harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
^{(0,0,0)}_{(N_{j},N_{k},0)}\phi_{(N_{j},N_{k},0)}^{(0,0,0)}=\frac{(\frac{d-2}{2})_{N_{j}+N_{k}}}{(N_{j}+N_{k})!(d-2)_{N_{j}+N_{k}}(\frac{d}{2})_{N_{j}+N_{k}}}
ig\int_{0}^{1}d\tau{\calP}\left(U[A]\left[F_{ab}(\overline{{X}})\xi^{a}\partial_{\tau}\overline{{X}}^{b}+\frac{1}{2!}D_{b}F_{ac}(\overline{{X}})\xi^{a}\xi^{b}\partial_{\tau}\overline{{X}}^{c}+\frac{1}{2!}F_{ab}(\overline{{X}})\xi^{a}\partial_{\tau}\xi^{b}\right]\right).
\bigl\{\vec{\Theta}(u)\cdot\vec{n}_{+},\vec{\Theta}(v)\cdot\vec{n}_{-}\bigr\}=u_{\alpha}v_{\alpha}\;,\qquad\bigl\{\vec{\Theta}(u)\cdot\vec{n}_{\pm},\vec{\Theta}(v)\cdot\vec{n}_{\pm}\bigr\}=0\;.\nonumber
K_{12}\,\approx\,\,<K_{T12}(P_{0}-h_{3})>\,\,=\,\,{\frac{-1}{2i\pi}}\,\beta_{1}\beta_{2}\,\Lambda_{12}^{++}\,V_{12}(P_{0}\!-\!h_{3})\,\Lambda_{12}^{++}\,,\cdots.
3!F_{3}^{(4)}=[L_{3}^{2}L_{1}]^{(4)}+[L_{2}^{2}L_{1}]^{(4)}+[L_{3}^{2}L_{2}]^{(4)}+2[L_{3}L_{2}L_{1}]^{(4)}-[\SigmaF_{2}]^{(4)}
L_{quant}=L_{clas}+{\bfs}\biggl(\bar{c}^{\mu}\omega_{\mu\nu}\left(F^{\nu}-\frac{1}{2}b^{\nu}\right)+\bar{\chi}^{\mu}\zeta_{\mu\nu}\left(f^{\nu}-\frac{1}{2}\pi^{\nu}\right)+\bar{\eta}^{\mu}\varsigma_{\mu\nu}\left(\rho^{\nu}-\frac{1}{2}d^{\nu}\right)\biggr)
\frac{d^{2}f}{dr^{2}}-\left[\frac{l(l+1)-3(Gs)^{2}}{r^{2}}-\frac{2GsE}{r}-E^{2}\right]~f-\left[\frac{2GMl(l+1)}{r^{3}}+\frac{4GJmE}{r^{3}}\right]~f~=~0~.
W^{(1/2)}({\bfp},{\bfu})=\frac{(p_{0}+m)(u_{0}+1)-{\bfu}\cdot{\bfp}+i{\bf\sigma}({\bfp}\times{\bfu})}{\sqrt{2(u_{0}+1)(p_{0}+m)(p_{0}u_{0}-{\bfp}\cdot{\bfu}+m)}},\quad{W}^{(1/2)}({\bfp},{\bfu})\,{W}^{\dagger(1/2)}({\bfp},{\bfu})=1.
g_{\alpha\beta}V(\phi)=2g_{\alpha\beta}g^{\gamma\lambda}(\partial_{\gamma}\partial_{\lambda}-\Gamma_{\gamma\lambda}^{\sigma}\partial_{\sigma})\frac{1}{\phi}-2(\partial_{\alpha}\partial_{\beta}-\Gamma_{\alpha\beta}^{\gamma}\partial_{\gamma})\frac{1}{\phi}
Z_{p-1,p}^{n}(q)=\sum_{\substack{N_{1}\geq\ldots\geqN_{p-2}\geq0\,N_{1}+\ldots+N_{p-2}=n}}\frac{q^{\sum_{m=1}^{p-2}N_{m}^{2}}}{(q)_{N_{1}-N_{2}}(q)_{N_{2}-N_{3}}\dots(q)_{N_{p-2}}}.
\vec{p}_{-,t\rightarrow-\infty}=\left(\begin{array}{c}{+.06}\\{+.81}\\{+.13}\\\end{array}\right)\;,\qquad\vec{p}_{-,t\rightarrowt_{0}}=\left(\begin{array}{c}{+.48}\\{-.45}\\{-.08}\\\end{array}\right)\;.
{\bar{D}}^{({\mathbfm}}{}_{{\mathbfj}}D^{4}X^{{\mathbfi}){\mathbfj}}=-\frac{36i}{5}\partial_{\alpha\dot{\alpha}}[D_{\mathbfk}^{\alpha},{\bar{D}}_{\mathbfl}^{\dot{\alpha}}]D^{({\mathbfm}{\mathbfi}}X^{{\mathbfk}{\mathbfl})}\.
\zeta_{D}^{(+)}(0)=\frac{(-1)^{\frac{D}{2}}}{(4\pi)^{\frac{D}{2}}(\frac{D}{2})!}m^{D},\;\;\;\;\zeta_{D}^{(-)}(0)=-\frac{1}{2}\frac{(-1)^{\frac{D-1}{2}}}{(4\pi)^{\frac{D-1}{2}}(\frac{D-1}{2})!}\frac{m^{D-1}}{L}
\begin{array}{|c|c|c|c|}\hline{}&{P_{1}^{t_{1},\mu_{1}}}&{P_{2}^{t_{2},\mu_{2}}}&{P_{3}}\\\hline{\eta_{1}}&{e^{2\pii\mu_{1}/t_{1}}}&{\infty^{t_{2}h_{t_{2}}}}&{0^{N_{2}}}\\\hline{\eta_{2}}&{0^{t_{1}h_{t_{1}}}}&{e^{2\pii\mu_{2}/t_{2}}}&{\infty^{N_{1}}}\\\hline{\eta_{3}}&{\infty^{N_{2}t_{1}h_{t_{1}}}}&{0^{N_{1}t_{2}h_{t_{2}}}}&{1}\\\hline{}\\\end{array}
Q_{1}=-\frac{1}{2}\tau_{3}+R(12);\;Q_{2}=+\frac{1}{2}\tau_{3}+R(12);R(12)=\mathrm{diag}(-\frac{1}{2}+\frac{1}{N},-\frac{1}{2}+\frac{1}{N},\frac{1}{N},\dots).
B(x)=\frac{\lambda_{o}+2\lambda_{1}x+(\frac{\lambda_{1}^{2})}{4})e^{-2x}}{[e^{2x}-(\frac{\lambda_{1}^{2}}{16})e^{-2x}-\frac{1}{2}(\lambda_{o}+\lambda_{1})-\lambda_{1}x]},
g_{m}^{(0)}=\frac{\sum_{l=1}^{n_{m}}\lambda_{l,m}^{2}\Bigl(e^{k_{l,m}(r_{m}+it-ic_{l,m})}-1\Bigr)^{-1}}{\sum_{l=1}^{n_{m}}\lambda_{l,m}^{2}\Bigl(1-e^{-k_{l,m}(r_{m}+it-ic_{l,m})}\Bigr)^{-1}}
{\calM}_{0}^{(r)}\sim{\textstyle{\frac{1}{2}}}\,(n_{1}+n_{2})\hat{Q}_{o}(0)-{\textstyle{\frac{1}{2}}}\,\left[2\cdot2^{r/2}\cdotk_{1}k_{2}+2\right](m+\bar{m})\hat{Q}_{v}(\zeta\tau)\,.
K_{\mu\nu}=\bar{K}_{\mu\nu}^{(0)}+\frac{1}{2}e_{\mu}^{(0)M}e_{\nu}^{(0)N}({\calL}_{\deltan}g_{MN}^{(0)}-2n^{(0)L}\delta\Gamma_{LMN})-\frac{1}{2}{\calL}_{\deltaZ}(e_{\mu}^{(0)M}e_{\nu}^{(0)N}){\calL}_{n^{(0)}}g_{MN}^{(0)}
{\bfr}_{1}\wedge{\bfr}_{2}=x_{1}y_{2}-x_{2}y_{1}={\frac{-i}{2}}~(z_{1}^{*},z_{2}^{*})\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)~\left(\begin{array}{c}{z_{1}}\\{z_{2}}\\\end{array}\right)~.
\begin{array}{rl}{{\displaystyle\left\{p_{\mu},x^{\nu}\right\}=-i\delta_{\mu}^{\nu},}}&{{\displaystyle\left\{p_{g},g\right\}=-i,}}\\{{\displaystyle\left\{\pi_{\alpha}^{i},\theta_{j}^{\beta}\right\}=-i\delta_{j}^{i}\delta_{\alpha}^{\beta},}}&{{\displaystyle\left\{\bar{\pi}_{\dot{\alpha}i},\bar{\theta}^{\dot{\beta}j}\right\}=-i\delta_{i}^{j}\delta_{\dot{\alpha}}^{\dot{\beta}}.}}\\\end{array}
S\to{\frac{aS+b}{cS+d}},\\\\\left(\begin{matrix}{{\calF}_{\mu\nu}^{i}}\\{\tilde{\calF}_{\mu\nu}^{i}}\\\end{matrix}\right)\to\omega_{S}^{-1}\left(\begin{matrix}{{\calF}_{\mu\nu}^{i}}\\{\tilde{\calF}_{\mu\nu}^{i}}\\\end{matrix}\right);\\\\\\omega_{S}=\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right)\inSL(2,{\bfR})_{S},
P_{(u)A}^{-2}{}^{C}U_{(u)}{\bar{N}}^{uv}U_{(v)}P_{(u)C}^{-2}{}^{B}=\delta_{A}^{B}{\bar{N}}^{uv}+\mu\alpha(1-i\Omega\Pi)_{A}{}^{B}C_{(u)}^{1/2}{\bar{N}}^{u}(C_{(v)}^{1/2}{\bar{N}}^{v})^{T}
S=\frac{1}{2}\intd\tautr\biggl\{\biggl(D_{0}A^{I}\biggr)^{2}-\frac{1}{2}\biggl[A^{I},A^{J}\biggr]~\biggl[A^{J},A^{I}\biggr]+i{\bar{\lambda}}D_{0}\lambda+i{\bar{\lambda}}{\gamma}^{I}~\biggl[A^{I,}\lambda\biggr]\biggr\}
N_{n}^{(\mathrm{tot})}(x)=-\sum_{{\calS}^{\prime}}\mathrm{sign}\left(\sum_{i=1}^{n-1}\nu_{i}\alpha_{i}\right)N_{n}\left(x+\sum_{i=1}^{n}\nu_{i}\alpha_{i}^{(\mathrm{st})}-2Kl\right),
M=Nm_{0}\left(\frac{2}{3}\frac{\vertE\vert}{m_{0}}+\frac{1}{3}\frac{m_{0}}{\vertE\vert}+\frac{1}{3}\frac{\mu_{0}^{2}}{m_{0}g^{2}N}\int\mathrm{d}^{3}{\bfr}\;\phi^{2}\left(r\right)\right)>0\,.
p_{i}=\frac{\partialL}{\partial\dot{m}^{i}}=h_{ij}\,\dot{m}^{j}+\frac{i}{2}\,\eta_{j}\,\tilde{\Gamma}_{i\,l}^{j}\,\eta^{l}\,,\qquad[p_{i}\,,m^{j}]=-i\delta_{i}^{j}\,.
Z[\zeta^{\mu},\xi^{\mu},\eta^{\mu}]={\calN}\,\int\prod_{x}\{\mathrm{d}v_{\mu}\,\mathrm{d}j^{\mu}\,\mathrm{d}|h|^{2}\,\mathrm{d}\alpha\}\,e^{\mathrm{i}\int\mathrm{d}^{4}x\,{\calL}[\zeta^{\mu},\xi^{\mu},\eta^{\mu};v_{\mu},j^{\mu},|h|,\alpha]}\,
[\bar{E}_{\dot{\gamma}},E_{\alpha\dot{\beta}}]={\check{c}}_{\gamma,\alpha\dot{\beta}}^{\enspace\enspace\enspace\,\sigma\dot{\sigma}}E_{\sigma\dot{\sigma}}+{\check{c}}_{\gamma,\alpha\dot{\beta}}^{\enspace\enspace\enspace\,\sigma}E_{\sigma}+{\check{c}}_{\gamma,\alpha\dot{\beta}}^{\enspace\enspace\enspace\,\dot{\sigma}}\bar{E}_{\dot{\sigma}}
Z^{W}=\int{\calD}\gamma_{++}\,{\calD}c\,{\calD}b^{--}~\mathrm{exp}\left[i\Gamma^{W}(\gamma)+i\int\!d^{2}\xi~b^{--}\hat{\nabla}_{-}(g_{-\alpha}\epsilon^{\alpha\beta}\partial_{\beta}c)\right]~.
{\calK}_{n}(\omega)=\frac{a^{n-3}\Gamma(\frac{n-2}{2})}{2\pi^{\frac{n}{2}}\Gamma(n-2)}\int_{0}^{\infty}d\nu\frac{1}{-(\omega/a)^{2}+\nu^{2}\pmi\epsilon}\,\left|\frac{\Gamma(\frac{n}{2}-1+i\nu)}{\Gamma(i\nu)}\right|^{2},
\frac{\theta_{A_{N-1}}^{0,d}(\tau)}{\eta(\tau)^{N}}=\sum_{m\in{\bfZ}^{N-1}}q^{\frac{d}{2}({}^{t}mA_{N-1}m)}/(q^{\frac{N}{24}}\prod_{n=1}^{\infty}(1-q^{n})^{N}).
\psi(x_{1})=(2\pi)^{-3/2}\intd^{3}p_{1}(m/E_{1})^{1/2}[b_{p_{1}r_{1}}u^{r_{1}}(p_{1})e^{ix_{1}.p_{1}}+d_{p_{1}r_{1}}^{*}v^{r_{1}}(p_{1})e^{-ix_{1}.p_{1}}];
\begin{aligned}{\omega_{m}{^r}\omega_{r}{^n}}&{{}=-\delta_{m}^{n}+K_{m}K^{n},}\\{i_{K}\chi}&{{}=0,}\\{\i_{K}\omega}&{{}=0,}\\{(i_{K}*\chi)_{mnr}}&{{}=\omega_{m}{^l}\chi_{nrl}=\omega_{[m}{^l}\chi_{nr]l}.}\\\end{aligned}
\begin{array}{ccc}{K_{ab}^{-1}=}&{p\left(\begin{array}{cc}{\sqrt{g}}&{{\frac{1}{\sqrt{g}}}}\\{{\frac{1}{\sqrt{g}}}}&{\sqrt{g}}\\\end{array}\right)}&{+(1-p)\left(\begin{array}{cc}{1}&{1}\\{1}&{1}\\\end{array}\right)}\\\end{array}
\begin{array}{rl}{\mathrm{Bekenstein-Verlinde~bound}:}&{\tilde{S}_{BV}=\frac{2\pi}{n}\tilde{M}{\calR},}\\{\mathrm{Bekenstein-Hawking~bound}:}&{\tilde{S}_{BH}=(n-1)\frac{V}{4G_{n}{\calR}},}\\{\mathrm{Hubble~bound}:}&{\tilde{S}_{H}=(n-1)\frac{HV}{4G_{n}}.}\\\end{array}
T_{\phantom{\mu}\nu}^{\mu}=5G_{(5)}^{\mu\alpha\beta\gamma\delta}G_{(5)\nu\alpha\beta\gamma\delta}-\frac{1}{2}\delta_{\phantom{\mu}\nu}^{\mu}G_{(5)}^{2}\,\qquad\mathrm{and}\quadG_{(5)}=dC_{(4)},
\Psi_{\kappa}^{(R)}=\frac1{2^{3/2}\pi}\,e^{\pi\kappa/2}\left\{\Gamma(i\kappa)\left(\frac{mx_{+}}2\right)^{-i\kappa}+\Gamma(-i\kappa)\left(-\frac{mx_{-}}2\right)^{i\kappa}+...\right\}.
ds_{\mathrm{can}}^{2}=\left(1+{\frac{4M}{r}}-{\frac{4g^{2}q^{2}}{r^{2}}}\right)^{-{\frac{1}{2}}}dt^{2}-\left(1+{\frac{4M}{r}}-{\frac{4g^{2}q^{2}}{r^{2}}}\right)^{\frac{1}{2}}d\vec{x}^{2}\.
\begin{aligned}{}&{{}Z_{1}}&{}&{{}Z_{2}}&{}&{{}X}&{}&{{}Y}&{}&{{}V}&{}&{{}W}\\{}&{{}\lambda_{1}}&{}&{{}0}&{}&{{}\lambda_{2}}&{}&{{}\lambda_{3}}&{}&{{}\lambda_{4}}&{}&{{}0.}\\\end{aligned}
A_{4B}=\int_{0}^{1}dxx^{k^{(3)}k^{(4)}-\frac{1}{4}Q^{2}-1}(1-x)^{k^{(2)}k^{(3)}-\frac{1}{4}Q^{2}-1}K(\zeta^{(r)},k^{(r)},x)
{\bf1}_{J}=\int_{0}^{\infty}{\frac{\left(2J+1\right)du}{\left(1+u\right)^{2}}}\sum_{m=0}^{2J}{\binom{2J}{m}}{\frac{u^{m}}{\left(1+u\right)^{2J}}}\vertm\rangle\langlem\vert\,
\begin{array}{rll}{[C(s)]}&{=}&{\alpha_{s+1}\alpha_{s+2}\ldots\alpha_{2a}\beta_{s+1}\beta_{s+2}\ldots\beta_{2b}\\,}\\{\left[\dot{C}(\dot{t})\right]}&{=}&{\dot{\alpha}_{\dot{t}+1}\dot{\alpha}_{\dot{t}+2}\ldots\dot{\alpha}_{2\dot{a}}\dot{\beta}_{\dot{t}+1}\dot{\beta}_{\dot{t}+2}\ldots\dot{\beta}_{2\dot{b}}\\,}\\\end{array}
\left<T_{\mu\nu}(x)\right>_{\mu^{2}}^{{\mathbbH}^{11}}=\frac{m^{2}}{665280\pi^{5}a^{9}}\left(576-820a^{2}b+273a^{4}b^{2}-30a^{6}b^{3}+a^{8}b^{4}\right)a\sqrtb\,\,g_{\mu\nu}^{{\mathbbH}^{11}}(x).
\stackrel{\wedge}{H}=\frac{1}{2}\cdot\left\{Q,Q^{\dagger}\right\}_{q\left(x\right)}=\frac{1}{2}\cdot\left(Q\cdotq\left(x\right)^{-1}\cdotQ^{\dagger}+Q^{\dagger}\cdotq\left(x\right)\cdotQ\right)
M_{\lambda}:=-i\,\langle\vec{q},r;\vec{p},s)|S^{(1)}|\vec{k},\lambda)\,\rangle=\sum_{l,n}c_{l}^{(-\,e)\ast}c_{n}^{(-\,p)}\;\widetilde{M}_{\lambda}(j_{p},j_{q};\vec{k},\lambda).
w^{1}=\frac{\tilde{w}^{1}}{\tilde{w}^{3}}\,\qquadw^{2}=\frac{\tilde{w}^{2}}{\tilde{w}^{3}}\qquad\mathrm{and}\qquadw^{3}=\frac{1}{\tilde{w}^{3}}\.
(1-\gamma)(\partial_{y}+2\ell^{-1})\,h|_{0_{+}}-\frac{2}{3}\,\gamma\,\ell\,\big(\partial_{\lambda}\partial_{\rho}h^{\lambda\rho}-\partial_{\lambda}^{2}h\big)|_{0_{+}}=\frac{1}{6M_{*}^{3}}\,S_{\mu}^{\mu}\,.
\,T_{k}\mid\dots,j_{k},\dots\rangle=\delta_{j_{k-1}j_{k+1}}\sum_{j_{k}^{\prime}=j_{k+1}\pm1/2}\mid\dots,j_{k}^{\prime},\dots\rangle\frac{\sqrt{d_{j_{k}}d_{j_{k}^{\prime}}}}{d_{j_{k+1}}}.
\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(p)=\underbrace{\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)}_{\mathrm{{as\in\(\ref{gao}})}}+\frac{1}{2}\sum_{\mu,\nu=0}^{3}\,p_{\mu}\,p_{\nu}\,\partial_{\mu}\,\partial_{\nu}\,\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)
\mu_{k}^{\mathrm{(in)}}(\eta)=C^{(\mathrm{in)}}\left(1+\mathrm{e}^{-x}\right)^{d+{\frac{1}{2}}}{}_{2}F_{1}\left(b+d+{\frac{1}{2}},-b+d+{\frac{1}{2}};2d+1;1+\mathrm{e}^{-x}\right)
{\calZ}(a_{1},a_{2},\dots,a_{N})~\equiv~\prod_{\mu}\int{\calD}P^{\mu}e^{Tr[A(P^{\mu})^{2}]}\prod_{ij}\delta((1-D_{ij})P_{ij}^{\mu})\prod_{i}\delta(\sum_{j}P_{ij}^{\mu})~~,
S=\frac{2\pi}{e^{2}}\intd^{3}x(K^{-1})_{ab}\left(\frac{1}{2}\partial_{\mu}\phi^{ia}\partial_{\mu}\phi^{ib}+\frac{e^{4}}{\pi^{2}(8\pi)^{2}}\frac{1}{2}\partial_{\mu}\sigma^{a}\partial_{\mu}\sigma^{b}+\frac{i}{2}\chi^{am}\gamma_{\mu}\partial_{\mu}\chi^{bm}\right)
P(0)=-E+q^{2}(\partial\psi)^{2}(0)-\frac14\int_{0}^{1}d\rho\left(e^{\frac{1-\rho}{8}{\partial}^{2}}(-q^{2}{\partial}^{2}+1)\partial\psi\right)\left(e^{-\frac{1-\rho}{8}{\partial}^{2}}\partial\psi\right).
\chi^{(1)}=1/2+1/\sqrt2\,\sum_{s}{\frac{(P_{\mathrm{vec}})_{s}}{|\vec{x}-\vec{x}_{s}|}}\quad,\quad\chi^{(2)}=1/2-1/\sqrt2\,\sum_{s}{\frac{(P_{\mathrm{vec}})_{s}}{|\vec{x}-\vec{x}_{s}|}}
\Gamma^{(1/2)}(A)={\frac{1}{2}}\int_{0}^{\infty}{\frac{ds}{s}}(e^{-m^{2}s}-e^{-\Lambda^{2}s})\intd^{4}x\,\mathrm{tr}\left[<xs|x>-<xs|x>|_{A_{\mu}=0}\right].
{\calL}=\frac{2}{\kappa^{2}}\sqrt{-g}R+\frac{1}{\kappa^{2}\,\xi}\eta_{\mu\nu}\left(\partial_{\rho}\sqrt{-g}g^{\rho\mu}\right)\left(\partial_{\sigma}\sqrt{-g}g^{\sigma\nu}\right)+\partial_{\mu}\chi_{\nu}\frac{\delta\sqrt{-g}g^{\mu\nu}}{\delta\epsilon^{\lambda}}\eta^{\lambda};\;\kappa\equiv\sqrt{32\piG},
A_{i}^{(p)}(x_{0},{\bfx})=-2ix_{0}^{d/2}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}a_{0}({\bfk}){\frac{k_{i}}{|{\bfk}|^{2}}}H(|{\bfk}|x_{0}).
\begin{array}{c}{k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{(4)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{1}\gamma_{\rho\sigma\mu}^{(1)}(k_{2})}\\{+C_{2}\gamma_{\rho\sigma\mu}^{(2)}(k_{2})+C_{3}\gamma_{\rho\sigma\mu}^{(3)}(k_{2})}\\\end{array}
P_{a}=2\sqrt{\beta}\left(\Pi_{a}+\frac{1}{4}C_{\;\cdota}^{cc^{\prime}}M_{cc^{\prime}}\right)\;\;\;\;\leftrightarrow\;\;\;\;P_{0}=2\sqrt{\beta}\Pi_{0},\;P_{j}=2\sqrt{\beta}\left(\Pi_{j}-\frac{1}{2}J_{j}\right),
u^{T}\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)u=\left(\begin{matrix}{A_{0}}&{0}\\{0}&{A_{0}}\\\end{matrix}\right)+\frac{1}{N^{2}}\left(\begin{matrix}{A_{1}+B_{1}}&{0}\\{0}&{A_{1}-B_{1}}\\\end{matrix}\right)+\mathcal{O}(N^{-4})
{\mathrm{Tr}}(\phi^{I_{1}}\ldots\partial_{\mu}\phi^{I_{k}}\ldots\phi^{I_{l}}\cdots\phi^{I_{m}})-{\mathrm{Tr}}(\phi^{I_{1}}\ldots\phi^{I_{k}}\ldots\partial_{\mu}\phi^{I_{l}}\ldots\phi^{I_{m}})
\begin{array}{ccc}{\left(\begin{array}{c}{a}\\{b}\\{\varepsilon^{i}}\\{\rho^{i}}\\{\lambda}\\{\omega_{AB}}\\{T_{i}^{~j}}\\\end{array}\right)~~~~}&{\longrightarrow}&{~~~~\left(\begin{array}{c}{b}\\{a}\\{\rho^{i}}\\{\varepsilon^{i}}\\{-\lambda}\\{\omega_{AB}}\\{T_{i}^{~j}}\\\end{array}\right)}\\\end{array}
\mathrm{dim}_{q}{\tilde{\hat{\rho}}_{\ell}}={\frac{([N][N-1]\ldots[\ell+1])([N+1][N]\ldots[N-\ell+2])}{[N-2\ell]![\ell]!([N-2\ell+2][N-2\ell+3]\ldots[N-\ell+1])}}
I_{\mu_{1}\cdots\mu_{n}}(x_{1},\ldots,x_{N})\equiv\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{k_{\mu_{1}}\cdotsk_{\mu_{n}}}{\prod_{i=1}^{N}(k^{2}+x_{i})(q^{2}+x_{i})}.
\Omega_{\alpha\beta}^{(N)}=\sum_{\begin{array}{c}{{\scriptstyle\alpha_{i}}}\\{{\scriptstylei\in[2,N]}}\\\end{array}}\sum_{\begin{array}{c}{{\scriptstyle\beta_{j}}}\\{{\scriptstylej\in[1,N-1]}}\\\end{array}}\psi_{\alpha\beta_{1}}\epsilon_{\beta_{1}\alpha_{2}}\psi_{\alpha_{2}\beta_{2}}\cdots\epsilon_{\beta_{N-1}\alpha_{N}}\psi_{\alpha_{N}\beta}
\bar{S}_{a}^{b}(z)\bar{S}_{c}^{d}(w)=\frac{\check{g}\Sigma_{ac}^{bd}}{(z-w)^{2}}+\frac{\bar{S}_{(abc)}^{d}(w)}{(z-w)}-\frac{\bar{S}_{c}^{(bad)}(w)}{(z-w)}+...
{\calL}_{0}^{m}{}^{\prime}[A,\Lambda]=-{\frac{1}{8}}(\epsilon_{\mu\nu\rho\sigma}\partial^{\nu}A^{\rho\sigma})^{2}-{\frac{1}{4}}(mA_{\mu\nu}+\partial_{\mu}\Lambda_{\nu}-\partial_{\nu}\Lambda_{\mu})^{2}.
\begin{array}{ll}{{\displaystyle\xi_{1}=\frac{1}{\sqrt{{\mathcal{B}}}}(p_{y}+{\mathcal{B}}x),}}&{{\displaystyle\xi_{2}=\frac{1}{\sqrt{{\mathcal{B}}}}(p_{x}+{\mathcal{B}}y),}}\\{*[0.4cm]{\displaystyle\eta_{1}=\frac{1}{\sqrt{{\mathcal{B}}}}p_{x},}}&{{\displaystyle\eta_{2}=\frac{1}{\sqrt{{\mathcal{B}}}}p_{y}.}}\\\end{array}
\begin{array}{c}{8t_{2}<O_{2}O_{k_{1}}...O_{k_{p-1}}P>_{g}}\\{+4t_{0}<O_{k_{1}}...O_{k_{p-1}}PP>_{g}+(4(p-1)+4\Sigma)<O_{k_{1}}...O_{k_{p-1}}P>_{g}=0,}\\\end{array}
Z(\xi)=\int{\calD}A_{\mu}^{\omega}{\calD}B{\calD}\bar{c}{\calD}ce^{-S_{0}(A^{\omega})-\frac{\xi}{2}\intB^{2}dx+\int[-iB\partial_{\mu}A_{\mu}^{\omega}+\bar{c}(-\partial_{\mu}\partial_{\mu})c]dx}
\Pi^{\alpha}{}_{\beta}\equiv(\gamma^{1}\bar{\gamma}^{2}\gamma^{3}\bar{\gamma}^{4})^{\alpha}{}_{\beta}\,,\qquad(\Pi^{\prime})^{\alpha}{}_{\beta}\equiv(\gamma^{5}\bar{\gamma}^{6}\gamma^{7}\bar{\gamma}^{8})^{\alpha}{}_{\beta}\,.
\bar{W}(\tilde{\phi})\equiv{\tilde{W}}(\tilde{\phi})-{\frac{w_{0}}{\alpha}}=\sum_{n=0}^{\infty}\bar{W}^{(n)}(\tilde{\phi}),\quad\tilde{\phi}(\tilde{y})=\sum_{n=0}^{\infty}\tilde{\phi}^{(n)}(\tilde{y}),\quadA(\tilde{y})=\sum_{n=0}^{\infty}A^{(n)}(\tilde{y}).
^{\beta^{\prime}}\langle\,p^{\prime\,\mu},\,\eta^{\prime}\,\vert\,p^{\mu},\,\eta\,\rangle^{\beta}\,=\,(2\,\pi)^{3}\,2\,p_{0}\,\delta^{3}({\bfp}\,-\,{\bfp}^{\prime})\,\delta_{\eta,\,-\,\eta^{\prime}}\,\delta_{\beta\,\beta^{\prime}}\quad.
S^{\pm}[x]\simeqS_{\mathrm{qc}}[x_{\mathrm{qc}}]\mp\hbar\varphi[x_{\mathrm{qc}}]+\frac{1}{2}\int_{0}^{t}\mathrm{d}\tau\left[\dot{\eta}^{2}-\left(\Phi^{2}\right)^{\prime\prime}\Bigl(x_{\mathrm{qc}}(\tau)\Bigr)\eta^{2}\right].
T(\bar{E})=\left({\frac{\partialS}{\partial\bar{E}}}\right)^{-1}={\frac{E_{P}^{2}}{8\pi\bar{E}}}\left[1+{\calO}\left({\frac{E_{P}^{2}}{\bar{E}^{2}}}\right)\right]\cong{\frac{E_{P}^{2}}{8\piM}}
S=\intd^{4}x(-{\frac{1}{8}}F_{mn}^{\alpha}F^{\alphamn}+{\frac{1}{4(-u_{l}u^{l})}}u^{m}{\calF}_{mn}^{\alpha}{\calF}^{\alphanp}u_{p}-\epsilon^{mnpq}u_{m}\partial_{n}B_{pq}).
\tilde{\chi}(m,\bar{m};\tau_{2})=\int\int_{H}\frac{d^{2}m^{\prime}}{(m_{2}^{\prime})^{2}}<m,\bar{m};\tau_{2}|m^{\prime},\bar{m^{\prime}};\tau_{1}>\tilde{\chi}(m^{\prime},\bar{m^{\prime}};\tau_{1}).
D_{\mu\nu}=\frac{-i}{p^{2}-\mu^{2}}(g_{\mu\nu}-\frac{p_{\mu}n_{\nu}+p_{\mu}}{np}+\frac{n^{2}p_{\mu}p_{\nu}}{(np)^{2}}-\frac{i\mu}{np}\epsilon_{\mu\nu\lambda}n^{\lambda})-i\xi\frac{n^{2}p_{\nu}}{(np)^{2}}\.
\stackrel{(0)}{s}=\varepsilon^{ijk}F_{jk}^{b}k_{ba}\stackrel{\rightarrow}{{\frac{\partial}{\partialA_{a}^{i*}}}}-\partial_{i}A_{a}^{i*}\stackrel{\rightarrow}{{\frac{\partial}{\partialC_{a}^{*}}}}+\partial_{i}C^{a}\stackrel{\rightarrow}{{\frac{\partial}{\partialA_{i}^{a}}}}
B_{n}(F,G):=\sum\Theta^{i_{1}j_{1}}\Theta^{i_{2}j_{2}}\dots\Theta^{i_{n}j_{n}}\partial_{i_{1}}\partial_{i_{2}}\dots\partial_{i_{n}}F\\partial_{j_{1}}\partial_{j_{2}}\dots\partial_{j_{n}}G,
<\sigma_{0}\sigma_{1}>_{1}\left({\frac{1}{\lambda_{1}\lambda_{2}^{3}}}+{\frac{1}{\lambda_{1}^{3}\lambda_{2}}}\right)={\frac{1}{4}}\Gamma_{1}+{\frac{1}{4}}(\Gamma_{2}+\Gamma_{3}+\lambda_{1}\leftrightarrow\lambda_{2})={\frac{1}{2}}{\frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\lambda_{1}^{3}\lambda_{2}^{3}}}.
H_{C}=\intd^{3}x\Bigg[\frac{g^{2}}{2}\left(\Pi_{ai}-\frac{\theta}{8\pi^{2}}B_{ai}\right)^{2}+\frac{1}{2g^{2}}B_{ai}^{2}+\Pi_{ai}\left(D_{i}A_{0}\right)_{a}\Bigg]~,
=-\pi^{2}\beta\int\,d^{3}k\left[\left(\frac{k^{\mu}k^{\nu}}{k^{2}}-g^{\mu\nu}\right)\Pi_{1}^{(1)}(k)+im\varepsilon^{\mu\nu\alpha}k_{\alpha}\Pi_{1}^{(2)}(k)\right]A_{\mu}(k)A_{\nu}^{*}(k)\,\,,
h_{pp^{\prime}}(\infty)=h_{pp^{\prime}}(0)-\int_{0}^{\infty}\!\!dl\sum_{q}\left(\frac{dh_{pq}(l)}{dl}\frac{h_{qp^{\prime}}(l)}{E_{p}(l)-E_{q}(l)}+\frac{h_{pq}(l)}{E_{p^{\prime}}(l)-E_{q}(l)}\frac{dh_{qp^{\prime}}(l)}{dl}\right).
a(k,\rho)\vert2\colonp_{i},k_{i}\rangle=-{\frac{e}{(2\pi)^{3/2}}}{\frac{1}{\sqrt{2k^{+}}}}f(k,\rho\colonp_{i})\vert2\colonp_{i},k_{i}\rangle+\delta^{3}(k-k_{i})\delta_{\rho{\lambda_{i}}}\vert1\colonp_{i}\rangle\;,
\intd\tau\,\deltaM^{2}=-\frac{2\,G_{N}}{\pi\,\ell_{s}^{2}}\int\frac{d^{D}\,k}{(2\pi)^{D}}\,\frac{1}{k^{2}-i\varepsilon}\int\intd^{2}\sigma_{1}\,d^{2}\sigma_{2}\,e^{i\,\ell_{s}^{2}\,k\cdotp\,(\tau_{1}-\tau_{2})}\,W(k,1,2)\,,
\Lambda_{g}^{\mu(1)}(q)=\frac{N_{g}}{16\pi}\frac{e^{2}}{m}\epsilon^{\mu\nu\rho}\frac{q_{\nu}}{m}\,\,\bar{v}^{(+)}({\bfp}_{1}^{\prime})\gamma_{\rho}v^{(-)}({\bfp}_{1}),
\kappa(t)=\kappa+\frac{m^{2}}{(4\pi)^{2}}\left(\xi-\frac{1}{6}\right)A(t),\\\a_{1}(t)=a_{1}+\frac{1}{2(4\pi)^{2}}\left(\xi-\frac{1}{6}\right)^{2}A(t).
\Pi_{a,{\mu\nu}}^{odd}\equiv{\cal{\pi}}_{\mathrm{a},\mu\nu}(p)=\frac{8v^{2}}{\kappa}\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\epsilon_{\mu\nu\lambda}k^{\lambda}cos^{2}(k\wedgep)}{(k^{2}+M^{2})[(p-k)^{2}+m_{\sigma}^{2}]},
S=-m\intd\tau\sqrt{{\dot{x}}^{2}}-\alpha\intd\tau\frac{\sqrt{({\dot{x}}{\ddot{x}})^{2}-{\dot{x}}^{2}{\ddot{x}}^{2}}}{{\dot{x}}^{2}}-\beta\intd\tau\sqrt{{\dot{x}}^{2}}\frac{\varepsilon_{\mu\nu\sigma}\stackrel{.}{x}^{\mu}\stackrel{..}{x}^{\nu}\stackrel{\ldots}{x}^{\sigma}}{({\dot{x}}{\ddot{x}})^{2}-{\dot{x}}^{2}{\ddot{x}}^{2}},
\Gamma^{n}(A)=\intd^{3}y_{1}....d^{3}y_{n}F_{\mu_{1}\nu_{1}}(y_{1})....F_{\mu_{n}\nu_{n}}(y_{n})\Omega^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}(y_{1},...,y_{n})\;,
=\frac{1}{2}\hat{f}^{\mu}\hat{f}_{\mu}-\epsilon^{\alpha\beta\gamma}\hat{f}_{\alpha}\partial_{\beta}(\hat{A}_{\gamma}+\hat{a}_{\gamma})+\frac{m}{2}\epsilon^{\alpha\beta\gamma}(\hat{A}_{\alpha}\partial_{\beta}\hat{A}_{\gamma}+2\hat{a}_{\alpha}\partial_{\beta}\hat{A}_{\gamma})+~O(\theta^{2}).
\omega_{_{(2)}}=M_{_{(2)}}^{^T}f^{-1}M_{_{(2)}}=\left(\begin{array}{cc}{\omega_{(1)}}&{\left[\frac{\partial\Omega_{(2)}}{\partialq}\right]}\\{-\left[\frac{\partial\Omega_{(2)}}{\partialq}\right]^{T}}&{0}\\\end{array}\right)\;.
\partial_{i}Z=(t^{I})_{,i}q_{I}=0\qquad\Longrightarrow\qquadg^{ij}t^{I}{}_{,i}t^{J}{}_{,j}\,q_{I}\equiv\Pi^{IJ}q_{I}=-{\frac{1}{3}}((C^{IJ}-t^{I}t^{J})q_{I}=0
S={\frac{m({\bfx}_{n+1}-{\bfx}_{n})^{2}}{2\varepsilon}}+{\frac{e}{2}}({\bfx}_{n+1}-{\bfx}_{n})\cdot({\bfA}({\bfx}_{n+1})+{\bfA}({\bfx}_{n}))\,,
\int_{\Sigma}g(B^{-1}+iB^{-1}A)({\bf{1}}+iA)g-2\int_{\Sigma}g(B^{-1}+iB^{-1}A)g=-\int_{\Sigma}g(B^{-1}+B^{-1}A^{2})g=\int_{\Sigma}gDg
D^{0}\psi-\left(\beta^{0}\right)^{2}D_{0}\psi-\beta^{i}\beta^{0}D_{i}\psi-\frac{e}{2m}F_{\alpha\mu}\left(\beta^{\mu}\beta^{0}\beta^{\alpha}+\beta^{\mu}\eta^{0\alpha}\right)\psi=0,
\left\{\deltaS_{0}\left[{\frac{\delta}{\delta\eta_{a}}},{\frac{\delta}{\delta\bar{\eta}_{a}}},{\frac{\delta}{\deltaJ_{a}^{\mu}}}\right]+\deltaS_{T}\left[{\frac{\delta}{\delta\eta_{a}}},{\frac{\delta}{\delta\bar{\eta}_{a}}},{\frac{\delta}{\deltaJ_{a}^{\mu}}}\right]\right\}Z\left[J_{\mu},\eta,\bar{\eta}\right]\,=\,0\,.
+i[({\frac{a}{2\alpha^{2}}}-{\frac{1}{4\alpha^{2}g\partial^{2}}})g^{\mu\nu}\delta(x-y)-({\frac{a}{2\alpha^{2}}}+g){\frac{\partial^{\mu}\partial^{\nu}}{\partial^{2}}}\delta(x-y)+{\frac{1}{2\alpha\partial^{2}}}(1+{\frac{a}{2\alpha^{2}g}})\epsilon^{\mu\nu\lambda}\partial_{\lambda}\delta(x-y)].
a_{[N+2k]\times[2k]}=\left(\!\begin{array}{c}{u_{[N]\times[2k]}}\\{a_{[2k]\times[2k]}^{\prime}}\\\end{array}\!\right),\qquadb_{[N+2k]\times[2k]}=\left(\!\begin{array}{c}{0_{[N]\times[2k]}}\\{1_{[2k]\times[2k]}}\\\end{array}\!\right).
\Omega(a_{0},a_{1};b_{0},b_{1})=\left\langle{\frac{\mathrm{Det}(1-a_{1}U)\,\mathrm{Det}(1-b_{1}\bar{U})}{\mathrm{Det}(1-a_{0}U)\,\mathrm{Det}(1-b_{0}\bar{U})}}\right\rangle\;.
{\frac{\partialS_{0}}{\partial\Omega_{p+1}}}=-{\frac{B}{2\Omega_{p+1}\,(p+1)!}}\left(\,\sigma_{\mu_{1}\dots\mu_{p+1}}-\sigma_{0)\mu_{1}\dots\mu_{p+1}}\right)^{2}-\frac{m_{p+1}}{2}\,p\,\Omega_{p+1}\,
b_{\scriptscriptstyle{L-1,L+1}}/b_{\scriptscriptstyle{L,L+1}}=\frac{(ika-1)\,a\,\psi_{\scriptscriptstyle{L,L+1}}^{>}(k,a)}{(\gamma_{\scriptscriptstyle{L+1,L+1}}+\frac{1}{2})\,\psi_{\scriptscriptstyle{L+1,L+1}}^{>}(k,a)}\,(1+\mathrm{O}(ka)^{2})\,,
R_{\frac{1}{2}\frac{1}{2}}^{0}(\theta)=R_{\frac{1}{2}\frac{1}{2}}^{1}(\theta)=2^{-\theta/\pii}\prod_{k=1}^{\infty}\left[\frac{\Gamma(k-\frac{\theta}{2\pii})\Gamma(k-\frac{\theta}{2\pii})}{\Gamma(k-\frac{1}{4}-\frac{\theta}{2\pii})\Gamma(k+\frac{1}{4}-\frac{\theta}{2\pii})}/\left\{\theta\,\leftrightarrow-\theta\right\}\right]=2^{-\theta/\pii}P(\theta)
\begin{aligned}{ds^{2}}&{{}=2dx^{+}dx^{-}+\sum_{i,j}^{8}A_{ij}x^{i}x^{j}(dx^{-})^{2}+\sum_{i=1}^{8}(dx^{i})^{2}}\\{F_{5}}&{{}=\mudx^{-}\wedge(dx^{1234}+dx^{5678})~.}\\\end{aligned}
Z_{k}(\lambda_{1}^{2},\lambda_{2}^{2},q^{2};\lambda^{2})=\int_{0}^{1}dzz^{k}ln\left(\frac{q^{2}z(1-z)+(\lambda_{1}^{2}-\lambda_{2}^{2})z-\lambda_{1}^{2}}{(-\lambda^{2})}\right),
{\calL}={\calR}\sqrt{{\cal-G}}=-(k^{ij}-hg^{ij})\sqrt{g}\frac{dg_{ij}}{ds}-(N{\calH}_{0}+N^{i}{\calH}_{i})-2\frac{d}{ds}(h\sqrt{g})-\nabla_{i}\varphi^{i}.
\hat{G}_{\delta}^{Box}[a,b;E]=\hat{{\calG}}_{0}^{Box}[a,b;E]-\frac{\hat{{\calG}}_{0}^{Box}[a,L;E]\hat{{\calG}}_{0}^{Box}[L,b;E]}{\frac{1}{v_{2}}+\hat{{\calG}}_{0}^{Box}[L,L^{-};E]}
W_{2}(R)={\frac{\sqrt{2}}{R}}BJ_{1}B^{-1}=\left(\begin{matrix}{0}&{0}&{0}&{0}\\{0}&{0}&{-i/R^{2}}&{i}\\{0}&{i}&{0}&{0}\\{0}&{i/R^{2}}&{0}&{0}\\\end{matrix}\right),
F_{0,0,1}(0,d_{2};0,d_{2})={\frac{1}{d_{2}^{2}(d_{2}-1)!}}{\frac{1}{\lambda_{1}^{d_{2}-1}}}{\prod_{l=1}^{d_{2}-1}\left((\lambda_{2}-2\lambda_{1})d_{2}+{l}\lambda_{1}\right)}.
\hat{X}^{\dagger}=\left(\begin{array}{cc}{X^{*}}&{\mathrm{sgn}\,(M_{2})|M_{2}|^{1/2}}\\{-|M_{1}|^{1/2}}&{\mathrm{sgn}\,(M_{1})X|_{{\calM}_{2}}\mathrm{sgn}\,(M_{2})}\\\end{array}\right).
\delta_{\epsilon}b=-\bar{\epsilon}\Gamma_{11}\Gamma_{m}d\theta\left(dX^{m}+{\frac{1}{2}}\bar{\theta}\Gamma^{m}d\theta\right)+{\frac{1}{2}}\bar{\theta}\Gamma_{11}\Gamma_{m}d\theta\bar{\epsilon}\Gamma^{m}d\theta.
\left(\begin{matrix}{1}&{0}&{-\sqrt{2}\xi}&{\sqrt{2}\xi}\\{0}&{1}&{0}&{0}\\{\sqrt{2}\xi}&{0}&{1-\xi^{2}}&{\xi^{2}}\\{\sqrt{2}\xi}&{0}&{-\xi^{2}}&{1+\xi^{2}}\\\end{matrix}\right).