harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\deltaR_{AB}=\partial_{C}\delta\Gamma_{AB}^{C}-\partial_{B}\delta\Gamma_{AC}^{C}+\overline{{\Gamma}}_{AB}^{C}\delta\Gamma_{CD}^{D}+\delta\Gamma_{AB}^{C}\overline{{\Gamma}}_{CD}^{D}-\delta\Gamma_{BC}^{D}\overline{{\Gamma}}_{AD}^{C}-\overline{{\Gamma}}_{BC}^{D}\delta\Gamma_{AD}^{C}.
\chi_{1}^{m}\tog^{m\bar{n}}\frac{\partial}{\partial(d\bar{z}^{\bar{n}})},\quad\chi_{2}^{m}\todz^{m},\quad\bar{\chi}^{\bar{m},1}\tod\bar{z}^{\bar{n}},\quad\bar{\chi}^{\bar{m},2}\tog^{n\bar{m}}\frac{\partial}{\partial(dz^{n})}
E=\left[\begin{array}{ccccc}{1}&{0}&{0}&{0}&{0}\\{-4}&{3}&{-7}&{1}&{-2}\\{6}&{-2}&{18}&{-1}&{6}\\{-4}&{-2}&{-17}&{-1}&{-6}\\{1}&{3}&{6}&{1}&{2}\\\end{array}\right]
\Gamma=\Gamma\left[<A_{\mu}(\bar{x})>,<A_{\mu,\theta}>,<c_{5}(\bar{x})>,<c_{5,\theta}>,<A_{\mu,\lambda}^{\prime}>,<c_{5,\lambda}>,<c_{4,\lambda}>,<\zeta>,<\zeta_{,\theta}>\right]
\hat{H}=\varepsilon(\vec{\pi})+\frac{e}{2}\left(\frac{1}{\sqrt{\varepsilon({\bf\hat{p}})}}\Phi\sqrt{\varepsilon({\bf\hat{p}})}+\sqrt{\varepsilon({\bf\hat{p}})}\Phi\frac{1}{\sqrt{\varepsilon({\bf\hat{p}})}}\right).
m={\frac{2^{\frac{1}{3}}\pi^{4/3}\Gamma(5/9)}{\Gamma(2/3)\Gamma(8/9)}}\,\left({\frac{2\Gamma(11/16)}{\Gamma(5/16)\Gamma(3/4)}}\right)^{8/9}\,g^{8/9}\approx4.9277912244\ldotsg^{8/9}.
\tilde{G}_{ab}=G_{MN}\frac{\partialX^{M}}{\partial\sigma^{a}}\frac{\partialX^{N}}{\partial\sigma^{b}},\,\,\,\tilde{B}_{ab}=B_{MN}\frac{\partialX^{M}}{\partial\sigma^{a}}\frac{\partialX^{N}}{\partial\sigma^{b}}~,
{\frac{1}{c^{2}}}\left(1-{\frac{c^{2}B}{I^{3/2}}}\varrho\right)\ddot{\Phi}+{\frac{3}{c^{2}}}{\frac{\dot{a}_{0}}{a_{0}}}\dot{\Phi}\left(1+{\frac{B}{\sqrt{I}}}\wp\right)+V^{\prime}(\Phi)=0.
S_{4*}=\intd^{4}x\sqrt{-g}\left[R+\frac{1}{4}\mathrm{Tr}\left[\nablaM\nablaM^{-1}\right]-\frac{1}{2}\left(\nablav\right)^{2}-\frac{1}{2}e^{\sqrt{3}v}\nabla\sigma^{T}M\nabla\sigma\right].
{\calS}(\Gamma)\equiv\int\left\{\partial_{\mu}c\frac{\delta\Gamma}{\deltaA_{\mu}}+B\frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\underline{Y}}\frac{\delta\Gamma}{\delta\underline{\varphi}}+\underline{q}\frac{\delta\Gamma}{\delta\underline{{\hat{\varphi}}}}\right\}=0\;\;.
I_{0}+2I_{2}=-\frac{\Gamma\left(2-\frac{D}{2}\right)}{(4\pi)^{\frac{D}{2}}}\int_{0}^{1}dx2(1-x)\sum_{n}\left[\left(1-\frac{4}{D}\right)\frac{1}{[k_{0}^{2}+m^{2}]^{2-\frac{D}{2}}}+\frac{4}{D}\frac{2-D/2}{\Gamma(3)}\frac{1}{[k_{0}^{2}+m^{2}]^{2-\frac{D}{2}}}\right].
\lambda(q;n^{\prime}j^{\prime},nj)=\frac{4\pi}{M(zz_{d}e^{2})^{2}}\cdot\frac{(n^{\prime}+\mu^{\prime})^{2}(n+\mu)^{2}}{(n^{\prime}+\mu^{\prime})^{2}-(n+\mu)^{2}}
G_{\bar{1}\bar{1}}=\frac{2R^{4}}{3r^{6}}\Biggl[\bar{z}^{2}\bar{z}^{2}\frac{\partial_{3}\phi}{z^{3}}+\bar{z}^{3}\bar{z}^{3}\frac{\partial_{2}\phi}{z^{2}}\Biggr]+\partial_{\bar{1}}\partial_{\bar{1}}\psi
M_{--}^{2}=g^{2}v^{2}\left(\begin{array}{ccccccc}{2}&{-1}&{0}&{\cdots}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{\cdots}&{0}&{0}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}&{0}&{0}\\{}&{}&{}&{\ddots}&{}&{}&{}\\{0}&{0}&{0}&{\cdots}&{-1}&{2}&{-1}\\{0}&{0}&{0}&{\cdots}&{0}&{-1}&{2}\\\end{array}\right).
L_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})-\frac{1}{2}m^{2}\Psi^{2}
\begin{array}{l}{3(3,2)_{\frac{1}{6}}+3(\overline{{3}},1)_{\frac{1}{3}}+3(\overline{{3}},1)_{-\frac{2}{3}}+12(1,2)_{-\frac{1}{2}}+9(1,2)_{\frac{1}{2}}}\\{\\\+12(1,1)_{1}+9(1,1)_{0}+9(1,1)_{-1}.}\\\end{array}
{\hat{s}}^{2}={\frac{1}{4}}({\hat{a}}_{(1)(1)}-{\hat{a}}_{(2)(2)})^{2}+{\frac{1}{2}}({\hat{a}}_{(1)(2)}{\hat{a}}_{(2)(1)}+{\hat{a}}_{(2)(1)}{\hat{a}}_{(1)(2)}),
\frac{d^{2}\phi(t)}{dt^{2}}+3\frac{\dot{a}(t)}{a(t)}\frac{d\phi(t)}{dt}+V^{\prime}(\phi(t))+\frac{V^{\prime\prime\prime}(\phi(t))}{2\Omega}\intd^{3}x\langle\eta^{2}(\vec{x},t)\rangle+\cdots=0
A_{q}\:\vec{T}_{0}^{q}\:\vec{T}_{1}^{q}\:=\:B_{q}\:f_{-1}^{p}f_{q}^{p}\:\prod_{k=1}^{q}\Phi_{k}\:\vec{I}+C_{q}\:\vec{T}_{0}^{q+1}\:\vec{T}_{1}^{q-1}
\mu_{\mathrm{wkb}}=\frac{1}{2}\sqrt{\frac{\epsilon}{\pi}}\vert\eta_{\mathrm{i}}\vert^{1/2}\biggl(\biggl\vert\frac{\eta}{\eta_{\mathrm{i}}}\biggl\vert\biggl)^{-2\pii/\epsilon+1/2}.
V=\left(\begin{array}{cc}{V_{+}^{1}}&{V_{-}^{1}}\\{V_{+}^{2}}&{V_{-}^{2}}\\\end{array}\right)={\frac{1}{\sqrt{2i\rho_{2}}}}\left(\begin{array}{cc}{\rhoe^{i\phi}}&{\bar{\rho}e^{-i\phi}}\\{e^{i\phi}}&{e^{-i\phi}}\\\end{array}\right).
\phi(x^{+},x^{-},x^{I})=\int_{-\infty}^{\infty}dp\theta(p)\left[a(p)e^{-{\frac{i}{2p}}(m^{2}+8fp)x^{+}}e^{ipx^{-}}+a^{*}(p)e^{{\frac{i}{2p}}(m^{2}+8fp)x^{+}}e^{-ipx^{-}}\right]e^{-{\frac{fp}{2}}x^{2}}.
{\calE}={\frac{\pi\mu^{2}}{\lambda}}\int_{0}^{\infty}drr\left[h^{2}\!+(\varphi^{\prime})^{2}+{\frac{f^{2}\varphi^{2}}{r^{2}}}+{\frac{\lambda}{2}}\left(\varphi^{2}-1\right)^{2}\right].
\left(\begin{array}{c}{\mathbf{k}}\\{k_{iL}}\\{k_{iR}}\\\end{array}\right)=\sqrt{\frac{2}{\alpha^{\prime}}}\,W(\lbrace\mathbf{a}_{i}\rbrace;B_{ij})\left(\begin{array}{c}{\mathbf{q}}\\{\sqrt{\frac{\alpha^{\prime}}{2}}(\frac{n_{i}}{R_{i}}+\frac{w_{i}R_{i}}{\alpha^{\prime}})}\\{\sqrt{\frac{\alpha^{\prime}}{2}}(\frac{n_{i}}{R_{i}}-\frac{w_{i}R_{i}}{\alpha^{\prime}})}\\\end{array}\right).
{\calA}=\left[(-1)^{A}({\cal{\varepsilon}}^{2})_{A}{}^{B}\,J_{B}{}^{C}\,(\delta_{\Lambda})_{C}{}^{A}\right],\quad\quad\mathrm{where}\quad\quadJ_{A}{}^{B}=\frac{\partial_{r}R_{A}}{\partial\Phi_{B}},
E_{32}=F^{+}=\left(\begin{array}{cc|c}{0}&{0}&{0}\\{0}&{0}&{0}\\\hline{0}&{\nuj_{1}}&{0}\\\end{array}\right),\;E_{23}=\bar{F}^{-}=\left(\begin{array}{cc|c}{0}&{0}&{0}\\{0}&{0}&{\nuj_{1}}\\\hline{0}&{0}&{0}\\\end{array}\right)
\begin{array}{rl}{u_{10}}&{=u_{0}+\frac{p^{+}}{\sqrt{2}}}\\{v_{10}}&{=\frac{1}{u_{0}+\frac{p^{+}}{\sqrt{2}}}\left\{1-\frac{[1-(u_{0}+\frac{p^{+}}{\sqrt{2}})(v_{0}+p^{-}\sigma_{per}^{+})]\times[1-(u_{0}+\frac{p^{+}}{\sqrt{2}})(v_{0}+p^{-}\sigma_{per}^{-})]}{[1-(u_{0}+\frac{p^{+}}{\sqrt{2}})(v_{0}-\frac{p^{-}}{\sqrt{2}})]}\right\}}\\\end{array}
\frac{\partial}{\partial\Lambda}W[J]=-\frac{1}{2}\left\{\frac{\deltaW}{\deltaJ}.\frac{\partialC^{-1}}{\partial\Lambda}.\frac{\deltaW}{\deltaJ}+\mathrm{tr}\left(\frac{\partialC^{-1}}{\partial\Lambda}.\frac{\delta^{2}W}{\deltaJ\deltaJ}\right)\right\}
R=\left(\begin{array}{cccccccc}{1}&{0}&{0}&{0}&{p}&{0}&{0}&{q}\\{0}&{1}&{0}&{0}&{0}&{p}&{-p}&{0}\\{0}&{0}&{1}&{0}&{0}&{-p}&{p}&{0}\\{0}&{0}&{0}&{1}&{p^{2}q^{-1}}&{0}&{0}&{p}\\{0}&{0}&{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}&{0}&{1}\\\end{array}\right),
S_{\mathrm{sugra}}=\frac{1}{4(2\pi)^{5}g_{s}^{2}\alpha^{4}}\intd^{6}x\intdrr^{3}[(\partial_{r}\varphi)^{2}+H\eta^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi+H^{\prime}\delta^{ij}\partial_{i}\varphi\partial_{j}\varphi]
b_{n}^{R}=\frac{\sqrt{2}}{\pi}\frac{(-1)^{(n-1)}}{(2n-1)^{3/2}}p_{0}-\frac{1}{\sqrt{2}}a_{2n-1}+\frac{1}{\sqrt{2}}\sum_{m=1}^{\infty}M_{1\,m,n}a_{2m}-M_{2\,m,n}a_{2m}^{\dagger},
X_{k}^{i}(\sigma,\tau)=x^{(k,f)i}+\frac{1}{2}i\sum_{m\neq0}\left(\frac{\alpha_{m+\frac{k}{m}}}{m+\frac{k}{m}}e^{-i(m+\frac{k}{n})(\tau-\sigma)}+\frac{{\tilde{\alpha}}_{m-\frac{k}{n}}^{i}}{m-\frac{k}{n}}e^{-i(m-\frac{k}{n})(\tau+\sigma)}\right).
K_{ab}^{i}=\Bigl(\frac{\partial^{2}X^{N}}{\partial\sigma^{a}\partial\sigma^{b}}+\frac{\partialX^{P}}{\partial\sigma^{a}}\frac{\partialX^{Q}}{\partial\sigma^{b}}\Gamma_{PQ}^{N}\Bigr)n_{N}^{i}~.
\rho_{N}(p)\equiv\rho_{p\rightarrow1+2+\cdotsN}\equiv\prod_{i}\left(\int\bar{d}^{2\ell}\!p_{i}\,\theta(p_{i})\bar{\delta}(p_{i}^{2}-m_{i}^{2})\right)\bar{\delta}^{2\ell}(p-\sum_{i=1}^{N}p_{i}).
\frac{\bar{B}(p^{2})}{m}=PP\Big[{\frac{e^{-{\frac{m^{2}}{2\mu^{2}}}}}{p^{2}}}+\int_{0}^{\infty}d\lambda^{2}{\frac{\bar{\sigma}(\lambda^{2})}{p^{2}-\lambda^{2}}}\Big]\;.
\bar{S}_{1}=\bar{S}+\Phi_{A}^{*(2)}\mu_{2}^{A}+\Phi_{A}^{*(3)}\mu_{3}^{A}+\bar{\Phi}_{A}^{(1)}\nu_{1}^{A}+\varphi_{I}^{*(2)}m_{2}^{I}+\varphi_{I}^{*(3)}m_{3}^{I}+\bar{\varphi}
\zeta_{\mathrm{T}}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\,t^{s-1}\sum_{m=-\infty}^{\infty}e^{-\Omega_{m}^{2}t}\sum_{\{k\}}^{}e^{-\omega_{k}^{2}t}{.}
S_{eff}=\int\left(-\frac{1}{4}\sum_{n}(\partial_{\mu}b_{\rho\sigma}^{n}\partial^{\mu}b^{n,\rho\sigma}+{m_{n}^{2}}b_{\rho\sigma}^{n}b^{n,\rho\sigma})-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\right)dx.
\|c_{\alpha\beta}\|=\left(\begin{array}{ccc}{0}&{0}&{1}\\{0}&{2}&{0}\\{1}&{0}&{0}\\\end{array}\right),\qquad\|c^{\alpha\beta}\|=\left(\begin{array}{ccc}{0}&{0}&{1}\\{0}&{1/2}&{0}\\{1}&{0}&{0}\\\end{array}\right).
\begin{aligned}{\big(M^{(8)},Q,Q\big)}&{{}\rightarrowA=\frac{1}{8}\,,}&{\big(M^{(5)},Q,Q\big)}&{{}\rightarrowB=\frac{1}{8}\,,}\\{\big(W^{(8)},Q,Q\big)}&{{}\rightarrowA_{z}=\frac{\tilde{m}}{8}=\frac{\tilde{\mu}}{8}\,,}&{\big(W^{(5)},Q,Q\big)}&{{}\rightarrowB_{z}=\frac{\tilde{n}}{8}=\frac{\tilde{\nu}}{8}\,.}\\\end{aligned}
\lambda_{R}\equiv{\frac{1}{3!}}{\frac{d^{4}{\calV}(\Phi)}{d\Phi^{4}}}\biggr|_{\Phi=0}=\lambda{\frac{1-6\lambdaI_{(2)}[m_{R}^{2}]}{1+3\lambdaI_{(2)}[m_{R}^{2}]}},
m=m_{b}={\frac{-1}{6l^{2}}}[3{r_{b}}^{5}+(l^{2}-15n^{2}){r_{b}}^{3}-3n^{2}(2l^{2}-15n^{2}){r_{b}}-3n^{4}(l^{2}-5n^{2})/r_{b}]
v_{\Lambda\Omega}=\mathrm{{sgn}}(\sigma_{R_{ij}^{(\ell)}\Lambda}^{a})\,v_{\Lambda_{ij}\left(R_{ij}^{(\ell)}\Lambda\right)_{ij}}n_{\omega^{s}}(\Lambda_{i}-\ell)n_{\omega^{s}}(\Lambda_{j}+\ell)\,,
{\calM}\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{dx}{2\pi}\displaystyle\frac{1}{2}e^{x}\,{\calQ}(x)\doteq\displaystyle\frac{1}{L}\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{dx}{2\pi}e^{x}\,{\calQ}_{+}(x)\,,
ds^{2}=\left(1+{\frac{r_{0}^{2}}{r^{2}}}\right)^{-1}\left[-dudv+\left({\frac{p(u)}{r^{2}}}-2\ddot{f}_{i}(u)y^{i}-2\ddot{h}_{i}(u)x^{i}\right)du^{2}\right]+\left(1+{\frac{r_{0}^{2}}{r^{2}}}\right)dx_{i}dx^{i}+dy_{i}dy^{i}
H_{c}=\frac{\pi_{c}^{2}}{2R^{3}}+R^{3}\Bigl(\frac{m^{2}}{2}+\frac{3\lambda}{2}\langle\hat{\phi}_{f}^{2}\rangle\Bigr)\phi_{c}^{2}+\frac{\lambda}{4}\phi_{c}^{4}.
\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[\tilde{a}_{n}(\omega)~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+\tilde{b}_{n}(\omega)~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right],
Z(T,U)=\int_{\cal{M}}\frac{d^{2}\tau}{\tau_{2}^{2}}4T_{2}\sum_{M}e^{8\pii|detM|T}e^{-\frac{4\piT_{2}}{\tau_{2}U_{2}}\big|(1,U)M\big({\tau\atop-1}\big)\big|^{2}}
\begin{array}{c}{<j_{1}-m_{1}\j_{2}-m_{2}|J-M>_{q}=<j_{2}m_{2}\j_{1}m_{1}|JM>_{q}=}\\{=(-)^{j_{1}+j_{2}-J}<j_{1}m_{1}\j_{2}m_{2}|JM>_{q^{-1}}}\\\end{array}
\frac{dx(y)}{dy}=G(\Phi_{s}(y),\bar{\Phi}_{s}(y))\frac{d\Phi_{s}}{dy}e^{-i\delta}\left(\left.\frac{d\bar{W}}{d\bar{\Phi}}\right|_{\Phi=\Phi_{s}(y)}\right)^{-1}\,.
\varphi_{-}(x^{+},x^{-})={\frac{1}{\sqrt{4\pi}}}\Bigl(\sum_{n=0}^{\infty}b_{n}u_{n}\\mathrm{e}^{-ik_{n}x^{-}}+\sum_{n=1}^{\infty}d_{n}^{\dagger}\v_{n}\mathrm{e}^{+ik_{n}x^{-}}\Bigr)\.
2\pi{\rho}_{a,j}(\beta)=\sigma_{j}^{(\infty)}*\rho_{a,0}(\beta)-\sum_{b=1}^{P}\sum_{l=1}^{\infty}A_{jl}^{(\infty)}*{\calK}_{ab}^{(P)}*\widetilde\rho_{b,l}(\beta)
{\overline{{V_{0}+V_{1}+V_{2}}}}=(-{\frac{1}{2}}({\frac{1}{6}}~{\frac{1}{3}})(-{\frac{1}{3}}{\frac{5}{6}})^{2}\vert\vert{\frac{e_{1}}{2}}~({\frac{5}{6}})^{2}\vert({\frac{1}{3}})^{5}~0_{r}^{5}~({\frac{2}{3}})_{r})\,
[{\hat{\psi}_{n}}^{\dagger},{\hat{\psi}_{m}}]_{+}~=~\delta_{n,m},~~[{\hat{\psi}_{n}},{\hat{\psi}_{m}}]_{+}~=~[{\hat{\psi}_{n}}^{\dagger},{\hat{\psi}_{m}}^{\dagger}]_{+}~=~0.
K_{\chi}^{\chi^{+}}\Bigl|_{\Sigma_{2}}-K_{\chi}^{\chi^{-}}\Bigl|_{\Sigma_{1}}+K_{\chi}^{\chi^{+}}\Bigl|_{\Sigma_{1}}-K_{\chi}^{\chi^{-}}\Bigl|_{\Sigma_{2}}=0.
P(0,0;t_{f})=exp\left[C({\bar{\beta}})+\frac{2\bar{V}}{2(2\pi^{2})}\int_{\bar{m}^{2}}^{\infty}d\Lambda\int_{-\bar{t_{f}}^{2}}^{0}d{\bar{p}}^{3}ln\frac{1+th(2-4e^{-\pi\Lambda})}{(1+th)^{2}}\right]
emb^{*}\left(WZ\right)=\frac1{\left(p+1\right)!}dx^{\mu_{1}}\cdotsdx^{\mu_{p+1}}\cdot\Pi_{,\mu_{1}}^{A_{1}}\cdots\Pi_{,\mu_{p+1}}^{A_{p+1}}\cdot\left(WZ\right)_{A_{p+1}\cdotsA_{1}}\left(\phi\right)\quad;
\delta_{1}\omega_{\nu}{}^{mn}=D_{\phi}\Big(\bar{\epsilon}\Gamma_{\nu}{}^{\phi\rhomn}\psi_{\rho}\Big)+4\,\bar{\epsilon}\Gamma^{mnp}\hat{\psi}_{\nup}-\frac{4}{d-2}\,\,\bar{\epsilon}\Gamma^{rs[m}\hat{\psi}_{rs}e_{\nu}{}^{n]}+{\calO}(\epsilon\psi^{3})\,,
\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}(\theta,CC)=\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}(\theta,\frac{1}{2}[C+A]\frac{1}{2}[C+A])-\pii\int_{0}^{1}dx\Theta[B]
V_{\beta}(r^{\prime},r)=\tilde{g}(r^{2}\beta^{2})\left[\sum_{i,j=0}^{N}(-\stackrel{\leftarrow}{\nabla^{\prime}}{}^{2})^{i}\lambda_{ij}(-\stackrel{\rightarrow}{\nabla}{}^{2})^{j}\right]{\tilde{g}}(r^{2}\beta^{2}),
\Gamma=\Gamma_{0}+\Gamma_{1}=\beta\sum_{N=1}^{K}\frac{N_{N}v_{N}^{2}}{2R}-\frac{15l_{P}^{9}}{16R^{3}}\sum_{M<N,\N=1}^{K}\int_{0}^{\beta}dt\frac{N_{M}N_{M}v_{MN}^{4}}{(v_{MN}^{2}{t}^{2}+r_{MN}^{2})^{\frac{7}{2}}}
\mathrm{either}\quad\tilde{\gamma}^{a}:=i\;dx^{0}\;\tilde{\tilde{\vee}}\;dx^{a}\;\tilde{\vee},\quad\mathrm{or}\qquad\tilde{\gamma}^{a}=i\;\tilde{\tilde{a}}{}^{0}\;\tilde{a}^{a}
\sum_{n=0}^{\infty}e(X^{[n]})(t\tilde{t})^{-n}q^{n}=\frac{1}{((t\tilde{t})^{-1}q)_{\infty}(t\tilde{t}^{-1}q)_{\infty}(q)_{\infty}^{20}(t^{-1}\tilde{t}q)_{\infty}(t\tilde{t}q)_{\infty}}
\left|{\calP}_{ee}^{r}({\bfk},t)\right|^{2}+\left|{\calP}_{\bar{e}e}^{r}({\bfk},t)\right|^{2}+\left|{\calP}_{\mue}^{r}({\bfk},t)\right|^{2}+\left|{\calP}_{\bar{\mu}e}^{r}({\bfk},t)\right|^{2}=1\,,
\hat{\calH}^{n}=\Biggl(\Bigl(\bigoplus_{\stackrel{\scriptstylen=N^{c}-N^{b}}{N^{c},N^{b}\geq0}}{\calF}^{-N^{b}}(\alpha_{-m}^{0},b_{-m};k^{0})\otimes{\calF}^{N^{c}}(c_{-m})\Bigr)\otimes{\calH}_{K}\Biggr)^{L_{0}}.
{\calO}_{2}{}^{i}{}_{j}\!\!\equiv\!\!\left[\begin{matrix}{-{\frac{\partial^{2}{\calW}}{\partialA_{\mathrm{cl}}^{i}\partialA_{\mathrm{cl}}^{j}}}}&{-i\left(\partial_{1}-i\partial_{2}\right)\delta_{j}^{i}}\\{-i\left(-\partial_{1}-i\partial_{2}\right)\delta_{j}^{i}}&{-{\frac{\partial^{2}{\calW}^{*}}{\partialA_{\mathrm{cl}}^{*i}\partialA_{\mathrm{cl}}^{*j}}}}\\\end{matrix}\right]
E=+E_{n}\,,\;n\geq1\;\;\psi_{n,p}^{(+)}=\sqrt{\frac{E_{n}+m}{2E_{n}}}\frac{e^{ipx_{2}}}{\sqrt{2\pi}}e^{-\frac{1}{2}{\zeta}^{2}}\left(\begin{array}{c}{N_{n}H_{n}(\zeta)}\\{-\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}+m}N_{n-1}H_{n-1}(\zeta)}\\\end{array}\right)
\sqrt{\left(1+\frac{b_{1}^{2}}{e^{\lambdaT_{1}}}+b_{2}^{2}+b_{3}^{2}\right)}\sqrt{\left(\frac{c_{1}^{2}}{e^{\lambdaT_{1}}}-\frac{c_{1}^{2}}{e^{\lambdaT_{0}}}\right)}=\frac{b_{1}c_{1}}{e^{\lambdaT_{1}}}+b_{2}c_{2}+b_{3}c_{3}.
L_{0}=p^{+}p^{-}+\frac{1}{k-2}\left(p_{2}^{2}+\frac{1}{4}\right)+\sum_{m>0}\left(\alpha_{-m}^{+}\alpha_{m}^{-}+\alpha_{-m}^{-}\alpha_{m}^{+}+s_{n-m}^{\prime}s_{m}^{\prime}\right)
{\calN}\prod_{0<\tau<\beta}D\mu[M_{k}(\tau)]={\calN}\prod_{0<\tau<\beta}\prod_{\gamma=1}^{N}d(M_{k}(\tau))_{\gamma\gamma}\prod_{\gamma>\nu=1}^{N}d\mathrm{Re}(M_{k}(\tau))_{\gamma\nu}\;d\mathrm{Im}(M_{k}(\tau))_{\gamma\nu}\;.
[\alpha^{I}\,\underline{{\alpha}}_{I}+\beta_{I}\,\underline{{\beta}}^{I},\hat{\alpha}^{I}\,\underline{{\hat{\alpha}}}_{I}+\hat{\beta}_{I}\,\underline{{\hat{\beta}}}{}^{I}]=(\alpha^{I}\hat{\beta}_{I}-\hat{\alpha}^{I}\beta_{I})\,\underline{{\epsilon}}_{0}+\omega^{i}(\alpha,\beta,\hat{\alpha},\hat{\beta})\,\underline{{\omega}}_{i}\.
\zeta^{T}(z)=\sum_{n=0}^{\infty}\sum_{m=1}^{\infty}D_{n,m}^{T}(\lambda_{n,m}^{T})^{-z}+\sum_{n=1}^{\infty}D_{n,0}^{T}(\lambda_{n,0}^{T})^{-z}~~~,
{\calT}(\lambda)={\calT}(2,\lambda){\calT}(1,\lambda)\,\qquad{\calT}(i,\lambda)=\left(\begin{array}{cc}{A_{i}(\lambda)}&{B(\lambda_{i})}\\{C_{i}(\lambda)}&{D(\lambda_{i})}\\\end{array}\right)\,\qquadi=1,2\.
{\calA}_{{\frac{1}{\sqrt{2}}}(x^{0}+iy),{\frac{1}{\sqrt{2}}}(x^{0}+iy)}={\frac{1}{2}}{\calT}_{p}\left({\frac{1}{1+2\pi^{2}\lambda^{2}e^{x^{0}+iy}}}-2\right){\frac{1}{1+2\pi^{2}\lambda^{2}e^{x^{0}-iy}}}~.
G(z,w)=\sum_{l=0}^{m-1}\,\rho^{l}G_{0}(z^{\frac{1}{m}},\rho^{l}w^{\frac{1}{m}})+\rho^{l}G_{0}(z^{\frac{1}{m}},\rho^{l}\overline{{w}}^{\frac{1}{m}})
\Theta_{2}^{\prime}\Theta_{3}-\Theta_{2}\Theta_{3}^{\prime}={\frac{\Theta_{2}^{5}\Theta_{3}-\Theta_{2}\Theta_{3}^{5}}{8\pii}}\left[{\int_{-1}^{1}}dx\sqrt{{\frac{1}{(x^{2}-1)(x\Theta_{3}^{4}+\Theta_{2}^{4}-\Theta_{4}^{4})}}}\right]^{2},
D_{-}^{\alpha}\equiv\partial_{z}+\frac{i}{2}(A_{z}^{1}+\frac{1}{\sqrt{3}}A_{z}^{2}+\cdots+\frac{1}{\sqrt{(\alpha-1)(\alpha-2)/2}}A_{z}^{\alpha-2}+\frac{\alpha}{\sqrt{\alpha(\alpha-1)/2}}A_{z}^{\alpha-1}),
[\hat{x}_{i},\hat{x}_{j}]=i\hbar\,\frac{(2\beta-\beta^{\prime})+(2\beta+\beta^{\prime})\beta\hat{p}^{2}}{(1+\beta\hat{p}^{2})}\left(\hat{p}_{i}\hat{x}_{j}-\hat{p}_{j}\hat{x}_{i}\right)\;.
H_{C}=\intd^{3}x\left(\Pi_{i}^{\dagger}\Pi^{i}+\partial_{i}\Pi^{i}W_{0}+\partial_{i}\Pi^{\daggeri}W_{0}^{\dagger}+{\textstyle\frac{1}{2}}F^{ij}F_{ij}^{\dagger}-m^{2}W_{\mu}^{\dagger}W^{\mu}\right)
\langle\Psi_{0}(z_{0})\cdots\Psi_{n}(z_{n})\rangle_{\Sigma}=Sp_{\partial\Sigma_{\mathrm{ext}}}\bigotimes_{i=0}^{n}{\langle\Psi_{i}(z_{i})\rangle}_{D_{i}}\otimes{\langle0\rangle}_{\Sigma_{\mathrm{ext}}}.
M^{\mu\alpha\beta}=x^{\alpha}T^{\mu\beta}-x^{\beta}T^{\mu\alpha}+\pi^{\rho\tau(\mu)}\left(\Sigma^{\alpha\beta}\right)_{\rho\tau}{}^{\lambda\sigma}h_{\lambda\sigma}+s^{ij(\mu)}\left(\Sigma^{\alpha\beta0\lambda}\right)_{ij}{}^{\rho\tau}\partial_{\lambda}h_{\rho\tau}\;\,,
\left\langle\begin{array}{l}{R_{1}}\\{J_{1}}\\\end{array}\right|{\hat{\calT}}_{J}^{R}\left|\begin{array}{l}{R_{2}}\\{J_{2}}\\\end{array}\right\rangle=\left\langle\begin{array}{ll}{R_{1}}&{R}\\{J_{1}}&{J}\\\end{array}\right.\left|\begin{array}{l}{R_{2}}\\{J_{2}}\\\end{array}\right\rangle\langleR_{1}\|{\hat{\calT}}^{R}\|R_{2}\rangle,
H_{\mu\nu\rho}=\epsilon_{\mu\nu\rho}\frac{r~r_{1}^{2}}{\left(r^{2}+r_{1}^{2}\right)^{2}}\left(f_{1}f_{5}\right)^{1/4};\;\;\;\;H_{abc}=\epsilon_{abc}\frac{r_{5}^{2}}{r^{3}}\left(f_{1}f_{5}\right)^{-3/4}
I_{1}\=\int\delta^{3}\!\left({\vec{r}_{1}-\vec{r}}\right)\,\delta\left(r_{1}-t\right)\frac{d^{3}r_{1}}{4\pi\,r_{1}}=\frac{1}{4\pir}\delta\left(r-t\right)=\frac{1}{2\pi}\delta\left(\tau^{2}\right)
\begin{array}{c}{F=u_{0}+v=\alpha+\bar{\beta},\quad\leftrightarrow\quadv=W=\alpha+\bar{\beta}-u_{0},}\\{\bar{F}=u+v_{0}=\bar{\alpha}+\beta,\quad\leftrightarrow\quadu=\bar{W}=\bar{\alpha}+\beta-v_{0}.}\\\end{array}
\intd^{10}x\,sqrt{-\hat{g}}\\left[{\textstyle\frac{3}{4}}\hat{G}^{2}\right]=\intd^{9}x\sqrt{g}\\left[{\textstyle\frac{3}{4}}kG^{2}-{\textstyle\frac{3}{4}}k^{-1}\left(H^{(2)}-\ellH^{(1)}\right)^{2}\right]\,.
{\calL}_{\mathrm{bulk}}+{\calL}_{\mathrm{source}}\sim\intd^{n}\xi\;\Bigl[{\frac{1}{g_{\mathrm{I}}^{2}}}(\partial_{\xi}m_{a})^{2}+{\frac{1}{g_{\mathrm{I}}}}\sum_{s}f_{s}(m_{a})\delta(\xi-\xi_{s})\Bigr].
(I\otimes\breve{R}^{\Phi_{1}\Phi_{2}})(\breve{R}^{\Phi_{1}\Phi_{3}}\otimesI)(I\otimes\breve{R}^{\Phi_{2}\Phi_{3}})=(\breve{R}^{\Phi_{2}\Phi_{3}}\otimesI)(I\otimes\breve{R}^{\Phi_{1}\Phi_{3}})(\breve{R}^{\Phi_{1}\Phi_{2}}\otimesI).
2i\z_{12}^{-1}z_{13}^{-3/4}z_{14}^{-3/4}z_{15}^{-1/2}z_{23}^{-3/4}z_{24}^{-3/4}z_{25}^{-1/2}z_{34}^{-1}z_{35}^{-1/2}z_{45}^{-1/2}\cdot(z_{14}z_{23}-z_{13}z_{24})
\prod_{i=1}^{s}{\frac{1}{(k+p_{i})^{2}+M_{j}^{2}}}=\mathrm{const.}\int_{0}^{1}\left(\prod_{i=1}^{s}dx_{i}\right)\delta(1-\sum_{i=1}^{s}x_{i}){\frac{1}{({k^{\prime}}^{2}+{M_{j}^{\prime}}^{2})^{s}}},
<\Phi(x,t)\overline{{\Phi}}(x^{\prime},t^{\prime})>_{q\approx0}=\frac{q}{M_{1}}\sum_{n=1}^{\infty}q^{-n}sin\left(\sqrt{q};\kappa_{n}x\right)sin\left(\sqrt{q};\kappa_{n}x^{\prime}\right)e^{i\kappa_{n}^{2}(t^{\prime}-t)}
=\sum_{j=1}^{|n\oplusm/2|}\sum_{j^{\prime}=1}^{|n|}c_{2j-1}c_{2j^{\prime}-1}\left(\frac{1}{2}\{\gamma_{2j-1},\gamma_{2j^{\prime}-1}\}+\frac{1}{2}[\gamma_{2j-1},\gamma_{2j^{\prime}-1}]\right)
\lambda_{1}=\left(\int_{0}^{y_{B}}dy\,\alpha^{2}\chi\varphi_{1}\right)^{-1}\left[\alpha^{4}\chi\varphi_{1}\left({\frac{\chi^{\prime}}{\chi}}-{\frac{\varphi_{1}^{\prime}}{\varphi_{1}}}\right)\right]_{y=y_{B}}.
S=\int_{{\calM}_{\tau}}\left[\Lambda\gamma_{a}\Lambda(e^{a}-i\theta\gamma^{a}{\calD}\theta)+\Lambda\gamma_{ab}\Lambda\left(R\omega^{ab}+{\frac{i}{2}}\theta\gamma^{ab}{\calD}\theta+tr(dvv^{-1}\gamma^{ab})\right)\right].
\delta_{\kappa}Z^{\underline{M}}E_{\underline{M}}^{~\underline{a}}=0,\quad\delta_{\kappa}Z^{\underline{M}}E_{\underline{M}}^{~\underline{\alpha}}=(1+\bar{\Gamma})_{~\underline{\beta}}^{\underline{\alpha}}\kappa^{\underline{\beta}}(\xi),
\delta\dot{\Lambda}\,\left[1+\frac{\alpha^{\prime}e^{\phi}\phi^{\prime}}{2g^{2}r}\,(1-3e^{-\Lambda})\right]=\frac{r\phi^{\prime}}{2}\,\delta\dot{\phi}+\frac{\alpha^{\prime}e^{\phi}}{g^{2}r}\,(1-e^{-\Lambda})\,\left(\delta\dot{\phi}\,\phi^{\prime}+\delta\dot{\phi^{\prime}}-\delta\dot{\phi}\,\frac{\Gamma^{\prime}}{2}\right)
\int\!\delta\xi\,dsTr\left[{\bar{\eta}}_{\mu}(g^{\mu\nu}\delta_{\rho}\delta^{\rho}-\delta^{\mu}\delta^{\nu})\eta_{\nu}\right]=-\int\!\delta\xi\,dsTr\left[(\delta_{\nu}{\bar{\eta}}_{\mu}-\delta_{\mu}{\bar{\eta}}_{\nu})(\delta^{\nu}\eta^{\mu}-\delta^{\mu}\eta^{\nu})\right],
S_{E}[X,g]=\int_{0}^{\pi}d^{2}\sigma{\frac{1}{2\pi}}\left\{\sqrtgg^{\alpha\beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{I}-iB^{IJ}\varepsilon^{\alpha\beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{J}\right\}\,
\beta_{00}=\sum_{\rho=0}^{N}\frac{\left(\eta_{0\rho}\right)^{2}}{4}\left[\left(\frac{\tilde{\omega}}{\Omega_{\rho}}-\frac{\Omega_{\rho}}{\omega}\right)\mathrm{e}^{i\Omega_{\rho}t}+\left(\frac{\Omega_{\rho}}{\tilde{\omega}}-\frac{\tilde{\omega}}{\Omega_{\rho}}\right)\mathrm{e}^{-i\Omega_{\rho}t}\right]\;.
\frac{\pi_{~~zz}^{TT}}{\sqrth}=-\frac{1}{4\pi}\sum_{k=2}^{\calN}t_{k}Q_{kzz},~~~~\frac{\pi_{~~\bar{z}\bar{z}}^{TT}}{\sqrth}=-\frac{1}{4\pi}\sum_{k=2}^{\calN}\bar{t}_{k}\bar{Q}_{k\bar{z}\bar{z}}.
ds^{2}=\frac{1-\left(\frac{t_{-}}{t}\right)^{2}}{-k+\left(\frac{t_{+}}{t}\right)^{2}}dy^{2}-\frac{dt^{2}}{\left(-k+\left(\frac{t_{+}}{t}\right)^{2}\right)\left(1-\left(\frac{t_{-}}{t}\right)^{2}\right)}+t^{2}d\Omega_{k}^{2}
\begin{array}{c}{[N_{i},b_{j}]=-b_{i}\delta_{ij},\quad\foralli,j\inS}\\{[N_{i},b_{j}^{+}]=b_{i}^{+}\delta_{ij}}\\{N_{i}=b_{i}^{+}b_{i},\quad\foralli,j\inS.}\\\end{array}
G_{\tilde{\lambda}}(l_{1},l_{2};t=1)=\langlel_{2}|\hat{T}|l_{1}\rangle=\mathrm{e}^{-\tilde{\lambda}(l_{1}+l_{2})}\sum_{T:l_{1}\rightarrowl_{2}}1\equiv\mathrm{e}^{-\tilde{\lambda}(l_{1}+l_{2})}\frac{1}{l_{1}+l_{2}}\Biggl({l_{1}+l_{2}\atopl_{1}}\Biggr).
i\gamma^{\mu}\partial_{\mu}\left(\begin{array}{c}{\psi_{L}}\\{\psi_{R}}\\{\psi_{\nu}}\\\end{array}\right)=\left(\begin{array}{ccc}{0}&{\phi^{2}}&{0}\\{\overline{{\phi^{2}}}}&{0}&{\overline{{\phi^{1}}}}\\{0}&{\phi^{1}}&{0}\\\end{array}\right)\left(\begin{array}{c}{\psi_{L}}\\{\psi_{R}}\\{\psi_{\nu}}\\\end{array}\right).
\alpha_{i}^{\pm}=\frac{1}{\sqrt{2}}\left[\alpha_{i}^{1}\mpi\alpha_{i}^{2}\right]\\,\\{\alpha_{i}^{\pm}}^{\dagger}=\frac{1}{\sqrt{2}}\left[{\alpha_{i}^{1}}^{\dagger}\pmi{\alpha_{i}^{2}}^{\dagger}\right]\\\,
\{Q_{\beta},Q_{\beta^{\prime}}\}=2i\delta_{\beta\beta^{\prime}}\overline{{z}}_{E}J_{E}\quad\quad\{Q_{\beta},Q_{\beta^{\prime}}^{\dagger}\}=\delta_{\beta\beta^{\prime}}H\+\2\Gamma_{\beta\beta^{\prime}}^{j}x_{jE}J_{E}
\left\langle\sigma(x)\right\rangle_{\mathrm{inst}}=\frac{1}{\sqrt{2}\,|v|}\left[\bar{\phi}_{\mathrm{inst}}(x)\phi_{\mathrm{inst}}(x)-|v|^{2}\right]=-\frac{|v|}{\sqrt{2}}\,\frac{\rho^{2}}{x^{2}+\rho^{2}}\,.
\Upsilon_{\sigma^{(l)}}(U)\Upsilon_{\tau^{(l)}}(U^{\dagger})=\sum_{k=0}^{\mathrm{min\}(\sigma_{l},\tau_{l})}{\binom{\sigma_{l}}{k}}{\binom{\tau_{l}}{k}}l^{k}k!\Upsilon_{\overline{{\tau^{(l)}\setminus[l^{k}]}}\,sigma^{(l)}\setminus[l^{k}]}(U,U^{\dagger})
D_{\!x}^{\sigma}\sum_{k=0}^{\infty}\frac{1}{\Gamma(1\!+\!k)}\,x^{k}=\!\!\sum_{k\,=\left\lceil\sigma\right\rceil}^{\infty}\!\frac{1}{\Gamma(1\!+\!k\!-\!\sigma)}\,x^{k-\sigma}\quad(\sigma>0)
f_{\;bc}^{a}\left(\overline{{\lambda^{a}}}\gamma^{\mu}\lambda^{c}\right)\left(\overline{{\xi}}\gamma_{\mu}\gamma^{5}\lambda^{b}\right)=+\frac{1}{2}f_{\;bc}^{a}\left(\overline{{\lambda^{a}}}\gamma^{\mu}\lambda^{c}\right)\left(\overline{{\xi}}\gamma_{\mu}\gamma^{5}\lambda^{b}\right)=0,
\deltaE\!\simeq\!{\frac{M|{\frac{|\delta\omega|}{2\pi}}}{r\!+\!{\frac{|\deltax|}{2}}}}\!-\!{\frac{M{\frac{|\delta\omega|}{2\pi}}}{r\!-\!{\frac{|\deltax|}{2}}}}\!\simeq\!-\!{\frac{M|\delta\omega|}{2\pir^{2}}}|\deltax|\!\simeq\!-\!{\frac{M}{2\pir^{2}}},
q^{2}\xi_{02}(p,q)-p.q\xi_{11}(p,q)={\frac{1}{2}}\left\{-\left[{\frac{1}{2}}+m^{2}\xi_{00}(p,q)\right]+{\frac{p^{2}}{2}}\xi_{10}(p,q)+{\frac{3q^{2}}{2}}\xi_{01}(p,q)\right\}
D_{pj}=\tilde{S}_{pj}+s_{pj}^{2}(\tilde{S}_{pj}^{2}-\tilde{T}_{pj})-\bar{S}_{pj}+s_{pj}^{2}(\bar{S}_{pj}^{2}-2\tilde{S}_{pj}\bar{S}_{pj}+\bar{T}_{pj})\.