harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
<\psi\left(k\right)~\psi\left(q-k\right)\omega\left(f_{1}\right)\cdots\omega\left(f_{N}\right)>~=~\int{d^{2}Re^{ikR}}~<\psi\left(R\right)\psi\left(0\right)\omega\left(f_{1}\right)\cdots\omega\left(f_{N}\right)>
\tilde{H}_{n,k}=n^{-k}\,\prod_{s=0}^{k}(s\,n-1\,)\sum_{m=0}^{\lfloor\frac{k}{2}\rfloor}\frac{1}{m!\,(k-2\,m)!\,n^{m}}\prod_{l=0}^{m}{\frac{1}{l\,n-1}}\,v^{mn}\,u^{k-2m}
e^{\phi}={\frac{1}{N}}\,\left[(g_{f}^{2})^{\frac{1}{p-3}}\left({\frac{u}{\calR}}\right)\left({\frac{d_{p}^{1/(7-p)}}{c_{p}^{1/(p-3)}}}\right)\right]^{\frac{(p-3)(7-p)}{2(5-p)}}\,.
{\bf\Omega}(x_{1},\ldots,x_{N})=\prod_{i=1}^{N/4}\bar{f}(x_{4i-1}-x_{4i})f(x_{4i-3}-x_{4i-2})\,\,\delta\left(\sum_{i=1}^{N/4}(x_{4i-3}-x_{4i-1})\right)\,,
\psi_{\kappa}\left(\frac{1}{\zeta}\right)=F\left(\begin{array}{cccc}{g_{1,\kappa},}&{\dots,}&{g_{n-1,\kappa},}&{d_{n,\kappa}}\\{h_{1,\kappa},}&{\dots,}&{h_{n-1,\kappa},}&{\zeta}\\\end{array}\right)
[b_{Q}^{a}({\bf{k}}),\,{\cal{A}}_{n}]\approxg\,f^{a\beta\gamma}{\int}d{\bfr}\,e^{-i{\bf{k\cdotr}}}\,[a_{i}^{\beta}({\bf{r}})+x_{i}^{\beta}({\bf{r}})]\,[{\cal{A}}_{n-1},\,\Pi_{i}^{\gamma}({\bf{r}})]\;,
\sum_{M,N=1}^{n+4}\sigma_{N}\left(e^{i\sum_{I}T_{I}V_{I}}\right)_{MN}\left(\bar{z}_{+}^{M}\,\,\bar{z}_{-}^{M}\right)\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)\left(\begin{array}{c}{z_{+}^{N}}\\{z_{-}^{N}}\\\end{array}\right)-\sum_{M\neq1,5}\sigma_{M}G_{M}V_{M}~,
{\calL}=(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}\phi^{*}\phi-\frac\lambda4(\phi^{*}\phi)^{2}+\frac\Theta2\epsilon_{\sigma\mu\nu}A^{\sigma}\partial^{\mu}A^{\nu}-\frac\xi2(\partial_{i}A^{i})^{2}\;,
R_{|n_{1},n_{2},\dots,n_{k}\rangle}^{(n)}(\eta,\vartheta,u)=R^{(n)}(\eta,\vartheta,u)\prod_{i\textrm{}\mathrm{odd}}b_{n_{i}}^{n}(\eta,u)\prod_{i\textrm{}\mathrm{even}}b_{n_{i}}^{n}(\bar{\eta},u)\,\,\,
Z_{1/R}(0^{16})\to\int\frac{d^{2}\tau}{\tau_{2}}\tau_{2}^{-5}|\eta(\tau)|^{-16}Z_{left}(\tau)\left\{{Z_{0}^{1}(\tau)^{*}}^{4}+{Z_{1}^{0}(\tau)^{*}}^{4}-{Z_{0}^{0}(\tau)^{*}}^{4}\right\}\,
\frac{\lambda_{i}-(\beta-1)i}{\betaN}=\frac{w_{i}-1}{2},\\\frac{\beta\lambda_{i}^{\prime}-i}{\betaN}=\frac{1-v_{i}}{2},\\\frac{r}{\betaN}=-\frac{w_{0}+1}{2},\\\frac{\nu_{i}-(\beta-1)i}{\betaN}=\frac{\xi_{i}-1}{2},
\begin{array}{c}{\displaystyle{D_{\nu}(t)=\nuD^{(1)}(t)+D^{(0)}(t)+\nu^{-1}D^{(-1)}(t)=}}\\{\displaystyle{\frac{{\nu\,\sqrt{1+{t^{2}}}}}{t}}-\displaystyle{\frac{t}{2\,\left(1+{t^{2}}\right)}\,+\,}\displaystyle{\frac{4\,t-{t^{3}}}{8\,{\nu}\,{{\left(1+{t^{2}}\right)}^{{\frac{5}{2}}}}}}.}\\\end{array}
\breve{g}^{\mu\nu}p_{\mu}p_{\nu}=p_{\tau}^{2}-2\breve{\Omega}p_{\tilde{\varphi}}+(r^{-2}+\breve{\Omega}^{2})p_{\tilde{\varphi}}^{2}+p_{r}^{2}+p_{z}^{2}~.
\left(\begin{array}{c}{\xi^{t}}\\{\xi^{*}}\\\end{array}\right)\equiv{\calG}\left(\begin{array}{c}{\Sigma^{t}}\\{\Sigma^{*}}\\\end{array}\right);\;\;\;\;{\calG}=e^{-\int^{\tau}{\calA}(\tau^{\prime})d\tau^{\prime}},
2\frac{\ddot{b}}{b}+\frac{{\dot{b}}^{2}-1}{b^{2}}+\frac{f^{\prime\prime\prime}(R)}{f^{\prime}(R)}\dot{R}^{2}+\frac{f^{\prime\prime}(R)}{f^{\prime}(R)}\ddot{R}+2\frac{f^{\prime\prime}(R)}{f^{\prime}(R)}\frac{\dot{b}}{b}\dot{R}-\frac{1}{2}\frac{f(R)}{f^{\prime}(R)}+\frac{1}{2}R=0.
\int{\calD}\tilde{\vec{p}}{\calD}\tilde{\vec{x}}{\calD}\sqrt{h}{\calD}\tilde{c}{\calD}\bar{c}{\calD}Bexp\{i\int_{0}^{\tau}d\tau[(\vec{p}\dot{\vec{x}}-Hh)+B(\sqrt{h}-\sqrt{f})-i\frac{1}{2}\bar{c}\sqrt{h}\partial_{\tau}(\frac{1}{h^{3/2}}\tilde{c})]\}
G_{ret}(k)=\frac{1}{\omega^{*}\omega}\left(\frac{(i\tilde{\omega}k_{0}-\omega^{*}\omega+2\overline{\omega}^{2})(-k_{0}^{2}+\omega^{*}\omega-4ik_{0}\tilde{\omega})}{(-k_{0}^{2}+\omega^{*}\omega)^{2}+16k_{0}^{2}\tilde{\omega}^{2}}\right).
ds^{2}=e^{4\phi}(dt+\vec{\omega}\cdotd\vec{x})^{2}-d\vec{x}^{2}-\left(dx^{4}-e^{2\phi}(dt+\vec{\omega}\cdotd\vec{x})\right)^{2}-\sum_{5}^{9}dx^{i}dx^{i}\.
H=\left[\begin{array}{cc}{1\}&{\0}\\{0\}&{\-1}\\\end{array}\right]\,\\E_{+}=\left[\begin{array}{cc}{0\}&{\1}\\{0\}&{\0}\\\end{array}\right]\,\\E_{-}=\left[\begin{array}{cc}{0\}&{\0}\\{1\}&{\0}\\\end{array}\right]\.
{\cal{B}}(x_{j},y_{j})=\left<\mathrm{exp}\left[\sum_{j=1}^{n}(s_{j}\Phi(x_{j},t)+t_{j}\Phi(y_{j},t))\right]~\mathrm{exp}\left[-i\sum_{j=1}^{n}(\omega(x_{j},t)-\omega(y_{j},t))\right]\right>
S_{I,\phi}=-\frac{\sigma}{2}\int\frac{d\nu}{2\pi}\int_{\bar{R}_{1}}^{\bar{R}_{2}}dr\tilde{\phi}_{I}^{*}\left[-\frac{\partial}{\partialr}\gammar^{2}\frac{\partial}{\partialr}-\nu^{2}\gamma^{3}r^{2}\right]\tilde{\phi}_{I}\,.
\xi_{\mathrm{ext}}^{(m)}=\left(\frac{u}{\bar{u}}\right)^{\frac{m}{2}}(u\bar{u})^{\frac{\alpha-1}{2}}\left[a(\rho)J_{\gamma}(2\rho\mupv^{\alpha})+b(\rho)J_{-\gamma}(2\rho\mupv^{\alpha})\right]
51{\calL}=-\frac{1}{4}\left(F_{\mu\nu}\right)^{2}+\frac{1}{2}\left[\bar{\psi},(i\gamma^{\mu}D_{\mu})\psi\right]-\bar{\psi}(\sigma+i\gamma_{5}\pi)\psi-\frac{1}{2G}(\sigma^{2}+\pi^{2}),
\Gamma_{\alpha_{1}\alpha_{2}\dots\alpha_{n}}(p_{1},p_{2},\dots,p_{n})=\Bigl(\bbox{e}_{\alpha_{1}}(p_{1}){\otimes}\bbox{e}_{\alpha_{2}}(p_{2}){\otimes}\dots{\otimes}\bbox{e}_{\alpha_{n}}(p_{n})\Bigr)\bullet\Gamma(p_{1},p_{2},\dots,p_{n})
[\phi_{m},\pi_{m}]-i[\psi_{\dot{\alpha}},\psi_{\dot{\alpha}}^{\dagger}]_{+}=[\tilde{\phi}_{m},\tilde{\pi}_{m}]-i[\tilde{\psi}_{\dot{\alpha}},\tilde{\psi}_{\dot{\alpha}}^{\dagger}]_{+}=i[(B-F-1)+\mid0\rangle\langle0\mid]
I_{\mathrm{non-planar}}\sim{\frac{\pi^{\frac{n}{2}}}{(2\pi)^{n}}}\,\int_{0}^{1}dx\,{\frac{2^{{\frac{n}{2}}-1}}{(M^{2})^{2-{\frac{n}{2}}}}}\,(|\bar{\theta}|M)^{2-{\frac{n}{2}}}K_{2-{\frac{n}{2}}}(|\bar{\theta}|M)
\begin{array}{cc}{C_{h}^{ag}C_{g}^{br}C_{da}^{e}C_{eb}^{f}(2\delta_{rf}\delta^{hd}-\delta_{r}^{h}\delta_{f}^{d}-\delta_{r}^{d}\delta_{f}^{h})=}&{}\\{}&{}\\{=2d(d-1)(d-1)-d(d-1)(d-1)+d(d-1)(d-4)}&{}\\{}&{}\\{=d(d-1)(2d-5),}&{}\\\end{array}
\{\Psi^{i},\Psi^{k}\}=\frac{R^{2}}{(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})^{2}}\left[1-\frac{1}{4}\frac{\sum_{s=1}^{N}|\Pi^{s}|^{2}}{\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2}}\right](\Pi^{*i}\Pi^{k}-\Pi^{*k}\Pi^{i}).
\frac{1}{z_{2n}}\prod_{a\inA}\oint_{C_{a}}\frac{dw_{a}}{2\piiw_{a}}\Psi_{b}^{(n)}(w|\zeta)Q_{b}^{(n)}(w|\zeta)\left(\frac{1-rz_{2n}}{1-rz_{2n}^{-1}}-\frac{xw_{2n}-z_{2n}}{xw_{2n}-z_{2n}^{-1}}\right).
\langlej_{k}({\bfx})\rangle=\theta\epsilon^{kl}d_{l}\langle{\calA}_{0}({\bfx})\rangle+{\frac{\theta}{2}}\left(K^{lk}-K^{kl}\right)\partial_{0}\langle{\calA}_{l}({\bfx})\rangle-\theta^{2}\epsilon_{kl}\sum_{{\bfx}^{\prime}}V({\bfx}-{\bfx}^{\prime}){\hat{d}}_{l}\langle{\calB}({\bfx}^{\prime})\rangle
e^{\phi}=-{\frac{1}{\sqrt2}}{\frac{q_{1}}{q_{0}}}\,\qquadt^{1}q_{1}={\frac{1}{2^{1/6}3^{1/3}}}\left(q_{0}(q_{1})^{2}\right)^{1/3},\qquadt^{0}q_{0}={\frac{1}{2}}t^{1}q_{1}\.
m(x,\theta)=m_{2}(\theta)\left(\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)^{2}\left(1-3\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)+O\left(\frac{x-x_{\mathrm{H}}}{x_{\mathrm{H}}}\right)^{4}\,
{\frac{\partial\psi^{\beta}}{\partial_{\Gamma}\phi_{2}}}\Rightarrow\sqrt2{\frac{\partial\psi^{\beta}}{\partial_{\Gamma}\omega_{2}}}=\sqrt2{\frac{\partial\psi^{\beta}}{\partial\omega_{2}}}{\frac{\partial\omega_{2}}{\partialM_{\Gamma}}}=\sqrt2{\frac{\partial\psi^{\beta}}{\partial\omega_{2}}}{\frac{M_{\Gamma}}{\omega_{2}}}
N_{nm}^{w}=\frac1{n\,m}\oint_{\infty}\frac{dz}{2\pii}z^{n}\oint_{\infty}\frac{d\zeta}{2\pii}\zeta^{m}\frac{w^{\prime}(z)w^{\prime}(\zeta)}{(w(z)-w(\zeta))^{2}}
M_{km}\frac{\partial}{\partialu_{m}}\Big(a_{D}^{i}\frac{\partiala_{i}}{\partialu_{k}}-a_{i}\frac{\partiala_{D}^{i}}{\partialu_{k}}\Big)=a_{D}^{i}M_{km}\frac{\partial^{2}a_{i}}{\partialu_{k}\partialu_{m}}-a_{i}M_{km}\frac{\partial^{2}a_{D}^{i}}{\partialu_{k}\partialu_{m}}=0.
w(m,m^{\prime})=\frac{1}{\sqrt{2\piN|\Phi_{m}^{(2)}(t_{+})|}}\frac{1}{\sqrt{2\piN|\Phi_{m^{\prime}}^{(2)}(s_{+})|}}\mathrm{e}^{-N(\Phi_{m}(t_{+})+\Phi_{m^{\prime}}(s_{+}))}\mathrm{det}M(t_{+},s_{+}),
\Psi(a,\psi,\theta)\,=\,\psi-\frac{1}{2}\,\theta^{\alpha\beta}\,a_{\alpha}\mathrm{P}_{-}\partial_{\beta}\psi\,+\,\frac{i}{8}\,\theta^{\alpha\beta}\,[a_{\alpha},a_{\beta}]\,\mathrm{P}_{-}\psi\,+\,o(\theta^{2}).
\int(d\epsilon_{1})(d\epsilon_{2})(d\epsilon_{3})V_{3}(\tau_{1},\tau_{2},\tau_{3};\epsilon_{1},\epsilon_{2},\epsilon_{3})V_{3}(\tau_{1}^{\prime},\tau_{2}^{\prime},\tau_{3}^{\prime};-\epsilon_{1},-\epsilon_{2},-\epsilon_{3}).
\Bigl(n_{\nu_{l}}^{U_{l}}\Bigr)_{\rho_{l}}^{\rho_{l}^{\prime}}\=\\sum_{\Omega_{l}(\mu_{l})=\mu_{l}}\frac{\bigl(\bar{\psi}^{U_{l}})_{\rho_{l}^{\prime}}^{\mu_{l}}\bigl(\psi^{U_{l}}\bigr)_{\rho_{l}}^{\mu_{l}}S_{\mu_{l}\nu_{l}}^{U_{l}}}{S_{\mu_{l}0}^{U_{l}}}
\left[A_{ii_{1}...i_{n}}^{l;~j_{0}...j_{2l-1}}+A_{i_{1}i...i_{n}}^{l;~j_{0}...j_{2l-1}}+...+A_{i_{1}...i_{2l-1}ii_{2l}...i_{n}}^{l;~j_{0}...j_{2l-1}}\right]=o(\omega^{d-n-2})\qquad\qquadl=1,...,m
\begin{array}{c}{[s_{\alpha\beta},s_{\gamma\delta}]=\epsilon_{\alpha\delta}s_{\beta\gamma}+\epsilon_{\beta\delta}s_{\alpha\gamma}+\epsilon_{\alpha\gamma}s_{\beta\delta}+\epsilon_{\beta\gamma}s_{\alpha\delta}\,,}\\{{}}\\{s_{\alpha\beta}=s_{\alpha}{}^{\gamma}\epsilon_{\gamma\beta}=s_{\beta\alpha}\,.}\\\end{array}
S_{\alpha}(z)S_{\dot{\beta}}(w)=\frac{1}{(z-w)^{\frac{D}{8}-\frac{1}{2}}}\gamma_{\mu\alpha\beta{\beta}}\psi_{\mu}(w)~~+~~O((z-w)^{-\frac{D}{8}+\frac{3}{2}})~.
\overline{X}_{i}\Big\{f,g\Big\}_{\calR}-\Big\{\overline{X}_{i}f,g\Big\}_{\calR}-\Big\{f,\overline{X}_{i}g\Big\}_{\calR}=-{\calF}_{i\phantom{jl}}^{\phantom{i}jl}\Big(\overline{X}_{j}f\Big)\Big(\overline{X}_{l}g\Big).
\begin{array}{rcl}{\chi_{1}}&{=}&{A\Lambda_{1,2}+B(\partial\Lambda)_{1,2}\,,}\\{\chi_{2}}&{=}&{\Lambda_{0,2}\,,}\\{\chi_{3}}&{=}&{A^{\prime}\Lambda_{1,2}+B^{\prime}(\partial\Lambda)_{1,2}\,,}\\{\chi_{4}}&{=}&{\Lambda_{2,2}\,,}\\{\chi_{5}}&{=}&{\frac{1}{2}\Lambda_{1,2}-\frac{1}{2}(\partial\Lambda)_{1,2}\,.}\\\end{array}
\bar{V}(\bar{\phi})=\sum_{k=1}^{M}\sum_{a_{k}=1}^{L_{k}}\frac{1}{2}\left[-\mu^{2}+\left(\frac{\alpha_{k}}{R}\right)^{2}\right]\left(\bar{\phi}_{a_{k}}^{(\alpha_{k})}\right)^{2}+\frac{\lambda}{8}\left(\sum_{k=1}^{M}\sum_{a_{k}=1}^{L_{k}}\left(\bar{\phi}_{a_{k}}^{(\alpha_{k})}\right)^{2}\right)^{2}
{j^{W}}^{(L)}(x)=\left(\begin{array}{l}{\theta(x_{1}+L)\theta(L-x_{1})\;[\delta(x_{2}-L)-\delta(x_{2}+L)]}\\{\theta(x_{2}+L)\theta(L-x_{2})\;[\delta(x_{1}+L)-\delta(x_{1}-L)]}\\\end{array}\right)\;.
\left(\begin{matrix}{e{^a}_{\mu}}\\{t{^a}_{\mu}}\\{t{^b}_{\mu}e{_b}^{\nu}t{^a}_{\nu}}\\\end{matrix}\right)\rightarrow\left(\begin{matrix}{0}&{2(\alpha+1)}&{-2(\alpha^{2}-\beta)}\\{1}&{-(2\alpha+1)}&{2(\alpha^{2}-\beta)}\\{1}&{-(3\alpha+2)}&{2\alpha(2\alpha+1)-3\beta+1}\\\end{matrix}\right)\\left(\begin{matrix}{e{^a}_{\mu}}\\{t{^a}_{\mu}}\\{t{^b}_{\mu}e{_b}^{\nu}t{^a}_{\nu}}\\\end{matrix}\right).
\begin{array}{l}{[S_{1}(\vec{x}),S_{2}(\vec{y})]_{P.B.}=-\frac{1}{2}\delta(\vec{x}-\vec{y})\,,}\\{[S_{2}(\vec{x}),S_{3}(\vec{y})]_{P.B.}=S_{1}(\vec{x})\delta(\vec{x}-\vec{y})\,,}\\{[S_{3}(\vec{x}),S_{1}(\vec{y})]_{P.B.}=S_{2}(\vec{x})\delta(\vec{x}-\vec{y})\,.}\\\end{array}
\begin{array}{lcr}{\sum_{j=1}^{n+1}(|\varphi_{j}^{1}|^{2}-|\overline{{\varphi}}_{2j}|^{2})}&{=\xi^{3}}\\{\sum_{j=1}^{n+1}(\varphi_{j}^{1}\overline{{\varphi}}_{2j})}&{=\xi^{1}+i\xi^{2}.}\\\end{array}
\Gamma_{R}^{(n)}=\Gamma_{fin}^{1(n)}+\Gamma_{fin}^{3(n)}=(-i\lambda)^{2n+1}(i)^{3n}\prod_{i=1}^{n}\int\frac{d^{6}k_{i}}{(2\pi)^{6}}\Xi^{(n)}(k_{1},...,k_{n},p,p^{\prime},m^{2})\cdot
V_{i}\cdotV_{j}={\vec{U}}_{i}\cdot{\vec{U}}_{j}-\sum_{r}V_{i}^{r}V_{j}^{r}+\sum_{\ell:~right}T_{i}^{\ell}T_{j}^{\ell}-\sum_{\ell:~left}T_{i}^{\ell}T_{j}^{\ell}~.
S_{mm^{\prime},nn^{\prime}}^{SNS}=\frac{1}{u}(-1)^{G+m+n+u(u-m-1)}e^{-i\pi(u+1)(u-m-1)n/u}e^{-i\pi(u-1)(m-2m^{\prime}+u)(n-2n^{\prime}+u)/u};
{\calV}(\sigma,\omega,|\phi|)=\left(\frac{\beta}{g^{2}}\right)^{4}\sum_{n_{1},n_{2}=-\infty}^{+\infty}\frac{A^{4}}{\left[(X_{8}-2n_{1}\Omega_{1}-2\kappan_{2}\Omega_{2})^{2}+(X_{7}-2n_{2}\Omega_{2})^{2}+A^{2}\right]^{7}}
\alpha_{k}\,=\,\frac{i}{\sqrt{4k\omega_{k}}}\left[C_{+}\biggl(\gamma_{k}^{*}+\frac{a^{\prime}}{a}\biggr)-\frac{C_{-}}{\eta_{\mathrm{m}}}\right],\quad\quad\beta_{k}\,=\,\frac{-i}{\sqrt{4k\omega_{k}}}\left[C_{+}\biggl(\gamma_{k}+\frac{a^{\prime}}{a}\biggr)-\frac{C_{-}}{\eta_{\mathrm{m}}}\right]\,,
8\delta^{ab}(C_{1}C_{3}+{\frac{1}{2}}C_{1}C_{2}){\frac{g^{4}}{p^{2}}}\Lambda^{3-\omega}\int{\frac{d^{\omega}k}{(2\pi)^{\omega}}}{\frac{F(k,p)}{k^{2}(k+p)^{2}}},
\begin{array}{cc}{L_{3}D^{\pm}\left|l_{3},\sigma_{3}\right\rangle=}&{\left(l_{3}\pm1\right)D^{\pm}\left|l_{3},\sigma_{3}\right\rangle}\\{\frac{\Sigma_{3}}{2}D^{\pm}\left|l_{3},\sigma_{3}\right\rangle=}&{\left(\sigma_{3}\mp1\right)D^{\pm}\left|l_{3},\sigma_{3}\right\rangle.}\\\end{array}
R=R_{h}+\frac{e^{-2\Phi_{h}}}{8KR_{h}^{2}}\left(16R_{h}^{4}-4R_{h}^{2}(w_{h}^{2}-1)^{2}-(w_{h}^{3}-3w_{h}+4\kappa)^{2}\right)r+{\calO}(r^{2})\,.
\tilde{D}^{L}(X)_{ij}=\sum_{k}[Q,P_{I}(F_{ik}^{I})]^{L}(X_{kj}),\\tilde{D}^{R}(X)_{ij}=\sum_{k}[Q,P_{I}(F_{ik}^{I})]^{R}(X_{kj})\.
sB_{\mu\nu}^{a}=gf^{abc}B_{\mu\nu}^{b}\omega^{c}+\partial_{[\mu}\omega_{\nu]}^{a}+gf^{abc}A_{[\mu}^{b}\omega_{\nu]}^{c}+gf^{abc}F_{\mu\nu}^{b}\theta^{c}+\alphaF_{\mu\nu}^{a}.
\mathrm{\normalsize~q(t)=-\mathrm{sgn}(q(0))\Bigl(2\sqrt{E}t+\frac{q(0)p(0)}{\sqrt{E}}+~\frac{g}{2\sqrt{E}(2Et+q(0)p(0))}\Bigr)+0\Bigr(\frac{1}{t^2}\Bigl)~}
\begin{array}{c}{2\sum_{p=0}^{(m+n+5)/3}t_{3p+1}{\calM}^{-m-n-8+3p}+\sum_{p=0}^{(n+1)/3}t_{3p+2}{\calM}^{-m-n-7+3p}+}\\{+\sum_{p=0}^{(m+1)/3}t_{3p+2}{\calM}^{-m-n-7+3p}\equivA_{22}+B_{22}+C_{22}}\\\end{array}
\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigmaB_{MN}V^{(B)MN}=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigmaB_{MN}\epsilon^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}
S_{11}^{A_{2}^{(1)}}\left(\theta\right)=S_{22}^{A_{2}^{(1)}}\left(\theta\right)=\left\{1\right\}_{\theta}\qquadS_{12}^{A_{2}^{(1)}}\left(\theta\right)=\left\{2\right\}_{\theta}\qquadS^{A_{2}^{(2)}}\left(\theta\right)=\left\{1\right\}_{\theta}\left\{2\right\}_{\theta}\,.
L_{2}={\frac{\pi^{2}}{8a}}\int_{\epsilon}^{1}{dx{\frac{1}{\lambda^{2}(1-\lambda^{2})x^{2}\sqrt{1-x^{2}}}}}-{\frac{\pi^{2}}{8a}}\int_{0}^{1}dx{\frac{1}{\sqrt{1-x^{2}}}}+O(\lambda^{2}).
\beta_{A}(z_{2}^{\prime},z_{3}^{\prime},z_{A}^{\prime},\mu)=\frac{1}{z_{2}^{\prime}}\beta_{A}(1,\frac{z_{3}^{\prime}}{z_{2}^{\prime}},\frac{z_{A}^{\prime}}{z_{2}^{\prime}},\mu)\equiv\frac{1}{z_{2}^{\prime}}\beta_{Ar}(u,v,\mu)
\phi_{(p_{2},q_{2};r_{2},s_{2})}\otimes\phi_{(p_{1},q_{1};r_{1},s_{1})}=\sum_{(p,q)\in\Lambda(p_{1},q_{1},p_{2},q_{2})}\;\sum_{(r,s)\in\Lambda(r_{1},s_{1},r_{2},s_{2})}\left[\phi_{(p,q;r,s)}\right],
f^{(5)}(T)={\frac{512\pi^{2}}{(T-{\bar{T}})^{3}}}\sum_{\ell=1}^{6}\int\frac{d^{2}\tau}{\tau_{2}^{\;3/2}}\bar{C}_{\ell}(\bar{\tau})\,\partial_{\bar{\tau}}\left[\tau_{2}^{2}\partial_{\bar{\tau}}(\tau_{2}^{\;3/2}\sum_{p_{L},p_{R}\in\Gamma_{\ell}}p_{L}^{3}{p}_{R}\,e^{\pii\tau\,|p_{L}|^{2}}\,e^{-\pii\bar{\tau}\,p_{R}^{2}})\right]\,.
\tilde{S}=\tilde{S}(x_{out},x_{in})=\epsilon(x_{out}^{0},x_{in}^{0})\int_{\Gamma}\,ds\int\langlex_{\mathrm{out}}|e^{-i\hat{\calH}(s,\chi)}|x_{\mathrm{in}}\rangled\chi\,\,,
\vert\lambda\rangle={e^{\lambda{J_{+}}}}\left(\begin{matrix}{0}\\{1}\\\end{matrix}\right),{J_{+}}=\left(\begin{matrix}{0}&{1}\\{0}&{0}\\\end{matrix}\right)\equiv{e_{12}},\langle\lambda\vert\lambda\rangle=(1+{\lambda}{\bar{\lambda}})
G(\bar{z})=\sum_{s}\oint\frac{du}{2\pii}\frac{1}{(h_{r}(0)-u)}\oint\frac{\bar{dv}}{2\pii}\frac{1}{(\bar{v}-\bar{z})}\frac{\xi_{s}^{\prime}(v)}{\xi_{s}(v)(1-\bar{\xi}_{s}(u)\xi_{s}(v))}.
\left(\begin{array}{cc}{A_{12}}&{A_{22}}\\{-A_{11}}&{-A_{12}}\\\end{array}\right)=\left(\begin{array}{cc}{\partial_{\tau}a}&{\partial_{\tau}b}\\{\partial_{\tau}c}&{\partial_{\tau}d}\\\end{array}\right)\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)^{-1},
f_{9}^{A}=f_{9;\mathrm{pert}}^{A}+2\pi\sum_{m\ne0}\sum_{n\ne0}\frac{|mn|^{1/2}}{n^{2}}e^{-2\pi|mn|(e^{-\phi}R_{A}\pmi\mathcal{A})}\left(1+O(e^{\phi})\right)\nonumber
\mathrm{Tr}\,(W^{\alpha}W_{\alpha})={\calV}(x_{0},\rho,\bar{\omega},\theta_{0},\bar{\beta})\,\frac{96\,\theta^{2}}{(x_{L}^{2}+1)^{4}}=\frac{96\,\tilde{\theta}^{2}\,\rho^{4}}{[(x_{L}-x_{0})^{2}+\rho^{2}]^{4}}\,,
S_{CS}=\frack{4\pi}\int_{M^{3}}\{\epsilon^{ijk}(A_{i}^{a}(\partial_{j}A_{k}^{b}-\partial_{k}A_{j}^{b})g_{ab}+\frac23C_{abc}A_{i}^{a}A_{j}^{b}A_{k}^{c})\},
<M(C_{i})M(C_{f})>=\int_{0}^{\infty}dl{\frac{e^{-r^{2}l/2\pi\alpha^{\prime}}\eta(il)^{-9}}{i\Theta_{11}(ul/\pi,il)}}\sum_{(\beta,\alpha)\neq(1,1)}~C_{\alpha}^{\beta}~\Theta_{(\beta,\alpha)}^{3}(0,il)\Theta_{(\beta,\alpha)}(ul/\pi,il)\quad.
L^{\mathrm{red}}=2\pi\gamma\sum_{s=1}^{N}\left((b_{2}^{s}+\frac{1}{2}x_{2}^{s})\dot{x}_{1}^{s}-(b_{1}^{s}+\frac{1}{2}x_{1}^{s})\dot{x}_{2}^{s}\right)-V^{\mathrm{red}}
D_{J=0}=-i\left(\frac{1}{\sqrt{B(r)}}\frac{\partial}{\partialt}+i\tau^{2}a_{0}(r,t)\right)+i\tau^{2}\left(\frac{1}{\sqrt{A(r)}}\frac{\partial}{\partialr}-i\tau^{2}a_{1}(r,t)\right).
I_{6}=\frac{1}{2\kappa^{2}}\intd^{6}x\sqrt{-G}e^{-\Phi}\left[R_{G}+G^{MN}\partial_{M}\Phi\partial_{N}\Phi-\frac{1}{12}G^{MQ}G^{NR}G^{PS}H_{MNP}H_{QRS}\right]\
|(+);n_{k_{1}}...n_{k_{m}}\rangle:={\frac{(a_{(+)k_{1}}^{\dagger})^{n_{k_{1}}}}{\sqrt{n_{k_{1}}!}}}...{\frac{(a_{(+)k_{m}}^{\dagger})^{n_{k_{m}}}}{\sqrt{n_{k_{m}}!}}}|0\rangle\;,\qquad\sum_{i=1}^{m}n_{k_{i}}=N.
a(n,r)=\sum_{j_{1}+\cdots+j_{n-2r+2}=r\atopj_{1},\ldots,j_{n-2r+2}\geq0}\gamma_{j_{1}}^{}\gamma_{j_{2}}^{}\tau_{j_{3}}^{}\cdots\tau_{j_{n-2r+2}}^{},\quadn\geq2r.
W=W_{o}(h_{l})+\sum(\lambda_{e}^{ab}L^{a}E_{c}^{b}H_{1}+\lambda_{D}^{ab}Q^{a}D_{c}^{b}H_{1}+\lambda_{U}^{ab}Q^{a}U_{c}^{b}H_{2}+\muH_{1}H_{2}).
E_{f}={\frac{R_{\natural}}{\ell_{11}^{2}}}=E_{f}^{\prime}={\frac{R_{\star}}{(\ell_{11}^{\prime})^{2}}}\quad,\qquad\rho_{i}\equiv{\frac{R_{i}}{\ell_{11}}}=\rho_{i}^{\prime}={\frac{R_{i}^{\prime}}{\ell_{11}^{\prime}}}\quad.
F=-\frac{1}{16a^{2}N}\sum_{x=1}{N}\left\{(n_{x}^{(1)}-n_{x+1}^{(1)})(n_{x}^{(2)}-n_{x+1}^{(2)})+(L_{x}^{(1)}-R_{x}^{(1)})(L_{x}^{(2)}-R_{x}^{(2)})\right\}\quad.
R_{(f)s}^{(1)}(z,z^{\prime})=\frac{\partial_{z}\{\Theta[1/2,1/2](J_{(1)}|\omega_{s}^{(1)})\}}{\Theta[1/2,1/2](J_{(1)}|\omega_{s}^{(1)})}\sqrt{\frac{\partial_{z^{\prime}}J_{(0)s}^{(1)}(z^{\prime})}{\partial_{z}J_{(0)s}^{(1)}(z)}}
H=\frac{\sqrt{\alpha^{\prime}}}{2}g^{\mu\nu}(P_{\mu}-Q_{e}A_{\mu})(P_{\nu}-Q_{e}A_{\nu})+(\frac{m^{2}R^{2}}{2\sqrt{\alpha^{\prime}}\alpha^{\prime}}+\frac{\sqrt{\alpha^{\prime}}Q_{e}^{2}}{2k^{2}}-\frac{Q_{e}\lambdaR}{k}),
\left[-\,\frac{d^{2}}{dr^{2}}+\frac{\ell(\ell+1)}{r^{2}}-\alpha\,\sqrt{i\,r}+i\,r^{3}\right]\,\psi(r)=E\psi(r),\\\\\r=x-i\varepsilon,\\\x\in(-\infty,\infty)
e^{{\frac{i}{\hbar}}S_{\mathrm{IF}}[\phi,\phi^{\prime}]}=\intd\varphi\int_{\varphi_{i}}^{\varphi}{\calD}\varphi\int_{\varphi_{i}^{\prime}}^{\varphi}{\calD}\varphi^{\prime}e^{{\frac{i}{\hbar}}\int^{\calC}d^{4}s\left[{\calL}_{h}^{0}[\varphi]+{\calL}_{\mathrm{int}}^{\prime}[\phi,\varphi]\right]}\rho_{h}(\varphi_{i},\varphi_{i}^{\prime};t_{i}),
\Psi(x^{\prime},t^{\prime})=\int_{-\infty}^{+\infty}{\frac{dk}{2\pi}}\,\langlex^{\prime}|k^{+}\ranglee^{-2i\sqrt{m^{2}+k^{2}}\,t^{\prime}}\langlek^{+}|x\rangle\Psi(x,t=0).
p_{n}\left(x\right)=\left(-1\right)^{n}q^{n/2+1/2}\left\{\left(1-q\right)\cdot\cdot\cdot\left(1-q^{n}\right)\right\}^{-1/2}\sum_{j=0}^{n}\left[\begin{array}{l}{n}\\{j}\\\end{array}\right]q^{j^{2}}\left(-q^{1/2}x\right)^{j},
u_{\kappa_{2}}(a)u_{\kappa_{1}}^{\prime}(a)-u_{\kappa_{1}}(a)u_{\kappa_{2}}^{\prime}(a)=(\kappa_{2}^{2}-\kappa_{1}^{2})\int_{a}^{\infty}u_{\kappa_{1}}(r)u_{\kappa_{2}}(a)dr
{S^{\prime}}_{M}^{(1)}=-\frace4\int\!d^{6}\thetaW(\theta)\theta^{\rho\sigma}\int\!d^{4}x[F_{\mu\rho}F_{\nu\sigma}F^{\mu\nu}-\frac14F_{\rho\sigma}F_{\mu\nu}F^{\mu\nu}].
\hat{P}=\left(\begin{array}{cc}{D_{\mu\nu\alpha\beta}+\left(\frac{R}{6}+2\Lambda\right)\delta_{\mu\nu\alpha\beta}}&{p_{12}R_{\mu\nu}}\\{p_{21}R_{\alpha\beta}}&{p_{r}R+p_{l}}\\\end{array}\right)
(\varepsilon_{0,i})^{\beta\rightarrow0}=-\frac{|e|H}{(2\pi)^{2}}\sum_{n=0}^{\infty}\int_{-\infty}^{\infty}dp\left\{\frac{1}{\beta}-\epsilon_{n,i}-\frac{\beta\epsilon_{n,i}^{2}}{2}\right\}.
L_{eff}=\frac{1}{2}m^{2}(t)\Phi^{2}+\frac{\lambda(t)}{4!}\varphi^{4}+\frac{1}{2}\xi(t)R\varphi^{2}+\Lambda_{0}(t)+\kappa(t)R+\widetilde{a}_{1}(t)R^{2},
A_{N}^{(0)}=g^{N-2}\int_{0}^{1}\prod_{i=1}^{N-3}dx_{i}\,\rho^{(0)}(x_{i})\prod_{i=1}^{N-3}x_{i}^{-\alpha(-(p_{i+1}-p_{N})^{2})-1}\prod_{2\leqi<j\leqN-1}y_{i-1j}^{-\alpha(-(p_{i-1}-p_{j})^{2})-1},
F_{s_{1}s_{2}}(\mathbf{p})=\sum_{\ell_{s_{1}s_{2}}}\sum_{m_{s_{1}s_{2}}}f_{s_{1}s_{2}}^{\ell_{s_{1}s_{2}}m_{s_{1}s_{2}}}\left(p\right)Y_{\ell_{s_{1}s_{2}}}^{m_{s_{1}s_{2}}}(\hat{\mathbf{p}}),
\delta_{tot}|\phi\rangle=\Bigl(\xi\partial+\frac{d-2}{2z}\xi^{z}+\frac{1}{2}\partial^{I}\xi^{J}M^{IJ}+M^{IJ}\partial^{+}\xi^{I}\frac{\partial^{J}}{\partial^{+}}+\frac{2s+2-d}{2z\partial^{+}}\partial^{+}\xi^{I}R^{zI}-s\frac{\partial^{+}\xi^{z}}{z\partial^{+}}\Bigr)|\phi\rangle
\sigma_{\mathrm{abs}}={\frac{\pi^{3}R^{8}}{64}}(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2})\omega{\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{2T_{L}}}-1)(e^{\frac{\omega}{2T_{R}}}-1)}}\.
j_{q2}(\tilde{x},t)=\partial_{\tilde{x}}^{2}\mathcal{J}_{q2},\qquad\mathcal{J}_{q2}=-\frac{i}{12\mu}\bigl\{\psi^{*}(\tilde{x},t)\tilde{x}^{2}\partial_{\tilde{x}}\psi(\tilde{x},t)-\bigl[\partial_{\tilde{x}}\psi^{*}(\tilde{x},t)\bigr]\tilde{x}^{2}\psi(\tilde{x},t)\bigr\}.
\Omega_{_{(2)}}^{r}=v_{_{0i}}^{r}\left[{\frac{\partial\hat{L}(q,\alpha)}{\partialq_{i}}}-{\frac{\partial^{2}\hat{L}(q,\alpha)}{\partial\alpha_{i}\partialq_{j}}}\alpha_{j}\right]+u_{_i}^{r}\Omega_{_{(1)}}^{i}\;.
{\calL}_{M}=-\frac{1}{4}g^{\mu\rho}g^{\nu\lambda}\mathbf{F}_{\mu\nu}\mathbf{F}_{\rho\lambda}+\frac{1}{2}g^{\mu\nu}\left({\calD}_{\mu}\mathbf{\Phi}\right)\left(\mathbf{{\calD}}_{\nu}\mathbf{\Phi}\right)-\frac{\lambda}{4}\left(\mathbf{\Phi}^{2}-v^{2}\right)^{2},
g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\lambda(r)\mathrm{d}t^{2}-\lambda^{-1}(r)\mathrm{d}r^{2}-R(r)(\mathrm{d}\theta^{2}+\mathrm{sin}^{2}\theta\mathrm{d}\phi^{2}).
\sqrt{-g}L\equiv{\frac{1}{4\pi{\calA}_{P}}}\left(\sqrt{-g}{\calG}\right)={\frac{1}{4\pi{\calA}_{P}}}\left(\sqrt{-g}\,g^{ik}\left(\Gamma_{i\ell}^{m}\Gamma_{km}^{\ell}-\Gamma_{ik}^{\ell}\Gamma_{\ellm}^{m}\right)\right).
\Lambda=\int{\frac{d^{2}\mu}{T^{D/2}\prod_{i<j}|a_{ij}|^{2}}}\sum_{\alpha,\beta\in\Xi}C\Big[\begin{array}{c}{\alpha}\\{\beta}\\\end{array}\Big]{\langleJ(r+)J(r-)\tilde{J}(\bar{s}+)\tilde{J}(\bar{s}-)\rangle_{s(\psi,\tilde{\psi})}}\prod_{f\inF^{\prime}}Q_{s(f)}^{1/8},
\langle\rho_{\mathrm{tail}}\rangle=\frac{1}{2\pi^{2}}\int_{k_{\mathrm{H}}}^{+\infty}k_{\mathrm{phys}}\mathrm{d}k_{\mathrm{phys}}\int\omega_{\mathrm{phys}}\mathrm{d}\omega_{\mathrm{phys}}\vert\beta_{k_{\mathrm{phys}}}\vert^{2},
\delta\tilde{H}^{\alpha}=\deltaH_{c}^{\alpha}+\sum_{n=1}^{\infty}\left(\deltaG_{(n)}^{\alpha}(q_{i},p_{i})\theta_{\alpha}^{n}+nG_{(n)}^{\alpha}(q_{i},p_{i})\theta_{\alpha}^{(n-1)}\delta\theta_{\alpha}\right)=0,
{\cal{H}}=\frac{1}{2}\intd^{2}x\displaystyle\sum_{n=0}^{\infty}(-1)^{n}(\pi_{(n)}^{2}+\varphi_{(n)}^{2})=\frac{1}{4}(\pi_{(0)}+\varphi_{(0)}^{\prime})^{2}-\displaystyle\sum_{n=1}^{\infty}\Lambda_{(n)}\Psi_{(n)}\approx\frac{1}{4}(\pi_{(0)}+\varphi_{(0)}^{\prime})^{2}
U=\left(\begin{array}{c|c|c|c}{\strutU_{11}}&{{\bf0}}&{\cdots}&{{\bf0}}\\\hline{\strut{\bf0}}&{U_{rr}}&{\ddots}&{\vdots}\\\hline{\strut\vdots}&{\ddots}&{\ddots}&{{\bf0}}\\\hline{\strut{\bf0}}&{\cdots}&{{\bf0}}&{U_{MM}}\\\end{array}\right).
{\calH}_{\mathrm{coll}}=-\frac{1}{2}g^{\alpha\beta}\left(\nabla_{\alpha}+A_{\alpha}\right)\left(\partial_{\beta}+A_{\beta}\right)-\frac{1}{2}g^{\alpha\beta}\left[\left(\nabla_{\alpha}+A_{\alpha}\right)\tilde{A}_{\beta}+\tilde{A}_{\alpha}\left(\partial_{\beta}+A_{\beta}\right)\right].
ds^{2}=(r_{2}^{3}r_{5}^{6}r_{6}^{7})^{1/8}\left[4\left(R^{-2}[-dudv+dR^{2}]+{\frac{p(u)}{4r_{2}r_{5}r_{6}}}du^{2}\right)+d\Omega_{2}^{2}+{\frac{dy_{a}dy^{a}}{r_{5}r_{6}}}+{\frac{dw^{2}}{r_{2}r_{6}}}\right].
\frac{I_{4}(\alpha,\beta,\gamma,\delta)}{\gamma\,\delta\,G(1,\delta+1)}=\frac{G(\alpha,\gamma+1)}{2\mu-3}S(\mu-\alpha-1,\beta-1,\mu+\alpha-\delta-2,\delta-\beta)+(\alpha\leftrightarrow\beta)\,,
W={\frac{1}{64\pi^{{\frac{1}{2}}}}}{\frac{\tau}{a}}(\Phi_{0}^{2}a^{2})^{2}Log({\frac{1}{M^{2}a^{2}}})-{\frac{c}{192(2\pi)^{\frac{1}{2}}}}({\frac{\tau}{a}}(\Phi_{0}^{2}a^{2})^{3})
f\star_{\theta}g(u)\simfg(u)+i\frac{\theta}{2}\{f,g\}(u)+\sum_{k=2}^{\infty}\,\biggl(i\frac{\theta}{2}\biggr)^{k}\frac{1}{k!}\,D_{k}(f,g)(u),\quad\mathrm{as}\\theta\to0,