harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g(\underline{{x}})=\bigl(\prod_{s=1}^{l}\frac{\partial}{\partialy_{j_{s}}^{0}}\bigr)^{K}\sum_{|\underline{{m}}|\leqK-1,m_{j_{1}}^{0}=...=m_{j_{l}}^{0}=0}(\frac{\partial}{\partial\underline{{y}}})^{\underline{{m}}}\nu_{\underline{{m}}}(\underline{{y}}),
\frac{\partial}{\partial\theta}\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I(p,m,\theta,\varepsilon)\mid_{_{\theta_{0}^{-}}}\neq\frac{\partial}{\partial\theta}\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I(p,m,\theta,\varepsilon)\mid_{_{_{\theta_{0}^{+}}}}
\begin{array}{ccccc}{E_{\mu}^{(1)}}&{\equiv}&{(e_{y}{}^{4})^{-1}\partial_{y}e_{\mu}{}^{4}}&{\mathrm{for}}&{\xi^{(1)},}\\{\psi_{\mu}^{(1)}}&{\equiv}&{\partial_{y}\psi_{\mu-}-E_{\mu}{}^{(1)}\psi_{y-}}&{\mathrm{for}}&{\varepsilon^{(1)},}\\{V_{\mu}^{(1)1,2}}&{\equiv}&{\partial_{y}V_{\mu}^{1,2}-E_{\mu}^{(1)}V_{y}^{1,2}}&{\mathrm{for}}&{\theta_{1,2}^{(1)},}\\{\vdots}&{}&{\vdots}&{}&{\vdots}\\\end{array}
\bar{\Lambda}_{\gamma}H^{-1}=\bar{\Lambda}_{\gamma}D_{\gamma}^{\-1}\left(\begin{array}{cc}{2}&{-1-e^{-2\pi\gamma}\textit{\textbf{I}}}\\{-1-e^{2\pi\gamma}\textit{\textbf{I}}^{-1}}&{2}\\\end{array}\right),
\Gamma_{Q}[\phi]=\frac{1}{2}\overline{{m}}_{ij}\phi^{i}\phi^{j}\,,\qquad\Gamma_{I}[\phi]=\sum_{n\ge0}\frac{1}{n!}\overline{{g}}_{i_{1}\dotsi_{n}}\phi^{i_{1}}\cdots\phi^{i_{n}}\,,
\phi_{_L}^{\uparrow}+\Theta\phi_{_L}^{\downarrow\,\,\ast}\equiv\Theta{\calK}\xi_{odd}^{^L}=\left(\begin{matrix}{E^{2}+iE^{1}}\\{-E^{0}-iE^{3}}\\\end{matrix}\right)\quad,\quad\phi_{_L}^{\uparrow}-\Theta\phi_{_L}^{\downarrow\,\,\ast}\equiv-\Theta{\calK}\xi_{even}^{^L}=\left(\begin{matrix}{B^{1}-iB^{2}}\\{-B^{3}+iB^{0}}\\\end{matrix}\right)\quad.
S=\frac{1}{\beta^{2}}\intd^{4}x~d^{2}\theta~d^{2}\bar{\theta}~tr\left[-\frac{1}{2}(e^{-V}D^{\alpha}e^{V})\bar{D}^{2}(e^{-V}D_{\alpha}e^{V})-e^{-V}\bar{\Sigma}e^{V}\Sigma\right]\,.
\begin{array}{l}{J_{\pm}^{\prime}\tilde{\psi}_{JM}({\bfx})=\sqrt{[J\mpM]_{q}[J\pmM+1]_{q}}\tilde{\psi}_{J,M\pm1}({\bfx})~,}\\{J_{3}^{\prime}\tilde{\psi}_{JM}({\bfx})=M\tilde{\psi}_{JM}({\bfx})~.}\\\end{array}
(q^{2}+y^{2}M_{\ast}^{2})^{k}\,C_{2k}^{\frac{1}{2}d-1}\left(\frac{y\left(q^{2}+M_{\ast}^{2}\right)}{\sqrt{q^{2}+y^{2}M_{\ast}^{2}}}\right)=\sum_{l=0}^{2k}B_{l}\,(q^{2},M_{\ast}^{2})\,C_{l}^{\frac{1}{2}d-1}(y)
\sum_{r,r^{\prime}=0}^{\infty}d(r)d(r^{\prime})e^{-2\pi\tau_{2}(r+r^{\prime})}\int_{-1/2}^{1/2}{d\tau_{1}}e^{2\pii\tau_{1}(r-r^{\prime}+pl)}=\sum_{r=0}^{\infty}d(r)d(r+pl)e^{-2\pi\tau_{2}(2r+pl)}
\mathcal{L}^{\mathrm{SUGRA}}\,=\,\dots\,+\,\mathrm{const}\,\left(\,S_{AB}(\phi)\,\psi_{\mu}^{A}\,\gamma^{\mu\nu}\,\psi_{\nu}^{B}\,+\,S^{AB}(\phi)\,\psi_{A\vert\mu}\,\gamma^{\mu\nu}\,\psi_{B\vert\nu}\,\right)
\begin{array}{cl}{}&{\displaystyle\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r(5r+1)/2-2ar}\left[\begin{array}{c}{n}\\{\left[\frac{n-5r}{2}\right]+a}\\\end{array}\right]_{q}}\\{=}&{\displaystyle\sum_{j=0\atop2j\leqn-a}q^{j^{2}+aj}\left[\begin{array}{c}{n-j-a}\\{j}\\\end{array}\right]_{q},}\\\end{array}
\mathbf{s}_{m}^{a}C^{\alpha_{0}b}=Z_{\alpha_{1}}^{\alpha_{0}}C^{\alpha_{1}ab}+\epsilon^{ab}B^{\alpha_{0}}-\mathrm{~\frac{1}{2}~}F_{\beta_{0}\gamma_{0}}^{\alpha_{0}}C^{\beta_{0}a}C^{\gamma_{0}b},
\mathrm{R}e\:\frac{d}{ds}\zeta_{H}(-2m;ix)=-\pi\:\mathrm{Im}\:\frac{{B_{2m+1}(ix)}}{2(2m+1)}+\frac{(-1)^{m+1}x^{2m}}{2}\mathrm{ln}\:x+\frac{(-1)^{m}(2m)!}{2(2\pi)^{2m}}\sum_{n=1}^{+\infty}\frac{e^{-2n\pi{x}}}{n^{1+2m}}
\left(\begin{array}{cccc}{(2-E)}&{a_{-}}&{b_{+}}&{0}\\{a_{+}}&{-E}&{0}&{b_{-}}\\{b_{-}}&{0}&{-E}&{a_{+}}\\{0}&{b_{+}}&{a_{-}}&{(-2-E)}\\\end{array}\right)\left(\begin{array}{c}{\psi_{11}}\\{\psi_{21}}\\{\psi_{12}}\\{\psi_{22}}\\\end{array}\right)=0.
\left(\cdots+2g_{2}^{(\mu_{1}\mu_{2}}\varepsilon_{s-1,s-2}^{\mu_{3}\cdots\mu_{s})}-\varepsilon_{s-1,s-1}^{\mu(\mu_{3}\cdots\mu_{s}}\partial_{\mu}g_{2}^{\mu_{1}\mu_{2})}+\partial^{(\mu_{1}}\varepsilon_{s-1,s-1}^{\mu_{2}\cdots\mu_{s})}\right)\left(g_{2}\right)_{\mu_{1}\mu_{2}}\left(g_{2}\right)_{\mu_{3}\mu_{4}}=0.
\mathcal{G}_{\mu\nu}=\eta_{MN}\partial_{\mu}X^{M}\partial_{\nu}X^{N}-2\eta_{MN}\partial_{\mu}X^{M}\overline{{\theta}}\Gamma^{N}\partial_{\nu}\theta+\eta_{MN}(\overline{{\theta}}\Gamma^{M}\partial_{\mu}\theta)(\overline{{\theta}}\Gamma^{N}\partial_{\nu}\theta)
W^{\Omega_{z_{1}},\Omega_{z_{2}}^{*}}\left(\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right)=(-1)^{n-1+\lambda+\nu}W^{(1,n-1)}\left(\left.\begin{array}{cc}{c}&{d}\\{b}&{a}\\\end{array}\right|z_{1}-z_{2}\right),
{\frac{\ddot{a}}{a}}=\left(1+{\frac{g}{g^{\prime}}}{\frac{\partial}{\partialr}}\right){\frac{1}{4}}\dot{r}^{2}\left({\frac{g^{\prime}}{g}}\right)^{2}=\left[1+{\frac{1}{2}}a{\frac{\partial}{\partiala}}\right]{\frac{1}{4}}\dot{r}^{2}\left({\frac{g^{\prime}}{g}}\right)^{2}=-{\frac{4\pi}{3}}(\rho_{\mathrm{eff}}+3p_{\mathrm{eff}}),
R(u)=\left(\begin{array}{ccc|ccc|ccc}{c}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{b}&{}&{e}&{}&{}&{}&{}&{}\\{}&{}&{d}&{}&{g}&{}&{f}&{}&{}\\\hline{}&{\bar{e}}&{}&{b}&{}&{}&{}&{}&{}\\{}&{}&{\bar{g}}&{}&{a}&{}&{g}&{}&{}\\{}&{}&{}&{}&{}&{b}&{}&{e}&{}\\\hline{}&{}&{\bar{f}}&{}&{\bar{g}}&{}&{d}&{}&{}\\{}&{}&{}&{}&{}&{\bar{e}}&{}&{b}&{}\\{}&{}&{}&{}&{}&{}&{}&{}&{c}\\\end{array}\right),
C_{J}=12\sqrt{2}\piV_{6}N^{2}\lambda^{-3/2}\sqrt{r_{0}}r_{H}\frac{(r_{H}^{2}-l^{2})(r_{H}^{2}+l^{2})}{5l^{4}+8l^{2}r_{H}^{2}-r_{H}^{4}}
V(\theta)=(2N_{F}-2){\frac{4\pi^{2}{\bar{\beta}}^{2}}{L^{4}}}\biggl[{\frac{N}{12}}({\bar{\beta}}-1)^{2}+{\frac{N(N-1)}{12}}({\bar{\beta}}^{2}+1)-{\frac{1}{3}}(N-1)(N-2)\biggr],
{\mathbfb}=\left(\,\begin{array}{ccc}{0}&{b_{12}}&{0}\\{0}&{0}&{b_{23}}\\{b_{31}}&{0}&{0}\\\end{array}\,\right)\,\qquad{\mathbfd}=\left(\,\begin{array}{ccc}{0}&{d_{12}}&{0}\\{0}&{0}&{d_{23}}\\{d_{31}}&{0}&{0}\\\end{array}\,\right)
\begin{array}{lcl}{0}&{=}&{{{\mathcalL}_{n}}(u)\starK[a]-K[a]\star{{\mathcalL}_{n}}(V)}\\{}&{=}&{(u_{3}-V_{3}+2n\thetaa_{2}^{\prime})p^{n-3}}\\{}&{+}&{(u_{4}-V_{4}+(n-3)\theta(V_{3}^{\prime}-u_{3}^{\prime})+2n\thetaa_{3}^{\prime}+4n\theta^{2}a_{2}^{\prime\prime})p^{n-4}}\\{}&{}&{\vdots}\\\end{array}
\prod_{i<j}^{N}(\xi_{i}-\xi_{j})^{-\lambda}\frac{1}{\omega}\left[-\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial\xi_{i}^{2}}+\frac{1}{2}\sum_{i\neqj}^{N}\frac{\lambda(\lambda-1)}{(\xi_{i}-\xi_{j})^{2}}+\frac{\omega^{2}}{2}\sum_{i=1}^{N}\xi_{i}^{2}\right]\prod_{i<j}^{N}(\xi_{i}-\xi_{j})^{\lambda}=-\frac{1}{\omega}T_{+}+\omegaT_{-}\equiv2L_{0}.
\tilde{g}_{1}=-\xi\left[\frac{132.71}{\zeta^{a_{1}}}+\frac{10.19}{\zeta^{b_{1}}}+\frac{8.11}{\zeta^{c_{1}}}\right]+\frac{28.20}{\zeta^{a_{2}}}+\frac{2.26}{\zeta^{b_{2}}}+\frac{8.11}{\zeta^{c_{2}}}
\begin{array}{lll}{X\equivX^{i=5}\quad}&{X^{i}\sim\epsilon^{ijklm}\Omega_{jk}\Omega_{lm}}&{\qquad10\times10\rightarrow5^{*}}\\{Y\equivY_{i=5}\quad}&{Y_{i}\sim\Phi^{j}\Omega_{ji}}&{\qquad5^{*}\times10\rightarrow5}\\{S}&{S\sim\mathrm{tr}\left(W^{\dot{\alpha}}W_{\dot{\alpha}}\right)}&{\qquad24\times24\rightarrow1}\\\end{array}
<0|T\left(\hat{\phi}(x_{f})\hat{\phi}(x_{i})\right)|0>=\int_{(\infty)}\frac{d^{4}k^{\mu}}{(2\pi)^{4}}\,e^{-ik\cdot(x_{f}-x_{i})}\,\frac{i}{k^{2}-m^{2}+i\epsilon}\,
|\Psi_{\mathrm{one\particle}}^{\uparrow,\downarrow}\rangle=\sqrtZ\,\,\delta(1-x)\,\,\delta(\vec{k}_{\perp}=\vec{0}_{\perp})\,\,\left|s_{\mathrm{f}}^{z}=\pm{\frac{1}{2}}\right>\,
<A_{1}|z>=\left(\frac{k\tau_{2}}{2\pi^{2}}\right)^{1/4}\,e^{-\frac{1}{2}|z|^{2}}\,e^{\frac{ik\tau}{4\pi}A_{1}^{2}}\,e^{-\frac{1}{2}z^{2}+zA_{1}\sqrt{\frac{k\tau_{2}}{\pi}}}\\\,
\begin{array}{lcl}{p~\partial_{\sigma}V^{(J)}}&{=}&{{\displaystyle2~\partial_{\tau}X^{(J)}~\partial_{\sigma}X^{(J)}}}\\{}&{}&{}\\{p~\partial_{\tau}V^{(J)}}&{=}&{{\displaystyle\frac{\alpha}{\tau^{2}}~(X^{(J)})^{2}+(\partial_{\tau}X^{(J)})^{2}+(\partial_{\sigma}X^{(J)})^{2}}}\\\end{array}
\delta_{X}x_{\alpha\dot{\alpha}}\bar{u}^{\dot{\alpha}}=\epsilon_{X}u_{\alpha}(\bar{u}_{\dot{\alpha}}\bar{u}^{\dot{\alpha}})=0,\quad\delta_{T}z_{\alpha\beta}u^{\beta}=\epsilon_{T}u_{\alpha}(u_{\beta}u^{\beta})=0
\eta_{1}=\left(1+\sum_{j=2}^{r}|\xi_{j}|^{2}+\sum_{j=1}^{r}|\xi_{-j}|^{2}\right)^{-{\frac{1}{2}}},\quad\eta_{j}=\xi_{j}\eta_{1},\quad\eta_{-j}=\xi_{-j}\eta_{1},\qquad2\leqj\leqr,
{\calL}_{gauge}=-\frac{1}{4}\sum_{j=0}^{N}(\sum_{n=0}^{N}a_{jn}\partial_{[\mu}A_{\nu]}^{n}+g_{L}f^{abc}\sum_{n=0}^{N}\sum_{m=0}^{N}a_{jn}a_{jm}A_{\mu}^{n}A_{\nu}^{m})^{2}
E_{D}(B)-E_{D}(B=0)=\frac{eB}{2\pi}V\int_{0}^{\infty}\frac{ds}{\sqrt{\pis}}\frac{d}{ds}\left(\frac{e^{-m^{2}s}}{1-e^{-2eBs}}-\frac{e^{-m^{2}s}}{2eBs}\right)\;.
\left.E_{\Pi}^{\\N}\right|=\left[\begin{array}{ccc}{e_{\mu}^{\\m}}&{\frac{1}{2}\psi_{\mu}^{\\A}}&{\frac{1}{2}\psi_{\mu}^{\\\dot{A}}}\\{0}&{\delta_{B}^{\\A}}&{0}\\{0}&{0}&{\delta_{\dot{B}}^{\\\dot{A}}}\\\end{array}\right]
2PS\equiv\sum_{n}\int\frac{d^{d}k_{1}^{n}}{(2\pi)^{d-1}E_{1}^{n}}\frac{d^{d}k_{2}^{n}}{(2\pi)^{d-1}E_{2}^{n}}\delta^{(d)}(P-k_{1}^{n}-k_{2}^{n}),
S_{int}\,=\,e\,\sum_{a=1}^{N}\intdt\,A_{k}({\vec{x}}_{a}(t))\,\frac{dx_{a}^{k}(t)}{dt}\;=\;\frac{b}{2}\,\intdt\,\sum_{a=1}^{N}\,{\dot{x}}_{a}^{j}(t)\epsilon_{jk}x_{a}^{k}(t)
D^{4}{\calF}({\calA})={\calF}^{\prime}D^{4}{\calA}+3D^{\alpha{\mathbfi}}{\calF}^{\prime}D_{\alpha{\mathbfi}}^{3}{\calA}+\frac{3}{2}D^{{\mathbfi}{\mathbfj}}{\calF}^{\prime}D_{{\mathbfi}{\mathbfj}}{\calA}-\frac{3}{2}D^{\alpha\beta}{\calF}^{\prime}D_{\alpha\beta}{\calA}-D^{3\alpha{\mathbfi}}{\calF}^{\prime}D_{\alpha{\mathbfi}}{\calA},
\hat{a}\frac{d\rho_{i}}{d\hat{a}}=a\frac{\partial\rho_{i}}{\partiala}+\frac{\beta}{\beta^{\prime}}\frac{\partial\rho_{i}}{\partial\phi_{i}}=-3(\rho_{i}+p_{i}),\,\,\qquadi=\mathrm{M},\mathrm{R}\,\,,
ad\left(\hat{G}_{\mu\nu}\right)\rightarrow2\left(x_{\mu}\frac{\partial}{\partialx_{\nu}}-x_{\nu}\frac{\partial}{\partialx_{\mu}}-w_{\mu\lambda}\frac{\partial}{\partialw_{\lambda\nu}}+w_{\nu\lambda}\frac{\partial}{\partialw_{\lambda\mu}}\right).
\bar{\calL}_{c}=-\frac{1-\rho^{2}}4\left(\bar{f}_{\mu\nu}^{i}\right)^{2}-\frac14\left(\bar{B}_{\mu\nu}^{i}\right)^{2}+\fraci2\rho\bar{B}_{\mu\nu}^{i}{}^{\ast}\!\bar{f}^{\mu\nui}-\frac1{2\beta}\left(\partial^{\mu}\bar{A}_{\mu}^{i}\right)^{2}.
\Omega_{\nu}^{-1}\equiv\intd^{4}x\left(c^{a}\partial_{\nu}C^{*a}-A_{\mu}^{a}\partial_{\nu}A^{*a\mu}+\xi_{\mu}^{a}\partial_{\nu}\xi^{*a\mu}-\phi^{a}\partial_{\nu}\phi^{*a}-\frac12\;B_{\mu\tau}^{a}\partial_{\nu}B^{*a\mu\tau}\right)\;,
0\rightarrow{\calE}\rightarrow\bigoplus_{{\itcodim}\,\sigma=0}\sigma\otimes{\calE}\rightarrow\bigoplus_{{\itcodim}\,\sigma=1}\sigma\otimes{\calE}\rightarrow\cdots\rightarrow\bigoplus_{{\itcodim}\,\sigma=n-1}\sigma\otimes{\calE}\rightarrow\emptyset\otimes{\calE}\rightarrow0
\tilde{{\mathcalH}}^{\,^{\bullet}}\:=\,displaystyle\bigoplus_{k=0}^{\infty}\\tilde{{\mathcalH}}^{k}\,\quad\tilde{{\mathcalH}}^{k}\:=\\overline{{\tilde{\Omega}^{k}({\mathcalA})}}^{(\cdot,\cdot)}\.
Re\,\tilde{\Pi}_{F}^{(q)\,\nu\mu}(Q)=-\frac{g^{2}}{2}\,N_{f}\,Re\int\frac{d^{\,4}K}{(2\pi)^{4}}\,tr\left[\gamma^{\mu}\,S^{(11)}(K-Q)\,\gamma^{\nu}\,S^{(11)}(K)\right]\,.
K_{k}^{\mathrm{II}}(\xi)_{2a,2c}=\frac{\vartheta_{1}(\frac{\nu-\xi+2a}{k},\frac{\tau}{k})\,\vartheta_{1}^{\prime}(0,\frac{\tau}{k})}{k\,\vartheta_{1}(\frac{\nu+c+a}{k},\frac{\tau}{k})\,\vartheta_{1}(\frac{-\xi-c+a}{k},\frac{\tau}{k})}\cdot
{\frac{\alpha}{\pi}}<F_{\mu\nu}^{a}F_{\mu\nu}^{a}>=NM^{4}{\frac{1}{2}}\left(-{\frac{1}{40\pi^{2}}}+{\frac{1}{24\pi^{2}}}\right)={\frac{N}{120\pi^{2}}}M^{4}=0.008~Gev^{4}
ds^{2}=-2dx^{+}dx^{-}-\beta^{2}\sum_{i=1}^{2}(\tilde{x}^{i})^{2}(dx^{+})^{2}+\sum_{i=1}^{2}(d\tilde{x}^{i})^{2}+\sum_{i=3}^{8}(dx^{i})^{2},
{\calL}=\sum_{i=1}^{N}\left[{\frac{m}{2}}\dot{\mathrm{\bfx}}_{i}^{2}+q\dot{\mathrm{\bfx}}_{i}\cdot\mathrm{\bfA}_{i}+q\left\{-\mathrm{\bfa}_{0}(\mathrm{\bfx}_{i})+\dot{\mathrm{\bfx}}_{i}\cdot\mathrm{\bfa}_{i}\right\}\right]+{\frac{1}{2\alpha}}\intd^{2}x\epsilon_{\rho\sigma\tau}{a}_{\rho}\partial_{\sigma}{a}_{\tau},
\frac{\partial\mathrm{ln}Z_{1}(u,r_{0})}{\partialr_{0}}=-\frac{r_{0}}{\pi(1-r_{0}^{2})}\int_{r_{0}\le|z|\le1}d^{2}z\,\Bigl(\frac{T_{zz}}{\bar{z}^{2}}+\frac{T_{\bar{z}\bar{z}}}{z^{2}}\Bigr)
{\calL}_{1}=(X+\bar{X})R-{\frac{i}{2}}(X-\bar{X})F-2\lambda_{-}\rho^{-}+2\lambda_{+}\rho^{+}+2\tilde{\lambda}^{-}{\tilde{\rho}}_{--}2\tilde{\lambda}^{+}{\tilde{\rho}}_{+}+4i(M\bar{H}-\bar{M}H)E^{z}E^{\bar{z}}.
S[X]=\frac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d\taud\sigma\Bigl\{\partial_{a}X^{M}\partial^{a}X^{N}g_{MN}-i2\pi\alpha^{\prime}\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}\Bigr\}~,
q^{\Lambda}=\left(\begin{array}{l}{q^{1}=\mathrm{charge~with~respect~to~B^{[2]}~(Neveu~Schwarz~2~--form)}}\\{q^{2}=\mathrm{charge~with~respect~to~C^{[2]}~(Ramond~2~--form)}\}\\\end{array}\right)
\tilde{J}^{-{\frac{1}{n}}}\equiv\tilde{f}_{n}=\left|\begin{matrix}{\partial_{1}\tilde{s}_{1}}&{\cdots}&{\partial_{n-1}\tilde{s}_{1}}\\{\vdots}&{\ddots}&{\vdots}\\{\partial_{1}\tilde{s}_{n-1}}&{\cdots}&{\partial_{n-1}\tilde{s}_{n-1}}\\\end{matrix}\right|^{-{\frac{1}{n}}}\quad,
\hat{x}_{12}=x_{12}+{\frac{2i}{(12)}}\left[(1^{-}2)\theta_{1}^{+}\bar{\theta}_{1}^{+}-(12^{-})\theta_{2}^{+}\bar{\theta}_{2}^{+}+\theta_{1}^{+}\bar{\theta}_{2}^{+}+\theta_{2}^{+}\bar{\theta}_{1}^{+}\right]\.
{\bf\Sigma}(p_{f})={\frac{e^{2}}{\beta}}\sum_{n}\int{\frac{d^{2}k}{(2\pi)^{2}}}{\frac{\gamma_{\mu}\gamma_{\nu}A(k_{f})}{k\!\!\!/_{f}A(k_{f})+B(k_{f})}}\left[D_{L}(q_{b})A_{\mu\nu}+D_{T}(q_{b})B_{\mu\nu}+{\frac{\epsilon(q_{b})}{q_{b}^{2}}}{\frac{q_{\mub}q_{\nub}}{q_{b}^{2}}}\right].
O_{0}=\int_{0}^{1}{\frac{dx}{1-x^{2}}}\bigl(x^{-1/6}-x^{1/6}\bigr)={\frac{1}{2}}\Bigl(\psi\Bigl({\frac{7}{12}}\Bigr)-\psi\Bigl({\frac{5}{12}}\Bigr)\Bigr).
{\delta_{\rho}}{\hat{G}_{K}}(\xi,\xi^{\prime})=-{\frac{1}{2\oint{d}\hat{s}(\xi^{\prime\prime\prime})}}\oint{d}\hat{s}(\xi^{\prime\prime})\rho(\xi^{\prime\prime})\left[{\hat{G}_{K}}(\xi^{\prime\prime},\xi)+{\hat{G}_{K}}(\xi^{\prime\prime},\xi^{\prime})\right].
\displaystyle{E^{TE}(a)=\left.\frac{\hbar\,c}{a}\,\sum_{\nu=3/2}^{\nu_{0}}\nu\,\left[\zeta_{\nu}^{TE}(s,0^{+})-\zeta_{\nu}^{TE}(s,x)\right]\right|_{s\rightarrow-1}}.
c_{1}(\mathcal{N})=n\left(\frac{1}{2}+\lambda\right)\sigma+\left(\frac{1}{2}-\lambda\right)\pi_{C}^{*}\eta+\left(\frac{1}{2}+n\lambda\right)\pi_{C}^{*}c_{1}(B)+\textstyle{\sum_{i}}\kappa_{i}N_{i}
M^{ab*}(n_{1}n_{2}nk,n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime})=\frac{Norm(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime})}{Norm(n_{1}n_{2}nk)}\overline{{M^{ab}(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime},n_{1}n_{2}nk)}}
K(k_{1},k_{2},\omega\tau;k_{1}^{\prime},k_{2}^{\prime},\omega\tau^{\prime})=-\frac{g^{2}}{4m^{2}(Q^{2}+\mu^{2})}\left[\bar{u}_{\sigma_{2}}(k_{2})u_{\sigma_{2}^{\prime}}(k_{2}^{\prime})\right]\,\left[\bar{u}_{\sigma_{1}}(k_{1})u_{\sigma_{1}^{\prime}}(k_{1}^{\prime})\right],
\Gamma_{a}^{(13)}=\left(\begin{matrix}{0}&{\Gamma_{b}^{(11)}}\\{\Gamma_{b}^{(11)}}&{0}\\\end{matrix}\right)\,,\quad\Gamma_{0}^{(13)}=\left(\begin{matrix}{0}&{{\bf1}_{2}}\\{-{\bf1}_{2}}&{0}\\\end{matrix}\right)\,,\quad\Gamma_{12}^{(13)}=\left(\begin{matrix}{{\bf1}_{2}}&{0}\\{0}&{-{\bf1}_{2}}\\\end{matrix}\right)\,,
{\bfS}^{\alpha}\left(x,y\right)=G^{\alpha}\left(x,y\right)-\intd^{2}zG^{\alpha}\left(x,z\right){\bfP}_{0}^{\alpha}\left(z,y\right)-\intd^{2}z{\bfP}_{0}^{\alpha}\left(x,z\right)G^{\alpha}\left(z,y\right),
J_{p}(s)=\sqrt{\pi}e^{-2\piip\tilde{t}_{1}}\sum_{n_{2}}\int_{0}^{\infty}\frac{dt}{t}\frac{t^{s}}{\sqrt{t}}e^{-\frac{\tilde{n}_{2}^{2}}{|\tau|^{2}}t-\frac{\pi^{2}}{t}p^{2}}.
W_{{\mathrm{\scriptsize~{div}}}}=-\frac{1}{2s}\frac{1}{4\pi}\int_{\partial{\calM}}d\tau\,\left[-\dot{\bar{X}}^{\rho}(\partial_{\nu}F_{\mu\rho}+\partial_{\mu}F_{\nu\rho})(1+F^{2})_{\nu\mu}^{-1}+\frac{1}{3}k\delta_{\nu}^{\nu}\right].
\begin{array}{lcr}{A_{n-1}:\qquadU^{+{\frac{n(n+1)}{2}}}V^{+{\frac{n(n+1)}{2}}}=[Z^{+{(n+1)}}]^{n}}\\{D_{n}:\qquad(x^{++})^{n}+x^{++}(y^{+(n-1)})^{2}+(z^{+n})^{2}=0}\\{E_{6}:\qquad(x^{+6})^{2}+(y^{+4})^{3}+(z^{+3})^{4}=0}\\{E_{7}:\qquad(x^{+9})^{2}+(y^{+6})^{3}+y^{+6}(z^{+4})^{3}=0}\\{E_{8}:\qquad(x^{+15})^{2}+(y^{+10})^{3}+(z^{+6})^{5}=0.}\\\end{array}
f_{m}^{\pm}(ik)\=\kR\left[i^{\beta}\left(I_{m}^{\prime}K_{m-\beta}+I_{m}K_{m-\beta}^{\prime}\right)\+\i^{-\beta}\left(I_{m}^{\prime}K_{m+\beta}+I_{m}K_{m+\beta}^{\prime}\right)\right]\,
M^{ab}e(n_{1}n_{2}nk)=\sum_{n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}}M^{ab}(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime},n_{1}n_{2}nk)e(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime})
A^{(s)}(\rho)=g^{ab}(\rho)T_{a}^{(s)}T_{b}^{(s)}=\frac{1}{a^{2}(\rho)}T_{1}^{(s)}T_{1}^{(s)}+\frac{1}{b^{2}(\rho)}T_{2}^{(s)}T_{2}^{(s)}+\frac{1}{c^{2}(\rho)}T_{3}^{(s)}T_{3}^{(s)}.
A=2\pi^{2}(r_{0})^{3}={\frac{\pi^{2}}{3}}\left\{(C^{IJ}q_{I}q_{J})_{\partialiZ=0}\right\}^{3/4}\qquad\Longrightarrow\qquad8\pi\sqrt{\frac{Q_{H}(Q_{F})^{2}}{2}}\,
\sum_{n=0}^{\ell_{\mathrm{max}}}\langleC_{(n)}|B\rangle_{R}={\sqrt2}T_{p}\,N_{p}(F)k(F)\,\int_{V_{p+1}}\left[\sum_{\ell=0}^{\ell_{\mathrm{max}}}C_{(p+1-2\ell)}\,\wedge\,\mathrm{e}^{\hat{F}}\right]_{p+1}
Q=\alpha^{2}x\left[\left(r+\frac{\pi}{2}\frac{1}{\sqrt{x}}\right)^{2}+\frac{\bar{c}^{2}}{x}\right],\quadR=\frac{\alpha^{2}}{x}\,F\!\left(\frac{1}{\sqrt{x}};r\right),\quad{\bfS}^{2}=\frac{\alpha^{4}}{x}\,G\!\left(\frac{1}{\sqrt{x}};r\right),
\prod_{E}\int\frac{d^{4}p_{{}_{E_{V}}}}{(2\pi)^{4}}V(p_{{}_{E_{V}}})\int\frac{d^{4}p_{{}_{E_{\phi}}}}{(2\pi)^{4}}\phi(p_{{}_{E_{\phi}}})\cdot\ldots\cdot(2\pi)^{4}\delta\Big(\sum_{E}p_{{}_{E}}\Big),
\sigma_{abs}^{\phi_{\pm}}={\frac{4\pi}{\omega^{3}}}P_{abs}^{\phi_{\pm}}={\frac{\pi^{3}r_{1}^{6}r_{5}^{6}}{64r_{\pm}^{4}}}\omega(\omega^{2}+16\pi^{2}T_{L}^{2})(\omega^{2}+16\pi^{2}T_{R}^{2}){\frac{e^{\frac{\omega}{T_{H}}}-1}{(e^{\frac{\omega}{T_{L}}}-1)(e^{\frac{\omega}{T_{R}}}-1)}},
\int\,d\mu(w)d\mu(w^{\prime})\,G_{\lambda}(w,w^{\prime})\underset{i\in\{1,4\}}{\prod}K_{\mu_{i}}(w,x_{i})\,\underset{j\in\{2,3\}}{\prod}K_{\mu_{j}}(w^{\prime},x_{j})
\langleP_{\mu}P^{\mu}\psi^{-}|[j,{\mathsfs}],b^{-}\rangle={\mathsfs}\,\langle\psi^{-}|[j,{\mathsfs}],b^{-}\rangle\,,\quad\mathrm{for~all~}\psi^{-}\in{\mathbf\Phi}_{+}\,.
x_{d}(t)=c\,{\frac{\mathrm{dn}(ct,k)}{\mathrm{sn}(ct,k)}},\quady_{d}(t)=-c^{2}{\frac{\mathrm{cn}(ct,k)}{\mathrm{sn}^{2}(ct,k)}},\quadz_{d}(t)=-\wp(t).
\ell\,\neq\,5\quad\,SpD_{56}\left(E^{{\vec{\alpha}}_{\ell}}\right)=\left(\begin{matrix}{A[{{\vec{\alpha}}_{\ell}}]}&{{\bf0}}\\{{\bf0}}&{-A^{T}[{{\vec{\alpha}}_{\ell}}]}\\\end{matrix}\right)
{\calL}={\frac{1}{2}}g_{ij}{\frac{dx^{i}}{dt}}{\frac{dx^{j}}{dt}}+{\frac{1}{2}}g^{ij}{\frac{\partialh}{\partialx^{i}}}{\frac{\partialh}{\partialx^{j}}}+\psi^{*}{}^{i}({\frac{d}{dt}}\psi_{i}-\Gamma_{ij}^{k}\psi_{k}{\frac{dx^{j}}{dt}})+H_{ij}\psi^{*}{}^{j}\psi^{i}+{\frac{1}{4}}R_{ijkl}\psi^{i}\psi^{j}\psi^{*}{}^{k}\psi^{*}{}^{l}.
R=\frac{1}{\sqrt{\biggl[1+\displaystyle\frac{g}{2\pi}(\sigma-1)\biggr]\biggl[1+\displaystyle\frac{g}{2\pi}(\sigma+1)\biggr]}}\biggl[1\pm\frac{g}{2\pi}\sqrt{1-\frac{4\pi}{g}\sigma-\sigma^{2}}\;\biggr].
\dot{\eta}=\left\{\eta,H^{\left(1\right)}\right\},\;\Phi^{\left(1\right)}\left(\eta\right)=0,\;H^{\left(1\right)}=H\left(\eta\right)+\lambda\Phi^{\left(1\right)}\left(\eta\right),\;\dot{\eta}\equiv\frac{d\eta}{dt}.
\langle0|T(\Phi(x,t)\Phi(x^{\prime},t^{\prime}))|0\rangle=\int_{-\infty}^{\infty}\frac{d^{2}k}{(2\pi)^{2}}\frac{ie^{-ik_{0}(t-t^{\prime})}}{k^{2}-m^{2}+i\epsilon}(e^{ik_{1}(x-x^{\prime})}+e^{ik_{1}(x+x^{\prime})})
S=-\gamma\intd\tau\intd\sigma\sqrt{{\dot{x}\acute{x}}^{2}-{\dot{x}}^{2}{\acute{x}}^{2}}-\sum_{i=1}^{2}m_{i}\intd\tau\sqrt{{\left(\frac{dx^{\mu}({\tau}_{i},{\sigma}_{i}(\tau))}{d\tau}\right)}^{2}}
\Gamma^{\mu\nu}=\overline{{P}}^{\mu\nu}\,(p^{2}+\Pi_{\mathrm{T}})+Q^{\mu\nu}\,(p^{2}+\Pi_{\mathrm{L}})+R^{\mu\nu}\,(p^{2}+\tilde{\Pi}_{\mathrm{T}})+\frac{p_{\mu}p_{\nu}}{\xi}
h_{j,j+1}(q,J)=\frac{J}{4}[\sum_{j=1}^{N-1}\sigma_{j}^{x}\sigma_{j+1}^{x}+\sigma_{j}^{y}\sigma_{j+1}^{y}+\frac{q+q^{-1}}{2}\sigma_{j}^{z}\sigma_{j+1}^{z}-\frac{q-q^{-1}}{2}(\sigma_{j}^{z}-\sigma_{j+1}^{z})]
~~~~~~~~~~~~~~~~~~~\left.-\epsilon_{n}^{s}\rho_{n}f_{l}({\bf{x}}_{n})f_{r}({\bf{x}}_{n-1})\frac{\left(E-V\right)^{2}}{2Mc^{2}}+\epsilon_{n}^{s}\rho_{n}f_{l}({\bf{x}}_{n})f_{r}({\bf{x}}_{n-1})\frac{Mc^{2}}2\right].
\bar{E}_{n}\Bigl|_{n\to\infty}\sim\frac{c\xi^{2}}{4\pia^{2}}\left[\frac{10-3\xi^{2}}{960n^{2}}-\frac{28224-7344\xi^{2}+720\xi^{4}}{15482880n^{4}}+O\left(\frac{1}{n^{6}}\right)\right]\,{.}
\chi_{1}=\left(\begin{matrix}{\psi_{1}}\\{\bar{\psi}_{1}}\\\end{matrix}\right)=\left(\begin{matrix}{\psi_{1}}\\{0}\\\end{matrix}\right)+\left(\begin{matrix}{0}\\{\bar{\psi}_{1}}\\\end{matrix}\right)={\calP}_{L}\Psi+{\calP}_{R}\Psi^{c}
D_{q\cdotq^{\prime}}(\alpha)=-\frac12\sum_{\mathrm{paths~}\rho}\alpha_{j_{1}(\rho)}\alpha_{j_{2}(\rho)}\cdots\alpha_{j_{\left|\rho\right|}(\rho)}C(\alpha,i_{1}(\rho),\cdots,i_{\left|\rho\right|-1}(\rho))
\begin{array}{c}{<\phi_{q}^{2}(\eta)>=\frac1{8\pi^{3}C(\eta)}\intd^{3}k\chi_{k}(\eta)\chi_{k}^{*}(\eta),}\\{<\phi_{q}^{4}(\eta)>=\frac1{64\pi^{6}C^{2}(\eta)}\intd^{3}k(\chi_{k}(\eta)\chi_{k}^{*}(\eta))^{2}and}\\{<\phi_{q}^{6}(\eta)>=\frac1{512\pi^{9}C^{3}(\eta)}\intd^{3}k(\chi_{k}(\eta)\chi_{k}^{*}(\eta))^{3}}\\\end{array}
\rho(\nu)=\frac{(q^{2};q^{4})_{\infty}^{2}}{(q^{4};q^{4})_{\infty}^{2}}\sum_{k=-\infty}^{\infty}\frac{e^{i\nuk-\epsilonk}}{1-q^{4k+2}}=\frac{(q^{2}e^{i\nu-\epsilon};q^{4})_{\infty}(q^{2}e^{-i\nu+\epsilon};q^{4})_{\infty}}{(e^{i\nu-\epsilon};q^{4})_{\infty}(q^{4}e^{-i\nu+\epsilon};q^{4})_{\infty}},
M_{ADM}=\frac{\hat{e}}{\sqrt{2}\kappa}\left[S_{12}\left(N_{2}^{(1)}+\bar{N}_{2}^{(1)}\right)+S_{34}\left(N_{2}^{(2)}+\bar{N}_{2}^{(2)}\right)+S_{56}\left(N_{2}^{(3)}+\bar{N}_{2}^{(3)}\right)\right]
{\not\!\!D}^{2}=(D_{\mu})^{2}-\frac{i}{4}[\gamma^{\mu},\gamma^{\nu}]F_{\mu\nu}=(D_{\mu})^{2}+\left(\begin{array}{cc}{(E+B)\sigma^{3}}&{0}\\{0}&{(-E+B)\sigma^{3}}\\\end{array}\right)
S=-{\frac{\pi^{3}R^{8}}{4\kappa_{10}^{2}}}\!\intd^{4}x\!\int_{\epsilon}^{\infty}\!dz\,{\frac{dz}{z^{3}}}\left[(\partial_{z}\phi)^{2}+\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+{\frac{m^{2}R^{2}}{z^{2}}}\phi^{2}\right]\,,
\hat{C}=\left(\begin{array}{ll}{1_{M}}&{0}\\{0}&{C_{2N}}\\\end{array}\right),\quadg=\left(\begin{array}{ll}{\alpha}&{\tilde{f}_{1}}\\{f_{2}}&{A}\\\end{array}\right),\quadg^{st}=\left(\begin{array}{cc}{\alpha^{t}}&{f_{2}^{t}}\\{-\tilde{f}_{1}^{t}}&{A^{t}}\\\end{array}\right),\quadg^{-1}=\left(\begin{array}{cc}{\alpha^{t}}&{\tilde{f}_{2}}\\{f_{1}}&{\tilde{A}}\\\end{array}\right)
\phi_{l^{\prime}m^{\prime}}^{\prime}=\sum_{l,m}^{l=2j-1}\left[\phi_{lm}\sqrt{\frac{(2l+1)(2l^{\prime}+1)}{2j(2j+1)}}\left(\sum_{\nu,\nu^{\prime}}C_{\;\;\nu\;\;\;\;m\;\;\;\nu^{\prime}}^{j-\frac{1}{2}\;\;l\;\;j-\frac{1}{2}}C_{\nu+\frac{1}{2}\;\;m^{\prime}\;\;\nu^{\prime}+\frac{1}{2}}^{\;\;\;j\quadl^{\prime}\quadj}\right)\right].
R+\frac{\gamma}{4}\left[\frac{(D\Omega)^{2}}{\Omega^{2}}-2\frac{D\cdotD\Omega}{\Omega}\right]+(1-\lambda)\mu\Omega^{-\lambda}+\frac{1}{4}(1-\frac{\epsilon}{2})\Omega^{-\epsilon/2}F^{2}-\frac{\delta}{2}\Omega^{\delta-1}(Df)^{2}=0,
\zeta^{(3)}=\epsilon^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}\Biggl[{\frac{1}{4!}}\gamma_{\alpha_{1}}\gamma_{\alpha_{2}}\gamma_{\alpha_{3}}\gamma_{\alpha_{4}}+{\frac{1}{2!}}{\cal{F}}_{\alpha_{1}\alpha_{2}}\gamma_{\alpha_{3}}\gamma_{\alpha_{4}}\Biggr]\quad,
\mu(E)=\frac{C}{N_{1}N_{2}\cdot\dots\cdotN_{g}}\prod_{i=1}^{g}(\tilde{N}_{i}+N_{i}\alpha^{2i-1}\alpha^{2i})\equiv\frac{C}{N_{1}N_{2}\cdot\dots\cdotN_{g}}\mu_{0}(E)\,.
S=\intd^{D}x\sqrt{\left|\widetilde{G}\right|}\left[-\widetilde{F}_{R}\left(\varphi\right)\widetilde{R}-4\widetilde{F}_{\varphi}\left(\varphi\right)\partial_{M}\varphi\widetilde{\partial}^{M}\varphi+\widetilde{L}_{m}\left(\varphi,\widetilde{G}_{MN},\psi\right)\right],
\tilde{\Psi}_{\mathrm{far}}(r)={\frac{1}{r}}\left[\left(A_{\mathrm{in}}+A_{\mathrm{out}}\right)r^{\sqrt{1-s}}+i\left(A_{\mathrm{in}}-A_{\mathrm{out}}\right)r^{-\sqrt{1-s}}\right].
\begin{array}{ccc}{{[}X_{+},X_{+}]=0,}&{[X_{+},X_{0}]=-q^{-1}X_{+},}&{[X_{+},X_{-}]=(q+q^{-1})X_{0},}\\{{[}X_{0},X_{+}]=qX_{+},}&{[X_{0},X_{0}]=(q-q^{-1})X_{0},}&{[X_{0},X_{-}]=-q^{-1}X_{-},}\\{{[}X_{-},X_{+}]=-(q+q^{-1})X_{0},}&{\:[X_{-},X_{0}]=qX_{-},}&{[X_{-},X_{-}]=0,}\\\end{array}
\stackrel{(0)}{\bfg}=\eta_{ab}\,\underline{{\theta}}^{a}\otimes\underline{{\theta}}^{b}\quad,\quad\stackrel{(2)}{\bfg}\equiv\stackrel{(2)}g_{ab}\underline{{\theta}}^{a}\otimes\underline{{\theta}}^{b}=\left(\sigma_{ab}+\sigma_{ba}\right)\underline{{\theta}}^{a}\otimes\underline{{\theta}}^{b}\qquad.
{\calL}^{\prime}=-\frac{1}{g_{2}^{2}}tr\{K_{{\calA}1}\pi(\theta_{\calA}^{MM2})+K_{{\calA}2}\pi(\theta_{\calA}^{MF2})+K_{{\calA}3}\pi(\theta_{\calA}^{FF2})+K_{\calB}\pi(\theta_{\calB}^{MM2})\},