harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u_{n}^{1I}=\frac{n(n-1)\Lambda^{2N_{c}-N_{f}}}{4}\sum_{p=0}^{N_{f}}t_{p}\left(\sum_{k=1}^{N_{c}}\frac{a_{k}^{N_{f}-p+n-2}}{\prod_{l\neqk}^{N_{c}}(a_{k}-a_{l})^{2}}+\frac{1}{n-1}r_{n}^{(N_{f}-p)}\right),
\langle\dot{y}_{a}\cdot\dot{y}_{b}\rangle=\mathrm{tr}\biggl[\ddot{\calG}_{Bab}+{\frac{1}{2}}{\frac{(\dot{\calG}_{Baa}-\dot{\calG}_{Bab})(\dot{\calG}_{Bab}-\dot{\calG}_{Bbb})}{\bar{T}-{\frac{1}{2}}{\calC}_{ab}}}\biggr]\;,
K(t)=\frac{V}{2(4\pit)^{\frac{m-1}2}}\left(2S\sqrt{\fract\pi}\int_{0}^{1}d\xi\,e^{-(M^{2}+(\xi^{2}-1)S^{2})\,t}+e^{(S^{2}-M^{2})\,t}\right)\,.
S_{L}=\sum_{i=1}^{\infty}\sigma_{A_{i}}^{*}D_{\beta}\sigma^{(\betaA_{i})}=\sigma_{\alpha}^{*}D_{\beta}\sigma^{(\beta\alpha)}+\sigma_{\beta\alpha}^{*}D_{\gamma}\sigma^{(\gamma\beta\alpha)}+\ldots\,
\frac{g_{0}^{2}}{4\pim_{0}^{2}}=\alpha\left(\sigma_{0}\right)\,\frac{\mu_{0}}{\sigma_{0}m_{0}}\sqrt{1-\frac{1}{4}\left(\frac{\mu_{0}}{\sigma_{0}m_{0}}\right)^{2}}=\left(2\sqrt{\alpha_{0}\alpha_{2}}+\alpha_{1}\right)\frac{\mu_{0}}{m_{0}}+O\left(\left(\frac{\mu_{0}}{m_{0}}\right)^{3}\right)
T_{k}=\left(\begin{array}{cccccccc}{}&{}&{}&{}&{0}&{1}&{0}&{0}\\{}&{}&{}&{}&{1}&{0}&{0}&{0}\\{}&{}&{}&{}&{0}&{0}&{0}&{-1}\\{}&{}&{}&{}&{0}&{0}&{0}&{t}\\{0}&{1}&{0}&{0}&{}&{}&{}&{}\\{1}&{0}&{0}&{0}&{}&{}&{}&{}\\{0}&{0}&{0}&{0}&{}&{}&{}&{}\\{0}&{0}&{-1}&{t}&{}&{}&{}&{}\\\end{array}\right)
\mathcal{L}=\frac{r^{2}\partial_{r}W\partial_{r}W^{*}+\partial_{\theta}W\partial_{\theta}W^{*}-r^{2}\dot{\lambda}^{2}\partial_{\lambda}W\partial_{\lambda}W^{*}}{4r^{2}(1+|W|^{2})^{2}}.
\frac{1}{k^{2}(k.n)^{m}}\rightarrow\frac{1}{k^{2}+i\varepsilon}\left\{\frac{\Theta(-k^{0})}{(k.n-i\xi)^{m}}+\frac{\Theta(k^{0})}{(k.n+i\xi)^{m}}\right\},\,\,\begin{array}{c}{\varepsilon\equiv2\epsilonk_{0}^{2}\rightarrow0^{+}}\\{\xi\equiv\epsilon|k^{0}|n^{0}\rightarrow0^{+}}\\\end{array},
\begin{array}{llll}{\nonumber\phi(out)_{g}}&{=}&{Tr(W_{out}U_{g^{-1}})}&{}\\{}&{=}&{Tr(SW_{in}S^{-1}U_{g^{-1}})}&{}\\{}&{=}&{Tr(W_{in}S^{-1}U_{g^{-1}}S)}&{}\\{}&{=}&{Tr(W_{in}U_{g^{-1}})}&{=\phi(in)_{g}\quad.}\\\end{array}
S_{\mathrm{boundary}}^{GLSM}=\intdx^{0}\left\{{\frac{i\theta}{4\pir}}\sum_{i}(\phi_{i}D_{0}\overline{{\phi}}_{i}-\overline{{\phi}}_{i}D_{0}\phi_{i})+\eta{\frac{\theta}{\sqrt{2}\pir}}{\frac{D}{e^{2}}}\sigmae^{i\gamma}\right\},
{\bf\mathrm{P}_{2}}(n)=\left[\begin{array}{ccccccc}{0}&{\cdots}&{\cdots}&{\cdots}&{0}&{|}&{n-2}\\{1}&{0}&{\vdots}&{\vdots}&{\vdots}&{|}&{n-3}\\{0}&{1}&{0}&{\vdots}&{\vdots}&{|}&{\vdots}\\{\vdots}&{0}&{\ddots}&{0}&{\vdots}&{|}&{~1}\\{\vdots}&{\vdots}&{0}&{1}&{0}&{|}&{~0}\\{0}&{\cdots}&{\cdots}&{0}&{1}&{|}&{-1}\\\end{array}\right].
\theta_{1}\equiv\stackrel{o\}{E^{+}},\\\theta_{2}\equiv\tilde{E}^{+},\\\theta_{3}\equiv\pi_{R}^{\dagger}-i\sqrt2\psi_{R}^{\dagger},\\\theta_{4}\equiv\pi_{R},\\\theta_{5}\equiv\pi_{L}^{\dagger},\\\theta_{6}\equiv\pi_{L}.
\intd^{4}x\tr\left[\theta^{\prime\prime}a(A)\right]-\intd^{4}x\tr\left[\theta_{1}a(A)\right]-\intd^{4}x\tr\left[\theta_{2}a(A)\right]=\intd^{4}x\tr\left[\theta_{1}\delta_{\theta_{2}}a(A)\right].
I_{\pm}=4\pi\int_{0}^{\verta\vert^{3/2}}\frac{1+\vert\beta\vert^{2}\verta\vert^{2}}{(1+\vert\beta\vert^{2}\verta\vert^{2}+4\vert\lambda\vert^{2}\verta\vert^{2}\vertb\vert^{2})^{2}}d\vertb\vert^{2}.
S_{DBI}=\intd^{4}\xi-\sqrt{-g}f\,\f=\sqrt{1+\bar{\tilde{\calH}}_{i}\tilde{\calH}^{i}+\frac{1}{2}Im(\bar{\tilde{\calH}}_{i}\tilde{\calH}_{j})Im(\bar{\tilde{\calH}}^{i}\tilde{\calH}^{j})},
N_{nm}^{f}=\frac1{n\,m}\oint_{0}\frac{dz}{2\pii}z^{-n}\oint_{0}\frac{d\zeta}{2\pii}\zeta^{-m}\frac{f^{\prime}(z)f^{\prime}(\zeta)}{(f(z)-f(\zeta))^{2}}
J_{A}{}^{B}=\frac{\partial_{r}\left(\delta_{susy}\Phi_{A}\right)}{\partial\Phi_{B}}=({\calJ}P_{q})_{A}{}^{B}=\left(\begin{array}{cc}{{\mathrm{\small{~\frac12~}}}{\calD}^{2}\,{\calJ}_{a}{}^{b}}&{0}\\{0}&{{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}\,\bar{{\calJ}}^{\bar{a}}{}_{\bar{b}}}\\\end{array}\right),
F(a,b,c;z)\stackrel{z\sim\infty}{\rightarrow}{\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}}(-z)^{-a}\,+\,{\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}}(-z)^{-b}.
\begin{array}{lcr}{\{[\zeta(x_{1})+\zeta(x_{2})+\zeta(x_{3})]-i[\zeta(y_{1})+\zeta(y_{2})+\zeta(y_{3})]\}}&{}\\{\{[\zeta(x_{1})+\zeta(x_{2})+\zeta(x_{3})]+i[\zeta(y_{1})+\zeta(y_{2})+\zeta(y_{3})]\}}&{}\\{=[\zeta(x_{1})+\zeta(x_{2})+\zeta(x_{3})]^{2}+[\zeta(y_{1})+\zeta(y_{2})+\zeta(y_{3})]^{2}}\\\end{array}
4\delta^{ab}(C_{1}C_{3}+{\frac{1}{4}}C_{1}C_{2}){\frac{g^{4}}{p^{2}}}\Lambda^{3-\omega}\int{\frac{d^{\omega}k}{(2\pi)^{\omega}}}\int{\frac{d^{3}q}{(2\pi)^{3}}}{\frac{k\cdotp}{k^{2}q^{2}(k+p+q)^{2}}}.
\gamma_{\mu}=\left(\begin{array}{cc}{\sigma_{\mu}}&{0}\\{0}&{-\sigma_{\mu}}\\\end{array}\right)\,;\mu=1\sim3\,,\quad\gamma_{4}=\left(\begin{array}{cc}{0}&{{\bbox1}}\\{{\bbox1}}&{0}\\\end{array}\right)\,,\quad\gamma_{5}=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}=\left(\begin{array}{cc}{0}&{i{\bbox1}}\\{-i{\bbox1}}&{0}\\\end{array}\right)\.
f^{1}(u)\equivf^{(+1)}(u)=f^{i}u_{i}^{1}+f^{(ijk)}u_{i}^{1}u_{j}^{1}u_{k}^{2}+f^{(ijklm)}u_{i}^{1}u_{j}^{1}u_{k}^{1}u_{l}^{2}u_{m}^{2}+\ldots\;.
\delta\psi_{\mu}=\nabla_{\mu}(\omega)\epsilon+{\frac{1}{8}}\Bigl(\Gamma_{\mu}{}^{\rho\sigma}-4\delta_{\mu}{}^{\rho}\Gamma^{\sigma}\Bigr)e^{{\frac{\phi}{3}}}F_{\rho\sigma}=\nabla_{\mu}(\omega)\epsilon+{\frac{\lambda}{8}}\Bigl(\Gamma^{\rho\sigma}\Gamma_{\mu}+2\Gamma^{\rho}\delta_{\mu}{}^{\sigma}\Bigr)F_{\rho\sigma}\epsilon\.
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(2/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~-\frac{7}{2})}&{e_{\mu}^{a}}&{+2}\\{\mid\mathrm{1st}>}&{(1~-\frac{7}{2})}&{2\Psi_{\mu}}&{+\frac{3}{2}}\\{\mid\mathrm{2nd}>}&{(0~-\frac{5}{2})}&{A_{\mu}}&{+1}\\\end{array}
\bar{Q^{\dagger}}_{I}^{j}\bar{Q}_{i}^{I}-\frac{\delta_{i}^{j}}{6}\bar{Q^{\dagger}}_{K}^{k}\bar{Q}_{k}^{K}=2{X^{\dagger}}_{ik}X^{kj}-\frac{\delta_{i}^{j}}{3}{X^{\dagger}}_{kl}X^{lk}~,
s=a{\theta_{3}}^{4}+b{\frac{\theta_{3}^{\prime}\theta_{3}^{3}}{\eta^{12}}}\,\\-t=a{\theta_{4}}^{4}+b{\frac{\theta_{4}^{\prime}\theta_{4}^{3}}{\eta^{12}}}\,\\-u=a{\theta_{2}}^{4}+b{\frac{\theta_{2}^{\prime}\theta_{2}^{3}}{\eta^{12}}}\,
\langle\Psi_{2},\Psi_{1}\rangle\rightarrow\langle\langle\Psi_{2},\Psi_{1}\rangle\rangle=\langle\Psi_{2},\widehat\eta\Psi_{1}\rangle=\varphi_{2}^{\dagger}\varphi_{1}+\Phi_{2}^{\dagger}\Phi_{1},
L(\dot{x}^{\mu},e)=\sqrt{g}\left[\frac{1}{2}g^{-1}\dot{x}^{\mu}\dot{x}^{\nu}\eta_{\mu\nu}-\frac{1}{2}m^{2}\right]=\frac{1}{2}|e|^{-1}\dot{x}^{2}-\frac{1}{2}|e|m^{2}\.
{Y}_{[{j_{1}},\psi_{j_{1}}][{j_{2}},\psi_{j_{2}}]}^{\quad\quad\quad\quad[{j}_{3},\psi_{j_{3}}]}=\sum_{[m]}\sum_{\psi_{m}}{\frac{{S}_{[m,\psi_{m}][{j_{1}},\psi_{j_{1}}]}{P}_{[m,\psi_{m}][{j_{2}},\psi_{j_{2}}]}{P}_{[m,\psi_{m}][{j_{3}},\psi_{j_{3}}]}^{*}}{{S}_{[m,\psi_{m}][0]}}}\;\;\;\;.
J_{\alpha}=\frac{\pi}{2e}\frac{\deltaS}{\delta\bar{\chi}^{\alpha}}=\rho^{\beta}\rho_{\alpha}\bar{\Psi}^{\mu}\partial_{\beta}X_{\mu}-\frac{1}{4}\bar{\Psi}^{\mu}\Psi_{\mu}\rho_{\alpha}\rho^{\beta}\chi_{\beta}=0
w_{q}^{2}(t)=q^{2}\,\frac{1+Y_{\Lambda}}{1+Z_{\Lambda}}+\frac{V_{\Lambda}^{\prime\prime}}{1+Z_{\Lambda}}+\frac{1}{4}\frac{Z_{\Lambda}^{2}}{(1+Z_{\Lambda})^{2}}\,\dot{\phi}^{2}+\frac{1}{2}\frac{Z_{\Lambda}^{\prime}}{1+Z_{\Lambda}}\,\ddot{\phi}
S={\frac{1}{16\piG_{10}}}\intd^{10}x\sqrt{-G^{dual}}\left[e^{-{\frac{6}{7}}\phi}({\calR}^{dual}+{\frac{16}{49}}\partial_{M}\phi\partial^{M}\phi)-{\frac{1}{4}}e^{{\frac{6}{7}}\phi}F_{MN}F^{MN}\right].
\tilde{g}_{\alpha\beta}=F_{\alpha}^{\gamma}(z)G_{\gamma\delta}(h)F_{\beta}^{\delta}(z)\,\\\\G_{\alpha\beta}=\omega_{\alpha\gamma}\omega_{\beta}{}^{\gamma}\,\\\\,omega_{\alpha\beta}=h^{i}f_{i\alpha\beta}\,
F({\bfk}_{1},{\bfk}_{2},\cdots,{\bfk}_{n})\equiv\frac{\theta(\epsilon_{\Lambda_{n-1}}-\sum_{i=1}^{n}\epsilon_{{\bfk}_{i}})\theta(\sum_{i=1}^{n}\epsilon_{{\bfk}_{i}}-{\epsilon_{\Lambda}})}{E_{i}-\sum_{i=1}^{n}\epsilon_{{\bfk}_{i}}}.
{\frac{q^{c}\prod_{j=1}^{p}X_{l_{\sigma(j)}}^{b_{jj}+1}}{\prod_{j=1}^{p}\bigl(1-X_{l_{\sigma(1)}}\cdotsX_{l_{\sigma(j)}}q^{|\alpha_{l_{\sigma(1)}}+\cdots+\alpha_{l_{\sigma(j)}}|^{2}}\bigr)}}.
\Omega^{-1}\,\left[\partial_{\lambda}\left(\Omega\,e_{\mu}^{a}\,e_{\nu}^{b}\,\hat{\xi}_{ab}\Omega^{-1}\right)\right]\Omega=0\quad\,\mathrm{or}\quad\left(\Omega\,e_{\mu}^{a}\,e_{\nu}^{b}\,\hat{\xi}_{ab}\Omega^{-1}\right)={\itconst.}
\sigma_{1}=\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right],\;\;\\\\sigma_{2}=\left[\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right],\;\;\;\sigma_{3}=\left[\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right]
{\calS}=\int_{0}^{T}dt\int_{\partial{\calV}}d^{3}x\,\nabla\cdot\left[\phi{\frac{\partialL}{\partial(\nabla\phi)}}\right]=T\int_{\partial\partial{\calV}}d^{2}x\\hat{\bfn}\cdot\phi{\frac{\partialL}{\partial(\nabla\phi)}}\equiv\int_{\partial\partial{\calV}}d^{2}x\hat{\bfn}\cdot{\bfP}
\deltaS^{-}=\int_{M}(\delta{\tilde{\overline{\Psi}}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}}+\delta{\tilde{\Psi}}_{L}{\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L}}}+\deltaE^{\muA}{\frac{\delta{S}^{-}}{\deltaE^{\muA}}}+\deltaA_{\muAB}{\frac{\delta{S}^{-}}{\deltaA_{\muAB}}}+\deltaW_{\mua}{\frac{\delta{S}^{-}}{\deltaW_{\mua}}})=0.
{\calL}_{0}(\phi,\partial_{\mu}\phi)=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^{2}\phi^{2}=\frac{1}{2}\left(\partial_{\mu}\phi\right)^{2}-\frac{1}{2}m^{2}\phi^{2}\.
\phi(t,x)=\sum_{n}\left[\frac{e^{-i\omega_{n}t}}{\sqrt{2\omega_{n}}}\hat{a}_{n}\epsilon_{n}(x)~+~\frac{e^{i\omega_{n}t}}{\sqrt{2\omega_{n}}}\hat{a}_{n}^{\dagger}\epsilon_{n}^{\dagger}(x)\right]
\begin{array}{c|ccc}{W_{4}}&{\mathrm{number~of~components}}&{\mathrm{genus}}&{\mathrm{moduli~space}}\\\hline{fF}&{1}&{1}&{dP_{8}\timesI}\\{f_{1}F+f_{2}F}&{2}&{1+1}&{\left(dP_{8}\timesI\right)^{2}}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{F+\dots+F}&{f}&{f\times1}&{\left(dP_{8}\timesI\right)^{f}}\\\end{array}
E=\left(\begin{array}{cc}{e}&{0}\\{0}&{e}\\\end{array}\right),N=\left(\begin{array}{cc}{n-1}&{0}\\{0}&{n}\\\end{array}\right),\psi=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right),\psi^{+}=\left(\begin{array}{cc}{0}&{0}\\{e}&{0}\\\end{array}\right)
M_{\ell}=\left(\begin{array}{cccc}{\frac{r_{0}}{r-b}}&{-\frac{r}{r-b}}&{0}&{0}\\{-\frac{r}{r-b}}&{\frac{b}{r_{0}}\frac{r}{r-b}}&{0}&{0}\\{0}&{0}&{-\frac{b}{r_{0}}}&{-1}\\{0}&{0}&{-1}&{-\frac{r_{0}}{r}}\\\end{array}\right)\,.
\begin{aligned}{}&{{}\partial_{x_{2}}=-x_{1}^{2}\partial_{x_{1}}-x_{1}y_{1}\partial_{y_{1}}+u_{1}x_{1}\partial_{u_{1}}+2x_{1}v_{1}\partial_{v_{1}},}&{}&{{}\qquad\partial_{y_{2}}=x_{1}\partial_{y_{1}},}\\{}&{{}\partial_{u_{2}}={\frac{1}{x_{1}}}\partial_{u_{1}},}&{}&{{}\qquad\partial_{v_{2}}={\frac{1}{x_{1}^{2}}}\partial_{v_{1}}.}\\\end{aligned}
E(B)={\frac{(k_{b}T)^{2}}{2\piv}}\{{\frac{1}{2}}x\mathrm{ln}y+L(1-e^{-x}\}={\frac{(k_{b}T)^{2}}{2\piv}}\{{\frac{B}{2}}x^{2}+{\frac{1}{2}}x\mathrm{ln}(1-e^{-x})+L(1-e^{-x})\}.
\nu^{-1}(\tilde{\varepsilon})={4}\,{\tilde{\varepsilon}^{-1}}-8\frac{n-4}{n-2}\,\tilde{\varepsilon}^{-2}+16\frac{n-4}{n-2}\,\tilde{\varepsilon}^{-3}-32\left\{\left[52+108\zeta(3)\right]-\left[16+36\zeta(3)\right]n+n^{2}\right\}\,\frac{\tilde{\varepsilon}^{-4}}{(n-2)^{3}}+\dots~.
\delta_{H}^{(1)}M_{A}^{\phantom{A}B}\,=\,\left[\left(\begin{matrix}{h}&{\chi}\\{-i\overline{{\chi}}}&{0}\\\end{matrix}\right),\left(\begin{matrix}{m}&{\Psi}\\{-i\overline{{\Psi}}}&{0}\\\end{matrix}\right)\right]_{A}^{\phantom{A}B},
S_{\partial\Sigma}=\frac{1}{2\pi\alpha^{\prime}}\int_{\partial\Sigma}d\theta\bigl\lbrack\partial_{r}{\tilde{X}}^{i}\Phi_{i}+\alpha^{\prime}(\psi^{\alpha}{\bar{\psi}}^{i}-\psi^{i}{\bar{\psi}}^{\alpha})\partial_{\alpha}\Phi_{i}\bigr\rbrack
<0|N|0>_{\mathrm{nonl}}=\int{\frac{d^{3}\vec{q}}{(2\pi)^{3}}}{\frac{1}{2\omega(\vec{q})}}\left[|J_{\mu}^{\mathrm{cl}}(\vec{q})|^{2}+{\frac{1}{\omega(\vec{q})^{2}}}\left(\sum_{i=1}|J_{i}^{\mathrm{cl}}(\vec{q})|^{2}q_{i}^{2}-|J_{0}^{\mathrm{cl}}(\vec{q})|^{2}\omega(\vec{q})^{2}\right)\right]
i\frac{\partialF^{(\pm)}}{\partialt}=-\frac{1}{2}\frac{\partial^{2}F^{(\pm)}}{\partialq^{2}}+\frac{q^{\prime}\pmq}{it}\frac{\partialF^{(\pm)}}{\partialq^{\prime}}+V(q^{\prime})F^{(\pm)},
ds_{\mathrm{rel}}^{2}=ds_{\mathrm{EH}}^{2}=\frac{du^{2}}{1-\zeta^{2}/u^{4}}+\frac{u^{2}}{4}\left\{\sigma_{1}^{2}+\sigma_{2}^{2}+\left(1-\frac{\zeta^{2}}{u^{4}}\right)\sigma_{3}^{2}\right\}\,.
\left[n(\tau,\vec{\sigma})\,U^{\check{A}}(\tau,\vec{\sigma})\right]_{;\check{A}}=\frac{1}{\sqrt{g(\tau,\vec{\sigma})}}\,\frac{\partial}{\partial\sigma^{\check{A}}}\,\Big[\sqrt{g(\tau,\vec{\sigma})}\,n(\tau,\vec{\sigma})\,U^{\check{A}}(\tau,\vec{\sigma})\Big]=0.
{j^{(b)}}_{\mu}^{T}\;:=\;T_{\mu\nu}\overline{{\psi}}^{(b)}\gamma_{\nu}\psi^{(b)}\;\;\;\;,\;\;\;\;T_{\mu\nu}\;:=\;\delta_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\triangle}\;,
\left[\frac{\nabla}{\nablax^{\mu}},\\frac{\nabla}{\nablax^{\nu}}\right]f=2\left(\frac{\partial{a_{\nu}}^{\lambda}}{\partialx^{\mu}}-\frac{\partial{a_{\mu}}^{\lambda}}{\partialx^{\nu}}\right)\\frac{\partialf}{\partialx^{\lambda}}+2\left(\frac{\partial{b_{\nu}}^{\lambda}}{\partialx^{\mu}}-\frac{\partial{b_{\mu}}^{\lambda}}{\partialx^{\nu}}\right)\\frac{\partialf}{\partialy^{\lambda}}
{\hat{f}}_{1}^{(0)}(t)=\int_{x_{1}=0}d^{D-1}x_{\parallel}\,S_{1,1}(0^{+},x_{\parallel})=-{\frac{1}{2}}\int_{x_{1}=0}d^{D-1}x_{\parallel}\,\left(\partial_{x_{1}}\phi_{0}(0^{+},x_{\parallel},t)\right)^{2},
M_{c}(\vec{\alpha})\geq\sum_{i=1}^{\ell}M_{c\cdot{\kappa}_{i}}(\vec{\alpha}_{i}),\,\,\,\,\,\,\,\,\,\,\,Q_{c}(\vec{\alpha})=\sum_{i=1}^{\ell}Q_{c\cdot{\kappa}_{i}}(\vec{\alpha}_{i}).
\begin{array}{l}{\bar{c}(r_{1}^{11},\ldots,r_{n}^{11})=c^{\prime}(n_{1},\ldots,n_{N};0;0;0),\quad\bar{c}(r_{1}^{12},\ldots,r_{n}^{12})=c^{\prime}(0;m_{1},\ldots,m_{M};0;0),}\\{\bar{c}(r_{1}^{21},\ldots,r_{n}^{21})=c^{\prime}(0;0;q_{1},\ldots,q_{Q};0),\quad\bar{c}(r_{1}^{22},\ldots,r_{n}^{22})=c^{\prime}(0;0;0;t_{1},\ldots,t_{T}).}\\\end{array}
J_{\Gamma}(x)=\bar{\psi}(x)\Gamma\psi(x)\equiv\frac{1}{8\pi}\intd\hat{\epsilon}e^{i\chi_{\Gamma}(\hat{\epsilon})}\left\{\left[\psi^{\dagger}(x+\epsilon),\gamma^{0}\Gamma\psi(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_{i}A_{i}^{tr}}\right\}_{|\epsilon|\propto|\eta|}
\intdU\U_{j_{1}}^{i_{1}}..U_{j_{n}}^{i_{n}}{U^{+}}_{l_{1}}^{k_{1}}..{U^{+}}_{l_{n}}^{k_{n}}=\sum_{\rho\sigma}\frac{\delta(\Omega_{n}^{-1}\rho)}{N^{n}}\\delta_{l_{\sigma(1)}}^{i_{1}}\delta_{j_{1}}^{k_{(\rho\sigma)(1)}}..\delta_{l_{\sigma(n)}}^{i_{n}}\delta_{j_{n}}^{k_{(\rho\sigma)(n)}}
e^{-\beta\lambda_{j}\cdot\phi}\rightarrow\frac{<\Lambda_{j}|e^{A_{i(1)}\widehat{F}^{i(1)}(\theta_{1})}|\Lambda_{j}>}{<\Lambda_{0}|e^{A_{i(1)}\widehat{F}^{i(1)}(\theta_{1})}|\Lambda_{0}>^{m_{j}}}
\begin{array}{ll}{X^{i}\sim\epsilon^{ijklm}\Omega_{jk}\Omega_{lm}}&{\qquad10\times10\rightarrow5^{*}}\\{Y_{i}\sim\Phi^{j}\Omega_{ji}}&{\qquad5^{*}\times10\rightarrow5}\\{S\sim\mathrm{tr}\left(W^{\dot{\alpha}}W_{\dot{\alpha}}\right)}&{\qquad24\times24\rightarrow1}\\\end{array}
\phi(w_{n}\prime,w_{m}\prime^{\ast})=\sum_{l=1}^{m}\left[A_{l}S\left(\frac{q^{-l+1/4}(q^{m},q^{n})}{1-q}\right)+B_{l}\tilde{C}\left(\frac{q^{-l+1/4}(q^{m},q^{n})}{1-q}\right)\right],
{\calL}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}\,+{\frac{\kappa}{4}}\epsilon^{\mu\nu\alpha}A_{\mu}F_{\nu\alpha}\,+{\frac{1}{2}}|{\calD}_{\mu}\phi|^{2}-{\frac{1}{2}}m^{2}|\phi|^{2}\,,
H=\frac{1}{2}\sum_{j=1}^{N}\left\{(1+\Gamma)\sigma_{j}^{x}\sigma_{j+1}^{x}+(1-\Gamma)\sigma_{j}^{y}\sigma_{j+1}^{y}+h(\sigma_{j}^{z}+\sigma_{j+1}^{z})\right\}\pm\frac{\mathrm{i}k^{\prime}\mathrm{sn}(\psi)}{2}(\sigma_{1}^{z}-\sigma_{N}^{z})
{\calL}=\bar{\psi}\gamma^{\mu}\left[i\partial_{\mu}+e{\calA}_{\mu}\right]\psi-\frac{1}{4}{\calF}_{\mu\nu}{\calF}^{\mu\nu}-\frac{\kappa}{2}\epsilon_{\mu\nu\rho}{\calA}^{\mu}\partial^{\nu}{\calA}^{\rho}-\frac{1}{2\xi}(\partial{\calA})^{2},
\hat{D}_{\mu\nu}=E_{0}^{2}(P^{2})D_{\mu\nu}=\int_{1}^{\infty}\frac{dx}{\Lambda^{2}}exp(x\frac{p^{2}}{\Lambda^{2}})(g_{\mu\nu}-(1-\xi)\frac{p_{\mu}p_{\nu}}{p^{2}}),
\Pi={\calC}^{a}P_{a}^{b}\bar{\calP}_{b}(-1)^{\epsilon_{b}}+\sum_{r=1}^{L}(-1)^{r}{\calC}^{a_{r}}P_{a_{r}}^{b_{r}}\bar{\calP}_{b_{r}}(-1)^{\epsilon_{b_{r}}}+...
S_{Q1}^{\prime}=\frac{1}{N}\mathrm{Tr}\,\left(\sum_{i=0}^{L-1}\frac{(M_{i+1}^{\prime}-M_{i}^{\prime})^{2}}{2}\frac{L}{T}+\tilde{g}\frac{T}{L}(M_{i}^{\prime})^{4}\right)
\begin{array}{ll}{D_{i}a=X_{i}a+[y_{i},a]\,,}&{}\\{D_{i}(a\circb)=(D_{i}a)\circb+a\circ(D_{i}b)\,,}&{\forall\,a,b\in{\cal{A}}\,,}\\{[D_{i},D_{j}]=f_{ij}^{k}D_{k}\,.}&{}\\\end{array}
I=\intd^{2}zd^{2}wz^{\alpha}(1-z)^{\beta}\bar{z}^{\bar{\alpha}}(1-\bar{z})^{\bar{\beta}}w^{\alpha^{\prime}}(1-w)^{\beta^{\prime}}\bar{w}^{\bar{\alpha}^{\prime}}(1-\bar{w})^{\bar{\beta}^{\prime}}|z-w|^{4\sigma}.
\Gamma=\intdt\Bigl[\sum_{a}{\frac{N_{a}}{2g_{s}}}v_{a}^{2}+{\frac{1}{2}}\sum_{a,b}{\frac{15N_{a}N_{b}}{16}}{\frac{v_{ab}^{4}}{r_{ab}^{7}}}-{\frac{1}{2}}\sum_{a,b}{\frac{15N_{a}N_{b}}{8}}{\frac{\dot{g}_{s}}{g_{s}}}{\frac{v_{ab}^{2}v_{ab}\cdotx_{ab}}{r_{ab}^{7}}}\Bigr].
{\frac{\deltaS_{\mathrm{loc};\,-d}}{\deltaG_{\mu\nu}}}={\frac{\sqrt{G}}{2}}\,W(\phi)\,G^{\mu\nu},\qquad{\frac{\deltaS_{\mathrm{loc};\,-d}}{\delta\phi^{i}}}=\sqrt{G}\,\,\partial_{i}W(\phi),
0=f_{A}\left(f_{B}^{\prime}+\frac{f_{B}f_{C}}{r}\right)-f_{B}\left(f_{A}^{\prime}-\frac{f_{B}f_{C}}{r}\right)+2\eta^{2}r^{2}\left[H\left(K^{\prime}+\frac{Hf_{C}}{r}\right)-K\left(H^{\prime}-\frac{Kf_{C}}{r}\right)\right]
\mathbf{Q}_{m}^{a}(\zeta)\Big(\frac{\deltaS_{m}(\zeta)}{\delta\eta_{I}}+\phi^{I}\Big)=0,\qquad\mathbf{Q}_{A}(\zeta)\Big(\frac{\deltaS_{m}(\zeta)}{\delta\eta_{I}}+\phi^{I}\Big)=0.
F/A=-\sum_{n=0}^{\infty}{\frac{(n-1+{\frac{D}{2}})\Gamma(n+D-2)}{2^{D-1}\pi^{\frac{D+1}{2}}a^{D+1}n!\,\Gamma\left({\frac{D-1}{2}}\right)}}\int_{0}^{\infty}dx\left[{\frac{xI_{n-1+{\frac{D}{2}}}^{\prime}(x)}{I_{n-1+{\frac{D}{2}}}(x)}}+{\frac{xK_{n-1+{\frac{D}{2}}}^{\prime}(x)}{K_{n-1+{\frac{D}{2}}}(x)}}+2-D\right].
\frac{H^{1}(TM)}{H^{0}(TM)}=\frac{\bigoplus_{i}H^{1}(C_{i},\psi^{*}TM)\bigoplus_{n}T_{\psi(x_{n})}M}{\bigoplus_{i}H^{0}(C_{i},\psi^{*}TM)}.
U=M_{P}^{2}|A|^{2}\Big(f-2f_{A}A-2f_{A^{\ast}}A^{\ast}+4f_{AA^{\ast}}|A|^{2}\Big)-f_{AA^{\ast}}^{-1}\Big|B-M_{P}f_{A}A+2M_{P}f_{AA^{\ast}}|A|^{2}\Big|^{2}
\deltax^{\mu}={\dot{x}}^{\mu}\xi\;,\;\deltae=\frac{d}{d\tau}\left(e\xi\right)\;,\;\deltaf_{ab}=\frac{d}{d\tau}\left(f_{ab}\xi\right)\;,\;\;\delta\psi_{a}^{n}=\dot{\psi}_{a}^{n}\xi\;,\;\;\delta{\chi}_{a}=\frac{d}{d\tau}\left(\chi_{a}\xi\right)\;,
ds^{2}=-\left[1-\left(\frac{r_{H}}{r}\right)^{7-p}\right]dw^{2}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}+dy^{2}+dy^{s}dy_{s}.
\{Q_{+{\frac{1}{2}}},Q_{+{\frac{1}{2}}}\}=f_{+{\frac{1}{2}}\;+{\frac{1}{2}}}^{\;\;+1A}B_{+1A}\qquad\rightarrow\qquad(f_{\delta_{+}\beta_{+}}^{+1A})_{sub}\neq0\.
[L_{1},L_{2},[{\textstyle\frac{k+1}{2}},+]]\oplus[L_{1},L_{2},[{\textstyle\frac{k+1}{2}},-]]\,longrightarrow\\bigoplus_{J}\N_{L_{1}\frac{k+1}{2}}^{(k)}{}^{J}\[J,L_{2},0]\\.
\frac{1}{4\pi^{2}}A(\vec{x})\wedgeF(\vec{x})=\frac{1}{2\pi^{2}}J^{3}(\vec{x})\wedgeJ^{1}(\vec{x})\wedgeJ^{2}(\vec{x})=\frac{1}{24\pi^{2}}\mathrm{tr}(g^{\dagger}(\vec{x})dg(\vec{x}))^{3},
S_{A}=-\frac{1}{4}\intF_{AB}F^{AB}d^{5}X=-\frac{1}{2}\int[\partial_{A}A_{B}(\partial^{A}A^{B}+\partial^{B}A^{A})-2\partial_{A}A^{A}\partial_{B}A^{B}]d^{5}X~.
\oint_{C}dx_{\mu}\alpha_{\mu}(S,x)=-\frac{1}{4}\int_{M(C)}d^{2}\tilde{\sigma}_{\alpha^{\prime}\beta^{\prime}}\int_{S}d^{D-2}\tilde{\sigma}_{\alpha\beta}\epsilon_{\mu\nu\alpha^{\prime}\beta^{\prime}}\epsilon_{\mu\kappa\alpha\beta}\partial_{\lambda}^{x}\partial_{\kappa}^{x}D(x-\bar{x}(\sigma))
S=\frac{ik}{4\pi}\int_{\partialM}dz\wedged\bar{z}\\mathrm{Tr}\left(h^{-1}\partial_{z}hh^{-1}\partial_{\bar{z}}h\right)+\frac{ik}{12\pi}\int_{M}\mathrm{Tr}\left(H^{-1}dH\right)^{3}.
S_{\mathrm{g.f.}}=\int_{M_{4}}\delta_{B}\{\bar{c}{\calL}_{c}^{\ast~k}\Sigma_{k}{}_{f}-\bar{\lambda}^{k}\delta\phi_{k}-\bar{\gamma}{\calL}_{\gamma}^{\ast~k}\phi_{k}+\bar{\eta}^{k}\delta\pi_{k}\}\ast1+\delta_{B}\{\pi^{ij}~{}^{\mathrm{t.f.}}\Sigma^{i}\wedge\Sigma^{j}\},
G_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})=\sum_{\varepsilon_{j}=\pm}v_{\varepsilon_{1}}\otimes\cdots\otimesv_{\varepsilon_{2n}}G_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n})^{\varepsilon_{1}\cdots\varepsilon_{2n}},
ds_{\mathrm{str}}^{2}=-dt^{2}+e^{-4U}d\vec{x}^{2}\,\qquade^{-4U}=e^{4\phi}=1-{\frac{1}{2}}\left({\frac{P_{\mathrm{vec}}}{r}}\right)^{2}\,\qquadM=0
\begin{array}{lll}{V}&{=\frac{1}{2}\int_{M}d^{2}x\Omega}&{\left[\frac{1}{2}F_{ij}F^{ij}+D_{i}\phi\overline{{D^{i}\phi}}\right.}\\{}&{}&{\left.+\frac{1}{4}(|\phi|^{2}-1)^{2}\right]}\\\end{array}
H^{(1)}=\intd^{2}x\left[m\Phi^{0}\partial_{i}A^{i}+\frac{m^{2}}{\sqrt{\kappa}}\epsilon_{ij}\Phi^{i}A^{j}+\sqrt{\kappa}\Phi^{i}\partial_{i}A^{0}-\Phi^{3}(mA^{0}-\frac{\kappa}{m}\epsilon_{ij}\partial^{i}A^{j})\right].
B_{i}^{v(1)}=-F_{i}^{\;\;k}({\bar{\nabla}}_{k}\Phi-\frac{1}{2}a_{k})+\frac{1}{2}{\bar{\nabla}}^{k}F_{ik}+\frac{1}{4a}{\hat{H}}_{ikm}G^{km}+\frac{1}{2}{\bar{\nabla}}_{i}S^{0},
E_{rot}\left[\bbox{\phi}\right]=\Lambda\int_{{\mathbbR}^{3}}d{\bfx}\;\frac{1}{2}\left(\partial_{t}\bbox{\phi}\right)^{2}+\frac{g_{1}}{8}\left(\partial_{t}\bbox{\phi}\times\partial_{i}\bbox{\phi}\right)^{2}+\frac{g_{2}}{8}\left(\partial_{t}\bbox{\phi}\right)^{2}\left(\partial_{i}\bbox{\phi}\right)^{2}+O\left(\left(\partial_{t}\bbox{\phi}\right)^{4}\right)\,.
R(\phi)=\sum_{n=0}^{\infty}r_{m_{1}\ldotsm_{n}}^{(n)}\phi_{m_{1}}\ldots\phi_{m_{n}},\;\;\;\;\;\;\tilde{R}(\tilde{\phi})=\sum_{n=0}^{\infty}r_{m_{1}\ldotsm_{n}}^{(n)}\tilde{\phi}_{m_{n}}\ldots\tilde{\phi}_{m_{1}}
\frac{i}{2}TrD^{-1}G=\frac{1}{12N}\intd^{D}x\left(\lambda_{0}+\frac{\eta_{0}\phi^{2}}{10N}\right)[G_{aa}(x,x)]^{2}+\frac{3\eta_{0}}{6!N^{2}}\intd^{D}x[G_{aa}(x,x)]^{3}+cte.
\sqrt{\frac{4m_{r}^{2}-k^{2}}{-k^{2}}}log\big[{\frac{\sqrt{4m_{r}^{2}-k^{2}}-\sqrt{-k^{2}}}{\sqrt{4m_{r}^{2}-k^{2}}+\sqrt{-k^{2}}}}\big]|_{k^{2}=0}=2.
x_{i}^{\gamma}=\beta^{\frac{\gamma}{n}}\nu^{i\,\gamma}\left[\psi_{0}\left(\frac{1}{\zeta}\right)+\frac{\gamma}{n}\sum_{\kappa=1}^{n-1}\nu^{i\,s\,\kappa}\,\theta_{\kappa}\,\rho^{\kappa}\psi_{\kappa}\left(\frac{1}{\zeta}\right)\right]\,\,\,\,\,\,\,\,\,(i=1,2,\dots,n),
\deltax^{\mu}=\epsilon\psi^{\mu}\,\quad\delta\psi^{\mu}=\epsilon{\frac{\dot{x}^{\mu}}{\sqrt{\dot{x}^{2}}}}-{\frac{1}{2{\dot{x}^{2}}}}(\dot{\epsilon}\psi\cdot\dot{x}+\epsilon\dot{\psi}\cdot\dot{x})\psi^{\mu}\.
\begin{array}{c}{b^{A}b_{A}=0\quad,\quadb^{A}\simab^{A}\quad,\quadh^{ABC}\simah^{ABC}\quad,\quada\inR\backslash\left\{0\right\}}\\{h^{ABC}=-\frac16\epsilon^{ABCDEF}h_{DEF}\qquad,\qquadb_{A}h^{ABC}=0}\\{h^{ABC}h_{CDE}=\frac14\delta^{[A}{}_{[D}b^{B]}b_{E]}}\\\end{array}
\times\Bigg|\frac{\prod_{i,j=1}^{n_{a}}\big(\hat{x}_{i}^{(a)}-\tilde{y}_{j}^{(a)}\big)\big(\tilde{x}_{i}^{(a)}-\hat{y}_{j}^{(a)}\big)}{\prod_{i,j=1}^{n_{a}}\big(\hat{x}_{i}^{(a)}-\tilde{x}_{j}^{(a)}\big)\big(\tilde{y}_{i}^{(a)}-\hat{y}_{j}^{(a)}\big)}\Bigg|\;,
\sum_{\alpha=0}^{\infty}\lambda^{\alpha}d\omega^{k_{s},\alpha}=\sum_{\alpha=0}^{\infty}\lambda^{\alpha}\left[-\frac{1}{2}c_{i_{p}j_{q}}^{k_{s}}\sum_{\beta=0}^{\alpha}\omega^{i_{p},\beta}\wedge\omega^{j_{q},\alpha-\beta}\right]\;.
\mu_{n}^{2}\phi_{n}(z)=\left(m^{2}-\frac{g^{2}N}{2\pi}\right)\left(\frac{1}{z}+\frac{1}{1-z}\right)\phi_{n}(z)-\frac{g^{2}N}{2\pi}\int_{0}^{1}dz^{\prime}\frac{\phi_{n}(z^{\prime})}{(z-z^{\prime})^{2}},
\begin{array}{lcl}{C^{\prime\prime}}&{=}&{\frac{1}{2}(X_{N-1,N}X^{N,N-1}+X_{N,N-1}X^{N-1,N})}\\{}&{}&{}\\{}&{=}&{-q^{N}q^{N-1}(X_{N-1,N})^{2}}\\{}&{}&{}\\{}&{=}&{(\eta_{N/2}X_{N-1,N})^{2}=(H_{N/2})^{2}}\\\end{array}
B_{i[j}^{\mu\nu}K_{klm]n}^{\rho\sigma\gamma\delta}+B_{n[j}^{\mu\nu}K_{klm]i}^{\rho\sigma\gamma\delta}+B_{i[j}^{\rho\sigma}K_{klm]n}^{\gamma\delta\mu\nu}+B_{n[j}^{\rho\sigma}K_{klm]i}^{\gamma\delta\mu\nu}+B_{i[j}^{\gamma\delta}K_{klm]n}^{\mu\nu\rho\sigma}+B_{n[j}^{\gamma\delta}K_{klm]i}^{\mu\nu\rho\sigma}\equiv0,
\tilde{f}_{0}(W_{i},g^{2})=1+\ldots+\left\langle\tilde{B}_{j}\right\rangle+\ldots+\left\langle\tilde{B}_{k}\right\rangle+\ldots+\left\langle\tilde{B}_{j}\tilde{B}_{k}\right\rangle+\ldots
x+\alpha+\beta+1-\frac{(\alpha+1)(\beta+1)}{x+\alpha+\beta+3-\frac{(\alpha+2)(\beta+2)}{x+\alpha+\beta+5-\ldots}}=\frac{\beta\int_{0}^{\infty}\frac{e^{-u}u^{\beta-1}}{(x+u)^{\alpha}}\du}{\int_{0}^{\infty}\frac{e^{-u}u^{\beta}}{(x+u)^{\alpha+1}}\du}
\beta_{\nuk}=i\int_{0}^{2\pi}\mathrm{d}\theta^{\prime}\int_{-\infty}^{\infty}\mathrm{d}z\int_{0}^{\infty}r\mathrm{d}r\left\{v_{\nu}(x^{\prime})\left[\partial_{t^{\prime}}\psi_{k}(x^{\prime})\right]-\left[\partial_{t^{\prime}}v_{\nu}(x^{\prime})\right]\psi_{k}(x^{\prime})\right\}.