harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\pi_{\phi}\rightarrow-i\frac{\partial}{\partial\phi}\simeq-i\frac{2M^{2}}{4M^{2}+Q^{2}}\frac{\partial}{\partialM},\pi_{\psi_{1}}\rightarrow-i\frac{\partial}{\partial\psi_{1}},\pi_{\chi}\rightarrow-i\frac{\partial}{\partial\chi},
{\calL}_{NSD}=\frac{q\,\beta^{2}}{p}\,\left(\frac{1}{\beta^{2}}\,A_{\mu}\,A^{\mu}\right)^{p/q}-\chi\;\frac{m}{2}\,\epsilon^{\mu\nu\lambda}\,A_{\mu}\,\partial_{\nu}A_{\lambda}\,,
\rho(z)\;=\;{\frac{\sqrt{(b-z)(z-a)}}{\pi}}\left[{\frac{1}{4\beta}}-{\frac{d-2}{2}}\int_{a}^{b}{\frac{dy}{y+z}}{\frac{\rho(y)}{\sqrt{(b+y)(y+a)}}}\right]\;\;\;\;\;\;\;\;\;\;\mathrm{for}\;\;\;\;\beta>\beta_{c}\;,
\begin{array}{c}{L^{2}\simK^{\otimes_{q}2}=K_{1}R_{21}K_{2}\quad\longmapstoT_{1}T_{2}K^{\otimes_{q}2}T_{1}^{\dagger}T_{2}^{\dagger}\quad,}\\{\,}\\{L^{3}\simK^{\otimes_{q}3}=K_{1}R_{21}K_{2}R_{31}R_{32}K_{3}\quad,}\\{\;}\\{L^{n}\simK^{\otimes_{q}n}=K_{1}\prod_{j=2}^{n}(R_{j1}R_{j2}...R_{j\,j-1}K_{j})\quad.}\\\end{array}
g^{2}({\frac{\partialS}{\partialT}})^{2}=\sum_{i}({\frac{\partialS}{\partial\psi_{i}}})^{2}+({\frac{\rho_{f}+p_{f}}{n_{f}^{2}}})({\frac{\partialS}{\partial\theta_{f}}})^{2}-g(2V_{eff}-\rho_{f}-p_{f})e^{6T},
q=\left(\begin{matrix}{}&{}&{}&{}&{}&{}\\{}&{q_{vv}}&{}&{}&{-{\calI}^{\prime}}&{}\\{}&{}&{}&{}&{}&{}\\{}&{{\calI}^{\prime}}&{}&{}&{0}&{}\\{}&{}&{}&{}&{}&{}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{0}&{1}&{2}&{0}&{-1}\\{0}&{0}&{-2}&{0}&{-4}&{0}\\{-1}&{2}&{0}&{-1}&{0}&{0}\\{-2}&{0}&{1}&{0}&{0}&{0}\\{0}&{4}&{0}&{0}&{0}&{0}\\{1}&{0}&{0}&{0}&{0}&{0}\\\end{matrix}\right)\.
\hat{W}^{\hat{a}\hat{b}\hat{c}}=2K\,\hat{P}^{\hat{a}\hat{b}\hat{c}}-\hat{\calH}^{\hat{a}\hat{b}\hat{c}}=2K\,^{\star}\hat{\calH}^{\hat{a}\hat{b}\hat{c}}-\hat{\calH}^{\hat{a}\hat{b}\hat{c}}\,.
\left(\begin{array}{cccccc}{0}&{1}&{0}&{0}&{0}&{0}\\{1}&{0}&{1}&{0}&{1}&{0}\\{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{1}&{0}&{1}&{0}&{1}\\{0}&{0}&{0}&{0}&{1}&{0}\\\end{array}\right)
{\mathcal{L}}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+\overline{\psi}\,i\gamma_{\mu}(\partial^{\mu}-ieA^{\mu})\psi+{\frac{g^{2}}{2}}\left[(\overline{\psi}\psi)^{2}+(\overline{{\psi}}i\gamma_{5}\psi)^{2}\right].
H(x_{1},x_{2})=\left(\begin{array}{cc}{\bar{q}_{D}(x_{1})q_{C}(x_{2})}&{D_{AB}(x_{1})q_{C}(x_{2})}\\{\bar{q}_{D}(x_{1})\bar{D}_{EF}(x_{2})}&{D_{AB}(x_{1})\bar{D}_{EF}(x_{2})}\\\end{array}\right)
\Sigma_{+}\:\-1+\tilde{\epsilon}\left(\tilde{x}^{3}-\frac{3}{4}\tilde{u}_{0}^{4}\tilde{x}-\frac{1}{4}(\tilde{u}_{0}^{6}+\tilde{u}_{1})-\frac{1}{4}(\zeta+\frac{1}{\zeta})\right)+{\calO}(\tilde{\epsilon}^{2})=0\,
\beta_{1}(a,b)\,\left(\partial_{\mu}\widehat{A}_{\nu}^{a}\right)\left(\partial^{\mu}\widehat{A}^{b\nu}\right)+\beta_{2}(a,b)\,\left(\partial_{\nu}\widehat{A}_{\mu}^{a}\right)\left(\partial^{\mu}\widehat{A}^{b\nu}\right)
{\frac{\ddot{a}_{0}}{a_{0}}}=-{\frac{16\pi^{2}G_{5}^{2}}{9c^{6}}}(2\varrho^{2}c^{4}+\sigma\varrhoc^{2}+3\sigma\wp+3\wp\varrhoc^{2})-{\frac{{\calC}c^{2}}{a_{0}^{4}}}+{\frac{4\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{4\piG_{5}}{3c^{4}}}\sigma^{2}\right),
k_{\;\;ab}^{\beta\;\mu\nu\;\lambda\rho}\partial_{\beta}\left(\eta_{\mu\nu}^{a}\eta_{\lambda\rho}^{b}\right)=\sigma_{\;\;ab}^{\nu\;\lambda\rho}\left(\partial^{\mu}\eta_{\mu\nu}^{a}\right)\eta_{\lambda\rho}^{b}+\sigma_{\;\;ab}^{\mu\nu\;\rho}\eta_{\mu\nu}^{a}\left(\partial^{\lambda}\eta_{\lambda\rho}^{b}\right),
\mathrm{Tr}(L_{1}\cdotsL_{n_{0}+2})=\mathrm{Tr}(L^{+})^{a}(L^{-})^{b}\,+\,\sum\mathrm{Tr}([L^{+},L^{-}]L_{1}^{\prime}\cdotsL_{n_{0}}^{\prime})
\beta_{{\bfk},{\bfp}}={\frac{1}{\sqrt{\omega(k)\omega(p)}}}\int\left(\sqrt{\gamma}n^{\alpha}k_{\alpha}+\omega({\bfp})\right)e^{-i({\bfp}\cdot{\bfx}+k_{\alpha}T^{\alpha}(x))}\,d^{n}x.
2\chi\;\frac{dA}{d\chi}=-1+\mathrm{sign}(\dot{\chi})\;\sqrt{K}\;,\;\;\;\;K\equiv\frac{\Omega}{3}\;\left(1+\frac{2\rho_{0}\chie^{3(1-\gamma)A}}{(\sigma(t)+c)^{2}}\right)\;.
\bar{\calX}_{,i}^{\mu}\bar{\calX}_{,j}^{\nu}{\calG}_{\mu\nu}=\bar{g}_{ij},\;\bar{\calX}_{,i}^{\mu}\eta_{A}^{\nu}{\calG}_{\mu\nu}=0,\;{\eta}_{A}^{\mu}{\eta}_{B}^{\nu}{\calG}_{\mu\nu}=g_{AB}
C|\psi>=\prod_{x=even}C\psi_{x}^{\dagger}e^{-i\sum_{x}A_{x}\frac{(-1)^{x}}{4}}C^{-1}C|0>=\prod_{x=even}\psi_{x+1}e^{-i\sum_{x}A_{x}\frac{(-1)^{x}}{4}}C|0>=|\psi>
f_{D0}^{A}=2\piV_{11}\int_{0}^{\infty}\frac{dt}{t^{5/2}}\sum_{n^{I}\ne0}e^{-\frac{\pi}{t}{\calM}^{2}}=V_{11}\sum_{n^{I}\ne0}\left({\calM}^{2}\right)^{-3/2},\quad{\calM}^{2}=n^{I}g_{IJ}n^{J},
\delta{\calE}_{1}^{1,0}=\frac{8K}{a_{0}^{6}}\left\{\left[\left(\frac{a_{0}}{\Lambda}\right)^{3}-\left(\frac{a_{0}}{\Lambda}\right)^{2}+2\left(\frac{a_{0}}{\Lambda}\right)\right]e^{-2\Lambda/a_{0}}-4E_{1}\left(\frac{2\Lambda}{a_{0}}\right)\right\}\,{,}
{\calT}_{0}^{\mu_{1}\nu_{1}\;\mu_{2}\nu_{2}}={\calT}_{A}^{\mu_{1}\nu_{1}\;\mu_{2}\nu_{2}}+{\calT}_{B}^{\mu_{1}\nu_{1}\;\mu_{2}\nu_{2}}+{\calT}_{C}^{\mu_{1}\nu_{1}\;\mu_{2}\nu_{2}},
\tilde{\rho}_{\pm}[\rho,\phi]=e^{iS_{0}^{(1)}-iS_{0}^{(0)}}\int{\calD}g\:{\calD}g^{*}\bar{\chi}_{R}\bar{\chi}_{0}^{*}\equiv\e^{iS_{0}^{(1)}-iS_{0}^{(0)}}\\tilde{D}_{\pm},
[\phi_{ij}(x),\pi_{kl}(y)]_{x^{+}=y^{+}}=[\phi_{ij}(x),D_{-}\phi_{lk}(y)]_{x^{+}=y^{+}}=\frac{i}{2}(\delta_{ik}\delta_{jl}-\frac{1}{N}\delta_{ij}\delta_{kl})\delta(x^{--}y^{-}).
{\calH}^{(0)}={\frac{1}{2}}\left[{n_{i}}^{\mu}(\Xi)\Pi_{\mu}\right]b(\Xi)\left[{n_{i}}^{\nu}(\Xi)\Pi_{\nu}\right]-{\frac{i}{4}}{n_{i}}^{\mu}(\Xi)b(\Xi){n_{i}}^{\nu}(\Xi)\varepsilon_{\mu\nu}
\epsilon_{\mathrm{FD}}^{(1)}(r)+\epsilon_{\mathrm{CT}}(r)=C_{d}\,r^{n-1}\int\frac{d^{d-1}q}{(2\pi)^{d-1}}\tilde{\sigma}(\vec{q})e^{i\vec{q}\cdot\vec{x}}\int_{0}^{1}d\zeta\frac{\zeta(1-\zeta){\vec{q}\,}^{2}}{\left[m^{2}+\zeta(1-\zeta){\vec{q}\,}^{2}\right]^{2-d/2}}\,,
{\calS}_{class.}\big[\underline{A}_{\mu}-\underline{{\tilde{A}}}_{\mu}-\tilde{\tilde{\underline{A}}}_{\mu}\big]=-{\frac{1}{4}}\intdxTr\big(F_{\mu\nu}^{\,(\,\underline{A}_{\mu}-\underline{{\tilde{A}}}_{\mu}-\tilde{\tilde{\underline{A}}}_{\mu})}F^{\mu\nu\,(\,\underline{A}_{\mu}-\underline{{\tilde{A}}}_{\mu}-\tilde{\tilde{\underline{A}}}_{\mu})}\big)
\betaF=-\frac{4}{3}\beta\intd^{3}x\int_{\mu\sqrt{\xi^{2}-\Omega_{0}^{2}r^{2}}}^{\infty}d\omega\frac{1}{e^{\beta\omega}-1}\frac{\xir}{\sqrt{(\xi^{2}-\Omega_{0}^{2}r^{2})}}\left(\frac{\omega^{2}}{\xi^{2}-\Omega_{0}^{2}r^{2}}-\mu^{2}\right)^{3/2}.
\nabla(V,V_{\bar{I}},\bar{V},\bar{V}_{I})=(V,V_{\bar{J}},\bar{V},\bar{V}_{J})\left(\begin{matrix}{0}&{0}&{0}&{\bar{P}_{I}}\\{0}&{0}&{\bar{P}^{\bar{J}}}&{\bar{P}_{\I}^{\bar{J}}}\\{0}&{P_{\bar{I}}}&{0}&{0}\\{P^{J}}&{P_{\\bar{I}}^{J}}&{0}&{0}\\\end{matrix}\right)
\epsilon^{ijk}\partial_{j}[ve^{\phi}(\delta_{kl}(1+e^{-\phi}B_{i}^{1}B_{i}^{1})-e^{-\phi}B_{k}^{1}B_{l}^{1})B_{l}^{2}-E_{k}^{1}-ve^{\phi}\psiB_{k}^{1}]=0.
\left({\frac{d\tilde{x}}{d\tilde{\lambda}}}\right)^{2}+\left(1+{\frac{\tilde{\calJ}^{2}}{\tilde{x}^{2}}}\right)\left(1-{\frac{\tilde{\mu}}{\tilde{x}^{D-p-4}}}\right)=\tilde{E}^{2}.
\beta_{2}\intdt\left(\frac1C\frac{\partialC}{\partialt}+\frac1C\frac{\sigma^{2}}{2}S^{2}\frac{\partial^{2}C}{\partialS^{2}}-r_{b}(1-\fracSC\frac{\partialC}{\partialS})\right)^{2}\.
\mathrm{\int_{0}^{1}dx\int_{0}^{x}dG\int_{G}^{1-x+G}dy\int_{0}^{x-G}dv=-\int_{0}^{1}dx\int_{0}^{x}dt\int_{1+t-x}^{t}dY\int_{0}^{t}dv~,}
|\Omega_{i_{1}\ldotsi_{n}}^{(n)\,\alpha_{1}\ldots\alpha_{n}}\rangle=\frac{1}{\sqrt{n!}}\left(a_{\alpha_{n}}^{i_{n}}\right)^{\dagger}\ldots\left(a_{\alpha_{1}}^{i_{1}}\right)^{\dagger}|\Omega\rangle\,.
ds^{2}=-\left(r^{2}/l^{2}-\frac{\mu}{r}-1\right)dt^{2}+\frac{dr^{2}}{\left(r^{2}/l^{2}-\frac{\mu}{r}-1\right)}+\frac{r^{2}}{z^{2}}\left(dz^{2}+d\varphi^{2}\right),
\psi({\vec{x}})=\sum_{n_{1},n_{2},n_{3}=-N}^{+N}\frac{1}{X^{3/2}}\sum_{s}\left(a_{\vec{n}}^{s}u_{\vec{n}}^{s}e^{+2\pii{\vec{n}}\cdot{\vec{x}}/X}+b_{\vec{n}}^{s\dagger}v_{\vec{n}}^{s}e^{-2\pii{\vec{n}}\cdot{\vec{x}}/X}\right),
\frac{d^{2}E}{d\omegad\Omega}=2{\calG}\frac{\omega^{D-2}}{(2\pi)^{D-4}}\left(T^{\mu\nu}(\omega,{\bfk})T_{\mu\nu}^{*}(\omega,{\bfk})-\frac{1}{D-2}|T_{\:\:\:\lambda}^{\lambda}(\omega,{\bfk})|^{2}\right)\,.
{\alpha}_{1}=\displaystyle\frac{g^{2}C_{V}}{4\pi},\quad{\alpha}_{2}=-\displaystyle\frac{g^{2}C_{V}}{32\pi},\quad{\beta}_{1}=-\displaystyle\frac{g^{2}C_{V}}{16\pi},\quad{\beta}_{2}=0\,.
\lambda=6{\frac{\kappa^{2}}{\widetilde\kappa^{4}}}\,,~~\Lambda={\frac{4\pi}{\widetilde{M}_{\mathrm{p}}^{3}}}\left[\widetilde{\Lambda}+\left({\frac{4\pi}{3\widetilde{M}_{\mathrm{p}}^{\,3}}}\right)\lambda^{2}\right]\,.
G_{N}(R)\approx{\frac{\kappa_{5}^{4}|\lambda_{0}|}{48\pi}}\left[1-{\frac{81}{2\kappa_{5}^{8}\lambda_{0}^{4}}}(\xi_{+}^{2}-\xi_{-}^{2})^{2}\right]R^{-\frac{k^{2}}{3}}
\intP(x,y)\:P(y,x)\:\eta(x)\:d^{4}x\;=\;\int\frac{d^{4}p}{(2\pi)^{4}}\:\hat{\eta}(p)\left[\int\frac{d^{4}q}{(2\pi)^{4}}\:\hat{P}(p+q)\:\hat{P}(q)\right]\;,
\frac{\delta\Lambda_{(0)}}{\deltaJ_{0}}J_{(2)}+\frac{1}{2}\frac{\delta^{2}\Lambda_{(0)}}{\deltaJ_{0}^{2}}(J_{(1)})^{2}+\frac{\delta\Lambda_{(0)}}{\deltaK_{0}}K_{(2)}+\frac{\delta^{2}\Lambda_{(0)}}{\deltaJ_{0}\deltaK_{0}}J_{(1)}K_{(1)}+\frac{\delta\Lambda_{(1)}}{\deltaJ_{0}}J_{(1)}+\Lambda_{(2)}=0,
\mu^{\lambda_{0}\sigma_{0}}\approxM^{\lambda_{0}\sigma_{0}}+Z_{\;\;\lambda_{1}}^{\lambda_{0}}\bar{D}_{\;\;\beta_{1}}^{\lambda_{1}}\omega^{\beta_{1}\gamma_{1}}\bar{D}_{\;\;\gamma_{1}}^{\sigma_{1}}Z_{\;\;\sigma_{1}}^{\sigma_{0}},
j_{a}^{\mu}(x)={\frac{1}{(2\pi)^{4}}}\intd^{4}ke^{-ik\cdotx}j_{a}^{\mu}(k)={\frac{1}{(2\pi)^{4}}}\intd^{4}ke^{-ik\cdotx}\sum_{n=1}^{\infty}j_{n,a}^{\mu}(k),
\sum_{\sigma=0,\pm1}\bigg(u_{\sigma}(\vec{p}\,)\,{\overline{u}}_{\sigma}(\vec{p}\,)+v_{\sigma}(-\,\vec{p}\,)\,{\overline{v}}_{\sigma}(-\,\vec{p}\,)\bigg)\,=\,\left(\begin{array}{cc}{{\calM}}&{0}\\{0}&{-{\calM}}\\\end{array}\right)\quad,
\Lambda_{L}=\left(\textstyle{\frac{6}{\mu}E_{0},\,{-\frac{1}{2}J_{z}}\,,\,{-\frac{1}{2}(J_{u}+2J_{v}+J_{w})}\,,\,{-\frac{1}{2}(J_{u}+J_{w})}\,,\,{-\frac{1}{2}(J_{u}-J_{w})}}\right)
\begin{array}{lcr}{{\sum_{p,q}}L_{p,q}\partial_{\sigma}^{2}X_{p,q}=\sum_{\stackrel{m_{i},n_{i}}{i=1,2,3}}8{\epsilon_{123}}F_{(123)}{\omega}(\{m_{i}\},\{n_{i}\})X_{{\sum{m_{i}}},{\sum{n_{i}}}}(\sigma)L_{\sum{m_{i}},\sum{n_{i}}}.}\\\end{array}
e^{-1}\mathcal{L}=\frac{1}{2}R+g^{2}V-\frac{1}{4}G_{IJ}F_{\mu\nu}^{I}F^{J\mu\nu}-\frac{1}{2}\mathcal{G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+\frac{e^{-1}}{48}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}\,,
A~=~{\frac{1}{2}}\;(\;X_{1}^{\dagger}~X_{1}~+~X_{2}^{\dagger}~X_{2}\;),~~B~=~{\frac{1}{3}}~\sqrt{X_{3}^{\dagger}~X_{3}},~~x~=~4~\sqrt{B}\;\mathrm{cos}\left(~{\frac{1}{3}}\;\mathrm{Arccos}~{\frac{~A~}{B^{3/2}}}~\right)\;.
\begin{array}{cc}{Z_{-}(z)=C^{-1}\bar{Z}_{+}^{t}(z)\left(\begin{array}{cc}{1}&{0}\\{0}&{\bar{\calE}}\\\end{array}\right)~~}&{~~\bar{Z}_{-}(z)=\left(\begin{array}{cc}{1}&{0}\\{0}&{-\bar{\calE}}\\\end{array}\right)Z_{+}(z){}^{t}C}\\\end{array}
\Phi^{I}=P^{I}+\frac{2i}{\sqrt{2p^{+}}}\left(\,(p^{+}-hP^{-})\delta^{IJ}+\frac{1}{2}hX^{-}s^{IJ}\,\right)\psi^{J}~\approx~0+{\calO}(\psi^{3})~,
r_{1\mu\nu}[F(z)]=\frac{\pi}{2}\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{k}=0}\left[F(z)\sum_{l=1}^{2}\frac{H_{\mu}^{(l)}(\lambdaz)}{H_{\nu}^{(l)}(z)}\right]+\pi\sum_{k}\sum_{l=1}^{2}{\mathrm{Res}}_{(-1)^{l}{\mathrm{Im}}z_{k}<0}\left[F(z)\frac{H_{\mu}^{(l)}(\lambdaz)}{H_{\nu}^{(l)}(z)}\right].
\times\prod_{a=1}^{N}\Big[c^{(a)}\Big]^{n_{a}+m_{a}}\;\;\prod_{b=1}^{N}\left[\left(\frac{2e^{-\gamma}}{M^{(1)}}\right)^{\frac{\pi}{\pi+gN}\frac{1}{N}}\prod_{I=2}^{N}\Big(\frac{2e^{-\gamma}}{M^{(I)}}\Big)^{(U_{Ib})^{2}}\right]^{n_{b}+m_{b}}\;.
m=\langle\sigma_{0}\rangle=\frac{e^{h}g_{n}^{\gamma}(+)-e^{-h}g_{n}^{\gamma}(-)}{e^{h}g_{n}^{\gamma}(+)+e^{-h}g_{n}^{\gamma}(-)}=\frac{e^{h}x_{n}^{\gamma}-1}{e^{h}x_{n}^{\gamma}+1}.
\omega^{(S)-1}(x)\{\tilde{A}_{i}^{(S)}(x)-S(x)\tilde{A}_{i}^{(N)}(x)S^{-1}(x)\}\omega^{(S)}(x)=2\sqrt{6}\{{\calT}_{i}^{(N)}(x)-{\calT}_{i}^{(S)}(x)\},
\widehat{B}=\frac{1}{6}\hat{e}^{\hat{r}}\wedge\hat{e}^{\hat{s}}\wedge\hat{e}^{\hat{t}}\widehat{C}_{\hat{r}\hat{s}\hat{t}}-\int_{0}^{1}dt\;\bar{\theta}\Gamma_{\hat{r}\hat{s}}\widehat{E}\wedge\widehat{E}^{\hat{r}}\wedge\widehat{E}^{\hat{s}}~,
\delta(v-\tau\sum_{j=1}^{K}\rho_{j}^{d})\approx\delta(-\taud\rho_{0}^{d-1}\sum_{j=1}^{K}\delta_{j})=\frac{\rho_{0}^{1-d}}{2\pi\taud}\int\!\!dq\\mathrm{e}^{iq\sum_{j=1}^{K}\delta_{j}}
{\tilde{\cal{H}}^{(1)}}=\eta^{1}\left[2\Theta_{3}-\phi^{\prime\prime}-\pi^{\prime}-2(A_{1}^{\prime}+E)\right]-\eta^{2}(2\Theta_{2}+\Theta_{4})+\eta^{3}\Theta_{3}-\eta^{4}\Theta_{2}
\widehat{\theta}_{ab}^{\alpha}=\frac{1}{\alpha!}\frac{i^{\alpha-1}}{2^{\alpha}}\sum_{i=1}^{\alpha}\{2\theta_{ab}\prod_{k\nei}(\theta^{\mu_{k}\nu_{k}}\partial_{\mu_{k}}\partial_{\nu_{k}})+(x_{a}\theta^{\mu_{i}}\,_{b}-x_{b}\theta^{\mu_{i}}\,_{a})\prod_{j\nei}(\theta^{\mu_{j}\nu_{j}}\partial_{\nu_{j}})\prod_{k=1}^{\alpha}\partial_{\mu_{k}}\}.
\left\langle\tilde{B}_{2}\tilde{B}_{5}\tilde{B}_{7}\tilde{B}_{9}\tilde{B}_{15}\right\rangle=\left\langle\tilde{B}_{2}\tilde{B}_{5}\tilde{B}_{7}\right\rangle\left\langle\tilde{B}_{9}\tilde{B}_{15}\right\rangle
\theta\left(\begin{array}{c}{\psi_{+}}\\{\psi_{-}}\\\end{array}\right)=D^{(s)}(i\sigma_{2})\left(\begin{array}{c}{\psi_{-}^{*}}\\{\psi_{+}^{*}}\\\end{array}\right),\,\,\,\,\,\pm\mathrm{=(anti)particle\thinspace~\thinspace~doubling}
+{\frac{1+g^{2}{\frac{p^{2}}{M^{2}}}}{g^{2}}}\left(arctan{\frac{1-g^{2}{\frac{m_{0}^{2}}{M^{2}}}}{g^{2}}}-arctan{\frac{1-g_{2}^{2}{\frac{\Lambda^{2}}{M^{2}}}}{g^{2}}}\right)
J_{C_{(p-1)}}=\frac{{\sqrt2}N_{p}(F)k(F)}{(p-1)!}\,V_{p+1}\,\frac{T_{p}}{2}\,C_{\alpha_{0}\ldots\alpha_{p-2}}\,{\hat{F}}_{\alpha_{p-1}\alpha_{p}}\,\varepsilon^{\alpha_{0}\ldots\alpha_{p}}~~.
Z_{f}=\left|\!\left|{\calD}{\calD}^{\dagger}\right|\!\right|=\left|\!\left|(i\partial_{\tau}-\frac{i}{2}(1-z)\dot{U}U^{+})(i\partial_{\tau}-\frac{i}{2}(1-z^{*})\dot{U}U^{+})\right|\!\right|,
\overline{{\psi}}^{(2)}(x)P_{+}\psi^{(2)}(x)\;\longleftrightarrow\;\frac{1}{2\pi}c^{(2)}:e^{-ia\Phi^{(1)}(x)}:_{M^{(1)}}:e^{+ib\Phi^{(2)}(x)}:_{M^{(2)}}\;e^{i\frac{\theta}{2}}\;,
{\calA}^{*}({\calE})=\sum_{i}\sum_{j}\int_{-\infty}^{\infty}d\tau\,e^{-i{\calE}\tau}\,u_{i}[{\tilde{x}}(\tau)]\,v_{j}^{*}[{\tilde{x}}(\tau)]\;\langle0\vert{\hat{a}}_{i}{\hat{b}}_{j}\vert\Psi\rangle.
\delta_{\alpha{\mathbfi}}J_{\beta{\mathbfj}}^{\mu}=\partial_{\nu}\left\{24i\varepsilon_{\alpha\beta}\varepsilon_{{\mathbfi}{\mathbfj}}Z^{\mu\nu}+96(a+24)\left[\varepsilon_{{\mathbfi}{\mathbfj}}\sigma_{(\alpha}^{\mu\nu\,\gamma}D_{\beta)\gamma}T|+\sigma_{\alpha\beta}^{\mu\nu}D_{{\mathbfi}{\mathbfj}}T|\right]\right\}
\{2+(D-2)d\}S[\Omega]=\int_{\Omega}d^{D}x\partial_{\mu}\tilde{J}^{\mu}=\tilde{Q}(T_{f})-\tilde{Q}(T_{i}),~~~\tilde{Q}\equiv\intd^{D-1}x~\tilde{J}^{0}(x).
f_{n}(\mu)=\frac{{\mu^{*}}^{n}e^{-\frac{1}{2}\mu^{*}\mu}}{2^{n+1}\pi|\mu|^{(2j+k-1)/2}}\sqrt{\frac{\Gamma(2j+k+n)}{\Gamma(n+1)}I_{2j+k-1}(2|\mu|)}
N_{S}(x,x^{\prime})=\left(\frac12\right)\int^{\prime}\frac{d^{3}k}{\left(2\pi\right)^{3}\Omega_{k}}\frac{d^{3}k^{\prime}}{\left(2\pi\right)^{3}\Omega_{k^{\prime}}}\;e^{i(k+k^{\prime})(x-x^{\prime})}\Theta_{kk^{\prime}}(t,t^{\prime})
(\xi_{j}T_{[r_{1}]}^{(n_{1})})(\xi^{j}T_{[r_{2}]}^{(n_{2})})\;\;,\;\;\;\;\;(\xi_{j}T_{[r_{1}]}^{(n_{1})})(\overline{{\xi^{j}T_{[r_{2}]}^{(n_{2})}}})\;\;\;\;\;{\mathrm{or}}\;\;\;\;\;(\overline{{\xi_{j}T_{[r_{1}]}^{(n_{1})}}})(\overline{{\xi^{j}T_{[r_{2}]}^{(n_{2})}}})\;.
\Pi(p^{2},\mu)=-\mu^{4-d}\sum_{\{x_{n}\}}\int_{0}^{1}\!\!dx\;(2x-1)^{2}\int\frac{d^{d}q}{(2\pi)^{d}}\,\frac{1}{[q^{2}+x_{n}^{2}+c^{2}(x)]^{2}}\;.
\zeta(s\midD_{0}D_{0}^{\dagger})=\frac{2|k||\tau|^{s}}{(4\pi|k|)^{s}}\sum_{n=1}^{\infty}\frac{1}{n^{s}}=\frac{2|k||\tau|^{s}}{(4\pi|k|)^{s}}\zeta_{R}(s),
X^{m}=\left(\begin{array}{ccc}{(v_{1}^{m}\beta+r_{1}^{m})\otimes1_{N_{1}\timesN_{1}}}&{0}\\{0}&{(v_{2}^{m}\beta+r_{2}^{m})\otimes1_{N_{2}\timesN_{2}}}&{0}\\{0}&{\dots}&{(v_{K}^{m}\beta+r_{K}^{m})\otimes1_{N_{K}\timesN_{K}}}\\\end{array}\right)
{\calL}=\mathrm{Tr}\;\left[\frac{1}{2R}\dot{X}^{a}\dot{X}_{a}+\frac{R}{8\pi^{2}}\sum_{a<b}[X^{a},X^{b}]^{2}+\frac{1}{4\pi}\theta^{T}(i\dot{\theta}-\frac{R}{2\pi}\;\Gamma_{a}[X^{a},\theta])\right].
M_{pl}^{2}={\frac{sign(V_{0})}{k}}M_{X}^{3}\left({\frac{1}{2}}e^{-2\sigma(y_{0})}+\sum_{i=1}^{n-1}(-1)^{i}e^{-2\sigma(y_{i})}+{\frac{(-1)^{n}}{2}}e^{-2\sigma(y_{n})}\right)~.~\,
\Omega_{i}\;=\;-\textstyle{\frac{1}{4\sqrt{3}}}\gamma^{+}\left[\lambda_{j}\gamma^{j}\gamma_{i}+\lambda_{i}\right]\;,\;\Omega^{-}\;=\;\textstyle{\frac{1}{4\sqrt{3}}}\lambda_{i}\gamma^{i}\left(\gamma^{+}\gamma^{-}+1\right)\;.
\begin{aligned}{(i,j)\;\in\;\{}&{{}(1,2)\;,\;(1,6)\;,\;(2,3)\;,\;(2,5)\;,\;(3,4)\;,\;(3,5)}\\{}&{{}(4,1)\;,\;(4,6)\;,\;(5,2)\;,\;(5,1)\;,\;(6,3)\;,\;(6,4)\;\}.}\\\end{aligned}
\delta\lambda_{i}=\Big({\frac{1}{3(Z_{m})_{\mathrm{cr}}^{2}}}\partial_{i}(q^{I}X_{I})\Gamma_{34}+{\frac{3i}{4(Z_{m})_{\mathrm{cr}}}}r(\partial_{r}X_{I}\partial_{i}X^{I})\Gamma_{2}\Big)\epsilon
\psi_{A}^{(\alpha\beta)}(x)\,\psi_{B}^{(\beta\gamma)}(y)\\sim\\sum_{C}\,(x-y)^{h_{C}-h_{A}-h_{B}}\,C_{AB}^{(\alpha\beta\gamma)\:C}\psi_{C}^{(\alpha\gamma)}(y)\quad\mathrm{~for~}\quadx<y
\begin{array}{rcl}{E_{0}=2\hbarg\omega\}&{:}&{\\\|E_{0},p=0>=|0>\\\,}\\{E_{1}=4\hbarg\omega\}&{:}&{\\\|E_{1},p=\pm1>={\alpha_{\pm}^{+}}^{\dagger}\,{\alpha_{\pm}^{-}}^{\dagger}\,|0>\\\,}\\{}&{}&{\\\|E_{1},p=0,\sigma=\pm1>=\frac{1}{\sqrt{2}}\,\left({\alpha_{+}^{+}}^{\dagger}\,{\alpha_{-}^{-}}^{\dagger}\,\pm\,{\alpha_{-}^{+}}^{\dagger}\,{\alpha_{+}^{-}}^{\dagger}\right)\,|0>\\\,}\\\end{array}
\frac{3[S_{0}^{\prime}]}{S_{0}}-4\alpha\frac{[{S^{\prime}}_{0}^{3}]}{S_{0}^{3}}+12\alpha\frac{k[S_{0}^{\prime}]}{S_{0}^{3}}+12\alpha\frac{[S_{0}^{\prime}]\dot{S}_{0}^{2}}{N_{0}^{2}S_{0}^{3}}=-\rho
\rho(x)=\frac{2}{x^{2}}+2\sum_{q\neq0}\{\frac{1}{(x-q)^{2}}-\frac{1}{q^{2}}-\frac{2q\cdotx}{q^{4}}-\frac{1}{q^{6}}(4(q\cdotx)^{2}-q^{2}x^{2})\}~.
T_{\frac{1}{2}\frac{3}{2}}(\lambda)(\lambda+\frac{i}{2})^{N}(\lambda-\frac{i}{2})^{N}=P_{\frac{3}{2}}(\lambda+i)Q_{\frac{3}{2}}(\lambda-i)-P_{\frac{3}{2}}(\lambda-i)Q_{\frac{3}{2}}(\lambda+i).
{\calS}=\frac14(\tau^{+})\intd^{4}x{}\epsilon^{\mu\nu\alpha\beta}{}^{+}{\calF}_{\mu\nu}^{\tau\lambda}{}^{+}{\calF}_{\alpha\beta}^{\sigma\rho}{}\epsilon_{\tau\lambda\sigma\rho}-\frac14(\tau^{-})\intd^{4}x{}\epsilon^{\mu\nu\alpha\beta}{}^{-}{\calF}_{\mu\nu}^{\tau\lambda}{}^{-}{\calF}_{\alpha\beta}^{\sigma\rho}{}\epsilon_{\tau\lambda\sigma\rho},
\Gamma_{\mu}=\int{\frac{d^{3}k}{(2\pi)^{3}}}{\frac{(2p-k)_{\lambda}(2p-2q-k)_{\nu}(2p-2k-q)_{\mu}}{\left[(p-k)^{2}-M^{2}\right]\left[(p-k-q)^{2}-M^{2}\right]}}G_{\lambda\nu}.
{\calG}_{nh}={\calE}[\varphi=0,h=h_{0}]-{\frac{2\pi\mu^{2}}{\lambda}}\int_{0}^{\infty}dr\,r\,h_{0}h_{\mathrm{ext}}={\frac{2\pi\mu^{2}}{\lambda}}\int_{0}^{\infty}dr\,r\,{\frac{1}{2}}\left[-h_{0}^{2}+{\frac{\lambda}{2}}\right].
\begin{array}{c}{\Omega=d{\bfp}\wedged{\bfx}+d{\bfp}_{+}\wedged{\bfe}_{+},}\\{{\calH}=(2\alpha)^{-1}\left({\bfp}_{+}^{2}+({\bfp}{\bfe}_{+}-2\alpha)^{2}+{\bfp}^{2}{\bfe}_{+}^{2}\right),}\\\end{array}
[\Psi_{p}({\bfr}^{\prime},t),a_{i}^{I}({\bfr},t)]=\epsilon^{ij}\nabla_{j}\frac{1}{\kappa_{I}}q_{p}^{I}G({\bfr}-{\bfr}^{\prime})\Psi_{p}({\bfr}^{\prime},t),
\left\langleN_{\nu_{\alpha_{1}}}^{(\alpha_{1})},\cdots,N_{\nu_{\alpha_{n}}}^{(\alpha_{n})}\right\vert\left.N_{\nu_{\alpha_{1}}}^{(\alpha_{1})},\dots,N_{\nu_{\alpha_{n}}}^{(\alpha_{n})}\right\rangle={\calA}^{\{{\calN}\}}\,.
\mathrm{Tr}\,d\Phi\,d\Phi^{\dagger}=\sum_{i}d\lambda_{i}^{2}+\sum_{\alpha,\beta}dt_{\alpha}\,dt_{\beta}\,\left(\sum_{i}\lambda_{i}\alpha_{i}\right)\left(\sum_{j}\lambda_{j}\beta_{j}\right)\,\mathrm{Tr}\,E_{\alpha}E_{\beta}.
{\calV}^{\prime\prime}+\left[\frac{\eta_{c}q^{2}}{(\eta_{c}-\eta)}+\frac{1}{4(\eta_{c}-\eta)^{2}}\right]{\calV}=0,~~~{\calW}^{\prime\prime}+\left[\frac{\eta_{c}q^{2}}{(\eta_{c}-\eta)}+\frac{1}{4(\eta_{c}-\eta)^{2}}\right]{\calW}=0
T_{\mathrm{scattering}}=-\frac{\alpha}{4\pi^{2}}\frac{1}{\sqrt{x(1-x)x^{\prime}(1-x^{\prime})}}\left(\frac{<j^{\mu}J_{\mu}>_{ex}}{Q^{2}}+\frac{<c^{\mu}C_{\mu}>_{an}}{M^{2}}\right)\,,
\dot{r}^{2}=\frac{A}{B}(1-\frac{A}{(C+E)^{2}}\frac{D+\ell^{2}}{D}),h_{ij}\dot{\varphi}^{i}\dot{\varphi}^{j}=\frac{A^{2}\ell^{2}}{D^{2}(C+E)^{2}}
H_{nm}^{\{t\sigma_{x}\}}(y_{1},y_{2})=\mu_{mn}!t^{\nu_{mn}}(-1)^{\mu_{mn}}y_{1}^{\frac{(m-n+|m-n|)}2}y_{2}^{\frac{(n-m+|n-m|)}2}L_{\mu_{mn}}^{|m-n|}(ty_{1}y_{2}),
\mu_{0}+\mu_{\rho}+2\mu_{\theta}=\int_{0}^{\infty}drM(r)^{4}\mathcal{L}(r)^{2}\left[J^{\prime}(r)^{2}+\frac{2\,J(r)^{2}K(r)^{2}}{\mathcal{L}(r)^{2}}+\alpha\left(J(r)^{2}-1\right)^{2}\right]\,.
\intd^{10}y\;d^{16}\epsilon_{0}\;\langle\psi^{1}(x_{1})\cdots\psi^{n}(x_{n})\rangle_{N}=e^{2\piiN\rho}\intd^{10}y\;d^{16}\epsilon\;\langle\psi\rangle_{s_{1}}\cdots\langle\psi\rangle_{s_{n}},
\begin{array}{ll}{C_{p}^{l}(\tau)_{m^{\prime},m^{\prime\prime}}=}&{\frac{1}{M}\sum_{k_{l+1}^{m}}e^{\tau(\epsilon^{l}(k_{l+1}^{m})-\mu)+ik_{l+1}^{m}(m^{\prime}-m^{\prime\prime})}}\\{}&{+\frac{q^{p}}{M}\sum_{k_{l}^{m}}e^{(\tau-\beta)(\epsilon^{l}(k_{l}^{m})-\mu)+ik_{l}^{m}(m^{\prime}-m^{\prime\prime})}(1+(q^{-1}-1)\theta(L-m^{\prime}))}\\\end{array}
\{f,h\}=\int_{S^{1}}d\varphi\mathrm{tr}\left(\frac{\deltaf}{\delta\phi}\frac{\deltah}{\deltaA}-\frac{\deltaf}{\deltaA}\frac{\deltah}{\delta\phi}\right)+\frac{\deltaf}{\deltac}\frac{\deltah}{\deltak}-\frac{\deltaf}{\deltak}\frac{\deltah}{\deltac}.
\left\{L_{\alpha},L_{\beta}\right\}=2i\left[(\hat{p}^{\mu}+\Pi_{1}^{\mu}){(C\Gamma^{\mu}S^{+})}_{\alpha\beta}-(\hat{p}^{\mu}-\Pi_{1}^{\mu}){(C\Gamma^{\mu}S^{-})}_{\alpha\beta}\right],
B_{m}(u)=\varphi(u)\frac{(t^{2}\omega^{m}u^{2/n};t^{2})_{\infty}(t^{2}\omega^{-m}u^{-2/n};t^{2})_{\infty}}{(q^{2}\omega^{m}u^{2/n};q^{2})_{\infty}(q^{2}\omega^{-m}u^{-2/n};q^{2})_{\infty}},
\int\mathrm{Tr}P\left[i_{\Phi}i_{\Phi}C^{(3)}\right]=\intdt\mathrm{Tr}\left[\Phi^{j}\Phi^{i}(C_{ijt}^{(3)}(\Phi,t)+(2\pi\alpha^{\prime})C_{ijk}^{(3)}(\Phi,t){\calD}_{t}\Phi^{k})\right]\,.
\left\{Q_{r},Q_{s}\right\}=\left(C^{(13)}\,\Gamma_{MN}^{(13)}\right)_{rs}\,Z_{\bf0}^{MN}+\left(C^{(13)}\,\Gamma_{M_{1}M_{2}M_{3}}\right)_{rs}Z_{\bf0}^{M_{1}M_{2}M_{3}}\,,
\left|<i_{l+1}^{1},k_{l+1}^{1},i_{l+1}^{2},k_{l+1}^{2}|e^{-\frac{\varepsilon}{2}H_{1,2}}|i_{l}^{1},k_{l}^{1},i_{l}^{2},k_{l}^{2}>\right|.
S=\intd^{3}x\left[-\left(\frac{\partial\phi^{A}}{\partialx^{i}}G_{AB}[\phi]\frac{\partial\phi^{B}}{\partialx^{j}}\right)+\frac{i}{2\kappa}\left|\frac{\partial\phi^{A}}{\partialx^{i}}\right|\sqrt{G[\phi]}R[\phi]\right],
H_{0}=\frac{1}{2}\intd^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left(\partial^{i}B_{0i}^{a}\right)\left(\partial_{j}B_{a}^{0j}\right)\right)\equiv\intd^{3}x\,h.
U_{c}N_{c}^{\mathrm{gh}}=(N_{1}+N_{2}-3)U_{1}GPU_{2}^{\dagger}+U_{1}\left[G,N_{1}^{\mathrm{gh}}\right]PU_{2}^{\dagger}-U_{1}N_{E}^{\mathrm{gh}}GPU_{2}^{\dagger}+U_{1}GPN_{F}^{\mathrm{gh}}U_{2}^{\dagger}\.
\varepsilon^{\rho\sigma\mu\nu}\varepsilon_{\rhoabc}=-2(\delta_{[a}^{\sigma}\delta_{b]}^{\mu}\delta_{c}^{\nu}+\delta_{[a}^{\mu}\delta_{b]}^{\nu}\delta_{c}^{\sigma}+\delta_{[a}^{\nu}\delta_{b]}^{\sigma}\delta_{c}^{\mu})
L_{VVW}=L_{0}(U,U^{+})+L(A_{\mu},\phi,\theta)+{\frac{1}{8}}i\epsilon_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}Tr[logU+logU^{+}]+{\frac{f_{\pi}}{2\sqrt{2}}}Tr[MU+MU^{+}],
T^{\rho\sigma}={\frac{\rho_{_\mathrm{S}}}{{\mu_{_\mathrm{N}}}^{2}}}\mu^{\rho}\mu^{\sigma}+{\frac{\rho_{_\mathrm{N}}}{{s_{_\mathrm{S}}}^{2}}}s^{\rho}s^{\sigma}+\Psig^{\rho\sigma}\,
I_{N}(p;\{\nu\})=i\left(\prod_{i=1}^{N}\int\frac{i\Gamma(\nu_{i})}{(p_{i}^{2}-m_{i}^{2})^{\nu_{i}}}\bar{d}^{2\ell}\!p_{i}\right)\bar{\delta}^{2\ell}(p-\sum_{i}p_{i}).
\|\widetilde{c}_{jl}\|=\left(\begin{array}{cccc}{1}&{\widetilde{c}^{\prime}}&{\widetilde{c}^{\prime}}&{\widetilde{c}^{\prime}}\\{\widetilde{c}^{\prime}}&{1}&{1/2}&{1/2}\\{\widetilde{c}^{\prime}}&{1/2}&{1}&{1/2}\\{\widetilde{c}^{\prime}}&{1/2}&{1/2}&{1}\\\end{array}\right),\quad\widetilde{c}^{\prime}=-\sqrt{\frac{1+3c}{2(1+2c)}}.
\chi_{top}\;:=\left(\frac{g^{2}}{16\pi^{2}}\right)^{2}\;\;\intd^{4}x\;\langleT\;q(x)\;q(0)\rangle\;\;,\;\;q(x)\;:=\;\mathrm{Tr}\Big(F(x)\tilde{F}(x)\Big)\;,
[\hat{x}_{0}^{\mu},\hat{p}_{0\nu}]=i\delta_{\nu}^{\mu}~,\quad[\hat{\phi}_{m}^{(i)\mu},\hat{\pi}_{n\nu}^{(j)}]=i\delta^{\mu\nu}\delta^{i,j}\delta_{m,n}~,\quad[\hat{\phi}_{m}^{(i)\mu},\hat{\varpi}_{n\nu}^{(j)}]=i\delta_{\nu}^{\mu}~,
{\frac{\sigma_{k}}{\sigma}}={\frac{k(N-k)}{N-1}}\left\{1+{\frac{D-2}{(g^{2}N)^{4}}}\left[e_{4}(k,N)-e_{4}(1,N)\right]+O\left({\frac{1}{(g^{2}N)^{6}}}\right)\right\}.
f\circ\mathrm{V}_{B}(w,k)=\left[f^{\prime}(w)\right]^{\alpha^{\,\prime}k^{2}}\mathrm{V}_{B}(f(w),k)-\left[f^{\prime}(w)\right]^{\alpha^{\,\prime}k^{2}}\frac{f^{\,\prime\prime}}{f^{\,\prime\,2}}c(f(w))e^{2ik\cdotX(f(w))}.