harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2{\frac{\partialA^{i}}{\partial\bar{z}}}=\mathrm{e}^{i\theta}{\frac{dA^{i}}{d\hat{x}}}=\mathrm{e}^{2i\theta}2{\frac{\partialA^{i}}{\partial{z}}}=\mathrm{e}^{2i\theta}\Omega_{-}K^{ij^{*}}\frac{\partial{\calW}^{*}}{\partialA^{*j}}.
\mathrm{det}\,{T_{(i,j)}}-\frac{\mathrm{det}\,{T_{(i,\,k-1)}}\,\mathrm{det}\,{T_{(k+1,\,j)}}}{\Omega_{(i,\,k-1)}\Omega_{(k+1,\,j)}}\Omega_{(i,\,j)}=0.
\frac{4}{\sqrt{-\triangle}}\frac{1}{\sqrt{4m^{2}-\triangle}}{\mathrmarctanh}\left(\frac{\sqrt{-\triangle}}{\sqrt{4m^{2}-\triangle}}\right)=\int_{m}^{\infty}\!{\mathrmd}\tilde{m}\,\frac{1}{\sqrt{{\tilde{m}}^{2}-m^{2}}}\,\frac{1}{4{\tilde{m}}^{2}-\triangle}.
\hat{R}^{\epsilon}M_{1}(\tilde{M}_{2}^{-1})^{t}=\tilde{M}_{1}(M_{2}^{-1})^{t}\hat{R}^{\epsilon}\quad\quad(\hat{R}^{\epsilon}M\otimes(\tilde{M}^{-1})^{t}=\tilde{M}\otimes(M^{-1})^{t}\hat{R}^{\epsilon})
\left(\begin{array}{c}{Q^{(1)}}\\{Q^{(2)}}\\\end{array}\right)\equiv{\calQ}\rightarrow\Lambda{\calQ};\qquad\left(\begin{array}{c}{P^{(1)}}\\{P^{(2)}}\\\end{array}\right)\equiv{\calP}\rightarrow(\Lambda^{-1})^{T}{\calP}
\gamma_{~\mu\nu}^{\sigma}\rightarrow\tilde{\gamma}_{~\mu\nu}^{\sigma}=\gamma_{~\mu\nu}^{\sigma}-\frac{f_{3}}{2}\tilde{F}_{~~~~~\mu\nu~\alpha\beta}^{-1~\sigma~~\lambda}\biggl(B_{\lambda~~~\eta\epsilon}^{~\alpha\beta\tau}\nabla_{\tau}-\triangle_{\lambda~~\eta\epsilon}^{~\alpha\beta}\biggr)\biggl(\frac{(1+f_{1})}{4}hg^{\eta\epsilon}-h^{\eta\epsilon}\biggr).
I_{B\mathrm{finite}}=-\frac{(n-1)(4\pi)^{(n-2)/2}\beta}{8\pi\ell^{2}}\sum_{i=0}^{(n-2)/2}\left({\frac{(n-2)}{2}\atopi}\right)(-1)^{i}N^{2i}\left[\frac{r_{+}^{n-2i-1}}{n-2i-1}\right]
\nabla_{+}x^{i}-\frac12\Omega_{+}^{+2i}x^{-2}-\frac12\Omega_{+}^{-2i}x^{+2}-i\lambda_{q}^{1}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-ih_{+\dot{q}}^{1-}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}-(1\leftrightarrow2)=0,
D\!\!\!\!/^{~2}=\left[e_{q}^{\mu}\left(\delta^{qp}+2\sigma^{qp}\right)\left(\Gamma_{\mu\kappa}^{\nu}e_{~p}^{\kappa}-e_{~p}^{\nu}\phi_{\mu}\right)+\left(g^{\mu\nu}+2\sigma^{\mu\nu}\right)D_{\mu}\right]D_{\nu}.
R={\frac{k_{1}-\sqrt{\left({\frac{n_{T}}{n_{I}}}\right)^{2}\omega^{2}-\left(k_{2}^{2}+\cdots+k_{m}^{2}\right)}}{k_{1}+\sqrt{\left({\frac{n_{T}}{n_{I}}}\right)^{2}\omega^{2}-\left(k_{2}^{2}+\cdots+k_{m}^{2}\right)}}},
U_{0}^{-1}(\xi_{1}^{i},\xi_{2}^{i})=\psi_{JM}=\sum_{M_{1},M_{2}}C(\frac{J_{1}}{2}M_{1}-\frac{J_{1}}{2},\frac{J_{2}}{2}M_{2}-\frac{J_{2}}{2};\frac{J}{2}M-\frac{J}{2})\psi_{J_{1}M_{1}}\psi_{J_{2}M_{2}}
\left[g_{\mu\nu}(u)\right]=\left(\begin{array}{c|c}{1}&{u^{0}\vec{u}^{\mathrm{T}}}\\\hline{u^{0}\vec{u}}&{-I+\vec{u}\otimes\vec{u}^{\mathrm{T}}(u^{0})^{2}}\\\end{array}\right),
\nabla_{\mu}\nabla^{\mu}T+\nabla_{\mu}T\nabla^{\mu}\Phi-\frac{e^{-2\Phi}\left(\nabla_{\mu}\nabla_{\nu}T-\nabla_{\mu}\Phi\nabla_{\nu}T\right)\nabla^{\mu}T\nabla^{\nu}T}{1+e^{-2\Phi}\nabla_{\alpha}T\nabla^{\alpha}T}-\frac{e^{2\Phi}}{V}\frac{dV}{dT}=0,
\begin{array}{rcl}{{[}\,\phi(t,{\bfx}),\pi(t,{\bfy})\,{]}}&{=}&{\mathrm{i}\,\delta^{(3)}({\bfx}-{\bfy})\,,}\\{{[}\,\phi(t,{\bfx}),\phi(t,{\bfy})\,{]}}&{=}&{{[}\,\pi(t,{\bfx}),\pi(t,{\bfy})\,{]}=0\,.}\\\end{array}
\widehat{H}=\displaystyle\frac{2\pi}{L}\left(\widehat{L_{0}}+\widehat{\overline{{L}}_{0}}-\displaystyle\frac{c}{12}\widehat{I}+\lambda\displaystyle\frac{L^{2-h}}{\left(2\pi\right)^{1-h}}\widehat{B}\right)\:,
\left[{\frac{dn_{s}(\omega,\mu)}{d\omega}}\right]_{\mathrm{div}}={\frac{1}{(4\pi)^{2}\kappa}}\int_{\Sigma}\left[2b+a\left({\frac{\omega^{2}}{\kappa^{2}}}P+2\left(\frac16-\xi\right)R\right)\right]~~~,
\frac{\partial\varphi_{c(x,x_{0})}}{\partialx^{\rho}}=A_{\mu}(c)\cdot\frac{\partialc^{\mu}}{\partialx^{\rho}}\vert_{c=c(x,x_{0},0)}^{c=c(x,x_{0},1)}-\int_{0}^{1}F_{\nu\mu}\frac{\partialc^{\mu}}{\partialx^{\rho}}\frac{\partialc^{\nu}}{\partial\tau}d\tau
ds^{2}=G_{\mu\nu}dx^{\mu}dx^{\nu}=-\left(1-\frac{1+\lambda}{r}\right)dt^{2}+\left(1-\frac{\lambda}{r}\right)dx^{2}+\frac{1}{4}\frac{dr^{2}}{(r-\lambda)(r-1-\lambda)}\,
F_{\mu}^{\\nu}=\left[\begin{array}{ccc}{0}&{\pm{\vec{\partial}}\phi^{9}}&{0}\\{\pm{\vec{\partial}}\phi^{9}}&{0}&{-{\vec{\partial}}\phi^{9}}\\{0}&{+{\vec{\partial}}\phi^{9}}&{0}\\\end{array}\right]\quad,
{\frac{\pi^{2}}{8a}}\int_{0}^{1}{dx{\frac{1}{\sqrt{1-x^{2}}\left(1-\lambda^{2}+\lambda^{2}x^{2}\right)}}\left[{\frac{(x^{4}-3\delta^{2}x^{2})}{\lambda^{2}(x^{2}+\delta^{2})^{3}}}+{\frac{\left(1+\lambda^{2}-\lambda^{2}x^{2}\right)}{\left(1-\lambda^{2}+\lambda^{2}x^{2}\right)}}{\frac{x^{2}}{(x^{2}+\delta^{2})}}\right]},
\sum_{j=0}^{i}q^{jr}\chi_{1,k(r+1)+1-i+2j}^{(p,kp+1)}(q)=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+2n_{k}^{2}+r(n_{1}+\cdots+n_{k-1}+2n_{k})-(n_{1}+\cdots+n_{i})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}\overline{{F}}_{p-1;r}^{(2n_{k}+r)}(q).
{\calO}_{\gamma_{1}}^{(1)}=2\int_{\gamma_{1}}\mathrm{tr}[\chi\eta],\quad{\calO}_{\gamma_{2}}^{(2)}=\int_{\gamma_{2}}\mathrm{tr}[\chi\chi+2R\eta],\quad{\calO}_{\gamma_{3}}^{(3)}=2\int_{\gamma_{3}}\mathrm{tr}[R\chi],
\tilde{\Phi}(N;N)=\left[(\prod_{p,q}{\frac{\bar{z}_{p}-\bar{w}_{q}}{z_{p}-w_{q}}})(\prod_{p>q}{\frac{(z_{p}-z_{q})(w_{p}-w_{q})}{(\bar{z}_{p}-\bar{z}_{q})(\bar{w}_{p}-\bar{w}_{q})}})\right]^{\frac{\nu}{2}}\Phi(N;N)
\left.\left(\begin{matrix}{P_{0}^{r}}\\{P_{1}^{r}}\\\end{matrix}\right)\right|_{\mathrm{line}}=\frac{\alpha\,Q^{r}}{4(1-\xi^{2})}\left(\begin{matrix}{\frac{1}{\sqrt{2}}\left(2-\xi^{2}-2\gamma\xi^{2}\right)}\\{2-\xi^{2}-\beta\xi^{2}}\\\end{matrix}\right)\,.
i_{v}^{\star}\omega^{p}:=\frac{1}{2(p-1)!}[v^{\rho}\star\omega_{\rho\mu_{1}\ldots\mu_{p-1}}^{p}+\omega_{\rho\mu_{1}\ldots\mu_{p-1}}^{p}\starv^{\rho}]\,dx^{\mu_{1}}\wedge\ldots\wedgedx^{\mu_{p-1}}\,,
Z_{2}=\Biggl(\frac{a_{1}+Z_{1}}{{\bar{a}}_{1}+Z_{1}^{-1}}\Biggr)=Z_{0}^{-2}\Biggl(\frac{a_{0}+Z_{0}}{{\bar{a}}_{0}+Z_{0}^{-1}}\Biggr)\Biggl(\frac{\mu_{+}+Z_{0}}{{\bar{\mu}}_{+}+Z_{0}^{-1}}\Biggr)\Biggl(\frac{\mu_{-}+Z_{0}}{{\bar{\mu}}_{-}+Z_{0}^{-1}}\Biggr)
\frac{\partial{a_{\nu}}^{\lambda}}{\partialx^{\mu}}=\frac{\partial{a_{\mu}}^{\lambda}}{\partialx^{\nu}},\qquad\frac{\partial{b_{\nu}}^{\lambda}}{\partialx^{\mu}}=\frac{\partial{b_{\mu}}^{\lambda}}{\partialx^{\nu}},\qquad\frac{\partial{\overline{{b}}_{\nu}}^{\lambda}}{\partialx^{\mu}}=\frac{\partial{a_{\mu}}^{\lambda}}{\partialy^{\nu}}
S=\int_{R}dt\int_{\Sigma}d^{2}x\epsilon^{0ab}\bigl[-e_{a}^{i}\dot{A}_{b}^{i}+n^{0}\pi_{n}\dot{\phi}_{n}-A_{0}^{i}\partial_{b}e_{a}^{i}+e_{0}^{i}(F_{ab}^{i}+2\epsilon_{ij}e_{a}^{j}\pi_{n}\partial_{b}\phi_{n})\bigl]
=\int_{0}^{\infty}\ldots\int_{0}^{\infty}\mathrm{d}s_{12}\ldots\mathrm{d}s_{(n-1)n}\tilde{F}(s_{12},\ldots,s_{(n-1)n},g^{2},d)e^{-(s_{12}r_{12}^{2}+\ldots+s_{(n-1)n}r_{(n-1)n}^{2})}
P_{i}=\intd^{3}x\:\left\{F_{0k}F_{ik}-{\frac{1}{2}}D_{0}\lambda\:D_{i}\phi-{\frac{1}{2}}D_{0}\phi\:D_{i}\lambda+iD_{i}\psi_{j}\:\chi_{j}+iD_{i}\eta\:\psi_{0}-A_{i}\:G\right\},
{\bfG^{(k)}}=\frac{1}{2}{\bfI}_{q}\oplus({\bfI}_{k}-\frac{1}{k+2-D}{\bf\Xi}_{k})~,~{\bfG_{(k)}^{-1}}=2\,{\bfI}_{q}\oplus({\bfI}_{k}-\frac{1}{D-2}{\bf\Xi}_{k})\,.
T_{\mu\nu}^{eff}=\phi_{,\mu}\phi_{,\nu}-K\tilde{g}_{\mu\nu}+\frac{b_{g}M^{4}}{(\zeta+b)^{2}}e^{-2\alpha\phi/M_{p}}\tilde{g}_{\mu\nu}+T_{\mu\nu}^{(f,canonical)}-mF(\zeta)\overline{{\Psi}}^{\prime}\Psi^{\prime}\tilde{g}_{\mu\nu},
\begin{array}{rl}{\phi_{j}(b(j))\,}&{=\,\frac{l^{2}}{(q-q^{-1})^{l}}\frac1{[d_{j}^{\pm}]^{2}}}\\{\phi_{j}^{\prime}(b(j))\,}&{=\,-\frac{l^{2}}{(q-q^{-1})^{(l+1)}}\frac{[2d_{j}^{\pm}]}{[d_{j}^{\pm}]^{5}}\qquad\qquad\mathrm{for}\quadj=0,\ldots,(m-1)}\\{\phi_{m}(b(m))\,}&{=\,\frac{l^{2}}{(q-q^{-1})^{(l-1)}}}\\\end{array}
R_{\mu\nu,\alpha\beta}^{L}(x)=\frac{1}{2}[\partial_{\alpha}\partial_{\nu}h_{\mu\beta}(x)-\partial_{\alpha}\partial_{\mu}h_{\nu\beta}(x)-\partial_{\beta}\partial_{\nu}h_{\mu\alpha}(x)+\partial_{\beta}\partial_{\mu}h_{\alpha\nu}(x)].
\frac{1}{2}[Q(\hat{K}_{L}(\hat{I}),\hat{K}_{L}(\hat{I})]=-\frac{1}{2\pi}\int_{0}^{\pi/2}d\sigmac^{0}(\sigma)\partial_{\sigma}F_{ik}^{I}(\sigma)\eta_{IJ}\partial_{\sigma}F_{kj}^{J}(\sigma)
S=-\frac{Lm^{3}\lambda}{8\pi}\sum_{j=0}^{\infty}\frac{(2B\lambda^{2})^{-j}}{j!}\sum_{k=1}^{\infty}\frac{(2k+j-1)!}{(2k)!}\frac{{\calB}_{2k+2j}}{(2k+j-\frac{1}{2})(2k+j-\frac{3}{2})}\left(\frac{2B}{m^{2}}\right)^{2k+j}
\begin{array}{c}{<0|C(\mu_{1})\cdotsC(\mu_{N})B(\lambda_{N})\cdotsB(\lambda_{1})|0>=}\\{=\sum_{\{\kappa\},\{\nu\}}K\left(\begin{array}{cc}{-}&{+}\\{\{\kappa\}}&{\{\nu\}}\\\end{array}\right)\tilde{K}\left(\begin{array}{cc}{\{\kappa\}}&{\{\nu\}}\\{+}&{-}\\\end{array}\right)\prod_{\{\kappa\}}a(\kappa)\prod_{\{\nu\}}a(\nu)}\\\end{array}
\begin{array}{rl}{\bar{\phi}_{ab}(z):=}&{(\bar{\phi}_{ab}^{(0)}(z),\cdots,\bar{\phi}_{ab}^{(n-1)}(z)),}\\{\bar{\phi}_{aa+\bar{\epsilon}_{j}}^{(i)}(z):=}&{(\tilde{\Phi}(z))_{j}^{i}/Ch(z-(n-1)/2),}\\\end{array}
{\calT}_{22}(q)=i\int_{0}^{\infty}\frac{d{q_{0}^{\prime}}^{2}}{2\pi}\rho(q_{0}^{\prime},\vec{q})\left[\frac{-i}{{q_{0}}^{2}-{q_{0}^{\prime}}^{2}-i\epsilon}+2\pi\delta({q_{0}^{\prime}}^{2}-{q_{0}}^{2})\frac{1}{e^{\beta|q_{0}|}-1}\right]
\mathrm{d}s^{2}=G_{\mu\nu}(\gamma,R)\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}=\mathrm{d}\rho^{2}+\rho^{2}\left(\mathrm{d}\phi+{\frac{\gamma}{R}}\mathrm{d}y\right)^{2}+\mathrm{d}y^{2}+\mathrm{d}x_{i}\mathrm{d}x^{i}\,,
R^{\mu}{}_{\nu,\lambda\rho}=\partial_{\lambda}\Gamma^{\mu}{}_{\nu,\rho}-\partial_{\rho}\Gamma^{\mu}{}_{\nu,\lambda}+\Gamma^{\mu}{}_{\sigma,\lambda}\Gamma^{\sigma}{}_{\nu,\rho}-\Gamma^{\mu}{}_{\sigma,\rho}\Gamma^{\sigma}{}_{\nu,\lambda},
N_{3V(c)k}^{(r)}\left(p_{1}^{2},p_{2}^{2},p_{3}^{2}\right)=\delta_{cf}\delta_{k0}\left(p_{1}^{2}p_{2}^{2}p_{3}^{2}\right)^{r}+\sum_{m_{1},m_{2},m_{3}\ge0}\,C_{m_{1}m_{2}m_{3}}^{(c)k}\left(p_{1}^{2}\right)^{m_{1}}\left(p_{2}^{2}\right)^{m_{2}}\left(p_{3}^{2}\right)^{m_{3}}\left(\Lambda^{2}\right)^{3r-k-(m_{1}+m_{2}+m_{3})}\,.
W_{L}=\sum_{s,r=1}^{N_{f}}(\lambda_{r,s}M+{m_{Q}}_{r,s})V^{2r-1,2s}+(\mathrm{Pf}V)^{\frac{1}{N_{f}-2}}(2-N_{f}){\Lambda_{L}}^{\frac{N_{f}-6}{N_{f}-2}}+\sum_{k=2}^{N_{c}}g_{r}{u^{cl}}_{r},\nonumber
S_{0}=\intd^{4}x~h\left[(2m)^{-1}D^{\mu}\stackrel{\sim}{\Phi}^{\dagger}D_{\mu}\stackrel{\sim}{\Phi}-\muN_{B}n_{A}+\frac{1}{2}(2m)^{-2}\bar{b}\left(N_{B}n_{A}\right)^{2}\right]
\deltam_{\lambda}^{2}=\deltam_{0}^{2}+\frac{1}{32\pi^{2}}\int_{0}^{1}dx\int_{{\calM}_{0}^{2}}^{\infty}d{\calM}^{2}\frac{f_{\lambda}^{2}-f_{\lambda_{0}}^{2}}{{\calM}^{2}-m^{2}}\;.
H(t)=\intd^{3}x\left\{-\frac{\hbar^{2}}{2a^{3}(t)}\frac{\delta^{2}}{\delta\eta^{2}}+\frac{a(t)}{2}\left(\vec{\nabla}\eta\right)^{2}+a^{3}(t)\left(V(\phi)+V^{\prime}(\phi)\eta+\frac{1}{2}V^{\prime\prime}(\phi)\eta^{2}+\cdots\right)\right\}
W_{5}(p,q,r;\psi,\phi,\eta)=\left(\begin{array}{ccccc}{1+\frac{p^{2}+q^{2}+r^{2}}{2}}&{p}&{q}&{r}&{-\frac{p^{2}+q^{2}+r^{2}}{2}}\\{p}&{}&{}&{}&{-p}\\{q}&{}&{R(\psi,\phi,\eta)}&{}&{-q}\\{r}&{}&{}&{}&{-r}\\{\frac{p^{2}+q^{2}+r^{2}}{2}}&{p}&{q}&{r}&{1-\frac{p^{2}+q^{2}+r^{2}}{2}}\\\end{array}\right)
A_{i}(s)={\frac{2}{\Gamma(s)}}\Bigr[r_{+}^{2s}+(-1)^{i}r_{-}^{2s}\Bigr]\zeta_{R}(2s+i)\sum_{l=0}^{i}x_{i,l}{\frac{\Gamma(s+l+{\frac{i}{2}})}{\Gamma(l+{\frac{i}{2}})}}.
\gamma^{0}=\sigma_{3},\quad\gamma^{1}=e^{{\frac{i}{2}}\sigma_{3}\chi_{s}}i\sigma_{1}e^{-{\frac{i}{2}}\sigma_{3}\chi_{s}},\quad\gamma^{2}=e^{{\frac{i}{2}}\sigma_{3}\chi_{s}}is\sigma_{2}e^{-{\frac{i}{2}}\sigma_{3}\chi_{s}},
F(p)=\int\frac{d^{3}q}{(2\pi)^{3}}\,\frac{1}{(p-q)^{2}}\frac{\bf{p.q}}{\left|{\bfp}\right|\left|{\bfq}\right|}\frac{F(q)}{\left|-\frac{1}{2}\delta+iq_{3}+\frac{\left|{\bfq}\right|^{2}}{2m}+\Sigma_{\pm}(q_{3},\left|{\bfq}\right|)\right|^{2}}
\Gamma\left(-N-\frac12,w\right)=\frac{(-1)^{N}}{\Gamma\left(N+\frac32\right)}\left[-\pi\,\mathrm{erfc}\left(\sqrt{w}\,\right)+e^{-w}\sum_{l=0}^{N}(-1)^{l}\Gamma\left(l+\frac12\right)w^{-l-\frac12}\right]\,,
<\rho_{-}>=\frac{k_{-}}{\sqrt{K}\alpha^{\prime}},\;\;\;\;\;<P_{-}>=\frac{1}{2\sqrt{K}\alpha^{\prime}}[\frac{k_{-}^{2}-2}{k_{-}}+\frac{2}{k_{-}}\;\frac{E(k_{-})}{K(k_{-})}],
\intd\xi_{0}^{(1)}d\xi_{0}^{(2)}d\xi_{0}^{(3)}~Tr\left[A^{(1)}(\xi_{0}^{(1)},\xi)\starA^{(2)}(\xi_{0}^{(2)},\xi)\starA^{(3)}(\xi_{0}^{(3)},\xi)\right]\sim\langle\Psi^{(1)}|\langle\Psi^{(2)}|\langle\Psi^{(3)}|V_{3}\rangle,
\tilde{\calL}_{\kappa=0}=-8(D-2)^{2}\intdr\left({\frac{\partialW}{\partial\phi}}\right)^{2}\left(1+{\frac{1}{2}}\partial_{\mu}\tilde{r}\partial^{\mu}\tilde{r}\-{\frac{1}{2}}\bar{\tilde{\eta}}_{-i}\tilde{\Gamma}^{\mu}\partial_{\mu}\tilde{\eta}_{-}^{i}\right)\.
X_{2}=\int_{\sqrt{\lambda^{2}-4}}^{\lambda}{dx{\frac{1}{(\lambda^{2}-x^{2})^{1/2}\left[(\lambda^{2}-x^{2})^{1/2}+1\right]}}\left[{\frac{1}{x^{2}}}+{\frac{1}{\left[(\lambda^{2}-x^{2})^{1/2}+1\right]}}\right]}.
{\mathcal{D}}^{\mu\nu}(k;\alpha)=\frac{1}{k^{2}}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)+\frac{\kappa\bar{k}^{2}\bar{g}^{\mu\nu}}{k^{2}\left(k^{2}\left(k^{2}-m_{\mathrm{A}}^{2}\right)-\kappa\bar{k}^{2}\right)}-\frac{\kappa\bar{k}^{\mu}\bar{k}^{\nu}}{k^{2}\left(k^{2}\left(k^{2}-m_{\mathrm{A}}^{2}\right)-\kappa\bar{k}^{2}\right)}+\alpha\frac{k^{\mu}k^{\nu}}{(k^{2})^{2}}.
T=\left(\begin{array}{ccccc}{\mathrm{rot}\theta_{1}}&{}&{}&{}&{}\\{}&{\mathrm{rot}\theta_{2}}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{\mathrm{rot}\theta_{N}}&{}\\{}&{}&{}&{}&{{\large1}}\\\end{array}\right)
L^{(+)}=\left(\begin{array}{cc}{(-1)^{F}q^{\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{(q-q^{-1})(-1)^{F}Y^{+}}\\{0}&{(-1)^{F}q^{\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
\tilde{\Gamma}_{\mu\nu}^{abc,r}=(G^{-1})_{\nu\alpha}^{r}(y)(G^{-1})^{r}(z)<\tilde{z}_{2}\tilde{g}t^{aed}B_{\mu}^{e}(x)c^{d}(x)\bar{c}^{c}(z)B_{\nu}^{b}(y)>^{r}
g=\Biggl({\frac{\frac{1}{2}(1+\epsilon)+e^{-k(r+it)}}{\frac{1}{2}(1+\epsilon)+e^{k(r+it)\hphantom{-}}}}\Biggr)\prod_{j=1}^{n}\Biggl({\frac{a_{j}+e^{-k(r+it)}}{\bar{a}_{j}+e^{k(r+it)}}}\Biggr)
I=\frac{1}{2}\intd^{2}x\sqrt{-g}e^{-2\phi}\Big(R+4\Lambdae^{2\phi}-(D\psi)^{2}-\frac{1}{4}(D\psi_{1})^{2}-\frac{1}{4}e^{-\psi_{1}-2\phi}F_{2}^{2}-\frac{1}{4}e^{\psi_{1}-2\phi}F^{2}\Big).
e^{\phi}={\frac{1}{N}}\left[(g_{f}^{2})^{1/x}\left({\frac{u}{\calR}}\right)\,\left({\frac{d_{p}^{1/\tilde{d}}}{c_{p}^{1/x}}}\right)\right]^{{\frac{-(D-2)a}{8}}\left({\frac{\beta+1}{\beta}}\right)}\,.
I_{se}=-4\intd^{4}k\left[\left({\frac{1}{\epsilon}}+2-C\right){\frac{1}{(k^{2})^{1+\epsilon}}}-{\frac{1}{\epsilon}}{\frac{1}{k^{2}}}\right]{\frac{1}{(k+p)^{2}}}={\frac{2}{\epsilon^{2}}}-{\frac{2}{\epsilon}}+(finite)
f\left(T_{a}(z),\zeta\right)=\left[T_{a}^{\prime}(z)\right]^{q-1}\left\{f(z,\zeta)+\sum_{b}\left[\frac{T_{a}^{\prime}T_{b}(\zeta)}{\left(T_{a}^{\prime}(z)\right)^{q-1/2}}-1\right]\left[z-T_{b}(\zeta)\right]^{-1}\left[T_{b}^{\prime}(\zeta)\right]^{q}\right\}\.
[X,Y]_{\beta\gamma}=-g^{2}\sum_{\kappa^{\prime}\cdot\gamma=1,\\kappa^{\prime}\cdot\beta=0}\left[x((\beta-\gamma+\kappa^{\prime})\cdotq)y(-\kappa^{\prime}\cdotq)-y((\beta-\gamma+\kappa^{\prime})\cdotq)x(-\kappa^{\prime}\cdotq)\right],
\mathrm{d}s^{2}=4\Big(\mathrm{d}|v_{1}|\,\mathrm{d}|v_{1}|+|v_{1}|^{2}(\Omega_{1})^{2}+\mathrm{d}|v_{2}|\,\mathrm{d}|v_{2}|+|v_{2}|^{2}(\Omega_{2})^{2}+(\mathrm{d}\rho)^{2}+(\mathrm{d}\tau)^{2}\Big)\,.
\begin{array}{c}{\sum_{i_{1}...i_{N}}d^{-N}\sum_{S}\prod_{j}{}\left(\varphi_{j},\psi_{S(j)}\right)\left(e_{i_{P(j)}},e_{i_{Q(j)}}\right)=}\\{=\sum_{S}\prod_{j}\left(\varphi_{j},\psi_{S(j)}\right)\varphi^{\left(N\right)}(PSQ^{-1})}\\\end{array}
\biggl[\alpha_{\mu}D_{\mu}+\fraci2\left(\kappa_{0}P_{0}+\kappa_{1}\overline{{P}}+\kappa_{2}\overline{{\overline{{P}}}}\right)\alpha_{\mu\nu}\mathcal{F}_{\mu\nu}+mP_{1}+\frac{m_{0}^{2}}{m}P_{0}\biggr]\Psi(x)=0,
+[({\frac{1}{2}}\theta_{2}^{\prime}+{\frac{1}{2}}\pi_{2})^{2}+2eA_{-}({\frac{1}{2}}\theta_{2}^{\prime}+{\frac{1}{2}}\pi_{2})+{\frac{e^{2}}{2}}(A_{-}^{2}-A_{+}A_{-})]={\calH}_{R}+{\calH}_{L},
q(t^{\prime})^{-(1+r)}+2r\frac{t^{\prime\prime\prime}t^{\prime}-(t^{\prime\prime})^{2}}{(t^{\prime})^{2}}+\frac{(1+r)^{2}+4r^{2}}{4}\left(\frac{t^{\prime\prime}}{t^{\prime}}\right)^{2}-1=0\,.
4^{4}\frac{\partial}{\partialB^{\prime}}\frac{\partial}{\partialb_{3}}\frac{\partial}{\partialb_{4}}\frac{\partial}{\partialb_{5}}\hat{\vartheta}(\mu)=\left(\psi_{0}\frac{\partial^{4}}{\partial\psi_{0}^{4}}\right)\frac{1}{\psi_{0}}\hat{\vartheta}(\mu)\.
d_{Q}\Big(T_{n}+N_{n}\Big)(x_{1},\ldots,x_{n})=\sum_{l=1}^{n}\frac{\partial}{\partialx_{l}^{\nu}}\,\Big(T_{n/l}^{\nu}+N_{n/l}^{\nu}\Big)(x_{1},\ldots,x_{l},\ldots,x_{n})
{\frac{\delta\Gamma}{\deltaV}}=-{\frac{\deltaW}{\deltaV}};\;\;\;\;{\frac{\delta\Gamma}{\deltaK}}=-{\frac{\deltaW}{\deltaK}};\;\;\;\;{\frac{\delta\Gamma}{\deltaL}}=-{\frac{\deltaW}{\deltaL}};\;\;\;\;{\frac{\delta\Gamma}{\deltaM}}=-{\frac{\deltaW}{\deltaM}};\;\;\;\;\,
\mathrm{Tr}\left[\left({\frac{\partial}{\partial\Phi}}\right)U^{\prime}(\Phi)\right]=\sum_{m,n=1}^{\nu}{\frac{U^{\prime}(x_{m})-U^{\prime}(x_{n})}{x_{m}-x_{n}}}=\sum_{m=1}^{\nu}U^{\prime\prime}(x_{m})+\sum_{m\neqn=1}{\frac{U^{\prime}(x_{m})-U^{\prime}(x_{n})}{x_{m}-x_{n}}}
K_{UD}(p)=\frac{1}{p^{2}+i\varepsilon}\frac{4d_{Q}(R)}{m_{U}^{2}+m_{D}^{2}}\int\frac{id^{4}l}{(2\pi)^{4}}\frac{(m_{D}^{2}-m_{U}^{2})l\cdotp-m_{U}^{2}(p^{2}+i\varepsilon)}{(l^{2}-m_{U}^{2}+i\varepsilon)[(l+p)^{2}-m_{D}^{2}+i\varepsilon]}
{\frac{d\tilde{n}^{(\lambda)}(\omega)}{d\omega}}={\frac{dn^{(\lambda)}(\omega)}{d\omega}}-{\frac{1}{4\lambda}}\sum_{k}\left[(\phi_{\omega,k}^{(\lambda)},\partial_{\lambda}H^{2}(\lambda)\phi_{\omega,k}^{(\lambda)})+(\phi_{\omega,k}^{(\lambda)},\partial_{\lambda}H^{2}(\lambda)\phi_{\omega,k}^{(\lambda)})^{*}\right]~~~,
\hat{H}={\frac{1}{2}}V^{\frac{1}{2}}f^{-{\frac{1}{2}}}\hat{p}_{i}V^{-1}G^{ij}f\hat{p}_{j}f^{-{\frac{1}{2}}}V^{\frac{1}{2}}+{\frac{i}{2}}V^{-{\frac{1}{2}}}f^{\frac{1}{2}}C_{oa}^{aj}\hat{p}_{j}f^{-{\frac{1}{2}}}V^{\frac{1}{2}}+V,
w_{n}^{\mathrm{susy}}\left(F_{n}\right)=\int\prod_{i=1}^{n}d^{2}x_{i}\prod_{i=1}^{n}d\xi_{i}\,\,w_{n}^{\mathrm{susy}}\left(x_{1},\xi_{1};\ldots;x_{n},\xi_{n}\right)F_{n}\left(x_{1},\xi_{1};\ldots;x_{n},\xi_{n}\right)\,\,.
\mathrm{~H_{nn}^{\{{\bf~R}\}}(0,0)~}=\left(\frac{R_{11}R_{22}}4\right)^{\fracn2}\sum_{l=0}^{\left[\fracn2\right]}\frac{(-1)^{n}(n!)^{2}}{(l!)^{2}\left(n-2l\right)!}(2r)^{n-2l}.
\langle\Psi_{1}^{\primeI_{1}}(z_{1})\Psi_{2}^{\primeI_{2}}(z_{2})\cdots\Psi_{n}^{\primeI_{n}}(z_{n})\rangle=\langle\Psi_{1}^{I_{1}}(z_{1})\Psi_{2}^{I_{2}}(z_{2})\cdots\Psi_{n}^{I_{n}}(z_{n})\rangle\,.
R\equivR_{\alpha}^{\alpha}=\frac{1}{2~A}\Biggl\{2(\frac{\dot{A}}{A})^{2}+(\frac{\dot{B}}{B})^{2}-(\frac{\dot{C}}{C})^{2}-2\frac{\ddot{A}}{A}-4\frac{\ddot{B}}{B}-2(\frac{A^{\prime}}{A})^{2}-(\frac{B^{\prime}}{B})^{2}+(\frac{C^{\prime}}{C})^{2}+2\frac{A^{\prime\prime}}{A}+4\frac{B^{\prime\prime}}{B}\Biggr\}.
T_{B}^{A}=(-1)^{p+1}\left(\frac{1}{4}\delta_{B}^{A}F_{\mu_{1}\dots\mu_{p+1}}F^{\mu_{1}\dots\mu_{p+1}}-\frac{p+1}{2}F_{\quad\mu_{1}\dots\mu_{p}}^{A}F_{B}^{\quad\mu_{1}\dots\mu_{p}}\right)~.
\partial^{\mu}\,F_{\mu\nu}\,=\,\Pi_{\nu0}\,(\phi^{N}(t)\,\delta(z))\,=\,\eta_{\nu0}\,\phi^{N}(t)\,\delta(z)\,-\int\,{\frac{dp^{2}}{(2\pi)^{2}}}{\frac{p_{\nu}p_{0}}{p^{2}}}\,\mathrm{e}^{-ipx}\,\tilde{\phi_{N}}(p_{0}),
\lambda^{(3)}=\left(\begin{matrix}{A_{1}}&{A_{2}}&{A_{3}}&{S}\\{A_{2}}&{-A_{1}}&{-S}&{A_{3}}\\{A_{3}}&{S}&{-A_{1}}&{-A_{2}}\\{-S}&{A_{3}}&{-A_{2}}&{A_{1}}\\\end{matrix}\right)
\frac{(\varepsilon-i)^{-\frac{1}{2}l}\,(\varepsilon+i)^{-\frac{3}{2}l}\,i^{\,d_{j}+s}}{\Bigl(\,\sum_{k,v}A_{kv}^{j}(\vec{\beta},\vec{\lambda})\,i\,p_{k}\etap_{v}+\,(i\varepsilon-1)\,m^{2}\sum_{k=1}^{s}\beta_{k}\Bigr)^{\,d_{j}+s}}\\Biggr]\quad.
{\int}d{\bfx}\,\left\{{\bar{\calF}}^{ba}({\bfr},{\bfx}_{0})\,+X_{i}\partial_{i}{\bar{\calF}}^{ba}({\bfr},{\bfx}_{0})+\frac{1}{2}\,X_{i}X_{j}\,\partial_{i}\partial_{j}{\bar{\calF}}^{ba}({\bfr},{\bfx}_{0})\;+\;\;\cdots\right\}{j}_{0}^{a}({\bfx})
\prod_{j}\frac{1}{i}\frac{\delta}{\deltaS_{\alpha_{j}}(y_{j})}Z_{0}\mid_{J=0}=-\frac{1}{e^{2}\xi}\frac{(\Box\partial_{\mu})_{x}}{E_{0}^{2}}<T^{*}[A^{\mu}(x)\prod_{j}A_{j}^{\alpha}(y_{j})]>=0,
n_{\pm}=\sqrt{1-\frac{\rho_{*}^{2}}{\rho_{p}^{2}}}\left[-\frac{1}{F(2\rho_{p})}\partial_{\Theta}\pm\frac{2\rho_{p}}{\ell\sqrt{1-\rho_{*}^{2}/\rho_{p}^{2}}}\partial_{R}\right].
\partial{\bar{\partial}}(\hat{\cal\phi}^{0},\hat{A_{z}^{0}})\sim-6(-1+a_{\Phi,z})a_{\Phi,z}^{3}{\frac{(((c_{\Phi,z}e_{\Phi,z})^{2}b_{\Phi,z}f_{\Phi,z}+(b_{\Phi,z}f_{\Phi,z})^{2}c_{\Phi,z}e_{\Phi,z})}{|z|^{2}}}.
\begin{array}{ccc}{{\calW}_{+}^{\dagger}}&{=}&{{\calW}_{-}}\\{{\calW}_{-}^{\dagger}}&{=}&{{\calW}_{+}}\\\end{array}\quad\begin{array}{ccc}{{\calW}_{3}^{\dagger}}&{=}&{{\calW}_{3}}\\{{\calW}_{4}^{\dagger}}&{=}&{{\calW}_{4}}\\\end{array}
|\frac{1}{2},-\frac{1}{2}\rangle_{0}=\frac{1}{\sqrt{2(q+q^{-1}+1)}}(-(q^{1/2}+q^{-1/2})|\downarrow\uparrow\downarrow\rangle-q^{1/2}|\downarrow\downarrow\uparrow\rangle-q^{1/2}|\downarrow\downarrow\uparrow\rangle)
I(p)=\intd^{d}q\frac{q_{\nu}q_{\rho}}{(p+q)^{2}q^{2}\left[-q^{2}-2q.p\frac{b^{\prime}}{a^{\prime}}-p^{2}\frac{c^{\prime}}{a^{\prime}}\right]^{2-\frac{d}{2}}}\;.
\epsilon_{FL}\nabla_{AA^{\prime}}\nabla^{B^{\prime}(F}\;\rho_{B^{\prime}}^{\;\;\;A)L}+{\frac{1}{2}}\nabla_{\;\;A^{\prime}}^{A}\nabla^{B^{\prime}M}\rho_{B^{\prime}(AM)}+{\Box}_{AM}\;\rho_{A^{\prime}}^{\;\;\;(AM)}+{\frac{3}{8}}\Box\rho_{A^{\prime}}=0\;,
\begin{array}{l}{[y,\bar{y}]={\frac{\mu}{3}}y^{3}}\\{[y^{3},y]={\frac{\mu}{6}}(y-\overline{{W^{\prime}}}(\overline{{y}}))}\\{[y^{3},\bar{y}]=-{\frac{\mu}{6}}(\bar{y}-W^{\prime}(y))}\\\end{array}
R_{\alpha_{0},j}^{i}R_{\beta_{0}}^{j}-(-1)^{\epsilon_{\alpha_{0}}\epsilon_{\beta_{0}}}R_{\beta_{0},j}^{i}R_{\alpha_{0}}^{j}=-R_{\gamma_{0}}^{i}F_{\alpha_{0}\beta_{0}}^{\gamma_{0}}-M_{\alpha_{0}\beta_{0}}^{ij}S_{\mathrm{cl},j},
e^{-i\left(k_{+}y^{+}+k_{-}y^{-}\right)}\left[\phi_{k}\left(x\right),\psi_{k\alpha}\left(x\right),V_{+k}\left(x\right)\right],\quadwith{\thinspace\thinspace\thinspace\thinspace}k_{+}k_{-}=\left|A\right|^{2}.
S_{p}=-\frac{V_{p-2}\tilde{V}_{2}}{(2\pi)^{p}\tilde{g}\tilde{b}}\intd\tau\left(\frac{u}{R}\right)^{7-p}\left[\sqrt{\tilde{f}}-1+\frac{u_{0}^{7-p}}{2u^{7-p}}\right].
W\,=\,\frac{\rhoF}{2}\int_{0}^{l}dx\,\left(\dot{y}^{2}\,+\,r^{2}y_{1}^{2}\right)\,-\,\frac{EI}{2}\int_{0}^{l}dx\,y_{1}^{2}\,-\,\frac{kFG}{2}\int_{0}^{l}dx\,y_{2}^{2}.
X_{(3t_{2})}={\frac{e^{{\frac{5}{2}}i\lambda_{2}}}{(2\mathrm{sin}{\frac{g_{s}}{2}})^{2}}}(e^{i\lambda_{2}}-1)(2e^{i\lambda_{2}}-1)+{\frac{1}{3(2\mathrm{sin}{\frac{3g_{s}}{2}})^{2}}}(-e^{{\frac{3}{2}}i\lambda_{2}}+e^{{\frac{9}{2}}i\lambda_{2}})+e^{{\frac{7}{2}}i\lambda_{2}}-e^{{\frac{9}{2}}i\lambda_{2}}.
S_{|P_{T_{l}}|}^{WZW}(\eta)\doteq\frac{\kappa}{4\pi\sqrt{-1}}\int_{((M;N_{0}),\mathcal{C})}tr\left(\widehat{h}^{-1}\partial\widehat{h}\right)\left(\widehat{h}^{-1}\overline{{\partial}}\widehat{h}\right)+S^{WZ}(\eta),
\beta^{(a)}(\tau)={2\pii\large\frac{{\int_{-1}^{1}}dx\sqrt{{\frac{x\Theta_{3}^{4}+\Theta_{2}^{4}-\Theta_{4}^{4}}{x^{2}-1}}}}{(\Theta_{2}^{8}-\Theta_{2}^{4}\Theta_{3}^{4})\left[\large{\int_{-1}^{1}}dx\sqrt{{\frac{1}{(x^{2}-1)(x\Theta_{3}^{4}+\Theta_{2}^{4}-\Theta_{4}^{4})}}}\right]^{3}}}.
f(z)_{a}\equiv\sum_{n=0}^{\infty}(-1)^{n}{\frac{a^{2n}z^{2n+1}}{(2n+1)^{2}}}=z\F\left(1,{\frac{1}{2}},{\frac{1}{2}};{\frac{3}{2}},{\frac{3}{2}};-a^{2}z^{2}\right).
\mathrm{Re}\,\Pi_{00}^{\mathrm{time}-\mathrm{like}}(0,0)={\frac{e^{2}}{\pi^{2}}}\int_{2m}^{\infty}d\omega\,\omega\sqrt{1-{\frac{4m^{2}}{\omega^{2}}}}\,n_{F}({\frac{\omega}{2}})
\langle~\delta_{S}{\bar{Q}}_{\dot{\alpha}}({\calI}^{+})~\rangle~=~\epsilon^{\beta}~\sum_{k}k_{\beta{\dot{\alpha}}}~\sum_{n_{k}^{B},n_{k}^{F}}~\rho_{n_{k}^{B}n_{k}^{F},n_{k}^{B}n_{k}^{F}}~(-)^{n_{k}^{F}}~\left(~n_{k}^{B}~+~n_{k}^{F}~\right)~.
\,\,\,\,\partial^{\mu}A_{\mu}\,=\,0\,\,\,\,\,;\,\,\,F_{\mu\nu}\,+\,{\tilde{F}}_{\mu\nu}\,=\,0\,\,\,\,\,;\,\,\,\partial^{\mu}\Psi_{\mu}\,=\,0
\begin{array}{rcl}{}&{}&{A^{\prime\prime}\equivA_{1}^{\prime\prime}+A_{2}^{\prime\prime}~,}\\{}&{}&{(\phi^{\prime})^{2}\equiv(\phi_{1}^{\prime})^{2}+(\phi_{2}^{\prime})^{2}~,}\\{}&{}&{\displaystyle\bar{\Lambda}=0~,}\\\end{array}
\left(\sum_{{\bf{\calN}}\in{\bfZ}^{2g}/\{\bf0\}}({\bf{\calN}}^{t}\Omega{\bf{\calN}})^{-s}\right)_{(\Omega_{2}\rightarrow0)}\longrightarrow\Omega_{2}^{s}\sum_{{\bf{\calN}}\in{\bfZ}^{\mathrm{g}}/\{\bf0\}}({\bf{\calN}}^{t}{\bf{\calN}})^{-s}=\Omega_{2}^{s}Z_{\mathrm{g}}|_{\bf0}^{\bf0}|(2s)\,,
\sum_{i,j=1,2}h_{ij}J_{L}^{i}{\bar{J}}_{R}^{j}=\Phi_{T}^{(1)}\partialZ_{1}^{-}{\bar{\partial}}Z_{1}^{+}+{\bar{\Phi}}_{T}^{(1)}\partialZ_{1}^{+}{\bar{\partial}}Z_{1}^{-}+\Phi_{U}^{(1)}\partialZ_{1}^{-}{\bar{\partial}}Z_{1}^{-}+{\bar{\Phi}}_{U}^{(1)}\partialZ_{1}^{+}{\bar{\partial}}Z_{1}^{+}\;,