harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}=\left(F_{\mu\nu}+{\frac{1}{4}}\varepsilon_{\mu\nu}{\calP}\right)^{2}+{\frac{1}{4}}\left(D_{\mu}q^{i}-\varepsilon_{\mu\nu}D_{\nu}q^{j}\varepsilon_{ji}\right)^{2}-{\frac{1}{2}}\varepsilon_{\mu\nu}\Omega_{\mu\nu}.
[W,e^{\hat{K}}]=\left[W,\sum_{N=0}^{\infty}\frac{1}{N!}\hat{K}^{N}\right]=\sum_{N=0}^{\infty}\frac{1}{N!}\hat{K}^{N}\left([W,\hat{K}]\right)=e^{\hat{K}}([W,\hat{K}])\.
\frac{1}{\Box_{2}}F_{mn}(\Box_{1},\Box_{2})X_{1}X_{2}=-\int_{-\infty}^{0}dq\,q^{m+n}\biggl(\frac{d^{m+1}}{dq^{m+1}}{\calH}_{q}X_{1}\biggr)\biggl(\frac{d^{n}}{dq^{n}}\int_{-\infty}^{q}\frac{d{\bar{q}}}{{\bar{q}}}{\calH}_{\bar{q}}X_{2}\biggr)\quad,
\frac{\partialA_{ij}}{\partialq^{k}}-\frac{\partialA_{kj}}{\partialq^{i}}=-\frac{1}{2}\frac{\partial}{\partial\dot{q}^{j}}\left[\frac{\partial}{\partial\dot{q}^{k}}(A_{in}f^{n})-\frac{\partial}{\partial\dot{q}^{i}}(A_{kn}f^{n})\right].
\left[x_{e}^{\mu},x_{e^{\prime}}^{\nu}\right]_{\star}=i\theta\delta_{ee^{\prime}}\eta^{\mu\nu}\,,\quad\left\{x_{e}^{b},p_{e^{\prime}}^{b}\right\}_{\star}=\left\{x_{e}^{c},p_{e^{\prime}}^{c}\right\}_{\star}={\theta^{\prime}}\delta_{ee^{\prime}}\,\,.
H_{\mu|\nu}^{(3)}=\frac16\epsilon^{\mu\mu_{1}\mu_{2}\mu_{3}}\epsilon^{\nu\nu_{1}\nu_{2}\nu_{3}}H_{\mu_{1}\mu_{2}\mu_{3}|\nu_{1}\nu_{2}\nu_{3}}^{(3)}=-\delta_{\mu\nu}B_{\rho}A_{\rho}+B_{\nu}A_{\mu}\,.
W[X_{0}]=\int{\calD}\xie^{-\frac{1}{4\pi}\int_{y>0}dxdy~\eta^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}~\eta_{MN}}~e^{-\int_{-\infty}^{+\infty}\frac{dx}{a}\intdk{\tilde{T}}(k)e^{ik\cdotX_{0}}e^{ik\cdot\xi}}
=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}g^{2n}\partial_{\nu_{1}}B_{\mu_{1}}(y)\cdots\partial_{\nu_{2n}}B_{\mu_{2n}}(y)\intDz\mu[z]P_{\mu_{1}\nu_{1}}[z]\cdotsP_{\mu_{2n}\nu_{2n}}[z],
\begin{array}{l}{{\calX}^{-}\vertn,z\rangle_{-}=q^{2n}z\vertn,z\rangle_{-}~,}\\{{\calD}_{-}\vertn,z\rangle_{-}=-q^{-1-2n}\lambda^{-1}z^{-1}\vertn+1,z\rangle_{-}~,}\\{\mu_{-}\vertn,z\rangle_{-}=\vertn-1,z\rangle_{-}~.}\\\end{array}
I^{(2)}(q)=\frac{K_{D}}{2}\left\langle\gamma^{(4)}({\bf{q}},-{\bf{q}},\hat{\bf{q}}^{\prime},-\hat{\bf{q}}^{\prime})-\gamma^{(4)}(0,0,\hat{\bf{q}}^{\prime},-\hat{\bf{q}}^{\prime})\right\rangle_{\hat{\bf{q}}^{\prime}}\;.
\delta_{\alpha{\mathbfi}}{\calR}^{\mu}=16i\left\{\frac{1}{192}\left[\frac{3}{2}J_{\alpha{\mathbfi}}^{\mu}+\sigma_{\alpha}^{\mu\nu\beta}J_{\nu\beta{\mathbfi}}\right]-\frac{3}{4}(8+a)\partial^{\mu}D_{\alpha{\mathbfi}}T|+\frac{1}{2}(a-24)\sigma_{\alpha}^{\mu\nu\beta}\partial_{\nu}D_{\beta{\mathbfi}}T|\right\}.
S^{(2)}=-\frac{1}{4\pi\alpha^{\prime}}\intd\taud\sigma\;\eta^{ab}\delta_{RS}\left(\Phi_{,a}^{R}+A_{aU}^{R}\Phi^{U}\right)\left(\Phi_{,b}^{S}+A_{bV}^{S}\Phi^{V}\right),
{\mathbfO}\,\,{\mathbfR}\,\,{\mathbfO}^{-1}\:=\:-8m+{\calM}_{\nu}^{\mu}{\mathbfa}_{\nu}^{\dagger}{\mathbfa}^{\mu}+{\calN}_{\nu}^{\mu}{\mathbfb}^{\dagger\nu}{\mathbfb}_{\mu}.
\left(\begin{matrix}{-\frac{d^{2}}{dx^{2}}+V_{\ell}(x)}&{0}\\{0}&{-\frac{d^{2}}{dx^{2}}+\tilde{V}_{\ell}(x)}\\\end{matrix}\right)\psi_{k}(x)=k^{2}\psi_{k}(x)
{\hat{F}}[{S^{3}/T^{*}}]=\alpha^{-1}~\sum_{m,\ell,R}{\frac{(-)^{6\epsilon_{\ell}-3\epsilon_{R}}{S_{0\Lambda_{\ell}}~S_{0\Lambda_{R}}~S_{\Lambda_{\ell}\Lambda_{m}}~S_{\Lambda_{\ell}\Lambda_{m}}~~S_{\Lambda_{R}\Lambda_{m}}^{*}~q^{-3C_{\ell}+{\frac{3}{2}}C_{R}}}}{S_{00}~S_{0\Lambda_{m}}}}~.
N(x_{\mu};q)=\frac{1}{2}\left[\{(L^{+})^{-\frac{1}{2}}+(L^{-})^{-\frac{1}{2}}\}\cdot1+\{(L^{+})^{-\frac{1}{2}}-(L^{-})^{-\frac{1}{2}}\}\cdot\hat{x}\right],
ds^{2}=\frac{1}{1+u^{2}}(2d{x^{\prime}}^{+}du+d{y^{\prime}}^{2}+{y^{\prime}}^{2}d\Omega_{5}^{2}+\frac{(1-u^{2})^{2}}{1+u^{2}}(d{x^{\prime}}^{2}+{x^{\prime}}^{2}d\Omega_{2}^{2}))
F_{8}^{{\mathcal{O}}_{1,1}^{0,1}|+-------}=\frac{-iH^{1,1}(\sigma_{7}^{-})^{1/2}/(\sigma_{1}^{+})^{7/2}}{8\prod_{2\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\begin{aligned}{\phi^{m_{1}m_{2}p}\phi_{n_{1}n_{2}p}}&{{}=2\delta_{n_{1}n_{2}}^{m_{1}m_{2}}+{*\phi}^{m_{1}m_{2}}{}_{n_{1}n_{2}},}\\{\phi^{mp_{1}p_{2}}\phi_{np_{1}p_{2}}}&{{}=6\delta_{n}^{m},}\\\end{aligned}
\delta_{\lambda_{1}\rho_{1}\cdots\lambda_{m-1}\rho_{m-1}\lambda\rho}^{\lambda_{1}\rho_{1}\cdots\mu_{m-1}\nu_{m-1}\mu\nu}\;\;(g^{\perp})^{[\lambda}{}_{[\mu}(g^{\perp})^{\rho]}{}_{\nu]}={\tilde{\delta}}_{\lambda_{1}\rho_{1}\cdots\lambda_{m-1}\rho_{m-1}}^{\mu_{1}\nu_{1}\cdots\mu_{m-1}\nu_{m-1}}.
\Psi^{(K)}|0\rangle\equivN^{-c/2}a_{\upsilon_{1}}^{\dagger\upsilon_{2}}(k_{1})a_{\upsilon_{2}}^{\dagger\upsilon_{3}}(k_{2})\cdotsa_{\upsilon_{c}}^{\dagger\upsilon_{1}}(k_{c})|0\rangle,
\begin{aligned}{\eta_{B}(0)}&{{}=-\frac{1}{6}-\frac{\lambda}{3}\bigg[\frac{16\,Z^{3}}{S^{2}}-4\,Z\bigg]+\frac{\lambda^{2}}{3}\bigg[12\,Z^{2}-\frac{12\,Z^{2}}{S^{2}}+1\,\bigg]}\\{}&{{}\quad-\frac{\lambda^{3}}{3}\bigg[\frac{3\,Z}{S^{2}}+4\,S^{2}Z-6\,Z\bigg]+\frac{\lambda^{4}}{4}\bigg[1-\frac{4\,S^{2}}{3}-\frac{1}{3\,S^{2}}\bigg]\,.}\\\end{aligned}
\tilde{G}(p)=2ie^{-(p_{\perp}l)^{2}}\frac{A(p_{\parallel}^{2})\hat{p}_{\parallel}+B(p_{\parallel}^{2})}{A^{2}(p_{\parallel}^{2})p_{\parallel}^{2}-B^{2}(p_{\parallel}^{2})}O^{(-)}
\left.|S_{2}\,(x,q\,;\,y,-q)|\leq\mathrm{sup}_{\tau}\,\left\{z(\beta)+z(\beta,\rho^{\tau})\right\}\times\,\intd\mu_{I_{\beta}}\,(A)\,\sum_{\tau}\,c_{\tau}\,\prod_{\rho\in{\calE}_{\tau}}\,\left[1+z(\beta,\rho)\,c_{\beta}\,(A,\rho)\right]\\\.\right.
\begin{array}{l}{\tilde{K}(\tilde{z};g)=I(z)^{-1}K(z;g)I(z^{\prime})~~~~~~~~~~}\\{{}}\\{\bar{\tilde{K}}(\tilde{z};g)=\bar{I}(z^{\prime})\bar{K}(z;g)\bar{I}(z)^{-1}=\left(\begin{array}{cc}{\gamma^{0}}&{0}\\{0}&{-1}\\\end{array}\right)\tilde{K}(\tilde{z};g)^{\dagger}\left(\begin{array}{cc}{\tilde{\gamma}^{0}}&{0}\\{0}&{-1}\\\end{array}\right)}\\\end{array}
\left(n^{A}({\calR}_{AB}-\frac{1}{2}{\calR}G_{AB}-8\pi{\bfG}{\calT}_{AB})n^{B}\right)^{(1)}-\left(n^{A}({\calR}_{AB}-\frac{1}{2}{\calR}G_{AB}-8\pi{\bfG}{\calT}_{AB})n^{B}\right)^{(2)}=0~.
\delta_{a}\tilde{\pi}_{b\alpha_{1}}=\left(-\right)^{\epsilon_{\alpha_{1}}+1}\bar{\pi}_{b\alpha_{1}}^{(a)},\;\delta_{a}\tilde{\lambda}_{\alpha_{1}}=\left(-\right)^{\epsilon_{\alpha_{1}}}\bar{\lambda}_{\alpha_{1}}^{(a)},
\langleq^{3}|\tilde{q}^{3}|q^{2}\tilde{q}|q\tilde{q}^{2}\rangle=R_{{\calN}=2}^{2222}\\left[F(s,t)\,\frac{[12][34]}{x_{12}^{2}\,x_{34}^{2}}+G(s,t)\,\frac{[14][23]}{x_{14}^{2}\,x_{23}^{2}}\right]\;.
\left[\M^{2}-\left(\frac{m_{i}^{2}}{x}+\frac{m_{j}^{2}}{1-x}\right)\\right]\psi_{ij}(xP^{+})=-\frac{N_{c}g^{2}}{2\pi}{\calP}\int_{0}^{1}dy\frac{\psi_{ij}(yP^{+})-{\psi_{ij}(xP^{+})}}{(x-y)^{2}}.
\bfE_{\mathrm{L}}\mathrm{=\frac{\bf\nabla\mathrm{}}{\bf\nabla\mathrm{^2}}\rho^{(2)}\equiv-\frac{1}{4\pi}\int\!d^{3}\!y\frac{1}{\mid\bfy\mathrm{-\bfx\mathrm{\mid}}}\nabla\rho^{(2)}(y)\;\;.}
<u^{a}(t,r)u^{b}(t^{\prime},r^{\prime})>=\frac{2T\delta^{ab}}{M^{2}R}\sum_{n=1}^{\infty}\sum_{m=-\infty}^{\infty}\frac{e^{i\omega_{m}(t-t^{\prime})}}{k_{n}^{2}+\omega_{m}^{2}}sink_{n}rsink_{n}r^{\prime},
P\Theta=\left(\begin{matrix}{p_{10}+p_{9}}&{p_{8}-\sum_{j=1}^{7}p_{j}e_{j}}\\{p_{8}+\sum_{j=1}^{7}p_{j}e_{j}}&{p_{10}-p_{9}}\\\end{matrix}\right)\left(\begin{matrix}{0_{1}}\\{0_{2}}\\\end{matrix}\right)=0\,,
\sum_{l,n}\frac{\mu_{p-1}\lambda^{k+n+l}i^{k}p!}{k!n!l!(p-l)!}\partial_{x^{i_{1}}}\ldots\partial_{x^{i_{n}}}C_{i_{1}^{\prime}\ldotsi_{2k}^{\prime}j_{1}\ldotsj_{l}[a_{l+1}\ldotsa_{p}}^{0}Str\left(\partial_{a_{1}}\phi^{j_{1}}\ldots\partial_{a_{l}]}\phi^{j_{l}}\phi^{i_{1}}\ldots\phi^{i_{n}}\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}\ldots\right)
\rho_{\mu_{1}\dots\mu_{p-1}\alpha_{1}}^{(1)}=(C\Gamma_{\nu\mu_{1}\dots\mu_{p-1}})_{\beta\alpha_{1}}\Pi^{\nu}\Pi^{\beta}\,\\rho_{\mu_{1}\dots\mu_{p-1}\alpha_{1}}^{(2)}=(C\Gamma^{\nu})_{\beta\alpha_{1}}\Pi_{\nu\mu_{1}\dots\mu_{p-1}}\Pi^{\beta}\quad.
S=-\frac{1}{2}\int_{M_{3}}d^{3}\xi\sqrt{-\gamma}\left(\gamma^{ij}L_{i}^{r}L_{j}^{s}\eta_{rs}+\gamma^{ij}\partial_{i}X^{m}\partial_{j}X^{n}\eta_{mn}-1\right)-\int_{M_{3}}d^{3}\xi\partial_{j}A_{k}^{r}\epsilon^{ijk}L_{i}^{r}
G_{\,+}^{\ellk}(q^{0})=i\int_{-\infty}^{\infty}dx^{0}e^{iq^{0}x^{0}}\intd\vec{x}\left(\Phi_{p},\left(j^{\ell}(\vec{x},x^{0})j^{k}(\vec{y},0)\right)_{+}\Phi_{p}\right)
D^{L}C_{\alpha_{1}\ldots\alpha_{n},\,\dot{\beta}_{1}\ldots\dot{\beta}_{n}}=h^{\gamma\dot{\delta}}C_{\alpha_{1}\ldots\alpha_{n}\gamma,\,\dot{\beta}_{1}\ldots\dot{\beta}_{n}\dot{\delta}}-n^{2}\lambda^{2}h_{\{\alpha_{1}\dot{\beta}_{1}}C_{\alpha_{2}\ldots\alpha_{n},\,\dot{\beta}_{2}\ldots\dot{\beta}_{n}\}_{\alpha,\dot{\beta}}}\,.
P[E_{ab}+E_{ai}(X^{ij}-\delta^{ij})E_{jb}]=-1+\lambda^{2}(\dot{\Phi}^{i})^{2}+\lambda^{2}\dot{\Phi}^{k}\delta_{ki}(X^{ij}-\delta^{ij})\delta_{jl}\dot{\Phi}^{l}\.
|\psi\rangle=\left\{\alpha(b^{\dagger})f_{1}^{\dagger}f_{2}^{\dagger}+\phi_{1}(b^{\dagger})f_{1}^{\dagger}+\phi_{2}(b^{\dagger})f_{2}^{\dagger}+\Psi(b^{\dagger})\right\}|0\rangle.
\vert\phi\vert^{2}A_{\circ}=-\frac{1}{\sqrt{2}(\eta^{2}-\vert\phi\vert^{2})}\epsilon_{ij}[(\eta^{2}-\vert\phi\vert^{2})F_{ij}-i\{D_{i}\phi(D_{j}\phi)^{*}-D_{j}\phi(D_{i}\phi)^{*}\}]
Z_{g}=\int[dg_{\alpha\beta}dXd\hat{X}]e^{-{\frac{R^{2}}{4\pi}}\int_{M_{g}}d\taud\sigma\sqrt{g}g^{\alpha\beta}\partial_{\alpha}X\partial_{\beta}X-\int_{M_{g}}d\taud\sigma\hat{L}}.
\delta_{\underline{{m}}}^{\underline{{n}}}=u_{\underline{{m}}}^{\underline{{a}}}u_{\underline{{a}}}^{\underline{{n}}}=u_{\underline{{m}}}^{a}u_{a}^{\underline{{n}}}-u_{\underline{{m}}}^{i}u^{i\underline{{n}}},
\begin{array}{ll}{D=3:}&{D_{\alpha}^{i}\;W_{a}=\frac{1}{8}(\Gamma^{i}\tilde{\Gamma}^{j})_{ab}D_{\alpha}^{j}\;W_{b}}\\{D=4:}&{D_{\alpha}^{(k}\;W^{[i)j]}=0\;,\quad\bar{D}_{\dot{\alpha}\{k}\;W^{[i\}j]}=0}\\{D=5:}&{D_{\alpha}^{(k}\;W^{i)}=0}\\{D=6:}&{D_{\alpha}^{(k}\;W^{[i)j]}=0}\\\end{array}
\vertJ\vert=\left\vert\mathrm{det}\left(\frac{\deltaA_{\mu}}{\delta(\eta,\xi)}\right)\right\vert=\left\vert\left[\mathrm{det}\left(\frac{\delta\eta}{\deltaA_{1}}\frac{\delta\xi}{\deltaA_{2}}\right)\right]^{-1}\right\vert=\left\vert\mathrm{det}\left(*D^{\dagger}D\right)^{-1}\right\vert.
C_{1}(\tau)_{free}=s\left(\frac{1}{2\pi}\right)^{2\sum_{b}(n_{n}+n_{b})}\prod_{b=1}^{N}\prod_{i,j=1}^{n_{b}+n_{b}^{\prime}}\Big(w_{i}^{(b)}-z_{j}^{(b)}\Big)^{-2}\prod_{1\leqi<j\leqn_{b}+n_{b}^{\prime}}\Big(w_{i}^{(b)}-w_{j}^{(b)}\Big)^{2}\Big(z_{i}^{(b)}-z_{j}^{(b)}\Big)^{2},
{\calZ}_{n}=\int\prod_{i,j,k}^{n}dx_{i}dy_{j}dz_{k}\int{\calD}\phi{\calD}\psi\left(\lambda\phi(x_{i})\psi(y_{j})\phi(z_{k})\right)^{n}e^{-\left(S_{free}^{E}\left[\phi\right]+S_{free}^{E}\left[\psi\right]\right)}
I[\hat{g},\hat{\phi}]=\frac{1}{\pi}\intd^{3}x\sqrt{-\hat{g}}\left(\frac{\hat{R}+2l^{-2}}{16}-\frac{1}{2}(\nabla\hat{\phi})^{2}-\frac{1}{16}\hat{R}\hat{\phi}^{2}+\frac{\nu}{8l^{2}}\hat{\phi}^{6}\right)\;,
\begin{aligned}{\xi_{I}\cup\xi_{J}}&{{}=-2C_{IJ}\eta^{0}}\\{\xi_{I}\cup\xi_{0}}&{{}=-2C_{IJ}\eta^{J}}\\\end{aligned}\qquad\begin{aligned}{\xi_{0}\cup\chi_{0}}&{{}=\phi}\\{\chi_{1}\cup\chi_{2}}&{{}=\phi}\\\end{aligned}\qquad\begin{aligned}{\xi_{0}\cup\eta^{0}}&{{}=\rho}\\{\xi_{I}\cup\eta^{J}}&{{}=\delta_{I}{}^{J}\rho}\\\end{aligned}
\left(\frac{M^{4}}{p_{T}^{6}}\right)^{N_{I}}\left(p_{T}\right)^{4N_{L}}=\left(\frac{M^{4}}{(t^{-1}l)^{6}}\right)^{N_{I}}\left(t^{-1}l\right)^{4N_{L}}.
\tilde{S}_{Q1f}^{pot}(B_{i})=g\left[4\sum_{j<k}\frac{1}{N^{2}}\mathrm{Tr}\,B_{j}^{2}\mathrm{Tr}\,B_{k}^{2}\left(\sum_{i}U_{ij}^{2}U_{ik}^{2}\right)+\sum_{j}\frac{1}{N}\mathrm{Tr}\,B_{j}^{4}\left(\sum_{i}U_{ij}^{4}\right)\right]
\dot{\varrho}_{m}+3\left(\varrho_{m}+{\frac{\wp_{m}}{\bar{c}_{4}^{2}}}\right){\frac{\dot{\bar{a}}_{R}}{\bar{a}_{R}}}=-\dot{\varrho}_{\delta\sigma}-\varrho{\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}+{\frac{3k\bar{c}_{4}\dot{\bar{c}}_{4}}{4\pi\bar{G}_{4}\bar{a}_{R}^{2}}}.
H_{LC}^{(AdS)}=\int_{0}^{l}\frac{d\sigma}{2\,\pi_{-}}\,\left(\pi_{i}\pi_{i}+\frac{1}{(2\pi\alpha^{\prime})^{2}}\frac{R^{4}}{Y^{4}}Z^{i\,\prime}Z^{i\,\prime}\right)\.
\vec{\phi}=(\underbrace{\frac{W+\bar{W}}{|W|^{2}+1}}_{\phi_{1}},\underbrace{i\frac{-W+\bar{W}}{|W|^{2}+1}}_{\phi_{2}},\underbrace{\frac{|W|^{2}-1}{|W|^{2}+1})}_{\phi_{3}},
+\frac{1}{2e^{2}\mid\varphi\mid^{2}}\{(\partial_{\nu}\Lambda^{\nu\mu})+\frac{e}{2}(\varphi^{\dagger}\partial_{\mu}\varphi-\partial_{\mu}\varphi^{\dagger}\varphi)\}^{2}+\frac{1}{2}\mid\partial_{\mu}\varphi\mid^{2}+\lambda(\mid\varphi\mid^{2})^{2}],
\frac{d\phi}{dr}=\frac{L}{r^{2}}\,\frac{1+2m\beta\left(E-V+r\frac{\partialV}{\partialr}\right)}{\sqrt{2m(E-V)-\frac{L^{2}\left[1+2m\beta(E-V)\right]^{2}}{r^{2}}}}\;,
P(\ell,\Phi)=\left[\ell^{3}-\frac{\ell^{2}\,\Phi}{\pi}+\ell\left(\frac{{\Phi}^{2}}{2\,{\pi}^{2}}-1\right)+\frac{2\,{\pi}^{2}\,\Phi-{\Phi}^{3}}{8\,{\pi}^{3}}\right],
I_{E}=\intd\tau\Biggl[\frac{3\pim_{P}^{2}}{4}\Biggl(\frac{a}{N}\Bigl(\frac{\partiala}{\partial\tau}\Bigr)^{2}+NV_{g}(a)\Biggr)-2\pi^{2}a^{3}(\tau)\Biggl(\frac{1}{2N}\Bigl(\frac{\partial\phi}{\partial\tau}\Bigr)^{2}+NV(\phi)\Biggr)\Biggr],
r_{0}^{2}(x)={\frac{56}{153}}x^{2}+{\frac{3}{2}}\left({\frac{175}{102}}\right)^{{\frac{1}{3}}}x^{{\frac{4}{3}}}\left\{\left[P(x)+Q(x)\right]^{{\frac{1}{3}}}-\left[P(x)-Q(x)\right]^{{\frac{1}{3}}}\right\},
{\calD}^{K}=\{{\calD},\;\;\}_{\calA}-t(\phi^{a}-tI_{a}g^{ab})\{I_{a},\;\;\}_{\calA}+\frac{t(1-t)}{2}f_{ab}^{c}c^{a}c^{b}\{I_{a},\;\;\}_{\calA}--tc^{a}\{Q_{a},\;\;\}_{\calA},
S_{\mathrm{min}}=\int_{t_{i}}^{t_{f}}dt\int_{0}^{1}d\lambda\Big[-({\frac{\partialu^{\mu}}{\partialt}}{\frac{\partialu_{\mu}}{\partialt}})({\frac{\partialu^{\nu}}{\partial\lambda}}{\frac{\partialu_{\nu}}{\partial\lambda}})+({\frac{\partialu^{\mu}}{\partialt}}{\frac{\partialu_{\mu}}{\partial\lambda}})^{2}\Big]^{\frac{1}{2}}\,,
i{\calM}~=~\frac{ip_{1}^{2}p_{2}^{2}}{-t}\frac{\Gamma(1-ip_{1}^{2}p_{2}^{2}/Ep)}{\Gamma(1+ip_{1}^{2}p_{2}^{2}/Ep)}\left(\frac{4}{-t}\right)^{-i\frac{p_{1}^{2}p_{2}^{2}}{Ep}}~.
{\calC}_{D}^{(i)}={\calC}_{N}{\frac{g_{D}^{(i)}}{\sqrt{3}}}\,,\qquad{\calC}_{R}^{(i)}={\calC}_{N}{\frac{g_{R}^{(i)}}{\sqrt{3}}}\,,\qquad{\calC}_{G}={\calC}_{N}{\frac{\Psi^{(0)}(y_{br})}{\sqrt{|k_{1}|}}}\;.
S_{D,p}^{0}=\int_{{\calM}^{p+1}}^{}{\calL}_{p+1}^{0}=\int_{{\calM}_{p+1}}^{}\left(-{\frac{(-1)^{p}}{p!}}(E^{a}e^{a_{1}}...e^{a_{p}}-{\frac{p}{(p+1)}}e^{a}e^{a_{1}}...e^{a_{p}})\varepsilon_{aa_{1}...a_{p}}\right)
\frac{(2\pi)^{3}\lambda_{\bfg}^{3/2}k^{-3/2}\widetilde{\tau}(S^{3})^{3/2}}{V(S^{3})^{3/2}V(SU(2))}=\frac{(2\pi)^{3}\lambda_{\bfg}^{3/2}k^{-3/2}(2\pi)^{3}}{(2\pi^{2})^{3/2}(\frac{2}{\lambda_{\bfg}})^{3/2}16\pi^{2}}=\sqrt{2}{\pi}k^{-3/2}\,(\frac{\lambda_{\bfg}}{\sqrt{2}})^{3}\,.
e^{\pmim_{n-1}\theta_{1}}\frac{a^{-\frac{n}{2}}}{asin{\theta_{1}}...sin{\theta_{n-1}}}B\prod_{k=1}^{n-2}(sin{\theta_{n-k}})^{m_{k+1}}C_{m_{k}-m_{k+1}}^{m_{k+1}+\frac{n-1}{2}-\frac{1}{2}k}(cos{\theta_{n-k}})
\big|\pi_{i=1\simN_{c}}\big|=\Lambda_{T}^{2},\\\big|\pi_{i=N_{c}+1\simN_{f}}\big|=\xi\big|\pi_{i=1\simN_{c}}\big|,\\|\pi_{B}|=|\pi_{\bar{B}}|\sim\Lambda_{T}^{N_{c}},\\,big|\pi_{\lambda}\big|\sim\Lambda^{3}\xi^{\frac{\rho+N_{f}-N_{c}}{N_{f}-N_{c}}},
\rho_{n,k}(u,w)=\sum_{m=0}^{\lfloor\frac{k}{2}\rfloor}\,\frac{(-1)^{m\,n}}{m!\,(k-2\,m)!\,n^{m}}\,\left(\prod_{l=m+1}^{k}\,l\,n+1\right)\,w^{mn+1}\,u^{k-2m}
d_{n}H_{n-1+{\frac{D}{2}}}^{(1)}(|\omega|r^{\prime})=e_{n}H_{n-1+{\frac{D}{2}}}^{(1)}(|\omega|r^{\prime})+f_{n}H_{n-1+{\frac{D}{2}}}^{(2)}(|\omega|r^{\prime}),
S_{5}=\intd^{5}\!x\sqrt{-g}\,\Bigl\{\bigl(-|\partial_{M}\phi|^{2}-|\partial_{M}\phi^{c}|^{2}-m_{\phi}^{2}|\phi|^{2}-m_{\phi^{c}}^{2}|\phi^{c}|^{2}+i\bar{\Psi}\gamma^{M}(\partial_{M}+\Gamma_{M})\Psi-im_{\Psi}\bar{\Psi}\Psi\bigr)\Bigr\}\,,
F_{+}^{(2)}-F_{-}^{(2)}={\frac{1}{8\pi^{2}}}\{\frac{\Lambda}{\vert\Lambda\vert}}\intd^{4}x\,varepsilon_{\mu\lambda\nu\sigma}\tr\Bigl(\theta(x)\partial_{\mu}\partial_{\sigma}A_{\nu}\Bigr)
\Phi_{\mathrm{div}}(\omega,D)={\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{m^{D-4}}{\kappa}}\int_{\Sigma}\left[2\left(m^{2}-\left(\frac16-\xi\right)R\right)-{\frac{\omega^{2}}{\kappa^{2}}}{\calP}\right],
{\calT}_{\mu\nu}^{\alpha\beta}=2{\itg_{1}}^{2}\epsilon_{~\mu}^{\alpha~\rho\sigma}\epsilon_{~\nu}^{\beta~\lambda\gamma}\bigg(F_{\rho\sigma}F_{\lambda\gamma}+\tilde{F}_{\rho\sigma}\tilde{F}_{\lambda\gamma}+{\frac{1}{\sqrt2}}\hat{\theta}_{5}\big(F_{\rho\sigma}\tilde{F}_{\lambda\gamma}+\tilde{F}_{\rho\sigma}F_{\lambda\gamma}\big)\bigg)\,,
Q_{\beta}\;=\;M_{a}^{(\beta)}\:\lambda_{a}\:+\:D_{a}^{(\beta)}\:\partial_{\lambda_{a}}\;\;,\;\;\;Q_{\beta}^{\dagger}\;=\;M_{a}^{(\beta)\:\dagger}\:\partial_{\lambda_{a}}\:+\:D_{a}^{(\beta)\:\dagger}\:\lambda_{a}
\mathrm{Area}=\sqrt{1+4|\partialy_{3}/\partialy|^{2}}dy\wedged\bar{y}={\frac{\sqrt{W+\overline{{W}}+|W^{\prime}|^{2}-yW^{\prime}-\bar{y}\overline{{W^{\prime}}}}}{y_{3}}}dy\wedged\bar{y}
U=\left[|x_{0}|^{2}+\ldots+|x_{4}|^{2}-5|p|^{2}-r\right]^{2}+|f(x_{i})|^{2}+|p|^{2}\sum_{i=0}^{4}\left|\frac{\partialf}{\partialx_{i}}\right|^{2}+2|\sigma|^{2}\left(\sum_{i=0}^{4}|x_{i}|^{2}+25|p|^{2}\right).
\bar{R}^{\lambda\nu\mu\rho}F_{\lambda\nu}F_{\mu\rho}-2\bar{R}_{\mu\nu}F^{\rho\mu}F_{\rho}~^{\nu}-\frac12(F_{\mu\nu}F^{\mu\nu})^{2}+\frac12F^{\lambda\nu}F^{\mu\rho}F_{\lambda\mu}F_{\nu\rho}~~~,
L({\tilde{z}},t)=-\tilde{\rho}-\frac{\Lambda_{{\scriptscriptstyle\Box}}}{2}(\mathrm{tr}\,{A}^{\dagger}{\dot{A}}T_{h{\scriptscriptstyle\Box}})^{2}-\frac{\Lambda_{{\scriptscriptstyle\Box^{\prime}}}}{2}{\scriptstyle\sum_{\Box^{\prime}}}(\mathrm{tr}\,{A}^{\dagger}{\dot{A}}T_{h{\scriptscriptstyle\Box^{\prime}}})^{2},
\nu(x^{\mu})=+{\frac{1}{2}}(\psi_{\downarrow}(x^{\mu})-\psi_{\uparrow}^{c}(x^{\mu}))-{\frac{\gamma^{5}}{2}}(\psi_{\uparrow}(x^{\mu})+\psi_{\downarrow}^{c}(x^{\mu}))\,.
\Omega_{1}=\left(\begin{array}{cc}{I}&{\gamma}\\{0}&{I}\\\end{array}\right)\quad\Omega_{2}=\left(\begin{array}{cc}{I}&{0}\\{\alpha}&{I}\\\end{array}\right)\quad\Omega_{3}=\left(\begin{array}{cc}{{A^{-1}}^{T}}&{0}\\{0}&{A}\\\end{array}\right)
\tilde{E}_{qnj}^{D}=E_{qnj}^{D}\left[1\pm\frac{\mu^{2}\lambda^{2}(R_{0}/a_{0})^{2}}{6(\mu^{2}-\lambda^{2})^{\frac12}[n+(\mu^{2}-\lambda^{2})^{\frac12}][n^{2}+\mu^{2}+2n(\mu^{2}-\lambda^{2})^{\frac12}]}\right]
[J_{I}^{K},J_{N}^{M}]=i\left(\delta_{N}^{K}J_{I}^{M}-\delta_{I}^{M}J_{N}^{K}+\eta_{N\!I}\,\eta^{M\!L}J_{L}^{K}-\eta_{N\!L}\,\eta^{M\!K}J_{I}^{L}\right)\.
M=\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}{\midm\mid}}\intd^{2}k(a^{+}(k)a(k)-b^{+}(k)b(k)),
V=\left(\begin{matrix}{V^{I}}\\{V^{II}}\\{V^{III}}\\\end{matrix}\right)=\left(\begin{matrix}{E^{-1}}&{-E^{-1}C}&{-E^{-1}a^{T}}\\{0}&{E}&{0}\\{0}&{a}&{I_{16}}\\\end{matrix}\right)
f\starg=\frac{1}{(2\pi)^{2}}\sum_{m,n}\int_{-\pi}^{+\pi}d\theta^{\prime}d\theta^{\prime\prime}\int_{-\infty}^{+\infty}dp^{\prime}dp^{\prime\prime}\delta(p_{\theta}-p^{\prime}+\frac{\hbarm}{2})\delta(p_{\theta}-p^{\prime\prime}-\frac{\hbarn}{2})e^{[-\frac{4i}{\hbar}A({\bfr},{\bfr}^{\prime},{\bfr}^{\prime\prime})]}f({\bfr}^{\prime})g({\bfr}^{\prime\prime}).
\Psi_{i,g_{1}}^{[I]_{a}[J]_{b}}(x_{1})\,\Psi_{j,g_{2}}^{[J]_{b}[K]_{c}}(x_{2})\=\\sum_{k,\tilde{g}\inS_{J}}\,(x_{1}-x_{2})^{h_{i}+h_{j}-h_{k}}\Psi_{k,g_{1}g_{2}\tilde{g}}^{[I]_{a}[K]_{c}}(x_{2})\,C_{(i,g_{1})(j,g_{2})(k,g_{12}\tilde{g})}^{[I]_{a}[J]_{b}[K]_{c}}+\dots
\chi_{0}(\tau)=\frac{1}{2\eta^{4}(\tau)}\left\{\Theta^{4}\left[\begin{array}{l}{0}\\{0}\\\end{array}\right](0\vert\tau)+\Theta^{4}\left[\begin{array}{l}{0}\\{1/2}\\\end{array}\right](0\vert\tau)\right\}
\zeta(s;{\calA})={\frac{C_{N}}{2}}{\frac{\Gamma({\frac{N}{2}})\Gamma(s-{\frac{N}{2}})}{\Gamma(s)}}({\frac{2\pi}{\beta}})^{-2s+N}\sum_{n=-\infty}^{+\infty}[\nu^{2}+(n+\chi)^{2}]^{-(s-{\frac{N}{2}})}\;,
=-\left(0.5836\frac{g^{2}}{4\pi}+0.2871\left(\frac{g^{2}}{4\pi}\right)^{2}+0.2133\left(\frac{g^{2}}{4\pi}\right)^{3}+0.2024\left(\frac{g^{2}}{4\pi}\right)^{4}+\textrm{order}\left(g^{10}\right)\right)
\begin{array}{rcl}{}&{}&{H\cdot{^m\Phi}_{hs}={^mE}_{h}~{^m\Phi}_{hs}~,}\\{}&{}&{{^m\Phi}_{hs}=\displaystyle\sum_{(i)}a_{(i)}^{h}~{^m\Psi}_{(i)s}~,~~~h=1,~2,~\cdots,~M_{n}.}\\\end{array}
\frac{n_{o}^{\prime}}{n_{o}}\frac{\dot{a}_{o}}{a_{o}}+\frac{a_{o}^{\prime}}{a_{o}}\frac{\dot{b}_{o}}{b_{o}}-\frac{\dot{a}_{o}^{\prime}}{a_{o}}=\frac{1}{6M^{3}}T_{05}
R(a_{1},a_{2})=2\pi\sqrt{a_{1}}\sum_{n=-\infty}^{\infty}\frac{1}{\sqrt{a_{1}n^{2}+c^{2}}\left(e^{2\pi\sqrt{(a_{1}n^{2}+c^{2})/a_{2}}}-1\right)}-f(a_{2})\,,
\intd\tau\intd\tau^{\prime}\partial_{\tau^{\prime}}f\left(\langleX^{i}\left(\tau\right)X^{j}\left(\tau^{\prime}\right)\rangle\right)=\left.\intd\tauf\left(\langleX^{i}\left(\tau\right)X^{j}\left(\tau^{\prime}\right)\rangle\right)\right|_{\tau^{\prime}=\tau+\delta}^{\tau^{\prime}=\tau-\delta}.
\frac{F(3,-\frac{i}{2}k^{\prime}+3,-\frac{i}{2}k^{\prime}+4,e^{-4x_{0}})}{k^{\prime}+6i}=\frac{1}{2!}\sum_{n=2}^{\infty}\frac{n(n-1)e^{-4(n-2)x_{0}}}{k^{\prime}+i(2+2n)}
\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{\partial^{+}\xi^{I}}{2z\partial^{+}}\{M^{zj},M^{jI}\}\Bigr)|\phi\rangle\,.
\left(\mu\frac{\partial}{\partial\mu}+\beta\left(g^{2}\right)\frac{\partial}{\partialg^{2}}-\gamma_{2}\left(g^{2}\right)\intd^{2}xJ\frac{\delta}{\deltaJ}+\eta\left(g^{2},\zeta\right)\frac{\partial}{\partial\zeta}\right)E=0
[\,:\!\hat{M}_{\hat{m}\hat{n}}\!:,:\!\hat{M}_{\hat{p}\hat{q}}\!:\,]=-i(\hat{\eta}_{\hat{m}[\hat{p}}:\!\hat{M}_{\hat{q}]\hat{n}}\!:-\hat{\eta}_{\hat{n}[\hat{p}}:\!\hat{M}_{\hat{q}]\hat{m}}\!:)\,.
\tilde{S}_{2f}=\frac{1}{N}\mathrm{Tr}\,(gB_{1}^{4}+gB_{2}^{4}+2B_{1}^{2}+2B_{2}^{2})-\frac{4g}{N^{2}}\mathrm{Tr}\,B_{1}^{2}\mathrm{Tr}\,B_{2}^{2}
\begin{array}{c}{\mathcal{U}(x)\star\mathcal{U}^{*}(x)=\1\=}\\{\mathcal{U}(x)\mathcal{U}^{*}(x)+\sum_{n=1}^{\infty}\left(\frac{i}{2}\right)^{n}\frac{1}{n!}\theta^{i_{1}j_{1}}\cdots\theta^{i_{n}j_{n}}\partial_{i_{1}}\cdots\partial_{i_{n}}\mathcal{U}(x)\partial_{j_{1}}\cdots\partial_{j_{n}}\mathcal{U}^{*}(x).}\\\end{array}
\deltaS=0=\intd\tau\Bigl(\delta[q^{1/2}p_{\mu}\dot{x}^{\mu}+\frac{i}{2}\psi\cdot\dot{\psi}-\frac{i}{2}\psi_{5}\dot{\psi}_{5}-{\calH}_{c}(x,p,\psi,\psi_{5})]-\frac{d}{d\tau}g(\tau)\Bigr),
{\calL}=\frac12\partial^{\mu}\varphi_{i}\partial_{\mu}\varphi_{i}+\frac{i}{2}\overline{\psi}_{i}\not\!\partial\psi_{i}+\frac12F_{i}F_{i}+\sigma\varphi_{i}F_{i}+\frac12\lambda\varphi_{i}\varphi_{i}-\frac12\sigma\overline{\psi}_{i}\psi_{i}-\overline{\xi}\psi_{i}\varphi_{i}-\frac{N}{2g}\lambda,
\gamma\alpha_{0}=\partial_{\mu}\left(\varepsilon^{\mu\nu\lambda\rho}f_{\;\;bc}^{a}\left(C_{\rhoa}A_{\nu}^{b}A_{\lambda}^{c}+C_{a}\partial_{\rho}\left(A_{\nu}^{b}A_{\lambda}^{c}\right)\right)\right).
V_{F}(p_{0}^{\prime})\,=-2i\pi\,\sum_{k^{\prime}}\,V_{i^{\prime}j^{\prime}}\,\delta(p_{k^{\prime}0}^{\prime}-h_{k^{\prime}})\,\beta_{k^{\prime}}\,\big[G_{k^{\prime}}^{0}(p_{k^{\prime}0}^{\prime})\big]^{-1}.
Z_{v}^{X}\left(-\frac{1}{\tau}\right)=N^{-\frac{b_{2}(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\cdot\sum_{u\inH^{2}(X,Z_{N})}\zeta_{N}^{u\cdotv}Z_{u}^{X}(\tau),
\Phi[C]=-\frac{1}{2\pi}\oint_{C}du^{\mu}\,\partial_{\mu}\theta\-\,frac{1}{2\pi}\oint_{C}du^{\mu}\,j_{\mu}=-\frac{1}{2\pi}\oint_{C}du^{\mu}\,\partial_{\mu}\theta\+\,frac{1}{2\pi}\oint_{C}du^{\mu}\,\frac{1}{f^{2}}J_{\mu}.
A_{i}=\frac{i}{2}[\frac{\phi_{1}}{s}\:\sigma_{i}\:\:+\frac{1}{s^{2}}(A_{1}-\frac{\phi_{1}}{s})\:x_{i}\vec{x}.\vec{\sigma}\:\:+(\frac{\phi_{2}+1}{s^{2}})\:\varepsilon_{ijk}\sigma_{j}x_{k}]
K_{M}(\sqrt{s}y,\sqrt{s}y^{\prime},s)|_{p_{0}}={\frac{e^{-(y-y^{\prime})^{2}/4}}{(4\pis)^{d/2}}}\left(1+\sum_{n=0}^{\infty}b_{(n+2)/2}(y,y^{\prime},p_{0})s^{(n+2)/2}\right)
\begin{array}{ll}{\sigma_{1}=\left(\begin{array}{rr}{1}&{0}\\{0}&{1}\\\end{array}\right)}&{\sigma_{x}=\left(\begin{array}{rr}{0}&{1}\\{1}&{0}\\\end{array}\right)}\\{\sigma_{z}=\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right)}&{\sigma_{e}=\left(\begin{array}{rr}{0}&{1}\\{-1}&{0}\\\end{array}\right)}\\\end{array}
W_{F_{4}}^{\underline{{27}}}={\frac{a_{1}(x)}{24}}-{\frac{1}{2}}\left\{\left(-q+\sqrt{q^{2}+4p^{3}}\right)^{1/3}+\left(-q-\sqrt{q^{2}+4p^{3}}\right)^{1/3}\right\},
\rho_{\mathrm{pol}}\simg^{2}V^{-1}\intd^{d}xA(x,t)^{2}A^{0}\simg^{2}A^{0}\intd^{d}k\verta(k)\vert^{2}\simg^{2}TA^{0}\intd^{d}k\omega_{k}^{-2}\,.
a_{m}a_{n}^{\dagger}=\tilde{a}_{m}\tilde{a}_{n}^{\dagger}=\delta_{mn},\;\;\;\;\;\;\psi_{\dot{\alpha}}\psi_{\dot{\beta}}^{\dagger}=\tilde{\psi}_{\dot{\alpha}}\tilde{\psi}_{\dot{\beta}}^{\dagger}=\delta_{{\dot{\alpha}},{\dot{\beta}}}
H_{1}=1+{\frac{R_{1}^{2}}{x^{2}}},~~~H_{5}=1+{\frac{R_{5}^{2}}{x^{2}}},~~~H_{1}^{\prime}=1+{\frac{{R^{\prime}}_{1}^{2}}{y^{2}}},~~~H_{5}^{\prime}=1+{\frac{{R^{\prime}}_{5}^{2}}{y^{2}}}.