b}^{\dagger}=\langle\Psi|\mu_{b}^{\ast}\,,\quad~\langle\Psi|\hat{c}^{\dagger}=\langle\Psi|\mu_{c}^{\ast}\,,\quad\langle\Psi|\hat{b}_{0}=\langle\Psi|\mu_{0}^{\ast}\,,
(F,G)\equivF{\frac{\overleftarrow{\partial}}{\partial\Phi^{A}}}{\frac{\overrightarrow{\partial}}{\partial\Phi_{A}^{*}}}G-F{\frac{\overleftarrow{\partial}}{\partial\Phi_{A}^{*}}}{\frac{\overrightarrow{\partial}}{\partial\Phi^{A}}}G.
E\left(\beta\right)=E_{0}+\widetilde{E}\left(\beta\right)=\frac{1}{2r}\sum_{\ell=1}^{\infty}\ell^{2}\left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}+\frac{1}{r}\sum_{\ell=1}^{\infty}\frac{\ell^{2}\left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}}{e^{\beta\left(\ell^{2}+\mu_{eff}^{2}\right)^{1/2}/r}-1},
\left(\frac{{\calP}_{L}^{\mu}+{\calP}_{L}^{\mu}}{8\sqrt{nn^{\prime}}}U^{i}V^{i^{\prime}}-\delta^{\mui}\sqrt{\frac{n}{n^{\prime}}}V^{i^{\prime}}-\delta^{\mui^{\prime}}\sqrt{\frac{n^{\prime}}{n}}U^{i}\right)
\{\frac{-l^{2}M^{2}}{2}+N^{(\alpha)}+N^{(b)}+\frac{1}{2}\sum_{i=1}^{D-1}\sum_{m=1}^{\infty}m\delta^{ii}-\frac{1}{2}\sum_{i=1}^{D-1}\sum_{k=1}^{\infty}k\delta^{ii}\}\mid\Psi_{phys}\rangle=0
L_{1}=2f_{G}^{d-1}\sum_{i}\epsilon^{\mu_{1}\mu_{2}\cdots\mu_{d}}\epsilon_{a_{1}a_{2}\cdotsa_{d+1}}{\calO}_{i,i+1}^{a_{1}n}(x)D_{\mu_{1}}{\calO}_{i,i+1;n}^{a_{2}}(x)e_{\mu_{2}}^{a_{3}}(x,i)e_{\mu_{3}}^{a_{4}}(x,i)\cdotse_{\mu_{d}}^{a_{d+1}}(x,i)\,;
\int_{r_{1}}^{r_{2}}dr\sqrt{2\mu(E-V_{\iteff}(r))}=\left(n+\frac{1}{2}+\frac{1}{2}|\alpha|-\frac{1}{2}\sqrt{\alpha^{2}-\textstyle\frac{1}{4}}\right)\pi+\Theta(E),\;\;\;\;\;(n=0,1,2,\cdots)
\beta_{\dot{K}}{}^{L{\dot{M}}}{}{}_{N}=\beta_{\dot{K}}{}^{L}{}_{N}{}^{\dot{M}}=\beta^{L}{}_{\dot{K}}{}^{\dot{M}}{}_{N}=\beta^{L}{}_{\dot{K}}{}_{N}{}^{\dot{M}}=-{\delta^{\dot{M}}}_{\dot{K}}\,{\delta^{L}}_{N}\,.
{\frac{\bar{A}_{1}^{T}(d)}{(4\pi)^{d/2}}}={\frac{\Gamma\left(\fracd2-1\right)}{12\gamma(d-3)!}}\left[d(d-7)+(d-1)\left(\gamma^{2}-1\right)+12\left(1-\gamma\right)\right]+\delta_{d,2}~~~,
\delta\phi_{k}\simeqC_{1}+{\tilde{C}}_{2}\,\int^{\,\eta}\frac{d\eta^{\prime}}{a^{2}}=C_{1}+C_{2}\,\mathrm{arctg}\left(\gamma\,\eta\right)\;\;,\;\;\vert\eta\vert\ll\vert\eta_{i}\vert\,\,.
f^{(n_{0}+1)}(\sigma^{2})=-\sum_{k>0}kC_{k}(f^{\prime}(\sigma^{2}))^{-k}\frac{f^{\prime\prime}(\sigma^{2})}{f^{\prime}(\sigma^{2})}+\sum_{l>0}C_{l}l(f^{\prime}(\sigma^{2}))^{l}\frac{f^{\prime\prime}(\sigma^{2})}{f^{\prime}(\sigma^{2})}
\int\frac{\mathrm{d}\Omega}{4\,\pi}\;\;{\calL}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left(k_{1},k_{2},k_{3},\hat{Q}\right)=-i\;f^{a_{1}\,a_{2}\,a_{3}}\;{\calA}_{\mu_{1}\mu_{2}\mu_{3}}\left(k_{1},k_{2},k_{3}\right).
S={\frac{1}{\pi}}\intd^{2}x[-{\frac{1}{\kappa}}\partial_{+}\chi\partial_{-}\chi+{\frac{1}{\kappa}}\partial_{+}\Omega\partial_{-}\Omega+\lambda^{2}e^{{\frac{2}{\kappa}}(\chi-\Omega)}+{\frac{1}{2}}\sum_{i=1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}]\\,