harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Omega_{\mu}\equiv{\frac{1}{8}}\Gamma_{11}\Gamma^{\rho\sigma}F_{\mu\rho\sigma}-{\frac{1}{16}}e^{\phi}(\Gamma_{11}\Gamma^{\rho\sigma}F_{\rho\sigma}-{\frac{1}{12}}\Gamma^{\rho\sigma\lambda\tau}F_{\rho\sigma\lambda\tau})\Gamma_{\mu}.
d_{\beta}\sim\gamma_{n_{1}}(-1)^{k_{1}+\dots+k_{N_{f}}}(2k_{0})^{4n_{1}-3}\prod_{i=1}^{N_{f}}{_{n_{1}}}C_{k_{i}}\qquad(2k_{0}\ggn_{1},\,0\leq{}^{\forall}k_{i}\leqn_{1}),
\Omega_{\rhoi_{2}\ldotsi_{2m-2}\sigma}^{(2m-1)}={\frac{1}{(2m-3)!}}k_{\rhol_{1}\ldotsl_{m-1}}{C_{j_{2}j_{3}}}^{l_{1}}\ldots{C_{j_{2m-2}\sigma}}^{l_{m-1}}\epsilon_{i_{2}\ldotsi_{2m-2}}^{j_{2}\ldotsj_{2m-2}}\,\,,
\psi_{a_{1}a_{2}}^{id}(x_{1},x_{2})=\sum_{\alpha,\beta=+,-}[e^{i(\alphak_{1}x_{1}+\betak_{2}x_{2})}A_{a_{1}a_{2}}(\alpha,\beta)-e^{i(\alphak_{1}x_{2}+\betak_{2}x_{1})}B_{a_{2}a_{1}}(\alpha,\beta)].
|{\bar{\varphi}}_{i}\,\rangle=|i\,\rangle+\alpha\left(E_{i}-{\calA}^{\dagger}H\alpha+i\epsilon\right)^{-1}{\calA}^{\dagger}H_{\mathrm{\scriptsize~I}}|i\,\rangle+i{\epsilon}\,\Xi\left(E_{i}-{\calX}{\bar{H}}+i\epsilon\right)^{-1}\left(1-\Xi^{\dagger}{\calA}^{\dagger}\right)|i\,\rangle.
W_{n}^{(s)}\psi_{p}=\sum_{{1\leqj_{k}\neqj_{l}\leqN\atop(0\leqk<l\leqs-1)}}\prod_{k=1}^{s-1}\frac{z_{j_{0}}^{n+s-1}}{z_{j_{0}}-z_{j_{k}}}\psi_{p}\,.
=\;8m^{2}\mathrm{Tr}\Big[\frac{1}{-\triangle+m^{2}}\not{\!\!B}\frac{1}{-\triangle+m^{2}}\not{\!\!B}\frac{1}{-\triangle+m^{2}}\not{\!\partial}\not{\!\!B}\frac{1}{-\triangle+m^{2}}\not{\!\partial}\not{\!\!B}\Big]\;,
\chi^{(\rho)}\cdot\pi^{(\sigma)}=\chi_{J}^{(\rho)}\:c_{i}^{J}\cdot\pi_{K}^{(\sigma)}\:c_{i}^{K}=\left[\chi_{J}^{(\rho)}\pi_{K}^{(\sigma)}\right]\;c_{i}^{J}\,c_{i}^{K}.
f=\int\frac{dt}{t^{3/2}}\int_{-\infty}^{+\infty}\!\!d\theta\,e^{-\pit(y^{2}+x_{0}^{2}+x_{1}^{2}+x_{2}^{2})-\frac{\pi}{t}\left(1+(\thetay)^{2}+(\thetax_{0}-\frac{x_{3}}{y})^{2}\right)-2\pii\thetax_{1}x_{2}}\,.
\frac{\delta\kappa}{\gamma}\;\;\;=\;\;\;-\frac{1}{2}\frac{\deltaI}{(I_{c})^{3}}\;-\;\frac{\lambda^{c}}{\gamma}\frac{M_{c}}{I_{c}}\left[\;\frac{\deltaM}{M_{c}}-\frac{\deltaI}{I_{c}}\right]\;\;\;=\;\;\;\left[\frac{-\pi^{2}}{256}+\frac{4}{3}\frac{\lambda^{c}}{\gamma}\right]\frac{\deltaI}{I_{c}}\;\;-\;\;\frac{\lambda^{c}}{\gamma}\frac{\varepsilon}{2}\;,
\begin{array}{ll}{\begin{array}{ll}{\varphi(r)=\eta,}&{r\rightarrow\infty,}\\{\varphi(r)=0,}&{r=0,}\\\end{array}}&{\begin{array}{ll}{P(r)=0,}&{r\rightarrow\infty,}\\{P(r)=1,}&{r=0.}\\\end{array}}\\\end{array}
\overline{{\Gamma}}(H;m)=-{\frac{VH^{2}}{8\pi^{2}}}\left({\frac{1}{240}}({\frac{H}{m^{2}}})^{2}-{\frac{1}{1008}}({\frac{H}{m^{2}}})^{4}+{\frac{1}{1440}}({\frac{H}{m^{2}}})^{6}+\cdots\right),
\theta_{SO(n,n)}=\frac{2\pi^{2}}{3}V+2V\!\sum_{m^{i}\neq0}\frac{1}{m^{i}g_{ij}m^{j}}+4\piV\!\!\!\!\sum_{(m^{i},n^{i})/Sl(2)}\!\!\!\!\frac{e^{-2\pi\,\sqrt{(m^{ij})^{2}}+2\piim^{ij}B_{ij}}}{\sqrt{(m^{ij})^{2}}}\,
\int_{\Omega}\delta({\calL})\sqrt{g}\;d^{4}x=\int_{\Omega}J^{\mu\nu}{}_{\lambda\rho}\left\{-2\nabla_{\mu}\nabla^{\lambda}(\deltag)_{\nu}{}^{\rho}+R_{\mu\nu}{}^{\sigma\lambda}(\deltag)_{\sigma}{}^{\rho}\right\}\sqrt{g}\;d^{4}x.
M_{\mathrm{{spin}}}^{kl}=ix^{k}\partial^{l}-ix^{l}\partial^{k}+\frac{1}{2}[\gamma^{k},\gamma^{l}]+\frac{1}{2}(Z_{\rho}^{kj}\partial^{l}-Z_{\rho}^{lj}\partial^{k})\gamma_{j}\gamma^{\rho}
e^{-S_{\mathrm{dual}}(A)\;}=\intDB_{\mu}DB_{\mu}^{\dagger}D\eta_{\mu}e^{-\left(S_{\mathrm{top}}(B,(\partial+i\eta)B)+\intd^{3}xA_{\mu}\epsilon^{\mu\nu\rho}\partial_{\nu}\eta_{\rho}\right)\;}\;.
S(\phi^{\prime},J^{\prime})=-\frac{1}{g^{2}}\sum_{n}\left\{\frac{1}{2}\sum_{\mu}(\phi_{n+\mu}^{\prime}-\phi_{n}^{\prime})^{2}+\frac{1}{2}m^{2}a^{2}\phi_{n}^{2}+\frac{1}{4!}\phi_{n}^{4}+J^{\prime}\phi_{n}^{\prime}\right\}
{\calH}=G_{00}^{1/2}STr_{\theta}\left\{\frac{-1}{G_{s}^{2}(2\pi)^{2p}\alpha^{p+1}}det_{(p)}(G+{\calF})+\tilde{\epsilon}^{t}(G-{\calF}G^{-1}{\calF})\tilde{\epsilon}\right\}^{1/2},
\Pi^{\mu\nu}(q^{2})=-\sum_{n}\int{\frac{d^{D}p}{(2\pi)^{D}}}\left[{\frac{(2p+q)^{\mu}(2p+q)^{\nu}}{(p^{2}+m_{n}^{2})\left((p+q)^{2}+m_{n}^{2}\right)}}-{\frac{2\eta^{\mu\nu}}{p^{2}+m_{n}^{2}}}\right].
F_{l_{i}}^{L}=\left(\begin{array}{c}{\nu_{i}^{L}}\\{e_{i}^{L}}\\\end{array}\right),\qquadF_{q_{i}}^{L}=\left(\begin{array}{c}{u_{i}^{L}}\\{d_{i}^{L}}\\\end{array}\right),\qquadi=1...N_{F},
R_{1/2,1/2}(x,y)=f_{q}(x,y)\cdot\left(\begin{array}{cccc}{1}&{{}}&{{}}&{{}}\\{{}}&{\frac{q^{-1}(y-x)}{y-q^{-2}x}}&{\frac{y(1-q^{-2})}{y-q^{-2}x}}&{{}}\\{{}}&{\frac{x(1-q^{-2})}{y-q^{-2}x}}&{\frac{q^{-1}(y-x)}{y-q^{-2}x}}&{{}}\\{{}}&{{}}&{{}}&{1}\\\end{array}\right)
\oint_{\mathrm{a}}\omega_{\alpha}=\left(\begin{matrix}{\frac{1}{2}-iT_{12}}\\{-iT_{22}}\\\end{matrix}\right)~~~,~~~\oint_{\mathrm{b}}\omega_{\alpha}=\left(\begin{matrix}{iT_{11}}\\{\frac{1}{2}+iT_{12}}\\\end{matrix}\right)~,
{{}\bf{\Lambda}}_{nm}=\left(\begin{array}{cccccc}{\ddots}&{\ddots}&{}&{}&{}&{}\\{\ddots}&{\ddots}&{\ddots}&{}&{}&{}\\{}&{\ddots}&{0}&{-1}&{}&{}\\{}&{}&{1}&{0}&{-1}&{}\\{}&{}&{}&{1}&{0}&{-1}\\{}&{}&{}&{}&{1}&{\ddots}\\\end{array}\right).
S=\frac{1}{\tilde{g}_{s}(2\pi)^{p}(\alpha^{\prime})^{\frac{p+1}{2}}}Tr[\sqrt{\mathrm{det}(\tilde{g}^{\mu\nu}+2\pi\alpha^{\prime}\tilde{b}^{\mu\nu}+i[X^{\mu},X^{\nu}])}].\nonumber
J_{+}=\frac{1}{4\sqrt{d}}Y_{10}\bar{X}_{10}=\frac{2\sqrt{d}}{2N+1}\left(\begin{matrix}{0}&{\sqrt{N-1}}&{0}&{\cdots}&{0}\\{\vdots}&{0}&{\sqrt{2(N-2)}}&{0}&{\vdots}\\{\vdots}&{\cdots}&{0}&{\ddots}&{0}\\{\vdots}&{\cdots}&{\cdots}&{0}&{\sqrt{N-1}}\\{0}&{\cdots}&{\cdots}&{\cdots}&{0}\\\end{matrix}\right)
M_{a\mu_{0}\mu_{1}\cdots\mu_{p+1}}\equiv\sum_{i=0}^{p+1}(-1)^{(p+1)\cdoti}\sigma_{a\mu_{i}\mu_{i+1}\cdots\mu_{p+1}\mu_{0}\mu_{1}\cdots\mu_{i-2}}\partial_{a\mu_{i-1}}.
\stackrel{(-1,-1)}{\bar{q}_{i}},\stackrel{(-2,-1)}{\bar{\varphi}_{\alpha_{0}}},\stackrel{(-1,-2)}{\bar{\varphi}_{\alpha_{0}}},\stackrel{(-2,-2)}{\bar{\pi}_{\alpha_{0}}},\stackrel{(-3,-1)}{\bar{\varphi}_{\alpha_{1}}},\stackrel{(-2,-2)}{\bar{\varphi}_{\alpha_{1}}},\stackrel{(-1,-3)}{\bar{\varphi}_{\alpha_{1}}},\stackrel{(-3,-2)}{\bar{\pi}_{\alpha_{1}}},\stackrel{(-2,-3)}{\bar{\pi}_{\alpha_{1}}}.
{\cal{J}}_{1}=\frac{x^{2}}{\varrho}\,;\quad{\cal{J}}_{2}=n\varrho\,;\quad{\cal{J}}_{3}=2\alphax^{2}-v^{2}\varrho+\frac{xv}{2}\varrho_{z}\,;\quad\varrho=\left(1-2\alphaz^{2}\right)\,.
\left(\tilde{\partial}_{+}\tilde{\partial}_{-}+\left|A\right|^{2}\right)\left[\phi^{\prime},\psi_{\alpha}^{\prime},V_{-}^{\prime}\right]=0,\,\quad\left(\tilde{\partial}_{+}\tilde{\partial}_{-}+\left|A\right|^{2}\right)\left[\phi,\psi_{\alpha},V_{+}\right]=0
f(x)=\int_{-\infty}^{\infty}\frac{d\xi}{2\pi}\mathrm{e}^{i(1-x)\,\xi}\langle\widetilde{H}(p)|W_{\scriptstyle\Pi}({v\!\cdot\!n}\,\xi\mu-i0)|\widetilde{H}(p)\rangle\,,
\left(\begin{array}{c}{\Phi_{1}}\\{\Phi_{2}}\\\end{array}\right)=\frac{1}{\sqrt{2}}\sum_{n}e^{-in\sigma}\left(\begin{array}{cc}{ie^{imH\alpha^{\prime}\tau}}&{-ie^{-imH\alpha^{\prime}\tau}}\\{e^{imH\alpha^{\prime}\tau}}&{e^{-imH\alpha^{\prime}\tau}}\\\end{array}\right)\left(\begin{array}{c}{C_{1n}}\\{C_{2n}}\\\end{array}\right).
\phi_{r}={\frac{\left(1-e^{2h}\right)\left[(c-e^{2h}+ce^{2h}+ce^{4h})(1-\phi)+2p_{1}{e^{2h}}g-2\phi^{2}q^{2}(1+e^{2h}+e^{4h})\right]}{cq+e^{2h}q+e^{4h}q+ce^{6h}q-2e^{3h}g}}\,
{\displaystyle\lambda_{\underline{\alpha}}\longrightarrow\lambda_{\underline{\alpha}}+\frac{ieg}{8F}D_{\underline{\alpha}}W,\\\bar{\lambda}_{\underline{{\dot{\alpha}}}}\longrightarrow\bar{\lambda}_{\underline{{\dot{\alpha}}}}-\frac{ieg}{8F}\bar{D}_{\underline{{\dot{\alpha}}}}\bar{W}.}
\langleT_{i}^{}\rangle_{D}^{(b)}\left(\xi_{1},\xi_{2},\xi\right)=\langleT_{i}^{k}\rangle_{D}^{(1b)}\left(\xi_{2},\xi\right)-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dk\,k^{d}\int_{0}^{\infty}d\omega\frac{I_{\omega}(k\xi_{1})}{I_{\omega}(k\xi_{2})}\frac{F^{(i)}[D_{\omega}(k\xi,k\xi_{2})]}{D_{\omega}(k\xi_{1},k\xi_{2})}.
\left.\begin{array}{ll}{\partial_{+}q_{1}={\displaystyle\frac{2}{\lambda}}(1-q_{1}q_{2})e^{\phi},\,\,}&{\partial_{+}q_{2}=-2q_{1}e^{-2\phi}-{\displaystyle\frac{2}{\lambda}}q_{2}^{2}e^{\phi},}\\{\partial_{-}q_{1}=-(q_{2}+\lambdaq_{1}^{2})+q_{1}\pi_{\phi},\,\,}&{\partial_{-}q_{2}=\lambda(1-q_{1}q_{2})-q_{2}\pi_{\phi},}\\\end{array}\right.
\left\langle\Psi_{X}(\xi^{\prime})\Psi_{X}(\xi)\right\rangle_{0}=\frac{1}{N^{2}}a^{-2}\hat{s}^{-2}5c[2w_{D}(y,y^{\prime})+u_{D}(y,y^{\prime})]+O(\frac{1}{N^{2}}a^{-5/3}),
\left(\begin{array}{cc}{u}&{i\alpha}\\{i\beta}&{\bar{u}}\\\end{array}\right)=\left(\begin{array}{cc}{e^{ir_{+}\phi}\sqrt{1-\alpha\beta}}&{ie^{r_{-}\phi}\alpha}\\{i\betae^{-r_{-}\phi}}&{e^{-ir_{+}\phi}\sqrt{1-\alpha\beta}}\\\end{array}\right)
S={\frac{1}{2k^{2}}}\intdz^{(-4)}du[\bar{q}_{1}^{+}D^{++}q_{1}^{+}+\bar{q}_{2}^{+}D^{++}q_{2}^{+}++V^{++}(\bar{q}_{1}^{+}q_{2}^{+}+\bar{q}_{2}^{+}q_{1}^{+}+\lambda^{++})]\qquad.
\left[(f_{1}f_{5})^{1/2}\gamma^{0}\partial_{t}+\gamma^{1}\partial_{r}+(f_{1}f_{5})^{1/2}\gamma^{2}\partial_{x_{5}}\right]\psi^{\prime\prime}=\left[\frac{(L+3/2)}{r}-g(r)\right]\psi^{\prime\prime}
S^{\mathrm{{matt}}}=\sum_{\pm}\intd^{D}x\,\sqrt{-g^{(D)\pm}}\,{\calL}^{\pm}\left(\chi_{(D)}^{\pm},g_{AB}^{(D)\,\pm}\right)=\intd^{4}x\sum_{\pm}\,\sqrt{-g_{\pm}}\,a_{\pm}^{D_{2}}{\calL}^{\pm}\left(\chi^{\pm},g_{\mu\nu}^{\pm}\right).
\displaystyle{\frac{P(a)}{\hbar\,\Omega}\,\left(\frac{{\Omega}}{c}\right)^{-3}=\frac{-5}{216\,{\pi}^{2}}+\frac{49}{5400\,\pi}\,\left(\frac{a\,\Omega}{c}\right)^{-1}+{\calO}\left(\frac{a\,\Omega}{c}\right)^{-2}\simeq-2\times10^{-3}},
\left[\frac{E(xb,x^{60})E(x^{11}b,x^{60})E(x^{31}/b,x^{60})E(x^{41}/b,x^{60})}{E(x/b,x^{60})E(x^{11}/b,x^{60})E(x^{31}b,x^{60})E(x^{41}b,x^{60})}\right]^{N}=b^{2N}.
{\calC}=\left(\begin{matrix}{1}&{0}&{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}&{1}\\{0}&{-1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{{\frac{1}{2}}}&{0}&{-{\frac{1}{2}}}\\{-{\frac{1}{2}}}&{0}&{-{\frac{1}{2}}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{-1}&{0}\\\end{matrix}\right)
(sin{\theta_{n}^{0}})^{m_{1}+l_{1}+1}(2m_{1}+n-1)C_{N-m_{1}-1}^{m_{1}+\frac{n-1}{2}+1}(cos{\theta_{n}^{0}})]C_{N^{\prime}-l_{1}}^{l_{1}+\frac{n-1}{2}}(cos{\theta_{n}^{0}})
\tilde{\nabla}_{a}\theta^{a}+{\Sigma_{a}}_{k}^{i}\Sigma^{aki}+{\Lambda_{a}}_{k}^{i}\Lambda^{aki}+\frac{1}{N-D}\theta_{a}\theta^{a}+(M^{2})_{i}^{i}-\epsilon^{ca}\Lambda_{c}^{ik}H_{ika}=0
(-\frac{1}{2}\frac{d^{2}}{{(dx^{1})}^{2}}-\Lambda_{1}^{3}+\frac{1}{2}{(\Lambda_{1}^{3})}^{2}(x^{1})^{2}+\frac{1}{(x^{1})^{2}}(8(m^{1})^{2}-\frac{1}{8}))Y_{1}^{2}(x^{1})=4E_{III}^{1}Y_{1}^{2}(x^{1}),
{\tilde{N}}(x,y)={\frac{1}{2}}\int_{0}^{1}d\gamma\left[\left(\xi-{\frac{(1-\gamma^{2})}{4}}\right)^{2}-{\frac{\gamma^{4}}{36}}\right]\int{\frac{d^{4}p}{(2\pi)^{4}}}cos[p(x-y)]\theta\left(-p^{2}-{\frac{4m^{2}}{1-\gamma^{2}}}\right).
\hat{\tilde{G}}=7\left\{D\hat{\tilde{C}}+10\hat{C}D\hat{C}-{\textstyle\frac{5}{4}}m\left(i_{\hat{k}}\hat{C}\right)\left(i_{\hat{k}}\hat{C}\right)\hat{C}+{\textstyle\frac{1}{14}}m\left(i_{\hat{k}}\hat{\tilde{N}}\right)\right\}\,,
{}_{\mathrm{FF}}\langlep^{\prime}|q^{L_{0}^{\mathrm{FF}}+{\tilde{L}}_{0}^{\mathrm{FF}}-{\frac{c_{\mathrm{FF}}}{12}}}|p\rangle_{\mathrm{FF}}=\delta(p^{\prime}-p)\q^{p^{2}}q^{-1/12}\prod_{n=1}^{\infty}\left(1-q^{2n}\right)^{-1}
ds^{2}=-\left(1-{\frac{\sqrt{A/4\pi}}{\tilde{r}}}\right)^{2}dt^{2}+\left(1-{\frac{\sqrt{A/4\pi}}{\tilde{r}}}\right)^{-2}d\tilde{r}^{2}+\tilde{r}^{2}d^{2}\Omega\;\;.
\begin{array}{ll}{{\calX}^{2}=}&{g^{2}\varphi^{2}M^{2}+\varphi^{2}e^{2gM};}\\{{\calY}=}&{-(hm\varphi^{2}+2ghmM\varphi^{2})e^{-2gM}+2m^{2}M-gh^{2}\varphi^{4}e^{-4gM}.}\\\end{array}
\Psi\left(x\right)=Q\left(\begin{array}{c}{\psi_{1}\left(x^{1},x^{2}\right)\left[m+F-ik_{1}\sigma_{2}\right]}\\{\psi_{2}\left(x^{1},x^{2}\right)\left[\left(m-F\right)\sigma_{3}-ik_{1}\sigma_{1}\right]}\\\end{array}\right)\upsilon\tilde{\Phi}\left(x^{0},x^{3}\right)\,,\;F=\hbar^{-1}\left(P_{0}+P_{3}\right)\,,
\delta_{\epsilon}{\mathbbA}=\left[\begin{array}{cc}{\epsilon^{k}\bar{\psi}_{k}-\psi^{k}\bar{\epsilon}_{k}}&{D\epsilon_{j}}\\{-D\bar{\epsilon}^{i}}&{\bar{\epsilon}^{i}\psi_{j}-\bar{\psi}^{i}\epsilon_{j}}\\\end{array}\right],
\hat{L}_{12}=i\hbar\left[{\alpha_{1}^{\pm}}^{\dagger}\alpha_{2}^{\pm}-{\alpha_{2}^{\pm}}^{\dagger}\alpha_{1}^{\pm}\right]=-\hbar\left[{\alpha_{+}^{\pm}}^{\dagger}\alpha_{+}^{\pm}-{\alpha_{-}^{\pm}}^{\dagger}\alpha_{-}^{\pm}\right]\\\,
T_{ab}=\frac{1}{4\pi\alpha^{\prime}}\,(\partial_{a}X^{\mu}\partial_{b}X_{\mu}-{\frac12}\,g_{ab}\,\partial_{c}X^{\mu}\partial^{c}X_{\mu})-\frac{1}{4\pik}\,(R_{ab}-{\frac12}\,g_{ab}\,R).
\Gamma_{NP}^{(1)}\,=\,\frac{g^{2}}{4}\,\frac{1}{16\pi^{2}}\int_{0}^{1}dx\int\frac{d\alpha}{\alpha}e^{i\alpha[x(1-x)p^{2}-m^{2}]+i\frac{p\circp}{4\alpha}-\alpha\epsilon}\Phi(-p,\theta)\bar{\Phi}(p,\theta)
\begin{array}{rl}{Q_{ij}}&{=\mathrm{diag}(n_{2}n_{3}m_{1},n_{1}n_{3}m_{2},n_{2}n_{1}m_{3})\,,}\\{P^{ij}}&{=\mathrm{diag}(m_{2}m_{3}n_{1},m_{1}m_{3}n_{2},m_{2}m_{1}n_{3})\,,}\\{q_{0}}&{=n_{1}n_{2}n_{3}\,,}\\{p^{0}}&{=-m_{1}m_{2}m_{3}\,.}\\\end{array}
\bar{N}_{m0}^{rs}=\bar{N}_{0m}^{sr}=\sqrt{2\mu\alpha_{s}}\,\epsilon^{st}\alpha_{t}\left[\left(C_{r}C^{-1}\right)^{1/2}A^{(r)T}\Gamma_{+}^{-1}B\right]_{m}\quadr,s,t=1,2
L_{5}^{10}=-\intd^{10}x\sqrtg_{5}e^{\frac{2\phi}{3}}\left[{\frac{4}{{\alpha^{\prime}}^{4}}}R+e^{-{\frac{2\phi}{3}}}{\frac{1}{{\alpha^{\prime}}^{3}}}trF^{2}+\cdots\right]\.
L^{(1)}:=||{\calV}_{1}^{(1)}||=\sqrt{\Bigl(\frac{l}{4d}(a_{l}+a_{r})+\frac{w_{1}}{2}\Bigr)^{2}+\Bigl(\frac{\sqrt{3}l}{4d}(a_{l}-a_{r})\Bigr)^{2}}\,a.
\sum_{l=-\infty}^{\infty}e^{-4\pi^{2}(l+w_{i})^{2}\beta^{-2}t}={\frac{\beta}{\sqrt{4\pit}}}\sum_{k=-\infty}^{\infty}\eta_{i}^{k}e^{-{\frac{\beta^{2}k^{2}}{4t}}}~~~.
E_{2}(n_{1},\ldots,n_{m},I)={\frac{(-1)^{n}}{n_{1}!\cdotsn_{m}!}}{\frac{\partial^{n}D(I;\lambda)}{\partial\lambda^{n_{1}}\cdots\partial\lambda^{n_{m}}}}\Bigr\vert_{\lambda_{1}=\cdots=\lambda_{m}=1}
S\=\int_{\tau_{1}}^{\tau_{2}}\!\!d\tau\\Bigl\{{\textstyle{\frac1{2e}}}(\dot{x}^{n}+\omega\epsilon^{nmkl}\psi_{m}\psi_{k}\psi_{l}+\chi\psi^{n})^{2}+\psi_{n}{\dot{\psi}}^{n}+\bar{\xi}\dot{\xi}+\phi\,(i\bar{\xi}\xi-{\textstyle{\frac23}}\epsilon^{nmkl}\psi_{n}\psi_{m}\psi_{k}\psi_{l})\Bigr\}\.
L_{\Sigma+}=\epsilon^{\mu\nu\rho\sigma}\bigg[\dots+\bigg(C-{\frac{B^{2}}{4A}}\bigg)\Sigma_{\mu\nu}^{+ab}\Sigma_{\rho\sigmaab}^{+}-{\frac{BE}{2A}}G_{\mu\nu}^{+ab}\Sigma_{\rho\sigmaab}^{+}+\dots\bigg],
c_{21}\leq\int_{1}^{\infty}d\alpha\alpha^{1/4}|\Phi_{1}^{(2)}(\alpha)|[\frac{128}{15}+\frac{64}{5}\delta],\leq(45).\int_{1}^{\infty}d\alpha\alpha^{1/4}|\Phi_{1}^{(2)}(\alpha)|.
s=-{\frac{1}{4\pic}}\varepsilon_{\alpha\beta\gamma}\int{d^{3}xn_{\alpha}A_{\beta}E_{\gamma}}={\frac{1}{4\pic}}\int{d^{3}xA_{\alpha}H_{\alpha}}={\frac{1}{4\pic}}\int{d^{3}x{{\bfA}}(\nabla\times{\bfA})},
T=-\frac{1}{2}(\partialX)^{2}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{\sqrt{2k}}\partial^{2}\phi-\frac{1}{2}(\partialu)^{2}-\frac{1}{2}(\partialv)^{2}-\frac{1}{2}(\partialH)^{2}-\eta\partial\xi,
S=-\sum{\mathrm{Tr}({{U}_{\mu}{U}_{\nu}{U}_{\mu}^{\dagger}{U}_{\nu}^{\dagger}})}+\frac{i}{2}\mathrm{Tr}{{\bar{\psi}}{\Gamma^{\mu}}(U_{\mu}\psiU_{\mu}^{\dagger}-U_{\mu}^{\dagger}\psiU_{\mu})},
P_{\mathrm{BH}}\simR_{\mathrm{BH}}^{-2}\sim\ell_{s}^{-2}\,\lambda^{-2/(d-2)}\,\\tau_{\mathrm{evap}}^{\mathrm{BH}}\simR_{\mathrm{BH}}\,S_{\mathrm{BH}}\sim\ell_{s}\,g^{-2}\,\lambda^{d/(d-2)}\,.
{\calV}(x,y,z)=\intd^{2\omega}\zeta{\frac{\partial}{\partialz^{\alpha}}}\left[{\frac{\partial}{\partialx^{\alpha}}}{\frac{\partial}{\partialy^{+}}}-{\frac{\partial}{\partialy^{\alpha}}}{\frac{\partial}{\partialx^{+}}}\right]F(x-\zeta)F(y-\zeta)G(z-\zeta)
S_{2}=\frac{4}{81}\sum_{n\ge0}\left(-\frac{1}{27}\right)^{n}\left\{\frac{9}{(6n+1)^{2}}-\frac{9}{(6n+2)^{2}}-\frac{12}{(6n+3)^{2}}-\frac{3}{(6n+4)^{2}}+\frac{1}{(6n+5)^{2}}\right\}
\delta_{\tilde{\epsilon}}x_{i}=-i{\tilde{\epsilon}}_{i}f_{ijk}x_{k}\psi_{j}\,\quad\delta_{\tilde{\epsilon}}\psi_{i}={\tilde{\epsilon}}{\dot{x}}_{j}f_{jik}x_{k}-i{\tilde{\epsilon}}f_{jki}\psi_{j}\psi_{k}
{\bfG}(t,\vec{x};t^{\prime},\vec{x}^{\prime})=\left\{\begin{array}{ll}{G^{++}(t,\vec{x};t^{\prime},\vec{x}^{\prime})}&{G^{+-}(t,\vec{x};t^{\prime},\vec{x}^{\prime})}\\{G^{-+}(t,\vec{x};t^{\prime},\vec{x}^{\prime})}&{G^{--}(t,\vec{x};t^{\prime},\vec{x}^{\prime})}\\\end{array}\right\}
Z_{(1,\Theta^{4})}^{torus}(\tau,\bar{\tau},T,U)=\sum_{m_{1},m_{2}\in2Z\atopn^{1},n^{2}\inZ}e^{2\pii\tau(m_{1}n^{1}+m_{2}n^{2})}e^{\frac{-\pi\tau_{2}}{T_{2}U_{2}}|TUn^{2}+Tn^{1}-Um_{1}+m_{2}|^{2}}\.
{\mathrm{}^{*}\calL}_{TM}=\beta^{2}{\calF}\left(\frac{-1}{16}F_{\mu\nu}F^{\mu\nu}+\frac1{24\beta}H_{\mu\nu\rho}H^{\mu\nu\rho}\right)-\frac{\chi}{2\theta}\varepsilon^{\mu\nu\rho\lambda}B_{\nu\rho}\partial_{\mu}A_{\lambda}\;.
m^{5}\;=\;\left(\begin{array}{rrr}{0}&{-1}&{0}\\{-1}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)\;,\qquadm^{6}\;=\;\left(\begin{array}{rrr}{0}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right)\;.
(\omega_{x^{+}})_{\pm}^{~\pm}=\pm{\frac{1}{2x^{+}}}{\frac{4(GM)^{2}}{r^{2}}},\;\;\;\;\;(\omega_{x^{-}})_{\pm}^{~\pm}=\pm{\frac{1}{2x^{-}}}{\frac{4(GM)^{2}}{r^{2}}}
J(Q^{2})=v_{4}k^{-2}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+2)!(n+1)}\left(\frac{Q^{2}}{k^{2}}\right)^{n+1}\left\{\int_{0}^{\frac{1}{2}}dz\frac{z^{n+1}}{(1-z)^{2}}-\int_{\frac{1}{2}}^{\frac{2}{3}}dz\frac{z^{n+1}}{(1-z)^{2}}\right\}.
C_{c.o.}:(m_{1}^{\prime}-m_{1})^{2}+\{m_{2}^{\prime}-\frac{m_{2}}{2}(\frac{\tau_{2}}{\tau_{1}}+\frac{\tau_{1}}{\tau_{2}})\}^{2}=\{\frac{m_{2}}{2}(\frac{\tau_{2}}{\tau_{1}}-\frac{\tau_{1}}{\tau_{2}})\}^{2}.
R_{bc}^{a}(u)={\calR}(u)\left(\frac{[b][c]}{[a][a]}\right)^{-u/2\gamma}\left[\delta_{b\neqc}X_{bc}^{a}(u)+\delta_{bc}\left\{\delta_{b,a+1}U_{a}(u)+\delta_{b,a-1}D_{a}(u)\right\}\right]
\begin{array}{l}{\partial_{j}^{}A_{k}^{\alpha}(q,u)\,-\,{\textstyle\frac{1}{2}}\,\alpha_{j}^{}\,\Bigl(\,\zeta(u-v)+\zeta(u+v)\Bigr)\,A_{k}^{\alpha}(q,u)}\\{\mathrm{}+\,{\textstyle\frac{1}{2}}\,\alpha_{j}^{}\,\Bigl(\,\Phi(\alpha(q),u-v)\,A_{k}^{\alpha}(q,v)\,+\,\Phi(\alpha(q),u+v)\,A_{k}^{\alpha}(q,-v)\Bigr)~=~0~.}\\\end{array}
ds_{\mathrm{6D}}^{2}=-\left(1+{\frac{\tilde{Q}_{1}}{r^{2}}}\right)^{-1}dudv+\left(1+{\frac{\tilde{Q}_{1}}{r^{2}}}\right)\left(dr^{2}+r^{2}d\Omega_{3}^{2}\right),
\langle\partial_{\mu}j_{R}^{\mu}\rangle=\frac{1}{16\pi^{2}}\mathrm{Im}\\mathrm{tr}\intd^{2}\theta_{1}d^{2}\theta_{2}\left[\Phi^{2}+\frac{\Lambda^{4}}{2\Phi^{2}}+O(\Lambda^{8})\right],
q(\mu,a,r)=\frac{1}{4\pi^{2}a^{4}}\sum_{l=1}^{\infty}\sum_{k=1}^{\infty}\sum_{\lambda=0}^{1}j_{\nu,k}^{(\lambda)3}T_{\nu}(j_{\nu,k}^{(\lambda)})\psi_{\mu}(j_{\nu,k}^{(\lambda)}/a)D_{l}^{(q)}(j_{\nu,k}^{(\lambda)}x),\quadx=r/a,
{\calL}=-{\textstyle{\frac{1}{2}}}(\partial{\phi^{\prime}})^{2}-{\bar{\lambda}}^{i}{\partial\llap/}\lambda_{i}+{\textstyle{\frac{1}{2}}}H^{2}-{\textstyle{\frac{1}{4}}}F^{2}+{\textstyle{\frac{1}{2}}}{\primeD}^{2}\,.
\int_{L_{\delta}}dk\;k^{\gamma-2}\;\leq\;\frac{1}{\delta^{2}}\int_{0}^{\infty}dk\;k^{\gamma-2}\left|\chi(k)-\phi_{0}\right|^{2}\;\leq\;\frac{q_{0}^{2}\varepsilon^{\prime}(q_{0})}{\delta^{2}}\.
{\calR}_{0}=\left(\begin{array}{cc}{\displaystyle\frac{{\mathfrakg}+{\mathfrakg}^{-1}}{2}}&{\displaystyle\frac{{\mathfrakg}^{-1}-{\mathfrakg}}{2\mathfrakt}}\\{\displaystyle\frac{{\mathfrakg}-{\mathfrakg}^{-1}}{2\mathfrakt}}&{\displaystyle\frac{{\mathfrakg}+{\mathfrakg}^{-1}}{2}}\\\end{array}\right)
\begin{array}{ccl}{{\calL}(A,B)}&{=}&{\int_{M}H_{[\mu\nu}^{a}K_{\lambda\rho]}^{b}}\\{mathrm{tr}(T^{a}I_{E})\mathrm{tr}(T^{b}I_{E})\d^{4}x}\\{}&{}&{-\frac12\int_{M}K_{[\mu\nu}^{a}K_{\lambda\rho]}^{b}\\mathrm{tr}(I_{E}I_{E})\\mathrm{tr}(T^{a}T^{b})\d^{4}x.}\\\end{array}
\begin{array}{c}{T_{fi}=ig^{2}f^{abc}f^{a^{\prime}b^{\prime}c}e^{\mu}(k_{1})e^{\nu}(k_{2})e^{\mu^{\prime}}(k_{1}^{\prime})^{*}e^{\nu^{\prime}}(k_{2}^{\prime})^{*}}\\{\times\Gamma_{\mu\nu\lambda}(k_{1},k_{2},q)D^{\lambda\lambda^{\prime}}(q)\Gamma_{\mu^{\prime}\nu^{\prime}\lambda^{\prime}}(k_{1}^{\prime},k_{2}^{\prime},q)}\\\end{array}
g_{nMN}=\sum_{k=0}^{\infty}C_{\xi}^{*k}C_{\eta}^{k}e_{kM}\otimese_{k-n,N}(\frac{q^{-n}}{1-q^{2}})^{k}q^{k^{2}}\sqrt{\frac{[-n]_{q^{2}}!}{[-n+k]_{q^{2}}![k]_{q^{2}}!}}
\epsilon\left(\begin{matrix}{a}&{b}&{u^{0}}\\{q^{-1}b^{\mathrm{T}}}&{a^{\mathrm{T}}}&{u^{1}}\\{0}&{0}&{{\bf1}}\\\end{matrix}\right)=\left(\begin{matrix}{1}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{1}\\\end{matrix}\right);
<k\cdot\dot{x}(T)\gamma\cdot\dot{x}(\tau_{i})>=\frac{1}{(4\piT)^{D/2}}[\gamma\cdotk\ddot{G}_{B}(\tau_{i},T)+k\cdot\dot{x}^{cl}(T)\gamma\cdot\dot{x}^{cl}(\tau_{i})],\quadi=1,2,
z_{2}\equiv\frac{\mathrm{tr}(\tilde{G}\tilde{{\calH}}\tilde{G}\tilde{{\calH}}\tilde{G}\tilde{{\calH}}\tilde{G}\tilde{{\calH}})}{4(-G)^{2}(1+C_{1}^{2})^{2}}=\frac{\tilde{{\calH}}^{4}-4y^{\mu}\tilde{{\calH}}_{\mu\alpha}\tilde{{\calH}}^{\alpha\lambda}y_{\lambda}+2(y^{\mu}y_{\mu})^{2}}{4(1+C_{1}^{2})^{2}},
A^{-}=\frac{g_{\mathrm{YM}}}{\partial_{-}^{2}}J=\frac{g_{\mathrm{YM}}}{\partial_{-}^{2}}\left(i[\phi,\partial_{-}\phi]+2\psi\psi\right),\quad\chi=-\frac{1}{\sqrt{2}\partial_{-}}D_{\perp}\psi.\nonumber
S=\intd\tau\left\{m\sqrt{-\stackrel{.}{x}{}^{2}+2(\stackrel{.}{x},z_{i})(D^{-1})^{ij}(\stackrel{.}{x},\overline{{z}}_{j})}-i(\stackrel{.}{z}\!_{i}^{A}\partial_{A}^{i}\Phi-c.c)\right\}
\chi_{m}^{(1)}(t)=-\frac{(u_{m}-v_{m})^{2}}{[(z_{m}-u_{m})(z_{m}-v_{m})Q_{m}^{2}(t)]^{3/2}}\,,\quad\phi_{m}^{(1)}(t)=\frac{\vartheta_{m}Q_{m}^{2}(t)\sqrt{(z_{m}-u_{m})(z_{m}-v_{m})]}}{(u_{m}-v_{m})}
G(f)_{\beta}^{(n)}=\Sigma_{m=0}^{n}\{\tilde{\Theta}_{\beta}^{(n-m)},\tilde{f}^{(m)}\}_{(q)}+\Sigma_{m=0}^{(n-2)}\{\tilde{\Theta}_{\beta}^{(n-m)},\tilde{f}^{(m+2)}\}_{(\phi)}+\{\tilde{\Theta}_{\beta}^{(n+1)},\tilde{f}^{(1)}\}_{(\phi)}
V_{F}\equivL^{3}\intd^{3}{\bfx}V(\phi)=\sum_{n=3}^{\infty}\frac{1}{L^{3n-6}}\sum_{m_{1},\cdots,m_{n-1}}\phi({\bfk}_{m_{1}},t)\cdots\phi({\bfk}_{m_{n-1}},t)\phi(-{\bfk}_{m_{1}}-\cdots-{\bfk}_{m_{n-1}},t).
\begin{array}{l}{L_{G}^{f}(x_{f})=J_{\psi}L_{G}(x)=-\displaystyle\frac{1}{2}\zeta_{0}^{-1}J_{\psi}(\nabla_{\mu}A^{\mu})^{+}\nabla_{\nu}A^{\nu}=}\\{=-\displaystyle\frac{1}{2}\zeta_{0}^{-1}J_{\psi}S(B_{f})^{2}(\lambda_{m}^{l}D_{l}A_{f}^{m})^{+}\lambda_{n}^{k}D_{k}A_{f}^{n},\quad\lambda_{m}^{l}=\displaystyle\frac{1}{2}(\delta_{m}^{l}+\omega_{m}^{l}),}\\\end{array}
\begin{array}{ccc}{(\gamma^{0})^{\dagger}=\gamma^{0},}&{(\gamma^{0})^{T}=-\gamma^{0},}&{(\gamma^{0})^{*}=-\gamma^{0},}\\{(\gamma^{1})^{\dagger}=-\gamma^{1},}&{(\gamma^{1})^{T}=\gamma^{1},}&{(\gamma^{1})^{*}=-\gamma^{1},}\\{(\gamma^{2})^{\dagger}=-\gamma^{2},}&{(\gamma^{2})^{T}=\gamma^{2},}&{(\gamma^{2})^{*}=-\gamma^{2}.}\\\end{array}
\phi=\frac{1}{4}P^{2}-\frac{1}{2}(m_{1}^{2}+m_{2}^{2})+\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{4P^{2}}+q_{\bot}^{2}-\frac{\alpha\bigl(P^{2}-(m_{1}-\kappam_{2})^{2}\bigr)}{\epsilonP\cdotr}=0,
\sigma_{{ab}}^{{[-2]}}\equiv(\sigma^{{0}}-\sigma^{{9}})_{{ab}}=\mathrm{~{\it~diag}}(0,2\delta_{{\dot{A}}{\dot{B}}})=(\tilde{\sigma}^{{0}}+\tilde{\sigma}^{{9}})^{{ab}}=\tilde{\sigma}^{{[+2]ab}},
\psi\left(m,z\right)=N_{m}\sqrt{\left|z\right|+\frac{1}{k}}\left[Y_{2}\left(m\left|z\right|+\frac{1}{k}\right)+\frac{4k^{2}}{\pim^{2}}J_{2}\left(m\left(\left|z\right|+\frac{1}{k}\right)\right)\right],
\tilde{\calL}^{(0)}=\pi_{a}\dot{\sigma}^{a}+\Psi\dot{\theta}-\frac{1}{2}\,{\sigma^{\prime}}_{a}^{2}-\frac{1}{2}\,\pi_{a}^{2}-\frac{1}{2}\,\lambda\,\bigl(\sigma_{a}^{2}-1\bigr)+\sigma_{a}\pi_{a}\theta.
{\calA}_{\mu}(x)=\left(\begin{matrix}{n_{1}{\calA}_{\mu}^{0}(x)}&{0}&{\ldots}&{0}\\{0}&{n_{2}{\calA}_{\mu}^{0}(x)}&{\ldots}&{0}\\{0}&{\ldots}&{\ldots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\ldots}&{n_{N}{\calA}_{\mu}^{0}(x)}\\\end{matrix}\right)
S_{OR}=\intd^{4}x\:\Biggl\{\frac{1}{2}\sum_{i=1}^{3}\left[\hat{\Phi}_{i}^{\dagger}\hat{\Phi}_{i}\right]_{D}+\left[\lambda\hat{\Phi}_{1}+m\hat{\Phi}_{2}\hat{\Phi}_{3}+g\hat{\Phi}_{1}\hat{\Phi}_{2}\hat{\Phi}_{2}\;\mathrm{+~h.c.}\right]_{F}\Biggr\}
\sum_{[b]}Y_{[b]}\sum_{(a)}(\bar{\psi}_{L})_{r}^{a\tilde{a}}\tilde{\phi}_{\tilde{a}}^{(a)}\phi^{(1)r}(\psi_{R})_{a}^{[b][1]}+\sum_{[b]}Y_{[b]}^{\prime}\sum_{(a)}(\bar{\psi}_{L})_{r}^{a\tilde{a}}\tilde{\phi}_{\tilde{a}}^{(a)}\phi^{(2)r}(\psi_{R})_{a}^{[b][2]},
\deltaL^{2}=\delta_{ab}\delta\Psi^{a}\delta\Psi^{*b}=G_{ik*}\delta\Pi^{i}\delta\Pi^{*k}=\sum_{a}\frac{\partial\Psi^{a}}{\partial\Pi^{i}}\frac{\partial\Psi^{*a}}{\partial\Pi^{*k}}\delta\Pi^{i}\delta\Pi^{*k}.
\sum_{n=-\infty}^{\infty}\frac{1}{\mid\vec{y}-2\pina\hat{z}\mid^{6}}\rightarrow\int_{-\infty}^{\infty}\frac{dz}{2\pia}\frac{1}{({\tilde{r}}^{2}+z^{2})^{3}}=\frac{3}{16a}\frac{1}{{\tilde{r}}^{5}}.
\left<j_{\mu}(x)j_{\nu}(y)\right>=\Biggl(\frac{\partial^{2}}{\partialx_{\lambda}\partialy_{\lambda}}\delta_{\mu\nu}-\frac{\partial^{2}}{\partialx_{\mu}\partialy_{\nu}}\Biggr){\calD}\left((x-y)^{2}\right),
h\phi_{n}(\varphi)\;=\;\lambda_{n}^{2}\phi_{n}(\varphi)\;\Rightarrow\;{\tilde{h}}\;[\frac{1}{\lambda_{n}}a\phi_{n}(\varphi)]\;=\;\lambda_{n}^{2}\;[\frac{1}{\lambda_{n}}a\phi_{n}(\varphi)]