harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_{hor}^{(b)}(r_{h})=\frac{r_{h}}{2}+\frac{b^{2}r_{h}}{3}(r_{h}^{2}-\sqrt{r_{h}^{4}+a^{4}})-\frac{2q^{2}}{3}\int_{0}^{r_{h}}{\frac{ds}{\sqrt{a^{4}+s^{4}}}},
ds^{2}=d\tilde{x}^{\tilde{m}}\tilde{g}_{\tilde{m}\tilde{n}}d\tilde{x}^{\tilde{n}}=e^{m}\eta_{mn}e^{n}+e^{\rho}e^{\rho}=\rho^{2}dx^{2}+R^{2}\,\left(\frac{d\rho}\rho\right)^{2}\,,
W^{\ast}=\frac{1}{12}(z+\frac{b}{z}+2)x_{0}^{6}+\frac{1}{6}x_{3}^{6}+\frac{1}{6}x_{4}^{6}+\frac{1}{2}x_{5}^{2}+c^{-1/6}x_{0}x_{3}x_{4}x_{5},
\left[{\mathcalJ}^{0}\left(t,\underline{{x}}\right)~,~~\widetilde{{\mathcalJ}}^{0}\left(t,\underline{{y}}\right)\right]\;=\;i~\frac{q^{2}}{2\pi}~\delta^{\prime}\left(\underline{{x}}-\underline{{y}}\right)~.
H=\left(p_{k}-\frac{e^{2}}{2\pi\kappa}\epsilon_{k\ell}\frac{x_{\ell}}{r^{2}}\right)\left(p_{k^{\prime}}-\frac{e^{2}}{2\pi\kappa}\epsilon_{k^{\prime}\ell^{\prime}}\frac{x_{\ell^{\prime}}}{r^{2}}\right)g^{kk^{\prime}}
\int_{-\infty}^{\infty}{\frac{dp}{2\pi}}\left({\frac{m^{2}-p(p+k)}{[m^{2}+p^{2}]\,[m^{2}+(p+k)^{2}]}}\right)={\frac{|m|}{k^{2}+4m^{2}}}-{\frac{|m|}{k^{2}+4m^{2}}}=0
\langlef|g\rangle_{{\calL}^{1|2}}=\langlef_{0}|g_{0}\rangle_{\calL}^{s}+\langlef_{1}|g_{1}\rangle_{\calL}^{s+1/2}+\langlef_{2}|g_{2}\rangle_{\calL}^{s+1/2}+\langlef_{3}|g_{3}\rangle_{\calL}^{s+1}\,,
\Gamma_{1}=\left(\begin{array}{cccc}{\cdot}&{-i}&{\cdot}&{\cdot}\\{i}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{i}\\{\cdot}&{\cdot}&{-i}&{\cdot}\\\end{array}\right)\;\;,\;\;\Gamma^{0}=-\Gamma_{0}\=\left(\begin{array}{cccc}{-i}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{i}&{\cdot}&{\cdot}\\{\cdot}&{\cdot}&{i}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{-i}\\\end{array}\right)
\Gamma_{f}[A_{+},A_{-}]={\frac{1}{2\pi}}\intd^{2}x\left(A_{+}A_{-}-{\frac{1}{2}}A_{+}{\frac{\partial_{-}}{\partial_{+}}}A_{+}-{\frac{1}{2}}A_{-}{\frac{\partial_{+}}{\partial_{-}}}A_{-}\right)\,,
\frac{\partial\mathcal{S}}{\partialT}=\frac{2l\pi^{2}\Xi^{2}r_{+}^{5}(\Xi^{2}+1)(r_{+}^{4}+q^{2}l^{4})}{3[4r_{+}^{8}(\Xi^{2}-1)+\Xi^{2}(q^{2}l^{4}-r_{+}^{4})^{2}+4q^{2}r_{+}^{4}l^{4}]}.
\frac{1}{24.8\pi^{2}}\int_{\partial{\calM}}\Big[\omega^{\sharp}\wedgeR^{\sharp}-\frac{1}{3}\omega^{\sharp}\wedge\omega^{\sharp}\wedge\omega^{\sharp}-\omega\wedgeR+\frac{1}{3}\omega\wedge\omega\wedge\omega\Big]=-\frac{2}{3}\frac{m^{4}r_{b}^{2}}{(r_{b}+2m)^{4}(r_{b}+m)^{2}}\,.
\psi=-i\int\left\{\sum_{a^{\prime}}\left[\left(\omega_{a^{\prime}}^{2}G_{a^{\prime}}+\sum_{b}\frac{\partial}{\partial\tau}\left(\dot{\alpha}_{b}\zeta_{ba^{\prime}}\right)\right)+\frac{i\omega_{a^{\prime}}^{2}}{2\Im_{a^{\prime}}}b_{a^{\prime}}\right]\bar{\eta}_{a^{\prime}}+\sum_{\bar{a},b}\dot{\alpha}_{b}\zeta_{b\bar{a}}\bar{\eta}_{\bar{a}}\right\},
\Gamma_{R(\tau)}^{(N)}(k_{i(\tau)},t,u_{\tau}^{*},\kappa_{\tau})=\kappa_{\tau}^{\frac{N\gamma_{\phi(\tau)}^{*}}{2}}F_{(\tau)}^{(N)}(k_{i(\tau)},\kappa_{\tau}t^{\frac{-1}{\gamma_{\phi^{2}(\tau)}^{*}}}).
\begin{array}{lllllll}{f_{0}}&{=}&{\tilde{f}_{0}}&{=}&{1}&{}&{\;,}\\{f_{i}}&{=}&{f_{k-i}}&{=}&{\tilde{f}_{i}}&{=}&{\tilde{f}_{k-i}\;\;\;;i=1,\ldots,(k-1)/2\;.}\\\end{array}
\delta_{\kappa}\pi^{ab}(x)={\frac{-{\frac12}\kappa^{2}}{1-\kappa^{2}}}\mathrm{Tr}~\Bigl(\Omega\chi[\infty,x]~\bigl([{\frac23}J_{0}(x)+{\frac1\kappa}C_{1}(x),M^{ab}]-{\frac13}
(T_{i\alpha\dot{\alpha}\ddot{\alpha}})^{*}=T^{i\alpha\dot{\alpha}\ddot{\alpha}}=\varepsilon^{ij}\varepsilon^{\alpha\beta}\varepsilon^{\dot{\alpha}\dot{\beta}}\varepsilon^{\ddot{\alpha}\ddot{\beta}}T_{j\beta\dot{\beta}\ddot{\beta}}.
\nabla_{\mu}\nabla_{\nu}K^{\mu}=R_{\nu\rho}K^{\rho}=(F_{\nu\rho}^{2}-\frac16g_{\nu\rho}F^{2})K^{\rho}+G_{IJ}\partial_{\nu}X^{I}\partial_{\rho}X^{J}K^{\rho}+\frac23g^{2}V(X)K_{\nu},
D_{\mu\nu}^{ab}(p,p_{3})=\frac{\delta^{ab}\delta^{a3}}{p^{2}-p_{3}^{2}+i\epsilon}\left[-g_{\mu\nu}+p_{\mu}p_{\nu}\left((1-\delta_{p_{3},0})\frac{1}{p_{3}^{2}}+\delta_{p_{3},0}(1-\xi)\frac{1}{p^{2}+i\epsilon}\right)\right]
UG^{-1}V,\,\,U=\left(\begin{array}{cc}{P_{L}}&{iP_{R}}\\{iP_{R}}&{P_{L}}\\\end{array}\right),\,\,V=\left(\begin{array}{cc}{P_{R}}&{iP_{L}}\\{iP_{L}}&{P_{R}}\\\end{array}\right).
\Lambda_{k}\;=\;\left(\frac{1}{4\alpha\beta}\,\mathrm{(\alpha-e^{\frac{i\pik}{L}})\,(\alpha-e^{-\frac{i\pik}{L}})\,(\beta-e^{\frac{i\pik}{L}})\,(\beta-e^{-\frac{i\pik}{L}})}\right)^{\frac12}
J_{r}^{(n)}(t;\{q\};L)=\tilde{J}_{r}^{(n)}(t;\{q\};L)-\frac{1}{2}\sum_{m,m^{\prime}}\Phi_{m}^{(r)}(L;\{q\})\hat{V}_{mm^{\prime}}^{-1}\Phi_{m^{\prime}}(t;L;\{q\})
f(x,p)=\frac{2^{2/3}}{2\pi}Ai(2^{2/3}(z-E))=\frac{2^{2/3}}{2\pi}Ai(2^{2/3}(x+p^{2}-E))={\frac{1}{(2\pi)^{2}}}\int\!dy~e^{iy(E-x-p^{2}-y^{2}/12)}\;.
Z^{int}(s)=\int{\calD}\psi{\calD}\bar{\psi}~e^{-\intd^{3}x~\bar{\psi}(\partial\!\!\!/+m+is\!\!\!/)\psi+\frac14(g^{2})^{1-2\alpha}J_{\mu}\Box^{-\alpha}J^{\mu}\;}\;,
S=\intd^{2}x\left[-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{1}{2}}\partial_{\mu}\phi\partial^{\mu}\phi+eA_{\nu}(\eta^{\mu\nu}-\epsilon^{\mu\nu})\partial_{\mu}\phi+{\frac{1}{2}}ae^{2}A_{\mu}A^{\mu}\right]
{}^{(D-1)}R_{\mu\nu}={}^{(D)}R_{\rho\sigma}q_{\mu}^{~\rho}q_{\nu}^{~\sigma}-{}^{(D)}R_{~\beta\gamma\delta}^{\alpha}n_{\alpha}q_{\mu}^{~\beta}n^{\gamma}q_{\nu}^{~\delta}+KK_{\mu\nu}-K_{\mu}^{~\alpha}K_{\nu\alpha}\,,
I_{n}^{\prime}(z)K_{n}^{\prime}(z)=-\frac{1}{2z}\left[1+\frac{4n^{2}-3}{2(2z)^{2}}+\frac{(4n^{2}-1)(4n^{2}-45)}{8(2z)^{4}}+O(z^{-6})\right]\,.
\hat{C}_{oa}^{b}={\frac{1}{2}}\left({\vartheta}^{-{\frac{1}{2}}}f^{\frac{1}{2}}C_{oa}^{bj}\hat{p}_{j}f^{-{\frac{1}{2}}}\vartheta^{\frac{1}{2}}+\vartheta^{\frac{1}{2}}f^{-{\frac{1}{2}}}\hat{p}_{j}C_{oa}^{bj}f^{\frac{1}{2}}{\vartheta}^{-{\frac{1}{2}}}\right),
{}^{L,R}\langlex|2k+1\rangle_{\Omega}^{L,R}\rightarrow-i\left({\frac{\Omega}{\pi}}\right)^{1/4}[2^{2k}(2k)!]^{-1/2}H_{2k}(0)\sqrt{\pi}\left({}^{L,R}\langlex|p\rangle_{-}^{L,R}\right).
0=\int\mathrm{d}\Phi\,{\frac{\partial_{r}}{\partial\Phi^{A}}}\left\{\mathrm{e}^{{\frac{i}{\hbar}}\Sigma+{\frac{i}{\hbar}}J_{A}\Phi^{A}}\,{\frac{\partial_{l}\Sigma}{\partialK_{A}}}\right\}.
\begin{array}{lclrrrr}{}&{}&{n_{1}}&{w^{1}}&{n_{2}}&{w^{2}}&{\vec{Q}}\\{{\bfx}_{1}}&{=}&{(0,}&{\!\!0,}&{\!\!1,}&{\!\!-1,}&{\!\!0),}\\{{\bfx}_{2}}&{=}&{(2,}&{\!\!-1,}&{\!\!1,}&{\!\!1,}&{\!\!0),}\\{{\bfx}_{2+j}}&{=}&{(0,}&{\!\!-1,}&{\!\!1,}&{\!\!1,}&{\!\!2e_{j}).}\\\end{array}
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})
\int{\calD\theta}\,e^{i\,\deltaS_{1}[\theta,\bar{A}_{\pm}]}=\int{\calD\theta}\,e^{\,{\mathrm{~\frac{ik}{8\pi}~}\int_{\partial{\calM}}\theta\epsilon^{ij}\partial_{i}\bar{A}_{j}}}\,\sim\delta\left(F_{+-}\right)\Big|_{\partialM}\,.
g_{i,j}(u^{1},\,u^{2})\,=\,x_{,i}^{\mu}x_{,j}^{\mu},\quadx_{,i}^{\mu}\,\equiv\,\frac{\partialx^{\mu}(u^{1},\,u^{2})}{\partialu^{i}},\quadi,\,j\,=\,1,\,2.
\int_{0}^{2\pi}d\tau^{\prime\prime}\triangle^{-1}(\tau-\tau^{\prime\prime})\triangle(\tau^{\prime\prime}-\tau^{\prime})=\delta(\tau-\tau^{\prime})-\frac{1}{2\pi},\\delta(\tau-\tau^{\prime})=\frac{1}{2\pi}\sum_{n}e^{in(\tau-\tau^{\prime})}\.
F_{01}=\frac{F_{01}^{c}}{\sqrt{1+\left(\frac{\alpha^{\prime}s}{R^{2}G_{o}^{2}}\right)^{2}}},\quadF_{01}^{c}=\frac{2\piR^{2}}{s}\frac{1}{G_{o}^{2}(2\pi\alpha^{\prime})^{2}}
{\calE}=\frac{1}{2}g_{mn}(\dot{z}^{m}-c_{I}a_{I}^{A}K_{A}^{m})(\dot{z}^{n}-c_{J}a_{J}^{B}K_{B}^{n})+\frac{1}{2}g_{mn}{a}_{\botI}^{A}K_{A}^{m}{a}_{\botJ}^{B}K_{B}^{n}+c_{I}g_{mn}\dot{z}^{m}a_{I}^{A}K_{A}^{n}
M=3D\intd^{2}k(a^{+}(k){\frac{1}{i}}{\frac{\partial}{\partial\theta}}a(k)+b^{+}(k){\frac{1}{i}}{\frac{\partial}{\partial\theta}}b(k))+{\frac{m}{\mid=m\mid}}\intd^{2}k(a^{+}(k)a(k)-b^{+}(k)b(k)),
\xi_{1}=\frac{-2\beta_{1}\beta_{2}}{\beta_{1}^{2}+\beta_{2}^{2}},\qquad\xi_{2}=0\quad\mathrm{and\quad}\xi_{3}=\frac{\beta_{2}^{2}-\beta_{1}^{2}}{\beta_{1}^{2}+\beta_{2}^{2}},
\frac{1}{4\pi\ell_{s}^{2}}~\int_{\Sigma}\sum_{I=0}^{3}\left(\epsilon^{2}{\overline{g}}_{Ii}^{C}+\epsilon{\overline{g}}_{Ii}X^{I}\right)e^{\epsilon(\phi_{(0)}-X_{(0)}^{I})}\Theta(X_{(0)}^{I})\partial_{\alpha}\phi\partial^{\alpha}y_{i}~
D_{\epsilon}=X^{0}\Theta_{\epsilon}(X^{0})=-i\int_{-\infty}^{\infty}\frac{dq}{q-i\epsilon}X^{0}e^{iqX^{0}}=-\int_{-\infty}^{\infty}\frac{dq}{(q-i\epsilon)^{2}}e^{iqX^{0}}
xy^{2}=\left[x\prod_{a=1}^{n_{c}}(x-\phi_{a}^{2})+\Lambda^{2n_{c}+2-n_{f}}\prod_{i=1}^{n_{f}}m_{i}\right]^{2}-\Lambda^{4n_{c}+4-2n_{f}}\prod_{i=1}^{n_{f}}(x+m_{i}^{2}),
\intd^{2}u\,d^{2}v\,e^{-F({\bfu},{\bfv})}\,P(u^{i},v^{j})\,L_{\ell}^{\alpha}\big(\frac{eB}{2}\,{\bfu}^{2}\big)\,L_{m}^{\beta}\big(\frac{eB}{2}\,({\bfu}+{\bfv})^{2}\big)\,L_{n}^{\gamma}\big(\frac{eB}{2}\,{\bfv}^{2}\big)\;,
{\bfB}^{\mathrm{em}}={\frac{M\omega^{2}}{g}}\mathrm{Re}\left[i\left({\bfa}-({\bf\hat{k}}\cdot{\bfa}){\bf\hat{k}}\right)\,e^{i{\bfk}\cdot{\bfx}-i\omegat}\right]\\\\(\omega=|{\bfk}|).
T(z)=-\frac{1}{2}:\!\partialX\partialX\!:+\,i\frac{\kappa}{2}\partial^{2}X,\qquad\tilde{T}(\bar{z})=-\frac{1}{2}:\!\bar{\partial}X\bar{\partial}X\!:-\,i\frac{\kappa}{2}\bar{\partial}^{2}X,
\left(R^{{\mathbbH}P^{n}}\right)_{XYWZ}=\frac12g_{Z[X}g_{Y]W}+\frac12J_{XY}^{\alpha}J_{ZW}^{\alpha}-\frac12J_{Z[X}^{\alpha}J_{Y]W}^{\alpha}=\frac12J_{XY}^{\alpha}J_{ZW}^{\alpha}+L_{[ZW]}{}^{AB}L_{[XY]AB}\,.
\begin{array}{l}{L_{1}=-\frac{1}{4}h_{\mu\nu}(\partial_{\lambda\rho}\partial^{\rho\lambda})h^{\mu\nu}-\frac{1}{2}h_{\mu\nu}(\partial^{\mu\lambda}\partial^{\nu\rho})h_{\lambda\rho}}\\{+h_{\mu\nu}(\partial^{\mu\lambda}\partial_{\lambda\rho})h^{\rho\nu}-h_{\mu}^{\nu}(\partial^{\mu\lambda}\partial_{\lambda\nu})h+\frac{1}{4}h(\partial^{\mu\nu}\partial_{\nu\mu})h}\\\end{array}
\frac{\left(\sum_{i_{\alpha_{1}}^{\prime}\inJ_{X_{i^{\prime}}^{\prime}}}L^{\beta}t_{i_{\alpha_{1}}^{\prime}}^{\prime}\right)^{2}}{2q_{1}(L^{d}-1)}\prod_{i_{\alpha_{1}}^{\prime}\inJ_{X_{i^{\prime}}^{\prime}}}\int_{0}^{1}\sum_{n_{i^{\prime}}}\left(n_{i^{\prime}}+1\right)\left(n_{i^{\prime}}\right)^{2}\times
Tr~(T_{x})=\frac{1}{4}\int_{\Theta}d\theta\int_{K_{1}}d\mu_{l}(k)~{\midk_{22}\mid}^{-2k}~x(\underline{{\theta}}^{-1}~\underline{{k}}~\underline{{\theta}})~+~\frac{1}{4}\int_{\Theta}d\theta\int_{K_{2}}d\mu_{l}(k)~{\midk_{22}\mid}^{-2k}~x(\underline{{\theta}}^{-1}~\underline{{k}}~\underline{{\theta}})
{\mathbbP}H^{0}(dP_{9},{\calO}_{dP_{9}}(3\sigma|_{dP_{9}}+4F))\simeq{\mathbbP}H^{0}({\calS},{\calO}_{{\calS}}(4)\oplus{\calO}_{{\calS}}(2)\oplus{\calO}_{{\calS}}(1))\simeq{\mathbbP}^{9}.
d\Theta^{1\underline{\mu}}=v_{q}^{-\underline{\mu}}\pi_{q}^{1+}+v_{\dot{q}}^{+\underline{\mu}}\pi_{\dot{q}}^{1-}=e^{+2}\psi_{+2\dot{q}}^{1-}v_{\dot{q}}^{\underline{\mu}+}+e_{q}^{+}v_{q}^{\underline{\mu}-}-\chie_{\dot{q}}^{-}v_{\dot{q}}^{\underline{\mu}+},
V(\rho)=\frac{1}{G}\rho^{2}-\frac{R}{(4\pi)^{2}}f(\xi_{1},\xi_{2})\left[\rho^{2}\mathrm{Ei}\left(-\frac{\rho^{2}}{\Lambda^{2}}\right)+\Lambda^{2}(e^{-\rho^{2}/\Lambda^{2}}-1)\right],
\langleM_{ij}M_{kl}\rangle=\frac{1}{N}[\delta_{il}\delta_{kj}+\mathrm{e}^{-\frac{3}{2}\beta}L_{il}(L^{2})_{kj}+\mathrm{e}^{-\frac{3}{2}\beta}(L^{2})_{il}L_{kj}],
\begin{array}{ccl}{\xi_{k}^{\alpha}(x)}&{=}&{\left(\left(\frac1{R\left(A(x)\right)R\left(A(x)\right)}\right)^{1/4}\right)_{kk^{^{\prime}}}^{\alpha\alpha^{^{\prime}}}B_{k^{^{\prime}}}^{\alpha^{^{\prime}}}(x)}\\{}&{=}&{\frac1{\pi\sqrt{2}}\int_{0}^{\infty}d\lambda\lambda^{-1/4}\frac1{\lambda+R\bulletR}\bulletB(x)}\\\end{array}
\left<\Omega\right|\phi_{H}(x)\phi_{H}(y)\left|\Omega\right>=\underset{\lambda}{\sum}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{i\;e^{-ip\cdot(x-y)}}{p^{2}-m_{\lambda}^{2}+i\epsilon}|\left<\Omega\right|\phi_{H}(0)\left|\lambda_{0}\right>|^{2}.
\Omega=i(\mathrm{d}z\frac{\partial}{\partialz}+\mathrm{d}\theta\frac{{\stackrel{\rightarrow}{\partial}}}{\partial\theta})\wedge(\mathrm{d}\bar{z}\frac{\partial}{\partial\bar{z}}+\mathrm{d}\bar{\theta}\frac{{\stackrel{\rightarrow}{\partial}}}{\partial\bar{\theta}})\Phi\;,
I_{rep}\equiv\mu_{0}\,\int_{\Sigma}d^{2}\sigma\,\Phi^{-1}\mathrm{det}\left[\,\gamma_{ab}(\sigma)\,\right]\,\qquad\Phi^{-1}\equiv\epsilon_{ij}\,\epsilon^{mn}\,\partial_{m}\,\phi^{i}\,\partial_{n}\,\phi^{j}
H(\{N_{0}\})=(z_{1}^{(0)}-z_{3}^{(0)})(z_{2}^{(0)}-z_{3}^{(0)})\left[1-\frac{\vartheta_{1}^{(0)}\vartheta_{3}}{2(z_{1}^{(0)}-z_{3}^{(0)})}-\frac{\vartheta_{2}^{(0)}\vartheta_{3}}{2(z_{2}^{(0)}-z_{3}^{(0)})}\right]\,.
{\mit\Sigma}(k^{2})={\mit\Sigma}(0)={\frac{-i\lambda}{2}}\int\!\!{\frac{d^{4}l}{(2\pi)^{4}}}{\frac{i}{l^{2}-\mu^{2}}}\left(\frac{-\Lambda^{2}}{l^{2}-\Lambda^{2}}\right)^{2}={\frac{-i\lambda}{2(4\pi)^{2}}}\mu^{2}\left[{\frac{\Lambda^{2}}{\mu^{2}}}-\mathrm{ln}\left({\frac{\Lambda^{2}}{\mu^{2}}}\right)+1\right]\,.
G_{Y}(y;y^{\prime})=\frac{3Na^{2}(y)a^{2}(y^{\prime})}{\kappa}\Bigl[u_{-}(y)u_{+}(y^{\prime})\theta(y^{\prime}-y)+u_{-}(y^{\prime})u_{+}(y)\theta(y-y^{\prime})\Bigr]\,.
\left({\frac{b^{\prime}}{b}}\right)^{2}+(kb^{2}-M){\frac{b^{2}}{\phi^{2}}}={\frac{1}{6}}\left(A^{2}+{\frac{B^{2}}{\phi}}+{\frac{3+2\omega}{2}}{\frac{\phi^{2}}{\phi^{2}}}\right),
\Psi_{q,j,m_{j}}^{-}=\frac{1}{\sqrt{2j+2}}\left(\begin{array}{l}{\sqrt{j-m_{j}+1}\,Y_{j+\frac{1}{2},m_{j}-\frac{1}{2}}^{q}}\\{-\sqrt{j+m_{j}+1}\,Y_{j+\frac{1}{2},m_{j}+\frac{1}{2}}^{q}}\\\end{array}\right)\,.
ds^{2}\,=\,\Big(1+{\frac{\Lambda^{4}}{r^{4}}}\Big)^{-1/2}(-dt^{2}+d{\vec{x}}^{2})+\Big(1+{\frac{\Lambda^{4}}{r^{4}}}\Big)^{1/2}(dr^{2}+r^{2}d\Omega_{5}^{2})
A_{Nop}^{(0)}=g^{N-2}\int_{0}^{1}\prod_{i=1}^{N-3}dx_{i}\,\prod_{i=1}^{N-3}x_{i}^{-\alpha({s_{i}})-1}\prod_{2\leqi<j\leqN-1}(1-x_{ij})^{2\alpha^{\prime}(P_{i}\cdotP_{j})},
\begin{array}{rcl}{\mathrm{tr}\left[w(1,e_{1},\ldots,e_{i-1})e_{i}\right]}&{=}&{Q^{-1/2}\mathrm{tr}\,w(1,e_{1},\ldots,e_{i-1})\;,}\\{\mathrm{tr}\left(e_{i}\right)}&{=}&{Q^{-1/2}\mathrm{tr}{\bf1}\;,\;\;\;\mathrm{and}}\\{\mathrm{tr}{\bf1}}&{=}&{Q^{l}\;,}\\\end{array}
S_{WZ}^{NM0}=\mu_{0}\int_{R}\mathrm{Tr}\left({\hat{k}}^{-2}{\hat{k}}^{(1)}+il_{p}^{2}i_{\hat{\Phi}}i_{\hat{\Phi}}({\hat{C}}-{\hat{k}}^{-2}{\hat{k}}^{(1)}\wedgei_{\hat{k}}{\hat{C}})-\frac12l_{p}^{4}(i_{\hat{\Phi}}i_{\hat{\Phi}})^{2}i_{\hat{k}}{\hat{\tilde{C}}}+\dots\right)\,.
R_{\alpha\,b}^{a}=-\fraci2B_{\alpha}^{mn}\,(L_{mn})_{b}^{a},\qquad\mathrm{where}\qquad(L_{mn})_{b}^{a}=-i\left(\delta_{m}^{a}\,\pi_{bn}-\delta_{n}^{a}\,\pi_{bm}\right).
\Gamma\left(p,P_{os}\right)=-\int\frac{d^{4}p^{\prime}}{\left(2\pi\right)^{4}}K\left(p,p^{\prime},P_{os}\right)G_{1}\left(p_{1}^{\prime}\right)G_{2}\left(p_{2}^{\prime}\right)\Gamma\left(p^{\prime},P_{os}\right).
\hat{V}_{\mu\nu\rho}^{(3\,z)\,ab}=\hat{V}_{\mu\nu\rho}^{(3\,x)}=\hat{V}_{\mu\nu\rho}^{(3\,y)\,ab}=-Y_{L}^{(e)\,3}\,\mathrm{Tr}\,\tau^{a}\tau^{b}\,V_{\mu\nu\rho}^{(3)}\,.
ds^{2}=(-H(\hat{r},z))^{-\frac{2}{3}}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+(-H(\hat{r},z))^{\frac{1}{3}}(d\hat{r}^{2}+\hat{r}^{2}d\Omega_{6}^{2}+dz^{2}).
\mathrm{J}_{j}(\lambda_{jk}r)=0~~\mathrm{or}~~{\frac{d\mathrm{J}_{j}(\lambda_{jk}r)}{\displaystyle{d\lambda_{jk}r}}}=0,~~~~~\mathrm{for}~~r=R_{1}~~\mathrm{and}~~R_{2}~.~\,
\varphi_{1}^{\prime}(N)=\left(\frac{(j+1)(N+j+2)}{(2j+1)(N+2)}\right)^{1/2}\varphi_{+}^{\prime}(N)+\left(\frac{j(N-j+1)}{(2j+1)(N+2)}\right)^{1/2}\varphi_{-}^{\prime}(N)
L_{D}=\int_{\calM}{\frac{1}{4\pi}}\mathrm{Tr}\bigg[\widetilde{d}\bigg({\frac{k}{2}}\widetilde{d}g^{-1}d_{0}g-{\frac{k}{2}}\widetilde{d}h^{-1}d_{0}h+\widetilde{d}\alphad_{0}hh^{-1}\bigg)-{\frac{k}{6}}(h^{-1}dh)^{3}+{\frac{k}{6}}(g^{-1}dg)^{3}\bigg].
{\calA}_{fi}^{(1)}=\intd{\bfx\,}\mathrm{e}^{\,-i\,{\bfp}^{^{\prime}}{\bf\cdot\,x}}\,\left[\frac{m}{w_{{\bfp}}}\,H_{1}({\bfx},{\bf\nabla})+V({\bfx)}\right]\,\mathrm{e}^{\,i\,{\bfp\,\cdot\,x}}\;.
\mathcal{W}_{ABC}{}^{D}=W_{ABC}{}^{D}-\frac{3}{2(r+1)}\delta_{(A}^{D}W_{BC)E}{}^{E}\,,\qquadR_{XY}=-{\calR}_{XYA}{}^{A}=\frac12\varepsilon_{ij}f_{X}^{iA}f_{Y}^{jB}W_{ABC}{}^{C}\,.
V_{\mu\nu\lambda}^{cde}(p,q,k)=\mathrm{i}\,\varepsilon^{cde}\,\Big[(q-k)_{\mu}\,\delta_{\nu\lambda}\,+\,(p-q)_{\lambda}\,\delta_{\mu\nu}\,+\,(k-p)_{\nu}\,\delta_{\lambda\mu}\Big]
C_{h}^{ag}C_{g}^{br}C_{da}^{e}C_{eb}^{f}(C_{f}^{ch}C_{rc}^{d})=C_{h}^{ag}C_{g}^{br}(C_{da}^{e}C_{eb}^{f}C_{f}^{ch}C_{rc}^{d})\equiv0.
\left(\begin{array}{cc}{k}&{j}\\{0}&{p}\\\end{array}\right)\,\\left(\begin{array}{cc}{0}&{-p}\\{k}&{j}\\\end{array}\right)\,\\left(\begin{array}{cc}{0}&{-p}\\{k}&{j+p}\\\end{array}\right)\,\\0\leqj<k,\\p\neq0\.
g_{\mu\nu}^{(1)}=-{\frac{l^{2}}{d-2}}\left({\frac{1}{a^{2}}}-1\right)\left(R_{\mu\nu}-{\frac{1}{2(d-1)}}h_{\mu\nu}R\right)-{\frac{2l}{d}}\left({\frac{1}{a^{d}}}-1\right)\chi_{\mu\nu}\,
\kappa^{2}=8\pi/M_{\mathrm{p}}^{2}\,,\lambda=6{\frac{\kappa^{2}}{\widetilde\kappa^{4}}}\,,~~\Lambda={\frac{4\pi}{\widetilde{M}_{\mathrm{p}}^{3}}}\left[\widetilde{\Lambda}+\left({\frac{4\pi}{3\widetilde{M}_{\mathrm{p}}^{\,3}}}\right)\lambda^{2}\right]\,.
\nu_{k}=\left[\frac{4i}{\mu_{k}-\bar{\mu}_{k}-\mu_{k}^{-1}+\bar{\mu}_{k}^{-1}}\,\frac{\mu_{k}-\mu_{k}^{-1}-2i}{\bar{\mu}_{k}-\bar{\mu}_{k}^{-1}+2i}\,\right]^{1/2}.
\Psi=\left[\begin{array}{c}{\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{0}\\{c_{1}\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{d_{1}\mathrm{L}_{N-1}^{\left|m\right|+1}F_{m+1}}\\\end{array}\right]\mathrm{~}
\frac{\partialU^{-1}}{\partialz_{\alpha}}+{\frac{1}{2\pi\kappa}}\sum_{\beta\neq\alpha}\frac{\frac{1}{2}(\hat{Q}^{+}(\xi_{\alpha})\hat{Q}^{-}(\xi_{\beta})+\hat{Q}^{-}(\xi_{\alpha})\hat{Q}^{+}(\xi_{\beta}))+\hat{Q}^{3}(\xi_{\alpha})\hat{Q}^{3}(\xi_{\beta})}{z_{\alpha}-z_{\beta}}U^{-1}=0
\int\frac{d^{n}k}{(2\pi)^{n}}\frac{(k^{2})^{a}}{(k^{2}+m^{2})^{b}}=\frac{1}{(4\pi)^{n/2}}\left(m^{2}\right)^{n/2+a-b}\frac{\Gamma(n/2+a)\Gamma(b-a-n/2)}{\Gamma(n/2)\Gamma(b)}
{\cal{G}}_{ij,kl}^{1,(\tilde{\eta}_{o})}(x_{1},x_{2};x_{3},x_{4})=G_{*}^{2}\frac{x_{24}^{2}}{(x_{12}^{2}x_{34}^{2})^{a}}\left[{\cal{A}}_{ij,kl}^{(\tilde{\eta}_{o})}(x_{1},..,x_{4})+C_{F}(d-\tilde{\eta}_{o})\,{\cal{A}}_{ij,kl}^{(d-\tilde{\eta}_{o})}(x_{1},..,x_{4})\right]\,,
\frac{2}{\pi}\int_{0}^{\infty}dr\left(\psi_{\ell}^{*}(k,r)\psi_{\ell}(k,r)-\psi_{\ell}^{(0)}{}^{*}(k,r)\psi_{\ell}^{(0)}(k,r)\right)=\frac{1}{\pi}\frac{d\delta_{\ell}}{dk}
{\calF^{\prime}}_{k_{1}k_{2}k_{3}}=\prod_{i,j=1}^{5}\left(\left[\left(\Phi_{i}\right)^{\left(l_{ij}^{-1}-l_{ji}^{-1}\right)q_{j}^{1}}\right]^{\k_{1}}\left[\left(\Phi_{i}\right)^{\left(l_{ij}^{-1}-l_{ji}^{-1}\right)q_{j}^{2}}\right]^{\k_{2}}\left[\left(\Phi_{i}\right)^{\left(l_{ij}^{-1}-l_{ji}^{-1}\right)q_{j}^{3}}\right]^{\k_{3}}\right)
\partial_{\sigma}\left(\frac{\chi^{2}e^{2\gamma}}{\sqrt{-G}V}\rho^{\prime}\right)={\frac{1}{\sqrt{-G}}}\left[{\frac{1}{2}}\frac{\partial}{\partial\rho}\left(\frac{\chi^{2}e^{2\gamma}}{V}\right)\left({\rho^{\prime}}^{2}+z^{\prime\;2}\right)-\rho\varphi^{\prime\;2}\right],
\begin{array}{lcr}{|X_{1}|^{2}+|X_{2}|^{2}-(|Y_{1}|^{2}+|Y_{2}|^{2})=R_{2}}&{(a)}\\{X_{1}\overline{{Y}}_{1}+X_{2}\overline{{Y}}_{2}=0}&{(b)}\\{\overline{{X}}_{1}{Y}_{1}+\overline{{X}}_{2}{Y}_{2}=0}&{(c)}\\\end{array}
T_{\mu\nu}=Tr({\frac{2}{3}}{\partial_{\mu}{\phi}}{\partial_{\nu}{\phi}}-{\frac{1}{6}}\eta_{\mu\nu}{(\partial_{\sigma}{\phi})}^{2}-{\frac{1}{3}}\phi{\partial_{\mu}{\partial_{\nu}{\phi}}}+{\frac{1}{12}}\eta_{\mu\nu}\phi\partial^{2}{\phi}).
\begin{array}{l}{\delta_{b}x^{\mu}={b^{\mu}}_{\nu}n^{\nu},}\\{\delta_{b}A_{a}^{\mu}=-{b^{\mu}}_{\nu}\displaystyle\left(\varepsilon_{ab}\frac{g^{bc}}{\sqrt{-g}}{\Pi_{c}}^{\nu}-\xi_{a}n^{\mu}+i(\bar{\theta}\Gamma^{\nu}\partial_{a}\theta)\right).}\\\end{array}
H^{(2)}=\intd^{3}{x}\,\biggl[\,\,\frac{1}{2}\mathrm{tr}\,\left(P-\frac{\theta}{8\pi^{2}}\,B^{(+)}\right)^{2}+\left(a_{s}^{(0)}(S,P)-\frac{\theta}{8\,\pi^{2}}\,B_{s}^{(-)}\right)^{2}+\,\frac{1}{2}V(S)\\biggr]\,,
H=v\frac{1}{t^{2}+\vec{x}^{2}+\rho^{2}}\left(\begin{array}{cc}{(t+iz)^{2}-(x-iy)^{2}}&{2i(tx-yz)}\\{2i(tx-yz)}&{(t-iz)^{2}-(x+iy)^{2}}\\\end{array}\right)
{\left(G_{ab}^{(1)}\right)}_{ij}=\left(\begin{array}{ccc}{\epsilon_{ab}\epsilon_{ijk}\partial^{k}}&{-\delta_{ab}\delta_{ij}}&{0}\\{\delta_{ab}\delta_{ij}}&{0}&{-\delta_{ab}\partial_{i}^{x}}\\{0}&{\delta_{ab}\partial_{j}^{x}}&{0}\\\end{array}\right)\delta(\vec{x}-\vec{y}).
{\calB}(\epsilon)=\langle\prod_{r=1}^{N}{\calO}_{r}(z_{r})\sum_{\alpha}O_{\alpha}^{\prime}(z_{0})\epsilon^{k_{\alpha}}\prod_{n=1}^{L}\oint\frac{dw_{n}}{2\pii}\frac{w_{n}}{d\rho}(w_{n}-z_{0})^{j_{\alpha,n}}\rangle.
\begin{array}{cc}{\alpha^{+}\alpha\mp\beta^{+}\beta=1,}&{\beta^{+}\alpha^{*}\mp\alpha^{+}\beta^{*}=0,}\\{\alpha\alpha^{+}\mp\beta^{*}\tilde{\beta}=1,}&{\alpha\beta^{+}\mp\beta^{*}\tilde{\alpha}=0.}\\\end{array}
{\frac{{\calW}_{na}\pi^{2\omega}e^{2i\pi\omega}}{C_{F}C_{A}(2T)^{4-4\omega}(LT)^{2}}}={\frac{1}{2(\omega-1)^{2}}}+{\frac{1-\gamma}{(\omega-1)}}-1-2\gamma+\gamma^{2}+{\frac{\pi^{2}}{12}}+{\calO}(\omega-1)\,
\begin{array}{l}{\triangle_{2i+1}^{\left(k\right)}=\left(1-\varepsilon\right)}\\{\triangle_{2i}^{\left(k\right)}=\varepsilon}\\{\alpha^{\left(k\right)}=\frac1{2\varepsilon^{\frac{k-1}2}\left(1-\varepsilon\right)^{\frac{k+1}2}}}\\\end{array}
\left(\begin{array}{c}{t^{\prime}}\\{z^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{-\frac{2\lambda}{\sqrt{\mu}}}\\{\frac{\sqrt{\mu}}{2\lambda}}&{\frac{2\lambda}{\sqrt{\mu}}}\\\end{array}\right)\left(\begin{array}{c}{t}\\{z}\\\end{array}\right).
S=\intd\taup_{\underline{a}}(\dot{x}^{\underline{a}}-i\dot{\bar{\theta}}\Gamma^{\underline{a}}\theta-\bar{\lambda}\Gamma^{\underline{a}}\lambda)+i\intd\tau\rho_{\underline{a}}(\chi^{\underline{a}}-\bar{\lambda}\Gamma^{\underline{a}}\theta).
A_{nm}^{(M)}=\gamma_{13}^{\Psi}\gamma_{24}^{\Psi}\;\pi^{\frac{d}{2}}2^{-l}\frac{\Gamma(d-\lambda-1)\Gamma(\frac{d}{2}-\lambda-l)}{\Gamma(d-\lambda-l-1)\Gamma(\frac{d}{2}-\frac{\lambda}{2})^{2}}\;\tilde{A}_{nm}^{(M)}.
T_{ab}{}^{r}=2\big(\left({\calC}\Gamma^{r_{1}}\right)_{ab}X^{r}{}_{r_{1}}+\frac{1}{2!}\left({\calC}\Gamma^{r_{1}r_{2}}\right)_{ab}\,X^{r}{}_{r_{1}r_{2}}+\frac{1}{5!}\left({\calC}\Gamma^{r_{1}\cdotsr_{5}}\right)_{ab}\,X^{r}{}_{r_{1}\cdotsr_{5}}\big)\,,
\tilde{\phi}^{(+)}(x)=\frac{i}{\sqrt{4{\pi}}}\int_{0}^{\infty}\frac{dp_{-}}{p_{-}}c(p_{-})(\mathrm{e}^{-ip{\cdot}x}-{\theta}({\tilde{\kappa}}-p_{-})),\quad\tilde{\phi}^{(-)}(x)=(\tilde{\phi}^{(+)}(x))^{\ast}
C=\frac{1}{n^{2}p^{4}}\sum_{k=0}^{n-1}\left(\sum_{i=k+1}^{n}\Gamma_{0}^{i}\Gamma^{n-k}\Gamma_{0}^{n+k-i}+\sum_{i=1}^{k}\Gamma_{0}^{i}\Gamma^{2n-k}\Gamma_{0}^{k-i}\right)
V^{ij}=\left(\begin{array}{cc|ccc}{0}&{v}&{0}&{\cdots}&{0}\\{-v}&{0}&{0}&{\cdots}&{0}\\\hline{0}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{}&{\vdots}\\{0}&{0}&{0}&{\cdots}&{0}\\\end{array}\right),
E_{\alpha}=(-1)^{\alpha_{1}}\sigma_{\alpha}(0)\sum_{j}\prod_{k\neqj}\frac{\sigma_{\alpha}(z_{jk})}{\sigma_{0}(z_{jk})}\left[\frac{l}{n}\sum_{i\neqj}\frac{\sigma_{\alpha}^{\prime}(z_{ji})}{\sigma_{\alpha}(z_{ji})}-\partial_{j}\right],\alpha\equiv(\alpha_{1},\alpha_{2})\neq(0,0)\equiv(n,n),
{\calL}=-{\frac{\kappa}{2}}\epsilon_{ij}A^{i}\dot{A}^{j}+i\phi^{*}\dot{\phi}-A^{0}(\phi^{*}\phi-\kappaF_{12})-{\frac{1}{2}}{|{\vec{D}}\phi|}^{2}+{\frac{1}{2\kappa}}(\phi^{*}\phi)^{2},
{\tilde{\calH}}^{\mu}(\tau)=p_{s}^{\mu}-u^{\mu}(p_{s})\sum_{i=1}^{N}\eta_{i}\sqrt{m_{i}^{2}+{\vec{\kappa}}_{i}^{2}(\tau)}-\epsilon_{r}^{\mu}(u(p_{s}))\sum_{i=1}^{N}\kappa_{i}^{r}(\tau)\approx0.
\begin{array}{cccc}{\xi_{1}=\phi_{1}+\psi_{1}}&{}&{}&{\zeta_{1}=\phi_{1}-\psi_{1}}\\{\vdots}&{}&{}&{\vdots}\\{\xi_{1}=\phi_{N}+\psi_{N}}&{}&{}&{\zeta_{N}=\phi_{N}-\psi_{N}}\\\end{array}
T_{m_{i}n_{i}}=\delta_{m_{i}n_{i}}e^{2(F_{i}-B)}[-\frac{h_{i}^{2}}{4}e^{-2qA-2rF_{i}+2C_{i}}(\partialC_{i})^{2}+\sum_{j\neqi}^{n}\frac{h_{j}^{2}}{4}e^{-2qA-2rF_{j}+2C_{j}}(\partialC_{j})^{2}]
{\bfg}(\Omega)={\frac{\partial^{2}\Omega}{\partialx^{i}\partial{\tilde{x}^{j}}}}dx^{i}d{\tilde{x}}^{j}\,,\quad\quad{\tilde{x}}^{i}={\tilde{x}}\,,{\tilde{y}}\,,\quadx^{j}=x\,,y\,.
\Psi=\Psi_{_{l,j=l\pm\frac{1}{2},m,q}}(\theta,\phi,\psi)=\Omega\left(\begin{array}{c}{\pm\sqrt{l\pmm+\frac{1}{2}}Y_{_{m-\frac{1}{2}q}}^{l}(\theta,\phi,\psi)}\\{\sqrt{l\mpm+\frac{1}{2}}Y_{_{m+\frac{1}{2}q}}^{l}(\theta,\phi,\psi)}\\\end{array}\right),
\left(f\bulletg\right)\bulleth-f\bullet\left(g\bulleth\right)=\frac{1}{6}\,\widetilde{\alpha}^{ia}\widetilde{\alpha}^{jb}\widetilde{\alpha}^{kc}\,H_{abc}\,\partial_{i}f\,\partial_{j}g\,\partial_{k}h+\cdots.
\frac{\partial}{\partial\Lambda}\Gamma_{\Lambda}[\phi^{c}]=-\frac{1}{2}\mathrm{tr}\left[\frac{1}{C}\frac{\partialC}{\partial\Lambda}.\left(1+C.\frac{\delta^{2}\Gamma_{\Lambda}}{\delta\phi^{c}\delta\phi^{c}}\right)^{-1}\right]\quad.
\mathrm{Re}\,\left[\Pi_{00}(k_{0},k)-\Pi_{00}(0,k)\right]={\frac{1}{\pi}}\int_{-\infty}^{\infty}d\omega\,\left({\frac{1}{\omega-k_{0}}}-{\frac{1}{\omega}}\right)\,\mathrm{Im}\,\Pi_{00}(\omega,k)
S_{EM}\rightarrow-{\frac{1}{4}}\intd^{4}x\sqrt{-g}{\lambda}^{2}\left[{\frac{1}{{\lambda}^{4}}}F_{\alpha\beta}F^{\alpha\beta}+{\frac{1}{{\lambda}^{2}}}2F_{\alphai}F^{\alphai}+F_{ij}F^{ij}\right]~~.