harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N_{2}=\phi_{1}^{2}g^{2}I_{2},\quad\mathrm{where~}\quadI_{2}=\frac{1}{16\pi^{3}}\int\frac{\sqrt{s}}{(s_{12}-m^{2})^{2}}\frac{d^{3}q}{\varepsilon(q,m)\varepsilon(q,\mu)}.
~~=\left(\begin{array}{cc}{u_{2}d_{1}^{2}-u_{1}d_{1}d_{2}}&{-u_{1}u_{2}d_{1}+u_{1}^{2}d_{2}}\\{u_{2}d_{1}d_{2}-u_{1}d_{2}^{2}}&{u_{1}u_{2}d_{2}-u_{2}^{2}d_{1}}\\\end{array}\right)
C_{(h_{i},h_{j};k_{i}+k_{j})}^{(h;k)}C_{(h,h_{l},h_{m};k+k_{l}+k_{m})}^{\vphantom{(}}=C_{(h_{i},h_{l};k_{i}+k_{l})}^{(h^{\prime};k^{\prime})}C_{(h^{\prime},h_{j},h_{m};k^{\prime}+k_{j}+k_{m})}^{\vphantom{(}}
f(x,\theta)=f_{2}(\theta)\left(\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)^{2}\left(1-\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)+O\left(\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)^{4}\,
{\calZ}(a_{1},a_{2},\dots,a_{N})~=~\int\prod_{\mu}{\calD}P^{\mu}e^{\sum_{i}[a_{i}\sum_{\mu}(p_{i}^{\mu})^{2}]}\prod_{\mu;i,j}\delta((1-D_{ij})P_{ij}^{\mu})\prod_{\mu;i}\delta(\sum_{j}P_{ij}^{\mu})~~,
{\calL}_{fermion}~=~\bar{\psi}~\left[i\gamma^{\mu}~\left(\partial_{\mu}~-~\sigma^{\rho\beta}v_{\rho}^{\nu}g_{\lambda\nu}\partial_{\mu}v_{\beta}^{\lambda}~-~g_{\alpha\delta}\sigma^{ab}v_{a}^{\beta}v_{b}^{\delta}\tilde{\Gamma}^{\alpha}\,_{\mu\beta}\right)\right]~\psi
\left.\begin{array}{lll}{\lambda_{1}=\gamma_{1,3,(0,0,0)}\lambda_{1}\gamma_{1,3,(0,0,0)}^{-1},\,\,\,}&{\lambda_{j}=\alpha\gamma_{1,3,(0,0,0)}\lambda_{j}\gamma_{1,3,(0,0,0)}^{-1}}&{(j=2,3,4)}\\\end{array}\right.
h_{\alpha}=\frac{1}{2}p_{r\alpha}^{2}-\frac{\hbar^{2}}{8r_{\alpha}^{2}}+\frac{\hbar^{2}\ell_{\alpha}^{2}}{2r_{\alpha}^{2}}+\frac{1}{2}p_{z\alpha}^{2}+\frac{1}{2}\omega^{2}r_{\alpha}^{2}\\\,\\\\gamma(\xi_{\alpha})=\epsilon\sum_{\alpha=0}^{N-1}\xi_{\alpha}\\\,
{\calD}_{\mu}\psi_{\nu}=\left((\partial_{\mu}+{\frac{1}{4}}\omega_{\mu}^{mn}\gamma_{mn}+\mathrm{\frac{i}{2}}\gamma_{5}A_{\mu})\delta_{\nu}^{\lambda}-\Gamma_{\mu\nu}^{\lambda}\right)\psi_{\lambda}\,.
\left({\calP}^{2}-m^{2}-\fracq2\sigma^{\mu\nu}{\calF}_{\mu\nu}\right)\phi(x)=0\;,\;\;\;{\calF}_{\mu\nu}=\partial_{\mu}A_{\nu}(x)-\partial_{\mu}(x)\;,\;\sigma^{\mu\nu}=\fraci2[\gamma^{\mu},\gamma^{\nu}]\;.
x_{d}(t)=c\,{\frac{\mathrm{cn}(ct,k)}{\mathrm{sn}(ct,k)}},\quady_{d}(t)=-c^{2}{\frac{\mathrm{dn}(ct,k)}{\mathrm{sn}^{2}(ct,k)}},\quadz_{d}(t)=-\wp(t),
y_{c}^{\prime}=\stackrel{-}{(+)}\frac{{r_{0}}^{p-1}}{\sqrt{r^{2p-2}-{r_{0}}^{2p-2}}},\;\;\;\sqrt{1+{y_{c}^{\prime}}^{2}}=\stackrel{-}{(+)}\frac{r^{p-1}}{\sqrt{r^{2p-2}-r_{0}^{2p-2}}}
\frac{1}{M_{PV}}\,\frac{1}{8\pi^{2}v^{2}\rho^{2}}\,\mathrm{d}^{2}\bar{\zeta}=\frac{1}{M_{PV}}\,\frac{1}{8\pi^{2}v^{2}\rho_{\mathrm{inv}}^{2}}\,\mathrm{d}^{2}\bar{\theta}_{0}\,.
l=-\frac{imk(cos\,\mu\,sin\,\lambda+e^{2iak}cos\,\lambda\,sin\,\mu)+mE(e^{2iak}-1)sin\,\lambda\,sin\,\mu}{m^{2}(e^{2iak}-1)sin\,\lambda\,sin\,\mu+k^{2}cos(\lambda+\mu)+iEk\,sin(\lambda+\mu)}
\left.|k|^{2}\right|_{\Phi_{H}}=\left.-\ell^{2}|\nabla\Phi|^{2}\right|_{\Phi_{H}}=\left.2\ellM-{\frac{\delta^{2}}{2\delta^{2}-\gamma}}\Phi^{2-{\frac{\gamma}{\delta^{2}}}}\right|_{\Phi_{H}}=\bar{g}_{\tau\tau}(\Phi_{H})=0.
{}~~~~~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{3},x_{13},x_{1}|r_{1}+\sigma)w(x_{4},x_{24},x_{2}|r_{2}+\sigma)}{\omega^{\sigmar_{5}}w(x_{4},x_{14},x_{1}|r_{3}+\sigma)w(x_{3}/\omega,x_{23},x_{2}|r_{4}+\sigma)}\Bigg\}_{0}
[\vec{\nabla^{2}}+\frac{1}{R^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+K^{2}-V(x,\phi)]G(K;\vec{x},\vec{x^{\prime}};\phi,\phi^{\prime})=\delta^{3}(\vec{x}-\vec{x^{\prime}})\delta(\phi-\phi^{\prime}),
f^{(n,0)}(r_{1}...r_{n-2},r_{n-1}+2,r_{n})\left(\frac{(r_{n}+1)^{2}}{(2r_{n}+3)(2r_{n}+1)}-\frac{(r_{n-1}+1)^{2}}{(2r_{n-1}+3)(2r_{n-1}+1)}\right)=0.
\mathrm{S}=\mathrm{S}_{0}+\mathrm{S}_{\mathrm{int}}={\frac{1}{2}}\int_{\Gamma^{*}}d\tau\left[{\frac{\omega_{\tau}^{\mu}\omega_{\tau\mu}}{g_{\tau}}}+g_{\tau}m^{2}\right]+i\int_{\Gamma^{*}}d\tau\,\omega_{\tau}^{M}e^{(q)}{\mathbbA}_{M}^{(q)}
(4\pi)^{2}\frac{d\Lambda_{vac}}{dt}\,=\beta_{\Lambda}\equiv\,2\,m^{4}\,-\,2\,\sum_{i}\,N_{i}\,m_{i}^{4}\,,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Lambda_{vac}(0)=\Lambda_{0}\,,
\theta(|\mathbf{p}+\mathbf{q}|-1)=\theta(\mathbf{\hat{p}_{.}q}+p/2)=\theta(\mathbf{q_{.}\hat{p}})+\sum_{m=1}^{\infty}\frac{(p/2)^{m}}{m!}\delta^{(m-1)}(\mathbf{q_{.}\hat{p}})
B(\alpha,\lambda_{0},\tau,\chi,c_{1}\chi,c_{2}\chi)=B(\alpha,\lambda_{0},\tau,0,0,0)+\chi\bar{\chi}\left[F_{0}+\sum_{m,n=1}^{2}F_{mn}c_{m}\bar{c}_{n}+(\sum_{m=1}^{2}F_{m}c_{m}+c.c.)\right]
S=\int_{\Sigma}d^{2}x\,{\sqrt{h}}\left[\frac{1}{8G_{B}}g^{\rho\tau}g^{\sigma\lambda}n_{\rho\sigma}n_{\tau\lambda}+2a_{B}R+b_{B}R_{\rho\sigma}g_{\perp}^{\rho\sigma}-c_{B}R^{\rho\sigma\tau\lambda}n_{\rho\sigma}n_{\tau\lambda}\right],
\frac{1}{2}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\delta_{n,d-2k}\frac{H_{n}}{k!}\frac{d^{k}}{dt^{k}}\left(\frac{d}{ds}K^{(1)}(t,s)\right)_{s,t=0}\pmm\delta_{n,(d-1)-2k}\frac{H_{n}}{k!}\frac{d^{k}}{dt^{k}}\left(\frac{d}{ds}K^{(2)}(t,s)\right)_{s,t=0}.
\begin{array}{rl}{n_{1}}&{=(k_{1}+k_{2}-k_{3})/3}\\{n_{2}}&{=(k_{1}-2k_{2}-k_{3})/3}\\{n_{3}}&{=(k_{1}+k_{2}+2k_{3})/3}\\\end{array}\qquad\begin{array}{rl}{k_{1}}&{=n_{1}+n_{2}+n_{3}}\\{k_{2}}&{=n_{1}-n_{2}}\\{k_{3}}&{=-n_{1}+n_{3}}\\\end{array}
f_{e}^{(2)}=\frac{g_{1}^{2}}{8\pi^{2}}\frac{g_{2}^{2}}{8\pi^{2}}\int_{\lambda_{1}^{2}}^{1}\frac{d\alpha_{1}}{\alpha_{1}}\int_{\lambda_{1}^{2}/\alpha_{1}}^{1}\frac{d\beta_{1}}{\beta_{1}}\int_{\alpha_{1}}^{1}\frac{d\alpha_{2}}{\alpha_{2}}\int_{\beta_{1}}^{1}\frac{d\beta_{2}}{\beta_{2}}.
L^{2}({\calA}_{\gamma}\times{\calF}_{\gamma})=\bigoplus_{\rho_{e}\in\mathrm{Rep}(G),\;\rho_{v}\inS}\quad\bigotimes_{v\inV}\bigl(\bigotimes_{t(e)=v}\rho_{e}^{*}\otimes\bigotimes_{s(e)=v}\rho_{e}\otimes\rho_{v}\bigr)
S_{N}^{(4)}(x,y)=K_{N}(x,y)-\psi_{3}(x)\sum_{l=1}^{4}[A_{0}C_{00}^{-1}C_{0}]_{3l}\\varepsilon\psi_{l}(y)-\psi_{4}(x)\sum_{k=1}^{4}[A_{0}C_{00}^{-1}C_{0}]_{4k}\\varepsilon\psi_{k}(y),
\vec{L}(\omega)=\left(\frac{e^{2}}{8\pi^{2}c}\right)^{1/2}\intdt^{\prime}e^{-i\omega(t^{\prime}-\vec{n}\cdot\vec{r}(t^{\prime}))}\vec{n}\times(\vec{n}\times\vec{\beta})[-i\omega(1+\frac{3\lambda}{2\zeta})+\frac{3\lambda}{2\zeta^{3}}\vec{n}\cdot\dot{\vec{\beta}}]\;.
\oint_{C}\frac{d\zeta}{\zeta^{j}}\hat{f}=\oint_{0}\frac{d\zeta}{\zeta^{j}}\chi+\oint_{\infty}\frac{d\zeta}{\zeta^{j}}\chi^{\prime}=\left(\oint_{0}+\oint_{\infty}\right)\frac{d\zeta}{\zeta^{j}}\chi-4\mu\oint_{\infty}\frac{d\zeta}{\zeta^{j+1}}\eta.
\mu^{2}=\mu_{0}^{2}{\frac{[1+(\xie^{2}/\pi\mu_{0}^{2})(1-r^{2})][1-(\etae^{2}/\pi\mu_{0}^{2})(1-r^{2})]}{1-(e^{2}/\pi\mu_{0}^{2})(\xir^{2}+\eta),}}
A\,\frac{d^{2}\delta\phi}{dr^{*2}}+\left[\,2B\,e^{(\Gamma-\Lambda)/2}-\frac{A}{2}\frac{d(\Gamma-\Lambda)}{dr^{*}}\,\right]\,\frac{d\delta\phi}{dr^{*}}+e^{\Gamma-\Lambda}(C+\sigma^{2}E)\,\delta\phi=0
S_{\mathrm{WZ}}^{N({M5}_{w},{\bar{M}5}_{w})}=\mu_{4}\int_{R^{4+1}}\mathrm{Tr}\left(l_{p}^{2}\left[{\hat{C}}D{\hat{X}}D{\hat{X}}D{\hat{X}}+il_{p}^{2}i_{\hat{\Phi}}i_{\hat{\Phi}}i_{\hat{k}}{\hat{\tilde{C}}}+\dots\right]\wedge({\hat{F}}^{(2)}-{\hat{F}}^{(2)\prime})\right)\,,
\left(i_{\hat{k}}\hat{\tilde{N}}\right)_{\mu_{1}\ldots\mu_{7}}=C^{(7)}{}_{\mu_{1}\ldots\mu_{7}}-5\cdot7C^{(3)}{}_{[\mu_{1}\mu_{2}\mu_{3}}B_{\mu_{4}\mu_{5}}B_{\mu_{6}\mu_{7}]}\,.
ds_{\mathrm{str}}^{2}=-e^{4U}dt^{2}+d\vec{x}^{2}\,\qquade^{4U}=e^{4\phi}\,\qquadE_{i}^{(a)}={\frac{1}{2}}e^{4U}({\calM}L)_{ab}\,\partial_{i}\chi^{b}\,
\begin{array}{ll}{\Gamma^{i+6}=\sigma^{i}\otimes\gamma^{5}\otimes\rho^{3},}&{i=1,2,3;}\\{\Gamma^{a}=1\otimes\gamma^{a}\otimes\rho^{3},}&{a=4,5,6,10;}\\{\Gamma^{1}=1\otimes1\otimes\rho^{1};}&{}\\{\Gamma^{2}=1\otimes1\otimes\rho^{2}}&{}\\\end{array}
\omega^{\ell}\;{\left(\frac{\vartheta_{1}(u_{j}-\lambda)}{\vartheta_{1}(u_{j}+\lambda)}\right)}^{N}\;\;=\;\;-\,\prod_{k=1}^{N}\frac{\vartheta_{1}(u_{j}-u_{k}-2\lambda)\,\vartheta_{1}(u_{j}-u_{k}+\lambda)}{\vartheta_{1}(u_{j}-u_{k}+2\lambda)\,\vartheta_{1}(u_{j}-u_{k}-\lambda)}
\begin{array}{ll}{R_{\mathrm{cr}}=0\,;}&{D=2\,,}\\{\displaystyleR_{\mathrm{cr}}=\frac{9}{2}m_{0}^{2}\,;}&{D=3\,,}\\{R_{\mathrm{cr}}=12m_{0}^{2}\,;}&{D=4\,.}\\\end{array}
{\frac{d^{2}t}{d\tilde{\lambda}^{2}}}=-{\frac{\kappa^{2}}{6}}\varrho\left[{\frac{\kappa^{4}}{36}}(2+3\omega)\varrho^{2}y_{0}^{2}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+y_{0}^{2}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-2(2+3\omega)\right]{\frac{dt}{d\tilde{\lambda}}},
A_{\calF}=-\tilde{A}^{T}K\tilde{A}-\frac{1}{4}\tilde{A}^{T}Z\tilde{A}-\tilde{A}^{T}J\tilde{A}+4{\calM}_{\mu\nu}^{-1\AB}\partial_{\rho}{\calH}_{B}^{\rho\nu}\partial_{\sigma}{\calH}_{A}^{\sigma\mu}
E_{\mathrm{ADM}}(t)-B(y^{-})=\frac{1}{2}\sum_{i=1}^{i=N}\int_{-\infty}^{y^{-}}dy^{-}(\partial_{-}f_{i})^{2}|_{y^{+}\to\infty}-2Q^{2}\int_{-\infty}^{y^{-}}dy^{-}t_{-}(y^{-}).
\kappa={\frac{1}{2\ell}}\Phi_{H}^{1-{\frac{\gamma}{\delta^{2}}}}={\frac{1}{2\ell}}\left[2\ellM\left(2-{\frac{\gamma}{\delta^{2}}}\right)\right]^{\frac{\delta^{2}-\gamma}{2\delta^{2}-\gamma}}.
\Pi^{i}=\nablax^{i}-\frac12\Omega^{+2i}x^{-2}-\frac12\Omega^{-2i}x^{+2}-i\pi_{q}^{1+}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-i\pi_{\dot{q}}^{1-}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}+i\pi_{q}^{2-}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{2+}+i\pi_{\dot{q}}^{2+}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{2-},
\varphi=\left(\begin{array}{c}{\lambda}\\{\pm\tau}\\\end{array}\right)\,,\qquad\lambda=\left(\begin{array}{c}{\partial_{z_{1}}}\\{\bar{z}_{2}}\\\end{array}\right)\,,\qquad\tau=\left(\begin{array}{c}{\bar{z}_{1}}\\{\partial_{z_{2}}}\\\end{array}\right)\,,
\begin{array}{lll}{s_{\;t}^{1}}&{=}&{\left(\frac{\textstyle{s^{3}}}{\textstyle{s_{\;x}^{1}}}\right)_{x}}\\{s_{\;t}^{2}}&{=}&{s_{\;x}^{1}}\\{s_{\;t}^{3}}&{=}&{\left[\frac{\textstyle{s^{3}}}{\textstyle{s_{\;x}^{1}}}\left(\frac{\textstyle{s^{3}}}{\textstyle{s_{\;x}^{1}}}\right)_{x}-s^{2}\right]_{x}}\\\end{array}
\left[\begin{array}{c}{\widehat{\mathbf{J}}^{2}}\\{\widehat{J}_{3}}\\{\widehat{\mathcal{P}}}\\{\widehat{\mathcal{C}}}\\\end{array}\right]\mide^{+}e^{-}\rangle=\left[\begin{array}{c}{J\left(J+1\right)}\\{m_{J}}\\{P}\\{C}\\\end{array}\right]\mide^{+}e^{-}\rangle\,,
M(\Phi,\Phi^{\prime},T)=\left(\begin{array}{cc}{-i\Phi_{I}({\bfX}^{0})\tilde{D}{\bfX}^{I}}&{\overline{{T}}({\bfX}^{0})}\\{T({\bfX}^{0})}&{-i\Phi_{I}^{\prime}({\bfX}^{0})\tilde{D}{\bfX}^{I}}\\\end{array}\right),\I=1,\dots,9\.
(f_{1}^{\mu},p_{1}^{\mu})\times(f_{2}^{\mu},p_{2}^{\mu})=f_{1}^{\mu}\timesf_{2}^{\mu}-v_{1}\frac{1}{1-M}(p_{1}^{\mu}f_{2}^{\mu}-p_{2}^{\mu}f_{1}^{\mu}).
S_{_\mathrm{Schild}}=\intd^{2}\sigma\left(\alpha\sqrt{g}(\frac{1}{4}\{X_{\mu},X_{\nu}\}_{_{PB}}^{2}-\frac{i}{2}\overline{{\psi}}\Gamma^{\mu}\{X_{\mu},\psi\}_{_{PB}})+\beta\sqrt{g}\right),
\frac{\partial\,\overset{(s_{\ell})}{e}{\!\!}_{A_{\ell}}}{\partial\,p}\;\frac{\overset{(r)}{e}{\!}_{A_{\ell}}\overset{(r)}{e}{\!}_{B_{\ell}}}{\mu_{\ell}\,Z_{\ell}}\;=\;-\overset{(s_{\ell})}{e}{\!\!}_{A_{\ell}}\;\frac{\partial\,\overset{(r)}{e}{\!}_{A_{\ell}}}{\partial\,p}\;\frac{\overset{(r)}{e}{\!}_{B_{\ell}}}{\mu_{\ell}\,Z_{\ell}}
V_{F}(S,z^{a})=\frac{e^{\alphaG}}{\kappa^{4}}\lbrack\alpha^{2}G_{\bar{S}}G^{\bar{S}S}G_{S}+\alpha^{2}G_{\bar{z}^{\bar{a}}}G^{\bar{z}^{\bar{a}}z^{a}}G_{z^{a}}-3\rbrack.
\begin{array}{c}{a^{-}(a^{+})^{n}~=~[{[n]}]q^{-N}(a^{+})^{n-1}+q^{N}(a^{+})^{n}a^{-}}\\{(a^{-})^{n}a^{+}~=~[{[n]}](a^{+})^{n-1}q^{-N}+q^{N}a^{+}(a^{-})^{n}}\\\end{array}
q_{1}^{+}=\etabe^{{\frac{3\phi}{2}}+{\frac{iH_{1}}{2}}+{\frac{iH_{C}}{2}}}+i\partialX^{+}e^{{\frac{\phi}{2}}-{\frac{iH_{1}}{2}}+{\frac{iH_{C}}{2}}}-G_{C}^{-}e^{{\frac{\phi}{2}}+{\frac{iH_{1}}{2}}+{\frac{iH_{C}}{2}}}.
\sum_{m\ge1}\frac{t^{m}}{m}\prod_{j=1}^{m-1}\bigg(1+\frac{x}{j}\bigg)=t\sum_{m\ge0}(-t)^{m}{\binom{-x-1}{m}}\int_{0}^{1}u^{m}\,du=\frac{(1-t)^{-x}-1}{x}.
({\bfp};l_{L},l_{R})\to({\bfp}^{\prime};l_{L}^{\prime},l_{R}^{\prime})=({\bfp}+wx{\bfA};u_{L}-{\bfp}\cdot{\bfA}-{\frac{wx}{2}}{\bfA}\cdot{\bfA},u_{R}-{\bfp}\cdot{\bfA}-{\frac{wx}{2}}{\bfA}\cdot{\bfA})\quad.
\psi(x_{1},x_{2})=f\left(\frac{x_{1}}{\sqrt{\gamma}}+i\sqrt{\gamma}x_{2}\right)e^{-\frac{1}{\theta}\left(\left(\frac{x_{1}^{2}}{\gamma}+\gammax_{2}^{2}\right)-z\left(\frac{x_{1}}{\sqrt{\gamma}}-i\sqrt{\gamma}x_{2}\right)\right)}
\begin{array}{lll}{{\displaystylex_{\pm}^{\prime}=\frac{x_{\mp}}{x_{\mp}^{2}}}~}&{~{\displaystyle\theta^{\prime}=-i\frac{\tilde{\bar{\theta}}~x_{-}\cdot\tilde{\sigma}}{x_{-}^{2}}}~}&{~{\displaystyle\bar{\theta}^{\prime}=i\frac{x_{+}\cdot\tilde{\sigma}\tilde{\theta}}{x_{+}^{2}}}}\\\end{array}
c_{\alpha}=(\varphi_{\alpha}^{(+)},\varphi)=i\left\langle\varphi_{\alpha}^{(-)}\pi-(D\varphi_{\alpha}^{(-)})\varphi\right\rangle\,,\qquadc_{\alpha}^{*}=-(\varphi_{\alpha}^{(-)},\varphi)=-i\left\langle\varphi_{\alpha}^{(+)}\pi-(D\varphi_{\alpha}^{(+)})\varphi\right\rangle\,.
\eta_{M_{1}M_{2}}(g_{1}g_{2})=\frac{\partialx^{N_{1}}(g_{2})}{\partialx^{M_{1}}(g_{1}g_{2})}\frac{\partialx^{N_{2}}(g_{2})}{\partialx^{M_{2}}(g_{1}g_{2})}\eta_{N_{1}N_{2}}(g_{2})\,.
\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}\!}&{}&{\!h^{0,1}\!}&{}\\{h^{2,0}\!}&{\!}&{h^{1,1}}&{\!}&{\!h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}.
\overrightarrow{\not\!\!D}_{R}\equive_{a}^{\mu}\gamma^{a}\left(\vec{\partial}_{\mu}+\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}+ig^{\prime}B_{\mu}\right);\quad\overleftarrow{\not\!\!D}_{R}\equiv\left(\overleftarrow{\partial}_{\mu}-\frac{1}{2}\omega_{\mu}^{cd}\sigma_{cd}-ig^{\prime}B_{\mu}\right)\gamma^{a}e_{a}^{\mu};
E_{r}=-\frac{c\hbar}{\left|e\right|}f\,_{1}^{\prime}\left(r\right),\,H_{\varphi}=\frac{c\hbar}{\left|e\right|}f_{2}^{\,\,\,\prime}\left(r\right),\,\,H_{z}=\frac{c\hbar}{\left|e\right|}\frac{A^{\prime}\left(r\right)}{r}\,,\;E_{\varphi}=E_{z}=H_{r}=0\,.
t_{\alpha}{}^{i}:=\;\stackrel{\mathrm{B}}{\Sigma}_{\alpha}{}^{\!i}(\stackrel{\mathrm{B}}{X})\qquad\mathrm{and}\qquad\stackrel{\mathrm{B}}{\tau}_{\alpha\beta}{}^{i}=0={\tau}_{\alpha\beta}{}^{\!i}-2\stackrel{\mathrm{B}}{X}_{[\alpha\beta]}{}^{\!i}
g(u,y^{p},\theta)=\left(\begin{array}{ccc}{-\frac{e}{2}u\,1_{2}}&{\frac{i}{\sqrt{2}}y^{p}\,\sigma_{p}}&{\sqrt{e\,}\theta_{N}}\\{.}&{\frac{e}{2}u\,1_{2}}&{.}\\{.}&{\sqrt{e}\,\vartheta^{\daggerN}}&{.}\\\end{array}\right)=u{\calC}+y^{p}{\calN}_{p}+\theta^{\daggerN}Q_{N}+\overline{{Q}}^{N}\theta_{N}
{\calL}_{p}={\calL}_{p}^{reg}+4p\epsilon_{\phi\rhoi_{1}...i_{2p-2}}\epsilon^{\phi\rhoj_{1}...j_{2p-2}}R_{(con)\\phi\rho}^{\phi\rho}R_{\\j_{1}j_{2}}^{i_{1}i_{2}}...R_{\\j_{2p-3}j_{2p-2}}^{i_{2p-3}i_{2p-2}}
E_{q-\mathrm{vib-rot}}(v,J)={\frac{1}{2}}\left([v+1+b\gamma(J)]_{q(J)}+[v+b\gamma(J)]_{q(J)}\right)hc\nu_{\mathrm{vib}}+\frac{h^{2}}{8\pi^{2}I}J(J+1)~.
U_{\alpha\beta}^{(a}{}^{\gamma}U_{\gamma\nu}^{b)}{}^{\rho}+U_{\beta\nu}^{(a}{}^{\gamma}U_{\gamma\alpha}^{b)}{}^{\rho}+U_{\nu\alpha}^{(a}{}^{\gamma}U_{\gamma\beta}^{b)}{}^{\rho}=0\,,
\intdx\left(\,{\frac{1}{2}}\,\tilde{g}^{ij}{\frac{\delta^{2}\Psi_{phys}}{\delta\phi^{i}\delta\phi^{j}}}\,+\,{\frac{1}{2}}\,g_{ij}\partial_{x}\phi^{i}\partial_{x}\phi^{j}\Psi_{phys}\,\right)\,=\,E\Psi_{phys}\,,
n={\frac{\alpha}{\pi}}(x_{0}\Omega)^{2}\int\intdtdt^{\prime}\bigl[{\frac{(sin\,t-sin\,t^{\prime})^{2}}{(t-t^{\prime}+i\epsilon)^{4}}}-{\frac{cos\,t~cos\,t^{\prime}}{(t-t^{\prime}+i\epsilon)^{2}}}\bigr]
\Big(\frac{1}{2\pi}\Big)^{2\sum_{b}n_{b}}\prod_{b=1}^{N}\prod_{i,j=1}^{n_{b}}\Big(x_{i}^{(b)}-y_{j}^{(b)}\Big)^{-2}\prod_{1\leqi<j\leqn_{b}}\Big(x_{i}^{(b)}-x_{j}^{(b)}\Big)^{2}\Big(y_{i}^{(b)}-y_{j}^{(b)}\Big)^{2}\;.
\mathcal{G}_{MN}=\frac{\vec{\Phi}\cdot\vec{G}_{MN}}{\vert\vec{\Phi}\vert}-\frac{1}{e\vert\vec{\Phi}\vert^{3}}\,\vec{\Phi}\cdot\left(\mathcal{D}_{M}\vec{\Phi}\times\mathcal{D}_{N}\vec{\Phi}\right).
i(\not\!p_{2}-m)_{\rho^{\prime}\alpha}\eta_{\rho\alpha}(p_{3},p_{2})(\not\!p_{3}-m)_{\rho\alpha^{\prime}}+{\frac{\lambda_{0}^{2}}{4\pi^{2}}}\intdp_{1}dp_{4}\delta(p_{1}+p_{3}-p_{2}-p_{4})\eta_{\alpha\alpha}(p_{4},p_{1})\delta_{\alpha^{\prime}\rho^{\prime}}=0.
<M(C_{i})M(C_{f})>=2\int_{0}^{\infty}dle^{-R^{2}l/2\pi\alpha^{\prime}}\eta(il)^{-22}{\frac{\eta(il)e^{-u^{2}l/\pi}}{i\Theta_{11}({\frac{ul}{\pi}},il)}}\quad.
I_{\mathrm{sing}}(D)={\biggl\lbrace\int_{0}^{\infty}dT\int_{0}^{1}du{\Bigl[T+u(1-u)\Bigr]}^{-{\frac{D}{2}}}\biggr\rbrace}^{2}=\biggl[{\frac{2B(2-{\frac{D}{2}},2-{\frac{D}{2}})}{D-2}}\biggr]^{2}
\frac{dF}{dt}=-e^{-t\Phi_{0}}\Phi_{0}Q_{0}(e^{t\Phi_{0}})+e^{-t\Phi_{0}}Q_{0}(\Phi_{0}e^{t\Phi_{0}})=e^{-t\Phi_{0}}Q_{0}(\Phi_{0})e^{t\Phi_{0}}\
\psi(x,g_{1})=\psi(x,g_{2})=\psi_{10}(x)=\left(\begin{array}{ccccc}{0}&{u_{3}^{c}}&{-u_{2}^{c}}&{-u^{1}}&{-d^{1}}\\{-u_{3}^{c}}&{0}&{u_{1}^{c}}&{-u^{2}}&{-d^{2}}\\{u_{2}^{c}}&{-u_{1}^{c}}&{0}&{-u^{3}}&{-d^{3}}\\{u^{1}}&{u^{2}}&{u^{3}}&{0}&{-e^{+}}\\{d^{1}}&{d^{2}}&{d^{3}}&{e^{+}}&{0}\\\end{array}\right)_{L},
\begin{aligned}{H=\int_{\Sigma}}&{{}(1/2\sqrt{W})[(P_{m})^{2}+(\Pi_{r})^{2}+(1/2)W\{X^{m},X^{n}\}^{2}+W(\mathcal{D}_{r}X^{m})^{2}+}\\{}&{{}+(1/2)W(\mathcal{F}_{rs})^{2}]+\int_{\Sigma}[(1/8)\sqrt{W}n^{2}-\Lambda(\mathcal{D}_{r}\Pi_{r}+\{X^{m},P_{m}\})]+}\\{}&{{}-(1/4)\int_{\Sigma}\sqrt{W}n^{*}\mathcal{F},\qquad\qquadn\not=0}\\\end{aligned}
\sum_{g\inG}\chi_{r}(a^{\sigma_{a}}b^{\sigma_{b}}i^{\sigma_{i}}g)\chi_{s}(gh^{\sigma_{h}}e^{\sigma_{e}}f^{\sigma_{f}})=\frac{|G|}{d_{r}}\delta_{r\bar{s}}\chi_{r}(a^{\sigma_{a}}b^{\sigma_{b}}c^{\sigma_{c}}f^{-\sigma_{f}}e^{-\sigma_{e}}h^{-\sigma_{h}}).
\begin{array}{rcl}{}&{}&{\delta(i_{h}i_{k}N^{(8)})=6\partial(i_{h}i_{k}\Omega^{(7)})}\\{}&{}&{}\\{}&{}&{\delta(i_{h}i_{k}{\calN}^{(8)})=6\partial(i_{h}i_{k}{\tilde{\Omega}}^{(7)})\,.}\\\end{array}
\chi(\alpha,\beta)=\int\frac{d^{2}m}{m_{2}^{\mathrm{~}2}}\frac{|\beta-m\alpha|}{\pi\tilde{\tau}\sqrt{m_{2}}}e^{-\frac{i}{m_{2}\tilde{\tau}}|\beta-m\alpha|^{2}}\chi(m_{1},m_{2}),
\left\langleV_{\lambda_{1},\bar{\lambda}_{1}}(z_{1},\bar{z}_{1})\cdotsV_{\lambda_{n},\bar{\lambda}_{n}}(z_{n},\bar{z}_{n})\right\rangle=0,\qquad\mathrm{unless~\Sigma~\lambda_i~=~\Sigma~\bar{\lambda}_i~=~1~}.
ds^{2}=(k+f_{0}\frac{R_{0}^{2}}{R^{2}})^{-1}dR^{2}+R^{2}d\Omega_{k}^{2}-(k+f_{0}\frac{R_{0}^{2}}{R^{2}})[dx^{5}+A_{R}(R)dR]^{2}
K_{12}=0,\;\;\;K_{11}=\frac{1}{\sqrt{1-(\vec{m}\dot{\vec{X}}_{0})^{2}}}\frac{1}{R_{1}(\tau)},\;\;\;K_{22}=\frac{1}{\sqrt{1-(\vec{m}\dot{\vec{X}}_{0})^{2}}}\frac{1}{R_{2}(\tau)},
A_{\epsilon}=l^{2}\int\!\!\!\!\int_{D^{\epsilon}}dxdy{\frac{1}{\epsilon^{2}}}e^{2\phi}\left(1+\epsilon^{2}e^{-2\phi}\left(\left({\frac{\partial\phi}{\partialx}}\right)^{2}+\left({\frac{\partial\phi}{\partialy}}\right)^{2}\right)\right)^{1/2},
\Gamma_{A}=\left(\begin{array}{cc}{0}&{(\sigma_{A})_{a\stackrel{.}{a}}}\\{(\widetilde{\sigma}_{A})^{\stackrel{.}{a}a}}&{0}\\\end{array}\right)\,\quad\begin{array}{c}{\sigma_{A}=\left\{1,\gamma_{0},i\gamma_{1},i\gamma_{2},i\gamma_{3},\gamma_{5}\right\}}\\{\widetilde{\sigma}_{A}=\left\{1,-\gamma_{0},-i\gamma_{1},-i\gamma_{2},-i\gamma_{3},-\gamma_{5}\right\}}\\\end{array}
{\left[\begin{matrix}{e_{1}}\\{e_{2}}\\\end{matrix}\right]}=-\sqrt{u}\mp\frac{1}{2}\Big({\frac{H(-\sqrt{u})}{u}}\Big)^{\frac{1}{2}}+{\calO}(\Lambda^{4-N_{f}}),\quad\left[\begin{matrix}{e_{3}}\\{e_{4}}\\\end{matrix}\right]=\sqrt{u}\mp\frac{1}{2}\Big({\frac{H(\sqrt{u})}{u}}\Big)^{\frac{1}{2}}+{\calO}(\Lambda^{4-N_{f}}),
S_{\left\langlep,\phi\right\rangle}^{\left\langleq,\psi\right\rangle}=\frac{1}{|\Omega_{p}||\Omega_{q}|}\sum_{z\in\Omega}\sum_{x,y\in\Omega_{p}\cap\Omega_{zq}}\overline{{\phi}}(x,y)\overline{{\psi}}^{z}(y,x)\prod_{\xi\in\mathcal{O}(x,y)}\Lambda_{p_{\xi}}^{(zq)_{\xi}}\left(\frac{\kappa_{\xi}}{\lambda_{\xi}}\right)
{W}_{\mathrm{ren}}^{\infty}=\mathrm{}-\frac12\frac{\partial}{\partial\epsilon}\left[\frac{\mu^{2\epsilon}}{\Gamma(\epsilon)}\int_{0}^{\infty}\!\frac{{\mathrm{d}}s}{s^{1-\epsilon}}\,{\mathrme}^{-sm^{2}}{\mathrm{Tr}}{K}(s)\right]_{\epsilon=0}.
G_{\beta}(\vec{x},\tau)={\frac{1}{V}}\sum_{\vec{p}}{\frac{1}{2E_{p}}}\left[{\frac{e^{i\vec{p}\cdot\vec{x}-E_{p}\tau}}{1-e^{-\betaE_{p}}}}+{\frac{e^{-i\vec{p}\cdot\vec{x}-E_{p}\tau}}{e^{\betaE_{p}}-1}}\right]\;.
ds^{2}=\left({\frac{1-|Z|^{2}}{1-|Z_{\infty}|^{2}}}\right)^{-1}\,dt^{2}-\left({\frac{1-|Z|^{2}}{1-|Z_{\infty}|^{2}}}\right)\,d\vec{x}^{2}\,\qquadI=5,6,7,8\.
\begin{array}{cc}{\left(\bar{C}_{0}\,\eta_{0}\,{\calA}...\right)\;\,\left(\bar{C}_{1}\,\eta_{0}\,B...\right)}\\{\left(\bar{C}_{1}\,\eta_{1}\,{\calA}...\right)\;\,\left(\bar{C}_{1}\,\eta_{0}\,{\calB}...\right)}\\\end{array}
{\frac{\langleO_{N}^{\dagger}\,O_{N/2}\,O_{N/2}\rangle}{\sqrt{O_{N}^{\dagger}\,O_{N}}\,\sqrt{O_{N/2}^{\dagger}\,O_{N/2}}\sqrt{O_{N/2}^{\dagger}\,O_{N/2}}}}\sim{\frac{1}{2^{N}}}\,.
\Psi_{I}^{(n)}(\hat{\alpha}_{a_{0}},\ldots,\hat{\alpha}_{a_{n}})=(-1)^{n+\sum_{i>0}|\alpha_{a_{0}}||\alpha_{a_{i}}|}\Psi_{I}^{(n)}(\hat{\alpha}_{a_{1}},\ldots,\hat{\alpha}_{a_{n}},\hat{\alpha}_{a_{0}}).
\chi_{\left\langlep,\phi\right\rangle}(\tau)=\frac{1}{|\Omega_{p}|}\sum_{(x,y)\in\Omega_{p}^{\left\{2\right\}}}\overline{{\phi}}(x,y)\prod_{\xi\in\mathcal{O}(x,y)}\omega_{p(\xi)}^{-\kappa_{\xi}/\lambda_{\xi}}\chi_{p(\xi)}(\tau_{\xi})
W(G)=\frac{1}{4}\intdx(\phi_{,\nu}^{2}G^{\mu\nu}\Gamma_{\mu\sigma}^{\sigma}-\phi_{,\sigma}^{2}G^{\mu\nu}\Gamma_{\mu\nu}^{\sigma}+\phi^{2}G^{\mu\nu}\Gamma_{\mu\nu,\sigma}^{\sigma})
\Psi=\sum_{i\;\bullet}c_{\bullet}\mathrm{\Large~e^{i{\bf~k~\cdot~r}_{i}}~}a_{i}^{+}\left|O\right\rangle+\sum_{i\;\circ}c_{\circ}\mathrm{\Large~e^{i{\bf~k~\cdot~r}_{i}}~}a_{i}^{+}\left|O\right\rangle
\mathrm{Re}\,\chi_{T}^{\mathrm{D}}(\omega)=-\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}dv\sqrt{v(\Omega-v)}\int_{0}^{\infty}du\,\frac{u^{2}}{e^{\frac{u}{2}}-1}\left[1+{\calO}(\Omega^{2})\right]
ie(\varphi\partial^{\mu}\varphi^{\ast}-\varphi^{\ast}\partial^{\mu}\varphi)+2e^{2}A^{\mu}|\varphi|^{2}+\mu\varepsilon^{\mu\nu\kappa\lambda}v_{\nu}\partial_{\kappa}A_{\lambda}=\partial_{\nu}F^{\mu\nu},
{\frac{1}{\tilde{g}}}={\frac{1}{32\pi^{3}R^{2}L}}\sum_{l=0}^{\infty}\sum_{n=-\infty}^{\infty}{\frac{2l+1}{(n+{\frac{\epsilon}{2}})^{2}/L^{2}+(l+1/2)^{2}/4\pi^{2}R^{2}+\lambda^{2}/4\pi^{2}}}.
N^{(2l-1)}=\delta_{l,1}\,\intdx\,\frac{\theta^{\prime}}{2\pi}-\sqrt{\frac{2mT}{\pi}}\,e^{-m/T}\,\frac{1}{(2l-1)!}\,\intdx\,\left(\frac{\theta^{\prime}}{2T}\right)^{2l-1}+\dots
\begin{aligned}{D_{+}^{2}}&{{}:}&{\rho}&{{}\geq\frac{\pi}{4}\qquad}&{\phi_{2}}&{{}=\phi_{+}\qquad}&{\phi_{1}=\phi_{+}-\Psi}\\{D_{-}^{2}}&{{}:}&{\rho}&{{}\leq\frac{\pi}{4}\qquad}&{\phi_{1}}&{{}=\phi_{-}\qquad}&{\phi_{2}=\phi_{-}+\Psi}\\\end{aligned}
det{\bigl\{\Phi_{A},\Phi_{B}\bigr\}}^{\frac{1}{2}}\delta(\Phi_{A})=det{\bigl\{\Theta_{a},\Upsilon_{b}\bigr\}}\delta(\theta)\delta({\theta}_{i}^{T})\delta(\Upsilon)\delta({\Upsilon}_{i}^{T})\;,
S_{f}={\frac{1}{\kappa}}\left\{A_{3}^{-}\psi_{1}^{a}\psi_{2}^{a}+A_{3}^{+}\dot{\psi}_{1}^{\dot{a}}\dot{\psi}_{2}^{\dot{b}}-{\frac{A_{1}^{i}A_{2}^{j}}{A_{3}^{+}}}\psi_{3}^{a}\gamma_{ab}^{ij}\psi_{3}^{b}\right\}.
\Phi=\,{\frac{-1}{2i\pi}}\,G_{1}^{0}G_{2}^{0}G_{3}^{0}\,\beta_{1}\beta_{2}\beta_{3}\,\left[\,V_{12}\,\psi_{12}\,+\,V_{23}\,\psi_{23}\,+\,V_{31}\,\psi_{31}\,\right]
I_{lc}^{\mu_{1}\cdots\mu_{n}}=\int\frac{d^{D}k_{i}}{A(k_{j},\;p_{l})}\frac{g(k_{j}^{\mu_{j}},\;p_{l}^{\mu_{l}})}{h(k_{j}\!\cdot\!n,\;p_{l}\!\cdot\!n)},
{\calL}=\frac{1}{2}h_{ij}(y)\dot{y}^{i}\dot{y}^{j}+\frac{1}{2}L_{AB}(y)(\dot{\xi}^{A}+w_{i}^{A}(y)\dot{y}^{i})(\dot{\xi}^{B}+w_{j}^{B}(y)\dot{y}^{j})\,,
H_{C}=\int{dx\left({-\frac{1}{2}\Pi_{1}\left({1+\frac{{g^{2}}}{\pi}\frac{a}{{a-1}}\frac{1}{{\partial^{2}}}}\right)^{-1}\Pi^{1}+\Pi^{1}\partial_{1}A_{0}+A_{0}J^{0}}\right)}.
H_{C}=\int[\Pi_{\alpha}^{*}\Pi_{\alpha}-\partial_{i}z_{\alpha}^{*}\partial^{i}z^{\alpha}+\frac{2g}{N}(z_{\alpha}^{*}\partial^{i}z_{\alpha})(z_{\beta}\partial_{i}z_{\beta}^{*})+\lambda(z_{\alpha}^{*}z_{\alpha}-\frac{N}{2g})]
\int\!dz\!\wedge\!d\bar{z}\,\partialX\bar{\partial}X=\sum_{\alpha}\left(\oint_{a_{\alpha}}\!\!dz\partialX\oint_{b_{\beta}}\!\!d\bar{z}\bar{\partial}X-\oint_{a_{\alpha}}\!\!d\bar{z}\bar{\partial}X\oint_{b_{\beta}}\!\!dz\partialX\right)~,
p_{M}\to\overline{{a}}_{M}^{N}\circp_{N}=\overline{{a}}_{M}^{N}p_{N}+\frac{1}{2}[p_{N},\overline{{a}}_{M}^{N}]=\overline{{a}}_{M}^{N}p_{N}-\frac{i\hbar}{2}(\partial_{N}\overline{{a}}_{M}^{N}+b_{M}^{P}\partial_{P}\overline{{a}}_{M}^{N})\,,
a_{1}^{\dagger}a_{2}^{\dagger}|0\rangle\langle0|a_{2}a_{1}=a_{2}^{\dagger}a_{1}^{\dagger}|0\rangle\langle0|a_{1}a_{2}=\frac{1+q}{2}|\phi_{s}\rangle\langle\phi_{s}|+\frac{1-q}{2}|\phi_{a}\rangle\langle\phi_{a}|.
\bar{d_{\dot{\alpha}}}=\bar{p}_{\dot{\alpha}}+\frac{i}{2}\theta^{\alpha}\partialx_{\alpha\dot{\alpha}}-\frac{1}{4}(\theta)^{2}\partial\bar{\theta}_{\dot{\alpha}}+\frac{1}{8}\bar{\theta}_{\dot{\alpha}}\partial(\theta)^{2}