harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{x}j(x;z)\partial_{x}\Phi_{1}(x,\bar{x};z,\bar{z})=:\partial_{x}j(x;z)\partial_{x}\Phi_{1}(x,\bar{x};z,\bar{z}):+\frac{2}{k+2}\partial_{x}^{2}:(j(x;z)\Phi_{1}(x,\bar{x};z,\bar{z})):,
\begin{array}{ccc}{F_{ij}}&{=}&{\partial_{i}a_{j}-\partial_{j}a_{i}+i(c_{j}^{*}c_{i}-c_{i}^{*}c_{j}),}\\{}&{=}&{\partial_{i}b_{j}-\partial_{j}b_{i}-i(c_{j}^{*}c_{i}-c_{i}^{*}c_{j}).}\\\end{array}
\begin{array}{lcll}{y_{1}(u,k)}&{=}&{\frac{1}{u^{3}}+\mathcal{O}(u^{-4})\}&{u\rightarrow\infty}\\{y_{2}(u,k)}&{=}&{1+\mathcal{O}(u-1)\}&{u\rightarrow1}\\\end{array}
U=(U_{\,\,\nu}^{\mu})=\left(\begin{array}{ccccc}{1}&{0}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{\frac1{\sqrt{2}}}&{\frac1{\sqrt{2}}}\\{0}&{0}&{0}&{\frac1{\sqrt{2}}}&{\frac{-1}{\sqrt{2}}}\\\end{array}\right),
\int_{-L}^{L}du^{-}\left[\epsilon(u^{--}v^{-})-\frac{u^{--}v^{-}}{L}\right]=\int_{-L}^{L}dv^{-}\left[\epsilon(u^{--}v^{-})-\frac{u^{--}v^{-}}{L}\right]=0\quad.
\left(\hat{q}_{i}^{a}\right)^{\dagger}=\hat{q}_{i}^{a}\\,\\\left(\hat{p}_{i}^{a}\right)^{\dagger}=\hat{p}_{i}^{a}\\,\\\left[\hat{q}_{i}^{a},\hat{p}_{j}^{b}\right]=i\hbar\delta^{ab}\delta_{ij}\\,
\tilde{B}_{d-1,i}=\frac{G_{d-1,i}}{G_{d-1,d-1}},\;\;\;\tilde{B}_{ij}=B_{ij}+\frac{G_{i,d-1}B_{d-1,j}+B_{i,d-1}G_{d-1,j}}{G_{d-1,d-1}}.
\zeta^{\prime}(z,t)=\sum_{l=0}^{\infty}\frac{(-t)^{l}}{l!}\frac{\Gamma(z+l)}{\Gamma(z)}\left\{\left[\psi(z+l)-\psi(z)\right]\zeta(z+l)+\zeta^{\prime}(z+l)\right\}-t^{z}\mathrm{ln}(z).
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[-\frac{1}{2}\frac{\partial\Gamma^{(2)}}{\partialv^{i}}\right]={e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[B_{i}^{v(2)}\right]+{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}K_{\mu\nu}^{g},
|M_{[\alpha,\psi_{\alpha}]i}^{\pm}|\leqA_{[\alpha,\psi_{\alpha}][\alpha,\psi_{\alpha}]i}^{\pm}\;\;\;\;\;\mathrm{and}\;\;\;\;\;M_{[\alpha,\psi_{\alpha}]i}^{\pm}=A_{[\alpha,\psi_{\alpha}][\alpha,\psi_{\alpha}]i}^{\pm}\;\;\;\;\mathrm{mod}\;\;2\;\;\;\;,
\Pi^{\alpha\dot{\alpha}}\=\\dot{x}^{\alpha\dot{\alpha}}-2\,\dot{\theta}^{\alpha}\tilde{\theta}^{\dot{\alpha}}+2\,\theta^{\alpha}\dot{{\tilde{\theta}}^{\dot{\alpha}}}+\omega\psi^{\alpha\dot{\beta}}\psi_{\beta\dot{\beta}}\psi^{\beta\dot{\alpha}}+\chi\psi^{\alpha\dot{\alpha}}+2\,\rho^{\alpha}\tilde{\mu}^{\dot{\alpha}}\quad.
[\tilde{\alpha}_{\omega_{1}},L_{\omega}]=\{\Theta(\omega)[3\Theta(\omega_{1})+2\Theta(-\omega_{1})]+\Theta(-\omega)[2\theta(\omega_{1})+3\theta(-\omega_{1})]\}\omega_{1}\sqrt{|\omega_{1}(\omega-\omega_{1})|}\:\tilde{\alpha}_{\omega_{1}-\omega}
i\frac{\partial}{\partialy^{-}}{\calT}(x,y)=\frac{m_{0}}{\sqrt{2}}\Big({\calM}(x,y)-{\calM}(y,x)\Big)-\frac{g^{2}}{2}\Big({\calM}(x,y){\calT}(y,y)-{\calT}(y,y){\calM}(y,x)\Big),
(\Gamma^{\underline{m}})_{~~\underline{\beta}}^{\underline{\alpha}}(C\Gamma_{\underline{m}})_{\underline{{\gamma\delta}}}+(\Gamma^{\underline{m}})_{~~\underline{\gamma}}^{\underline{\alpha}}(C\Gamma_{\underline{m}})_{\underline{{\delta\beta}}}+(\Gamma^{\underline{m}})_{~~\underline{\delta}}^{\underline{\alpha}}(C\Gamma_{\underline{m}})_{\underline{{\delta\beta}}}=0,
\alpha_{i}^{-1}(v^{2},R^{2})=\alpha_{i}^{-1}(M^{2},R^{2})+b_{i}\\elln\,\frac{M^{2}}{v^{2}}=b_{s}\\elln\,\frac{M^{2}}{\Lambda_{QCD}^{2}}+b_{i}\,elln\,\frac{M^{2}}{v^{2}}\.
\mu_{11}=-{\frac{1}{12}}{\frac{\pi^{-(D+1)/2}}{2^{D}}}\left[-{\frac{(D-3)(D-2)(D-1)}{D}}\Gamma\left({\frac{D-1}{2}}\right)\zeta(D-1)+2(D+1)\Gamma\left({\frac{D+1}{2}}\right)\zeta(D+1)\right]\,{\frac{\hbarL^{D-1}}{a^{D}}},
F_{j}(\omega,\vec{k})\;=\;\int\frac{d\omega^{\prime}}{2\pi}\int\frac{d\vec{k}^{\prime}}{(2\pi)^{3}}\;\tilde{B}(\omega-\omega^{\prime},\vec{k}-\vec{k}^{\prime})\:C_{j-1}(\omega^{\prime},\vec{k}^{\prime})\:F_{j-1}(\omega^{\prime},\vec{k}^{\prime})
S~=~\int\,d^{d}x\,\left[\,\intd^{4}\theta\,\left(\bar{\Phi}^{i}\Phi^{i}+\frac{\bar{\sigma}\sigma}{g^{2}}\right)~-~\frac{1}{2}\intd^{2}\theta\,\sigma\Phi^{2}~-~\frac{1}{2}\intd^{2}\bar{\theta}\,\bar{\sigma}\bar{\Phi}^{2}\right]
\frac{1}{2}\int_{\varepsilon^{2}}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}e^{-(k+1/2)^{2}\pit/R^{2}}=R^{-4}{\calJ}(\varepsilon^{2}/R^{2},\alpha=-2)=\frac{1}{R^{4}}\frac{-15}{16}\frac{3}{4\pi^{2}}\zeta(5)+\frac{1}{5}\varepsilon^{-5}R
d_{\tilde{\omega},\vec{k}}^{\,(\sigma)}=\int_{-\infty}^{+\infty}dk_{1}\left[\frac{1}{\sqrt{2\pi\omega_{k}}}\left(\frac{\omega_{k}+k_{1}}{\omega_{k}-k_{1}}\right)^{i\sigma\tilde{\omega}/2}\right]\,a_{k_{1},\vec{k}},
Q_{0}=-\sum_{\kappa=-\infty}^{\infty}\vert\kappa\vert\left\{N_{\kappa}^{+}-N_{\kappa}^{--}n_{l}^{+}+n_{\overline{l}}^{-}+\frac{2}{\pi}\int_{0}^{\infty}\mathrm{d}r\;\varphi\left(r\right)\right\}\,,
\dot{A}=-\frac{{\dot{\mu}}\,k}{2\,\omega^{2}}\,\mathrm{e}^{2\,i\,\int^{\eta}\omegad\eta^{\prime}}B\,,\qquad\dot{B}=\frac{{\dot{\mu}}\,k}{2\,\omega^{2}}\,\mathrm{e}^{-2\,i\,\int^{\eta}\omegad\eta^{\prime}}A\,.
\left\{Q_{\alphaA},Q_{\dot{\beta}\bar{B}}\right\}=\sigma_{\alpha\dot{\beta}}^{\mu}\,P_{\mu}\,\,\delta_{A\bar{B}},\quad\Leftrightarrow\,\,\left[T\left(\gamma^{m}v_{m}\right)T^{\dagger}\right]_{A\bar{B}}=\delta_{A\bar{B}}
\left.{\frac{\partialS}{\partial\Phi^{\primeA}}}\right|_{K^{\prime}}=\left.{\frac{\partial_{r}\Phi^{B}}{\partial\Phi^{\primeA}}}\right|_{K^{\prime}}{\frac{\partialS}{\partial\Phi^{B}}}={\frac{\partialS}{\partial\Phi^{B}}}{(M^{-1})^{B}}_{A}.
\hat{\omega}_{u|2}^{\phantom{\mu|2}1}{\mathcalU}_{I1}^{u}\bar{\zeta}^{I}\epsilon^{1}=\hat{\omega}_{\bar{s}|2}^{\phantom{\mu|A}1}{\mathcalU}_{I1}^{\bar{s}}\bar{\zeta}^{I}\epsilon^{1}=0
S_{j}(\lambda)t_{j}S_{i}(\lambda^{\prime})-S_{j}(\lambda^{\prime})t_{j}S_{i}(\lambda)={\frac{\prod_{k\neqi,j}(\lambda-\lambda_{k})\prod_{k\neqi,j}(\lambda^{\prime}-\lambda_{k})}{\prod_{k\neqj}(\lambda_{j}-\lambda_{k})\prod_{k\neqi}(\lambda_{i}-\lambda_{k})}}(\lambda_{i}-\lambda_{j})(\lambda-\lambda^{\prime})
{\calG}=\frac{-1}{(1+H^{2}r^{2})m\alpha^{\prime}}\left(\begin{array}{ccc}{\dot{r}}&{E}&{0}\\{E}&{\dot{r}}&{0}\\{0}&{0}&{-(1+H^{2}r^{2})m\alpha^{\prime}}\\\end{array}\right).
H=v\left(\begin{array}{cc}{U}&{0}\\{0}&{U^{*}}\\\end{array}\right)\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\left(\begin{array}{cc}{U^{T}}&{0}\\{0}&{U^{\dagger}}\\\end{array}\right)=v\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right).
\Omega_{jk}^{(\alpha\beta)}\,=\,\frac{2R_{j}^{(\alpha\beta)}R_{k}^{(\alpha\beta)}-(R^{(\alpha\beta)})^{2}\delta_{jk}}{(R^{(\alpha\beta)})^{4}}\;\;\;,\;\;\;R^{(\alpha\beta)}\equivR^{(\alpha)}-R^{(\beta)}\;.
\widetilde{R}_{\:\:\nu\rho\sigma}^{\mu}=\widehat{\kappa}_{i\:\rho}^{\mu}\widehat{\kappa}_{\nu\sigma}^{i}-\widehat{\kappa}_{i\:\sigma}^{\mu}\widehat{\kappa}_{\nu\rho}^{i}+\gamma_{\nu}^{\alpha}\gamma_{\rho}^{\beta}\gamma_{\sigma}^{\gamma}\widehat{R}_{i\:\alpha\beta\gamma}^{\mu}
\psi^{0}(T,\sigma)=(T+\tilde{\Lambda})^{\frac{-\alpha}{1+\alpha}}\sum_{n=-\infty}^{\infty}\left[\tilde{\alpha}\vphantom{\alpha}_{n}^{0}e^{in(\sigma-T)}+\tilde{\beta}\vphantom{\beta}_{n}^{0}e^{in(\sigma+T)}\right]+\tilde{b}\vphantom{b}_{0}^{0}(T+\tilde{\Lambda})^{\frac{1}{1+\alpha}}
\overline{{S}}^{\prime}=\intd^{4}x\left(\dot{A}^{ijk}\pi_{ijk}-3\pi_{ijk}\pi_{ijk}-\frac{M^{2}}{2\cdot3!}A^{ijk}A_{ijk}+\frac9{M^{2}}\left(\partial^{i}\pi_{ijk}\right)^{2}\right).
\|(\Lambda\partial_{\Lambda}+4-2m)A_{2m}^{(s)}(\Lambda)\|\leq\sum_{l=1}^{m}\sum_{t=1}^{s-1}\|A_{2l}^{(t)}\|\|A_{2m-2l+2}^{(s-t)}\|+\|A_{2m=2}^{(s)}\|
f(\omega)=\frac{1}{(2-\omega)^{3}}\left[\frac{\Gamma^{2}(3-\omega)\Gamma(2\omega-3)}{\Gamma(5-2\omega)}-{\frac{\Gamma(\omega-1)\Gamma(\omega)(10\omega^{2}-19\omega+10)}{4(2\omega-3)(2\omega-1)}}\right]\.
{\calL}=-\frac{1}{2(p+2)!}F^{\mu_{0}\mu_{1}\ldots\mu_{p+1}}F_{\mu_{0}\mu_{1}\ldots\mu_{p+1}}+\frac{1}{2(p+1)!}\frac{g^{2}}{e^{2}}\psi^{\mu_{0}\mu_{1}\ldots\mu_{p}}\psi_{\mu_{0}\mu_{1}\ldots\mu_{p}}.
\begin{array}{lcl}{v_{2}-u_{2}}&{=}&{0}\\{v_{3}-u_{3}}&{=}&{-2\thetau_{2}^{\prime}}\\{v_{4}-u_{4}}&{=}&{-\thetau_{3}^{\prime}+\theta^{2}u_{2}^{\prime\prime}}\\\end{array}
F^{+}(x)=\sum_{\kappa=-n}^{n}(-1)^{\kappa}\biggl(\begin{array}{c}{2n+\frac{2}{3}}\\{n-\kappa}\\\end{array}\biggr)\biggl(\begin{array}{c}{2n-\frac{2}{3}}\\{n+\kappa}\\\end{array}\biggr)x^{\kappa+\frac{1}{3}}\mathrm{~and~}F^{-}(x)=F^{+}(1/x).
\vec{\chi}^{L}=\frac{\vec{P}_{\mathrm{vec}}}{2|\vec{x}|}\,\qquad\vec{\chi}^{R}={\frac{\vec{n}}{\sqrt2}}+\frac{\vec{P}_{\mathrm{gr}}}{2|\vec{x}|}\,\qquad\vec{n}^{2}=1\,\qquad\vec{n}\cdot\vec{P}_{\mathrm{gr}}\geq0
G_{\mu\nu}^{ab}\,=\,[M^{-1}]_{\mu\nu}^{ab}\;\;,\;\;M_{\mu\nu}^{ab}\,=\,\delta_{\mu\nu}\delta^{ab}+ig^{-\frac{1}{2}}\epsilon_{\mu\lambda\nu}f^{acb}\theta_{\lambda}^{c}\;.
\left\{(1-x^{2})\frac{d^{2}}{dx^{2}}+\left[\fracm{|m|}\sigma_{z}-(2|m|+2)\,x\right]\fracd{dx}-m\,(m+1)+\left(\lambda^{2}-\frac14\right)\right\}\left(\begin{array}{c}{\xi_{\lambdam}}\\{\eta_{\lambdam}}\\\end{array}\right)=0.
\frac{\delta^{2}G}{\delta\eta^{\beta\alpha}(y,x)\delta\eta^{\beta^{\prime}\alpha^{\prime}}(y^{\prime},x^{\prime})}=-\frac{\delta^{2}G}{\delta\eta^{\beta^{\prime}\alpha}(y^{\prime},x)\delta\eta^{\beta\alpha^{\prime}}(y,x^{\prime})}.
\left(\begin{array}{ccc}{1}&{-\bar{t}_{a}R}&{-\bar{t}_{a}R}\\{-\bar{t}_{b}R}&{1}&{-\bar{t}_{b}R}\\{-\bar{t}_{c}R}&{-\bar{t}_{c}R}&{1}\\\end{array}\right)\left(\begin{array}{c}{l_{aa}(s_{a}^{\prime})}\\{l_{ba}(s_{a}^{\prime})}\\{l_{ca}(s_{a}^{\prime})}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{\bar{t}_{b}Rt_{a}(s_{a}^{\prime})}\\{\bar{t}_{c}Rt_{a}(s_{a}^{\prime})}\\\end{array}\right)
\begin{array}{lcllcl}{\delta_{\alpha}B_{\mu\nu}}&{=}&{0,}&{\delta_{\alpha}V_{\mu}^{1}}&{=}&{0,}\\{\delta_{\alpha}\psi_{\mu\nu}^{1}}&{=}&{\partial_{\alpha}B_{\mu\nu},}&{\delta_{\alpha}\phi_{\mu}^{2}}&{=}&{\partial_{\alpha}V_{\mu}^{1},}\\{\delta_{\alpha}\phi^{3}}&{=}&{\partial_{\alpha}m^{2},}&{\delta_{\alpha}m^{2}}&{=}&{0,}\\\end{array}
\widehat{h}_{0}=\frac{B}{4m^{*}}\big(\widehat{w}\,\widehat{\!\bar{w}}+\widehat{\!\bar{w}}\,\widehat{w}\big)=\frac{B}{2m^{*}}\big(\widehat{w}\,\widehat{\!\bar{w}}+1\big).
\begin{array}{c}{\nabla{\bfK}=\frac1{\sqrt{2\epsilon}}\{\left(-\frac1{\sqrt{2\epsilon}}{\bfK}+\frac1{2\epsilon}{\bfM}_{1}\right)\left(\vec{R}-i\frac12\vec{\Gamma}\right)+\frac1{2\Gamma}{\bfM}_{2}\vec{\Gamma}}\\{+i\frac1{\Gamma\left(\epsilon^{2}-\eta^{2}\right)}\frac{\xi^{*}}{\left|\xi\right|}M_{3}\left(\vec{\Gamma}\times\vec{R}\right)\}}\\\end{array}
\left(\begin{array}{c}{\varphi_{N}^{\prime}(x)}\\{\varphi_{N-1}^{\prime}(x)}\\\end{array}\right)=\left(\begin{array}{cc}{A(x)}&{B_{N}(x)}\\{-C(x)}&{-A(x)}\\\end{array}\right)\left(\begin{array}{c}{\varphi_{N}(x)}\\{\varphi_{N-1}(x)}\\\end{array}\right),
I_{ab}^{-1}=\left(\begin{matrix}{1}&{-4}&{5}&{0}&{-5}&{4}\\{0}&{1}&{-4}&{5}&{0}&{-5}\\{0}&{0}&{1}&{-4}&{5}&{0}\\{0}&{0}&{0}&{1}&{-4}&{5}\\{0}&{0}&{0}&{0}&{1}&{-4}\\{0}&{0}&{0}&{0}&{0}&{1}\\\end{matrix}\right)
(-)^{{\bar{\alpha}}+{\bar{\beta}}+\bar{\alpha}\bar{\beta}}\;\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]^{5}}{{\bar{\eta}}^{5}}\;{\frac{1}{2}}\sum_{\gamma,\delta}\frac{{\bar{\vartheta}[_{\delta}^{\gamma}]}^{8}}{{\bar{\eta}}^{8}}
B_{eff}^{(1)}(\vec{R}_{p_{1}})=\frac{2\pi\kappa\sigma_{1}}{q_{1}}\varepsilon_{ij}\partial_{i}v_{j}^{(2)}(\vec{R}_{p_{1}})\;\;,\;\;\partial_{i}=\frac{\partial}{\partialR_{p_{1}}^{i}}\;\;.
[i\hbar\gamma^{\mu}\partial_{\mu}-Mce^{-\eta-i\chi(x)}{\frac{1+\gamma^{5}}{2}}-Mce^{\eta+i\chi(x)}{\frac{1-\gamma^{5}}{2}}-\hbar\gamma_{-}^{\mu}\partial_{\mu}\chi(x)]\psi^{D}(x)=0\,.
{\frac{1}{2}}{\frac{\partialS_{cl}^{(h)}}{\partialz_{i}}}=c_{i}^{(h)},\qquad{\frac{\partialc_{i}^{(h)}}{\partial\bar{z}_{j}}}=-{\frac{1}{2}}\left\langle{\frac{\partial}{\partialz_{i}}}\,,{\frac{\partial}{\partialz_{j}}}\right\rangle_{WP},
\Gamma[\phi]={\frac{1}{2}}\int_{0}^{\infty}{\frac{dT}{T}}\,\mathrm{e}^{-m^{2}T}\int{\calD}x\,\mathrm{exp}\left[-\int_{0}^{T}\left({\frac{{\dot{x}}^{2}}{4}}+U^{\prime\prime}\left(\phi(x)\right)\right)\right]
\langleJ_{\mu\nu}\rangle=-\,\frac{\pi^{2}}{90L^{4}}\,\Lambda_{\mu}^{\rho}\,\Lambda_{\nu}^{\sigma}\,\left[g_{\rho\sigma}-4\,\frac{\ell_{\rho}\,\ell_{\sigma}}{\ell^{2}}\right]\;.
a_{0}(x)=\lambda+{\frac{1}{\sqrt{L_{1}L_{2}}}}\sum_{\vec{p}\ne0}{\frac{1}{\sqrt{2p_{0}}}}[{\frac{p_{0}}{\mu}}g(\vec{p})e^{-ip\cdotx}+c.c]_{p^{0}=|\vec{p}|}\,.
\psi^{+,\alpha}(P,x,t)=\,{\frac{\mathrm{Det}\left(\begin{matrix}{h^{+,\alpha}(P)}&{g_{j}^{+}(P)}\\{h^{+,\alpha}(Q_{i})}&{\Theta_{ij}^{+}(x,t)}\\\end{matrix}\right)}{\mathrm{Det}\left(\begin{matrix}{\Theta_{ij}^{+}(x,t)}\\\end{matrix}\right)}}\,e^{-kx-k^{2}t}
S=\intd^{6}x\left[{\frac16}F_{{\mu}{\nu}{\sigma}}F^{{\mu}{\nu}{\sigma}}+{\frac12}{\lambda}_{{\mu}{\nu}}{\calF}^{{\mu}{\rho}{\sigma}}{{\calF}^{\nu}}_{{\rho}{\sigma}}+{\frac13}G^{{\mu}{\nu}{\sigma}}\left(F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]}\right)\right],
\langle\bar{\xi}^{\prime\prime}\verte^{-i{\hat{H}}t}\vert\xi^{\prime}\rangle=\int\prod_{p=1}^{S-1}d\mu(p)N(p)\prod_{p=1}^{S}\langleZ(p)\verte^{-i{\hat{H}}\epsilon}\vertZ(p-1)\rangle
\Phi^{-1}(\lambda)=I+\frac{JA^{\mathrm{t}}J}{\lambda-\lambda_{0}}+\frac{JA^{\dagger}J}{\lambda-\lambda_{0}^{*}}+\frac{JB^{\mathrm{t}}J}{\lambda-\lambda_{0}^{-1}}+\frac{JB^{\dagger}J}{\lambda-\lambda_{0}^{*-1}}
\begin{array}{cccc}{R_{i}^{~j}=\left(\begin{array}{cc}{0}&{-\zeta^{ab}}\\{\bar{\zeta}_{ab}}&{0}\\\end{array}\right)~~~~}&{~~~~~\zeta^{ab}\bar{\zeta}_{bc}=\delta_{~c}^{a}~~}&{~~\zeta^{ab}=\zeta^{ba}~~}&{~~\bar{\zeta}_{ab}=(\zeta^{ab})^{\ast}}\\\end{array}
2\mathrm{Im}{\calL}^{(1)}=(2s+1)\frac{(e\epsilon)^{2}}{(2\pi)^{3}}\sum_{n=1}^{\infty}\frac{(\pm1)^{n+1}}{n^{2}}e^{-\pin/\beta},\qquad\beta=\frac{e\epsilon}{m^{2}}.
Z_{{\itlog}}[p/q]=\frac{1}{\eta\bar{\eta}}\left(|\chi_{0}|^{2}+|\chi_{pq}|^{2}+\sum_{1\leqs\leqpq-1}\left(\chi_{s}^{0}(\chi_{s^{\prime}}^{0})^{*}+\chi_{s}^{+}(\chi_{s^{\prime}}^{-})^{*}+\chi_{s}^{-}(\chi_{s^{\prime}}^{+})^{*}\right)\right)\,,
\hat{\Phi}(0)=\left(\begin{array}{cccc}{A_{11}u_{11}(0)}&{A_{12}u_{12}(0)}&{\dots}&{A_{1N}u_{1N}(0)}\\{A_{21}u_{21}(0)}&{A_{22}u_{22}(0)}&{\dots}&{A_{2N}u_{2N}(0)}\\{\dots}&{\dots}&{\dots}&{\dots}\\{A_{N1}u_{N1}(0)}&{\dots}&{A_{N,N-1}u_{N,N-1}(0)}&{A_{NN}u_{NN}(0)}\\\end{array}\right)\times\Phi(0)\.
\hat{K}_{\alpha+}(t)=\frac{1}{2\hbar}\hat{I}_{\alpha11}^{(2)}(t),\hat{K}_{\alpha-}(t)=\frac{1}{2\hbar}\hat{I}_{\alpha22}^{(2)}(t),\hat{K}_{\alpha0}(t)=\frac{1}{2\hbar}\hat{I}_{\alpha12}^{(2)}(t).
\begin{array}{c}{(T_{\pm})^{c}=T_{\mp},\quad(T_{3})^{c}=-T_{3},\quad(Y)^{c}=-Y,}\\{(\Sigma_{\pm})^{c}=\pm\Sigma_{\mp}^{\prime},\quad(\Sigma_{\pm}^{\prime})^{c}=\pm\Sigma_{\mp}.}\\\end{array}
:\prod_{q=1}^{N}U^{r}(z_{q}):\mid\lambda>=\prod_{q=1}^{N}z_{q}^{\lambda^{2}}\sum_{\{t\}}(-1)^{\midt\mid}J_{\{t\}}^{(r^{2})}(z_{q})\left(j_{t}^{r^{2}}\right)^{-1/2}\mid\{t\},Nr+\lambda>
[\widehat{G_{1}}(\vec{q}),\,\widehat{G_{2}}(\vec{p})]=i\hbar(\{G_{1}(\vec{q}),\,G_{2}(\vec{p})\})_{\mathrm{{op}}}=i\hbar(\nabla_{\vec{q}}\,G_{1}\cdot\nabla_{\vec{p}}\,G_{2})_{\mathrm{op}}.
\biggl[\begin{array}{cc}{A^{\mu},\,X^{4,5}}&{X^{6,7,8,9}}\\{X^{6,7,8,9}}&{A^{\mu},\,X^{4,5}}\\\end{array}\biggr]\to\biggl[\begin{array}{cc}{A^{\mu},\,\phi}&{A_{1,2}}\\{B_{1,2}}&{\tilde{A}^{\mu},\,\tilde{\phi}}\\\end{array}\biggr]\.
{W}\,(\varphi_{1},\varphi_{2})=\sqrt{2}\left[\frac{m^{2}}{\lambda}\,\varphi_{1}-\frac{\lambda}{6}\,\varphi_{1}^{3}+\frac{\lambda}{2}\,\varphi_{1}\varphi_{2}^{2}+\frac{\mu}{\sqrt{2}}\,\varphi_{2}^{2}\right]\,,
{\calL}=\gamma\frac{r}{v}\mathrm{Tr}_{D}T_{\mu\nu}=\gamma\frac{r}{v}\mathrm{Tr}_{4}T-\gamma\frac{\epsilon}{4}\frac{r}{v}F_{\mu\nu}F^{\mu\nu}+\gamma\frac{\epsilon}{2}\frac{r}{v}(2M_{W}^{2}W^{+}W^{-}+M_{Z}^{2}Z^{2}),
{\calW}_{a}=\tilde{I}_{aa^{\prime}}\int\!V_{b}^{\mu}z_{bb^{\prime}}\hat{\epsilon}_{b^{\prime}c^{\prime}a^{\prime}}(\theta_{W}+\theta_{V})\tilde{I}_{c^{\prime}c}z_{cd}\frac{\delta}{\deltaV_{d}^{\mu}}
ds_{5}^{2}=\Big[1-({\frac{\mu}{r}})^{2}\Big]^{\frac{1}{3}}\Big[-(1-({\frac{\mu}{r}})^{2})dt^{2}+(1-({\frac{\mu}{r}})^{2})^{-2}dr^{2}+r^{2}d\Omega_{3}^{2}\Big].
<\{t\},Nr+\lambda\mid\prod_{q=1}^{N}U^{r}(z_{q})\mid\lambda>=\prod_{l>l^{\prime}}(z_{l}-z_{l^{\prime}})^{r^{2}}\prod_{q=1}^{N}z_{q}^{r\cdot\lambda+\lambda^{2}}(-1)^{\midt\mid}J_{\{t\}}^{(r^{2})}(z_{q})\left(j_{t}^{r^{2}}\right)^{-1/2}
\hat{\calI}_{\mu\nu}^{\alpha\beta}={\itg_{2}}\bigg(\hat{\theta}_{\rho}\big(-i\tilde{F}_{\lambda\gamma,\sigma}+F_{\lambda\gamma}\hat{P}_{\sigma}-{\sqrt2}i\hat{\theta}_{5}F_{\lambda\gamma,\sigma}\big)+\hat{\theta}_{\lambda}\big(-i\tilde{F}_{\rho\sigma,\gamma}+F_{\rho\sigma}\hat{P}_{\gamma}-{\sqrt2}i\hat{\theta}_{5}F_{\rho\sigma,\gamma}\big)\bigg)\,,
5\partial_{[i_{1}}{\hat{\tilde{b}}}{}_{i_{2}\cdotsi_{5}]\5}+{\textstyle\frac{m}{2}}\hat{c}^{(6)}{}_{i_{1}\cdotsi_{5}\5}=-5\partial_{[i_{1}}v^{(4)}{}_{i_{2}\cdotsi_{5}]}+{\textstyle\frac{m}{2}}v^{(5)}{}_{i_{1}\cdotsi_{5}}\,,
\kappa^{2}c^{4}f^{2}\left(\frac{E}{\kappac^{2}},\frac{\vec{P}^{\,2}}{\kappa^{2}c^{2}}\right)-c^{2}\vec{P}^{\,2}g^{2}\left(\frac{E}{\kappac^{2}},\frac{\vec{P}^{\,2}}{\kappa^{2}c^{2}}\right)\=\inv.\=\\mu^{2}c^{4}.
\tilde{\Psi}_{L}^{+}=\tilde{\eta}^{+}\tilde{F}_{L}~~\Rightarrow~~\tilde{\omega}^{+}=0,~~\tilde{\Psi}_{R}^{-}=\tilde{\eta}^{-}\tilde{F}_{R}~~\Rightarrow~~\tilde{\omega}^{-}=0.
\alpha_{00}(t)=\frac{\mathrm{e}^{-\pigt/4}}{4\tilde{\omega}\tilde{\tilde{\omega}}}\left[\left(4\tilde{\omega}+4\tilde{\tilde{\omega}}+i\pig\right)^{2}\mathrm{e}^{i\tilde{\tilde{\omega}}t}-\left(4\tilde{\omega}-4\tilde{\tilde{\omega}}+i\pig\right)^{2}\mathrm{e}^{-i\tilde{\tilde{\omega}}t}\right]\;.
\begin{array}{ccc}{}&{U(1)_{4,5}}&{U(1)_{8,9}}\\{v}&{2}&{0}\\{w}&{0}&{2}\\{y}&{4N_{c}}&{0}\\{x}&{4N_{c}}&{0}\\{\mu}&{-2}&{2}\\{\Lambda_{N=2}^{2N_{c}+2-N_{f}}}&{4N_{c}+4-2N_{f}}&{0}\\\end{array}
\Gamma_{\underline{{\gamma}}\underline{{\delta}}}^{\underline{{b}}}(u^{-1})_{\underline{{b}}}^{~\underline{{a}}}\equiv(v^{-1})_{\underline{{\gamma}}}^{~\underline{{\alpha}}}\Gamma_{\underline{{\alpha}}\underline{{\beta}}}^{{\underline{a}}}(v^{-1})_{\underline{{\delta}}}^{~\underline{{\beta}}}.
(\phi_{n+\mu}^{\phantom{2}}-\phi_{n}^{\phantom{2}})_{\phantom{n}}^{2}\phi_{n}^{2},\\\\\\phi_{n+\mu}^{2}\phi_{n}^{4}\\\\mathrm{and}\\\(\phi_{n+\mu}-2\phi_{n}+\phi_{n+\mu})^{2}.
\int_{0}^{\beta}d\tau_{j}\int_{-\infty}^{\infty}dx_{j}\frac{\alpha(T)}{2\lambda^{2}},\;\;\;\alpha(T):=\alpha\left(\frac{T}{\rho}\right)^{\lambda^{2}/4\pi}=\alpha_{0}\left(\frac{T^{2}}{\Lambda^{2}}\right)^{\lambda^{2}/8\pi},
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\left(\frac{\lambda_{x}^{2}}{4}\,(\gamma_{2}+\frac{\gamma_{1}\lambda_{1}}{2\lambda_{x}})^{2}+\frac{\lambda_{1}^{2}}{2r_{x}^{2}}\right)\frac{1}{r-r_{x}}+{\cal{O}}(\frac{1}{\sqrt{r-r_{x}}})\rightarrow\infty
\frac{1}{J^{2}}\frac{J^{4}}{N^{2}}g_{YM}^{4}N^{2}\frac{1}{J^{2}}\frac{1}{J}=\left(\frac{g_{YM}^{4}N^{2}}{J^{4}}\right)\frac{J^{4}}{N^{2}}\frac{1}{J}
{\calP}_{\Lambda}=\left(\begin{array}{c}{8e^{(\tilde{K}+K)/2}e_{\Lambda}^{2}}\\{0}\\{2e^{\tilde{K}}e_{\Lambda}^{1}-2e^{\tilde{K}}[(C+\bar{C})R^{-1}]^{0}\,e_{\Lambda}^{2}}\\\end{array}\right).
|R_{1}^{(-)}(\nu,k)|\leq2\int_{1}^{\infty}d\alpha\{\sum_{\ell=1}^{5}(\frac{4}{3})^{\ell}|\sum_{j=0}^{5-\ell}|C^{(2)}(\ell,j)|\alpha^{\frac{-1}{2}+j}\psi^{(j)}{(\alpha)}|\}
\begin{array}{cl}{A{\circ_{X}}(Z^{\dagger}{\circ_{X}}B)}&{=(AX)[X^{\dagger}((Z^{\dagger}X)(X^{\dagger}B))]}\\{}&{=(AX)[X^{\dagger}(X((X^{\dagger}Z^{\dagger})B))]}\\{}&{=(AX)[(X^{\dagger}Z^{\dagger})B]}\\{}&{=(AX)(Y^{\dagger}B),}\\\end{array}
\begin{array}{rl}{c_{i}c_{j}=}&{\displaystyle(c_{i}c_{j}-{\frac{1}{3}}f_{ijk}d_{k})+{\frac{1}{3}}f_{ijk}d_{k}}\\{=}&{\displaystyle(P_{20})_{ij,pq}c_{p}c_{q}+(P_{8})_{ij,pq}c_{p}c_{q}}\\\end{array}
\phi^{A}\psi^{B}+\phi^{B}\psi^{A}=\frac{1}{8}\left((\phi^{T}\psi)(\delta^{AB}\pm{\Gamma_{9}}^{AB})+\frac{1}{4!}(\phi^{T}\Gamma^{\mu\nu\alpha\beta}\psi){\Gamma_{\mu\nu\alpha\beta}}^{AB}\right),
\tilde{L}^{(0)}=\pi_{i}\dot{a}_{i}+(a_{i}a_{i}-1){\eta}-M-\frac{1}{8\lambda}\pi_{i}\pi_{i}+\frac{1}{4\lambda}(a_{i}\pi_{i})\theta-\frac{1}{8\lambda}a_{i}^{2}\theta^{2},
t^{\sigma}=\sum_{k=0}t_{(k)}^{\sigma}{}_{\Sigma}^{\mu_{1}...\mu_{k}}{}_{\nu_{1}...\nu_{k}}^{\Sigma^{\prime}}\partial_{\mu_{1}...\mu_{k}}\Gamma^{\Sigma}\frac{\partial}{\partial(\partial_{\nu_{1}...\nu_{k}}\Gamma^{\Sigma^{\prime}})},
S_{q}=\intd^{4}x\,\,\,(\Pi^{ia}\dot{A}_{i}^{a}+\Pi^{0a}\dot{A}_{0}^{a}+i\overline{\psi}\gamma^{0}\dot{\psi}+\dot{\calP}^{a}\overline{{c}}^{a}+\dot{c}^{a}\overline{{\calP}}^{a}-H_{c}-\{\Psi,Q\})
\begin{array}{c}{C_{2}=\frac12L_{AB}L^{AB}=E_{o}(E_{o}+3)+s(s+1)}\\{C_{4}=W_{A}W^{A}=-s(s+1)(E_{o}-1)(E_{o}-2)}\\{W_{A}=\frac18\epsilon_{ABCDE}L^{BC}L^{DE}}\\\end{array}
\lambda_{ren}=\lambda_{B}+{\frac{1}{2}}~{\frac{1}{(4\pi)^{3/2}}}~\left(\int_{\epsilon^{2}}^{\infty}{\frac{ds}{s^{5/2}}}e^{-\mus/l^{2}}+l^{2}\int_{\epsilon^{2}}^{\infty}{\frac{ds}{s^{3/2}}}e^{-\mus/l^{2}}\right)~~.
e^{\,i\Gamma[A^{(gen)}[a,\theta],\theta]_{DR}}\,=\,\int\,d\bar{\psi}\,d{\psi}\;e^{\,iS_{nc,\,DR}^{(-)}(A^{(gen)}[a,\theta],\Psi^{(gen)}[a,\psi,\theta],\bar{\Psi}^{(gen)}[a,\bar{\psi},\theta])}.
\frac{1}{4}\frac{(\bar{\mathcal{F}}_{11}+\mathcal{F}_{11})}{4}F_{\mu\nu}^{1}F^{\mu\nu1}+\frac{1}{4}\frac{(\bar{\mathcal{F}}_{22}+\mathcal{F}_{22})}{4}F_{\mu\nu}^{2}F^{\mu\nu2}
F(z,m)-F(0,m)={\frac{m^{3}}{g_{4}}}\Biggl({\frac{z^{2}}{2}}+z^{4}+{\frac{g_{6}}{g_{4}^{2}}}z^{6}+{\frac{g_{8}}{g_{4}^{3}}}z^{8}+...\Biggr).\,
F(p)=\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{1}{(p-q)^{2}}\frac{F(q)}{\left|-\frac{1}{2}\delta+iq_{3}+\frac{\left|{\bfq}\right|^{2}}{2m}+\Sigma_{\pm}(q_{3},\left|{\bfq}\right|)\right|^{2}}.
J^{\mu\nu}{}_{\lambda\rho}\equiv{\frac{\partial{\calL}}{\partialR_{\mu\nu}{}^{\lambda\rho}}}={\tilde{J}}^{[\mu}{}_{[\lambda}g^{\nu]}{}_{\rho]}={\frac{1}{4}}\left[{\tilde{J}}^{\mu}{}_{\lambda}g^{\nu}{}_{\rho}-{\tilde{J}}^{\mu}{}_{\rho}g^{\nu}{}_{\lambda}+{\tilde{J}}^{\nu}{}_{\rho}g^{\mu}{}_{\lambda}-{\tilde{J}}^{\nu}{}_{\lambda}g^{\mu}{}_{\rho}\right]_{.}
\tilde{\theta}=\left(\begin{array}{ccccc}{L_{1}}&{0}&{0}&{\cdots}&{0}\\{0}&{L_{3}}&{0}&{\cdots}&{0}\\{0}&{0}&{L_{5}}&{}&{0}\\{\vdots}&{\vdots}&{}&{\ddots}&{\vdots}\\{0}&{0}&{0}&{\cdots}&{L_{2N-1}}\\\end{array}\right),\quadL_{2n-1}=\left(\begin{array}{cc}{0}&{\tilde{\theta}_{2n-1,2n}}\\{-\tilde{\theta}_{2n-1,2n}}&{0}\\\end{array}\right),
\left(\begin{matrix}{\hat{V}^{11}}&{\hat{V}^{12}}&{\hat{V}^{21}}\\{\hat{V}^{21}}&{\hat{V}^{11}}&{\hat{V}^{12}}\\{\hat{V}^{12}}&{\hat{V}^{21}}&{\hat{V}^{11}}\\\end{matrix}\right)
T_{\\sigma}^{\rho}=\Phi^{2}\left(\mu^{\rho}\mu_{\sigma}+{\calK}s^{\rho}s_{\sigma}\right)+\left[{\calL}+c^{2}\Phi^{2}\left(s^{2}{\calK}+y^{2}{\calA}\right)\right]g_{\\sigma}^{\rho}\.
\begin{array}{lcr}{\sum_{\nu=1}^{8}\{\{X^{\mu},X^{\nu}\},X^{\nu}\}=}\\{\sum_{\stackrel{m_{i},n_{i}}{i=1,2,3}}(-\gamma^{\mu})8{\epsilon_{123}}F_{(123)}{\omega}(\{m_{i}\},\{n_{i}\})X_{\sum_{i=1,2,3}m_{i},\sum_{i=1,2,3}n_{i}}(\sigma)L_{\sum_{i=1,2,3}m_{i},\sum_{i=1,2,3}n_{i}},}\\\end{array}
\mu_{0}+\mu_{\rho}+2\mu_{\theta}=\int_{0}^{\infty}drM(r)^{4}\mathcal{L}(r)^{2}\left[J^{\prime}(r)^{2}+\frac{\left[M(r)^{4}\mathcal{L}(r)^{2}J^{\prime}(r)\right]^{\prime}}{M(r)^{4}\mathcal{L}(r)^{2}}J(r)+\alpha\left(1-J(r)^{2}\right)\right]\,.
ln\Biggl(\Bigl(\frac{1+e^{i\chi_{0}}+\varepsilon}{1+e^{-i\chi_{0}}+\varepsilon}\Bigr)\Bigl(\frac{1+e^{-i\chi_{0}}-\varepsilon}{1+e^{i\chi_{0}}-\varepsilon}\Bigr)\Biggr)=ln\Biggl(\frac{cos(\chi_{0}/2)-i(\varepsilon/2)sin(\chi_{0}/2)}{cos(\chi_{0}/2)+i(\varepsilon/2)sin(\chi_{0}/2)}\Biggr)
Z\left(X(\frac{p_{1}}{q_{1}},\frac{p_{2}}{q_{2}},\frac{p_{3}}{q_{3}}),k\right)=\sum_{0<\alpha_{1},\alpha_{2},\alpha_{3}<K}\sum_{\lambda=0,\frac{1}{2}}\frac{1}{2}e^{2\pii\lambda(1-\alpha_{1}-\alpha_{2}-\alpha_{3})}\tilde{M}_{\alpha_{1}1}^{(p_{1},q_{1})}\tilde{M}_{\alpha_{2}1}^{(p_{2},q_{2})}\tilde{M}_{\alpha_{3}1}^{(p_{3},q_{3})}\tilde{N}_{\alpha_{1},\alpha_{2},\alpha_{3}}
<\phi_{\omega},\phi_{\sigma}>=\int_{\Sigma}\sqrt{h}d^{D-1}x\left[(\omega+\sigma)\phi_{\omega}^{*}\phi_{\sigma}+i\phi_{\omega}^{*}a^{i}(\nabla_{i}+i\sigmaa_{i})\phi_{\sigma}-i(\nabla_{i}-i\omegaa_{i})\phi_{\omega}^{*}a^{i}\phi_{\sigma}\right]~~~,
\left[\left({\frac{\dot{R}}{R}}\right)^{2}+\xi_{+}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}-\left[\left({\frac{\dot{R}}{R}}\right)^{2}+\xi_{-}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}=-{\frac{\kappa_{5}^{2}}{3}}[\lambda(\phi)+\rho].
\hat{Z}_{888}^{10,1}=\frac{(det\Im\\Omega)^{2}}{2^{3}}\left|\Theta\left[\begin{array}{lll}{1/2}&{1/2}&{1/2}\\{1/2}&{1/2}&{0}\\\end{array}\right](0\vert\Omega)\right|^{8}
Tr_{s+1}e^{-a_{s}^{\dagger}a_{s}/M^{2}}-Tr_{s}e^{-a_{s}a_{s}^{\dagger}/M^{2}}=Tr_{s+1}e^{-a_{s}a_{s}^{\dagger}}-Tr_{s}e^{-a_{s}a_{s}^{\dagger}}=1