harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0=2\sum_{a=1}^{\tilde{n}_{c}}(\delta_{\ell}-\phi_{a})+2\sum_{k=1}^{n_{c}-\tilde{n}_{c}}\frac{\delta_{\ell}-\gamma_{k}}{1-\omega^{2k-2\ell+1}}-\sum_{i=1}^{n_{f}}(\delta_{\ell}+m_{i}).
X_{12}=\omega_{1}^{2}X_{12}^{*},\,X_{23}=\omega_{2}^{2}X_{23}^{*},\,X_{13}=\omega_{1}^{2}\omega_{2}^{2}X_{13}^{*},\,Q_{\pm}=\omega_{1}\omega_{2}Q_{\pm}^{*},
\left[\left({\mathcal{P}}^{2}+\left(\frac{\overline{{g}}}{V}\right)^{2}\right)\overline{{g}}^{a1}\partial_{a}X_{\mu}+\frac{1}{3}\Pi^{(U)a1}A_{\mu\nu\lambda}{\mathcal{P}}^{\nu}\partial_{a}X^{\lambda}\right]_{\sigma_{1}=0,\pi}\approx0
K_{w}(t_{1},t_{2},\cdotst_{l})=\prod_{j=1}^{l}\Gamma(-\beta_{j})^{-1}\prod_{p=1}^{N}\prod_{j=1}^{n}\{P_{w}^{j}(x^{(p)},z_{p};t_{1},t_{2},\cdots,t_{l})\}^{\mu_{j}^{(p)}}\prod_{j=1}^{l}t_{j}^{-\beta_{j}-1},
G_{D}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=G_{D}^{(0)}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)-\frac{G_{D}^{(0)}({\bfr^{\prime\prime}},{\bf0};E)G_{D}^{(0)}({\bf0},{\bfr^{\prime}};E)}{G_{D}^{(0)}({\bf0},{\bf0};E)-1/\sigma}\;.
\int_{-1}^{1}du\,\frac{(1+u)^{n}(1-u)^{n-1}}{(1+\alphau)^{n}}=\sqrt{\pi}\frac{\Gamma(n)}{\Gamma(n+\frac{1}{2})}(1+\alpha)^{-n}F\left(n,n;1+2n;\frac{2\alpha}{1+\alpha}\right).
\int_{\calR}d^{n}p^{\prime}\,(\tilde{\omega}_{p^{\prime}}|\hat{e}_{p^{\prime}p}|\eta_{p})=\sum_{p^{\prime}\in{\calR}}\,\eta_{\alpha}(p)\,({\calP}\int_{e}A)_{\beta}^{\alpha}\,i[\eta]^{\beta}(p^{\prime})
A_{Ti}^{a}({\bf{r}})=\sum_{{\bf{k}};s=1,2}\frac{\epsilon_{i}^{s}({\bf{k}})}{\sqrt{2k}}\,[a_{s}^{a}({\bf{k}})e^{+i{\bf{k}}\cdot{\bf{r}}}+a_{s}^{a\dagger}({\bf{k}})e^{-i{\bf{k}}\cdot{\bf{r}}}]\;,
T^{\pm(n)}{}_{\mu\nu}=n\F^{\pm}{}_{(n)\mu}{}^{\rho_{1}\cdots\rho_{n-1}}F_{(n)\nu\rho_{1}\cdots\rho_{n-1}}^{\pm}-{\textstyle\frac{1}{2}}g_{\mu\nu}F_{(n)}^{\pm}{}^{2}\,.
\sum_{n=1}^{l_{\mathrm{max}}}\left[\sum_{l=1}^{l_{\mathrm{max}}}\left((X_{2,1,1}\,x^{2})^{l}y+\sum_{k=1}^{l}\sum_{i=1}^{E(l,k)}X_{l,k,i}\,x^{l}y^{k}\right)\right]^{n}
\Bigl(\prod_{p\inP}\bar{T}_{|p|}^{c}(...)\Bigr)\times_{\hbar}\Bigl(\prod_{q\inQ}T_{|q|}^{c}(...)\Bigr)=\sum_{n\geq0}\hbar^{n}M_{n}\Bigl(\prod_{p\inP}\bar{T}_{|p|}^{c}(...),\prod_{q\inQ}T_{|q|}^{c}(...)\Bigr)
\partial_{\mu}\Psi_{a}=U_{ext}\left(x^{\mu}\right)\widetilde{U}_{int}\left(x^{\mu},x^{m}\right)\left(\partial_{\mu}+imv_{\mu\alpha}\sigma^{\alpha}+imv_{\muc}\sigma^{c}\right)\psi_{a}^{r}\psi_{r}^{int}
\tilde{\calH}=\frac{1}{2}(\vec{\calE}^{\,2}+\vec{\calB}^{\,2})+\frac{1}{2}p^{\dagger}p+\frac{1}{2(e\chi)^{2}}(\vec{\nabla}\vec{\calE})^{2}+\frac{1}{2}(\vec{\nabla}\chi)^{2}+\frac{1}{2}(e\chi)^{2}\vec{\calA}^{\,2}+V(\chi^{2}/2)\.
D_{(n,j,m)\,M^{\prime}}^{J}(h^{-1}g)=\sum_{m^{\prime\prime}}D_{(n,j,m)\,(n,j,m^{\prime\prime})}^{J}(h^{-1})\,D_{(n,j,m^{\prime\prime})\,M^{\prime}}^{J}(g)\;.
\hat{H}_{b}\bigl(\hat{b}^{\pm}\bigr)^{n_{b}}u_{0}^{\pm}=\mpi\left(n_{b}+\frac{1}{2}\right)\hslash\gamma\bigl(\hat{b}^{\pm}\bigr)^{n_{b}}u_{0}^{\pm}\,\,\,\left(n_{b}=0,1,2,\cdots\right).
\widetilde{\Gamma}_{a}^{(11)}=\left(\begin{matrix}{0}&{\widetilde{\Gamma}_{b}^{(9)}}\\{\widetilde{\Gamma}_{b}^{(9)}}&{0}\\\end{matrix}\right)\,,\quad\widetilde{\Gamma}_{0}^{(11)}=\left(\begin{matrix}{0}&{{\bf1}_{2}}\\{-{\bf1}_{2}}&{0}\\\end{matrix}\right)\,,\quad\widetilde{\Gamma}_{10}^{(11)}=\left(\begin{matrix}{{\bf1}_{2}}&{0}\\{0}&{-{\bf1}_{2}}\\\end{matrix}\right)\,.
{\calC}\left[\begin{array}{c}{{\alpha}_{1}\\cdots\\alpha_{g}}\\{{\beta}_{1}\\cdots\\beta_{g}}\\\end{array}\right]={\calC}\left[\begin{array}{c}{{\alpha}_{1}}\\{{\beta}_{1}}\\\end{array}\right]{\calC}\left[\begin{array}{c}{{\alpha}_{2}}\\{{\beta}_{2}}\\\end{array}\right]\cdots{\calC}\left[\begin{array}{c}{{\alpha}_{g}}\\{{\beta}_{g}}\\\end{array}\right]
I=\intd^{4}x\sqrt{-g}\left[\frac{1}{16\piG}(R-2\Lambda)-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}\right]-\frac{1}{8\piG}\int_{\partial\mathcal{M}}d^{3}x\sqrt{-h}K.
G_{n}^{r}=\frac{1}{(\gamma+(r+1)^{2})^{n}}+\sum_{k=1}^{[(n-1)/2]}\\sum_{m=k+1}^{n-k}{\binom{m-1}{k}}\frac{1}{(\gamma+(r+1)^{2})^{m}}\,sum_{i_{1}=1}^{\infty}\ldots\sum_{i_{k}=1}^{\infty}\delta_{i_{1}+\ldots+i_{k},n-m}G_{i_{1}}^{r+2}\ldotsG_{i_{k}}^{r+2}
\int_{0}^{\infty}dx\,w^{(N_{f})}(x,M)p_{n}^{(N_{f})}(x,M)p_{m}^{(N_{f})}(x,M)\=\\frac{Z_{n+1}^{(N_{f})}(M)}{(n+1)\,Z_{n}^{(N_{f})}(M)}\,\delta_{nm}\.
\frac{\partial^{2}F}{\partialt^{2}}=\phi_{tt}^{3}+\sum_{j=2}^{3}[2\phi_{jt}^{3}f_{t}^{j}+\phi_{j}^{3}f_{tt}^{j}+\sum_{k=2}^{3}(\phi_{jk}^{3}f_{t}^{k})f_{t}^{j}]
M=\left(\begin{array}{cc}{\varphi_{k}(R/2)}&{\varphi_{-k}(R/2)}\\{\chi_{k}(R/2)}&{\chi_{-k}(R/2)}\\\end{array}\right)\,\N=\left(\begin{array}{rr}{-\chi_{k}(-R/2)}&{-\chi_{-k}(-R/2)}\\{\varphi_{k}(-R/2)}&{\varphi_{-k}(-R/2)}\\\end{array}\right)\,\v=\left(\begin{array}{c}{A}\\{B}\\\end{array}\right)\.
U_{\mathrm{PA}}^{\mathrm{bf}}(\vec{x};C)=C(\mathrm{i}\vec{\tau}\cdot\hat{n})U_{\mathrm{PA}}^{\mathrm{bf}}(D(\mathrm{i}\vec{\tau}\cdot\hat{n})\cdot\vec{x};C)(-\mathrm{i}\vec{\tau}\cdot\hat{n})C^{\dagger}
R=\left(\begin{array}{cccc}{R_{11}^{11}}&{R_{11}^{21}}&{R_{11}^{12}}&{R_{11}^{22}}\\{R_{21}^{11}}&{R_{21}^{21}}&{R_{21}^{12}}&{R_{21}^{22}}\\{R_{12}^{11}}&{R_{12}^{21}}&{R_{12}^{12}}&{R_{12}^{22}}\\{R_{22}^{11}}&{R_{22}^{21}}&{R_{22}^{12}}&{R_{22}^{22}}\\\end{array}\right)=\left(\begin{array}{cccc}{a}&{b}&{c}&{d}\\{f}&{g}&{h}&{j}\\{k}&{l}&{m}&{n}\\{p}&{q}&{u}&{v}\\\end{array}\right).
\{T(z),W_{\pm}(z^{\prime})\}=-\partial_{z^{\prime}}W_{\pm}(z^{\prime})\delta_{2\pi}(z-z^{\prime})+W_{\pm}(z^{\prime})\partial_{z}\delta_{2\pi}(z-z^{\prime})\mp\frac{i\gammap_{2}}{2\pi}W_{\pm}(z^{\prime})\delta_{2\pi}(z-z^{\prime}).
\alpha_{\mathrm{eff}}^{\prime}=\frac{\alpha^{\prime}}{\epsilon},\qquadu=\frac{r}{\epsilon\alpha_{\mathrm{eff}}^{\prime}},\qquadG_{o(1)}^{2}=\frac{g_{s}}{\epsilon},\qquadQ_{5}=\alpha_{\mathrm{eff}}^{\prime}N
(k_{1}-k_{2})_{\mu}T_{\mu}^{VS}=\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left\{\hat{1}\frac{1}{[\not{k}+\not{k}_{2}-m]}\right\}-\int\frac{d^{4}k}{(2\pi)^{4}}Tr\left\{\hat{1}\frac{1}{[\not{k}+\not{k}_{1}-m]}\right\}.
\begin{array}{ccc}{\mathrm{The~7~Cartan~generators}}&{C_{i}=H_{\alpha_{i}}}&{i=1,\dots,7}\\{\mathrm{The~28~positive~root~generators}}&{E^{\rho_{i}}}&{i=1,\dots,28}\\{\mathrm{The~28~negative~root~generators}}&{E^{-\rho_{i}}}&{i=1,\dots,28}\\\end{array}
f^{(1)}=\left(\begin{array}{cccc}{0}&{\partial_{i}^{y}}&{\delta_{ij}}&{0}\\{-\partial_{j}^{y}}&{0}&{0}&{0}\\{-\delta_{ij}}&{0}&{0}&{\partial_{i}^{y}}\\{0}&{0}&{-\partial_{j}^{x}}&{0}\\\end{array}\right)\delta(\vec{x}-\vec{y}).
U_{\mathrm{w}}\;=\;{\frac{4}{3n_{\mathrm{w}}(k^{2})^{2}}}\int_{0}^{k}d\zeta\,\rho_{\mathrm{w}}(\zeta)(1-\zeta^{2})-{\frac{1}{3}}-{\frac{2}{3}}{\frac{d_{\mathrm{w}}(k^{2})-p_{\mathrm{w}}(k^{2})}{n_{\mathrm{w}}(k^{2})^{2}}}\;,
{\frac{\ddot{\delta{\calR}}}{h^{2}}}+\left\{{\frac{{\frac{1}{l^{2}}}-6{\frac{M}{\bar{\calR}^{5}}}}{k+{\frac{\bar{\calR}^{2}}{l^{2}}}-{\frac{M}{\bar{\calR}^{3}}}}}-(7+8\omega)(3+4\omega){\frac{\kappa_{(6)}^{4}}{64}}\bar{\rho}^{2}\right\}\delta{\calR}=0\,.
\left(\begin{array}{c}{z^{\prime}}\\{\bar{z}^{\prime}}\\{\bf1}\\\end{array}\right)={\bfM}\left(\begin{array}{c}{z}\\{\bar{z}}\\{\bf1}\\\end{array}\right)\equiv\left(\begin{array}{ccc}{u}&{0}&{t}\\{0}&{\bar{u}}&{\bar{t}}\\{0}&{0}&{{\bf1}}\\\end{array}\right)\left(\begin{array}{c}{z}\\{\bar{z}}\\{\bf1}\\\end{array}\right)\.
L_{c}^{-1}[\psi_{1}\otimes\cdots\otimes\psi_{m}]_{x_{1}^{0},...,x_{l}^{0}}(\underline{{x}};\underline{{n}},\underline{{T}}))=\bigl(\prod_{i=1}^{l}L_{c}^{-1}[\psi_{i}]_{x_{i}^{0}}(x_{i};n_{i},T_{i})\bigr)\bigl(\prod_{i=l+1}^{m}\psi_{i}(x_{i})\bigr)
G^{{\mu}{\nu}{\sigma}}=-F^{{\mu}{\nu}{\sigma}}-{\lambda}^{{\rho}[{\mu}}{{\calF}_{\rho}}^{{\nu}{\sigma}]}-{\frac{1}{3!}}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\lambda}_{{\theta}[{\rho}}{{\calF}^{\theta}}_{{\eta}{\delta}]}.
\begin{array}{c}{\delta\psi_{m}=D_{m}\epsilon+\frac{1}{64}e^{\frac{3}{4}\phi}\left(\Gamma_{m}^{\\np}-14\delta_{m}^{n}\Gamma^{p}\right)\Gamma^{10}\epsilon{\calF}_{np},}\\{\delta\lambda=-\frac{1}{2\sqrt{2}}\left(D_{m}\phi\right)\Gamma^{m}\Gamma^{10}\epsilon+\frac{3}{16\sqrt{2}}e^{\frac{3}{4}\phi}\Gamma^{mn}\epsilon{\calF}_{mn},}\\\end{array}
f^{\prime\prime}={\frac{{f^{\prime}}^{2}}{2f}}+{\frac{3}{2}}f^{3}+2\omegaxf^{2}+(\mathrm{~\frac{1}{2}~}\omega^{2}x^{2}+k_{3}-k_{1})f-{\frac{k_{2}^{2}}{2f}}.
(-\tilde{g})^{-1/2}\partial_{\mu}\left(\sqrt{-\tilde{g}}\tilde{g}^{\mu\nu}\partial_{\nu}\phi\right)-\frac{\alphaM^{4}}{M_{p}(\zeta+b)}e^{-2\alpha\phi/M_{p}}+\frac{\alpham}{M_{p}}F(\zeta)\overline{{\Psi}}^{\prime}\Psi^{\prime}=0.
G\left({}_{\zeta}|{}^{\zeta^{\prime}}\right)_{p_{-},n,r,p_{-}^{\prime},n^{\prime},r^{\prime}}=\delta_{rr^{\prime}}\delta(p_{-}-p_{-}^{\prime})\delta(p_{1}-p_{1}^{\prime})\ldots\delta(p_{2\left[(d-1)/2\right]-1}-p_{2\left[(d-1)/2\right]-1}^{\prime})\;g\left({}_{\zeta}|{}^{\zeta^{\prime}}\right)_{nn^{\prime}}\;.
\sum_{n=1}^{\infty}F(2\pin)=-\sqrt{\frac{2}{\pi}}~\frac{1}{2\beta}~\frac{\partial}{\partialM}~\int_{0}^{\infty}\frac{dt}{t}~e^{-\frac{1}{2}(~t~+\frac{M^{2}\tilde{p}_{2}^{2}~}{t})}~\sum_{n=1}^{\infty}e^{-\frac{2\tilde{p}_{2}^{2}\pi^{2}}{\beta^{2}t}~n^{2}}~~.\,
\psi\leftrightarrow\left(\begin{array}{cccc}{\psi_{1}}&{-\psi_{2}^{*}}&{\psi_{3}}&{\psi_{4}^{*}}\\{\psi_{2}}&{\psi_{1}^{*}}&{\psi_{4}}&{-\psi_{3}^{*}}\\{\psi_{3}}&{\psi_{4}^{*}}&{\psi_{1}}&{-\psi_{2}^{*}}\\{\psi_{4}}&{-\psi_{3}^{*}}&{\psi_{2}}&{\psi_{1}^{*}}\\\end{array}\right).
|\tilde{\mu}_{I}>=(1-\tilde{\mu}^{*}\tilde{\mu})^{j+k/2}\sum_{n=0}\tilde{\mu}^{n}\sqrt{\frac{\Gamma(2j+k+n)}{\Gamma(n+1)\Gamma(2j+k)}}\;\intd^{2}\mu\;f_{n}(\mu)|\mu>
{\cal{F}}_{V}^{(\tilde{\eta}_{o})}(Y,W)=C_{\phi}^{2}W^{\frac{1}{3}\eta}\Bigl[(1-Y)^{-\frac{1}{2}\eta}-(1-Y)^{\frac{1}{2}\eta}\Bigl]+\frac{g_{*}^{2}}{C_{\tilde{O}}}{\cal{F}}_{V,1}^{(\tilde{\eta}_{o})}(Y,W)+O(\frac{1}{N^{2}}),
L_{BM}^{g}=\frac{1}{2e}(\dot{x}_{\mu}-i\lambda\xi_{\mu})^{2}-\frac{e}{2}m_{-}^{2}+i\lambdam_{+}\xi_{*}+iq\epsilon_{\mu\nu\lambda}\dot{x}{}^{\mu}{\calA}^{\nu}\xi^{\lambda}\xi_{*}-\frac{i}{2}\xi\dot{\xi}-\frac{i}{2}\xi_{*}\dot{\xi}_{*},
\frac{1}{24.8\pi^{2}}\int_{\partial\widetilde{\calM}}\Big[\omega^{\sharp}\wedgeR^{\sharp}-\frac{1}{3}\omega^{\sharp}\wedge\omega^{\sharp}\wedge\omega^{\sharp}-\omega\wedgeR+\frac{1}{3}\omega\wedge\omega\wedge\omega\Big]=-\frac{2}{3}\frac{m^{2}r_{b}^{2}}{(r_{b}+2m)^{4}}\,.
m^{2}\left\{\int{\frac{d^{n}p}{(2\pi)^{n}}}{\frac{1}{\left(p^{2}-m^{2}\right)^{2}}}\right\}_{\mathrm{dim}}=\left({\frac{n}{2}}-1\right)\left\{\int{\frac{d^{n}p}{(2\pi)^{n}}}{\frac{1}{p^{2}-m^{2}}}\right\}_{\mathrm{dim}}
\overline{{V}}\left(\left.a\begin{array}{c}{b}\\{c}\\\end{array}\right|z\right)=t_{a}^{*b}(-z)K(z)t_{c}^{a}(z)=\sum_{j,k}t_{a(j)}^{*b}(-z)K(z)_{k}^{j}t_{c}^{a(k)}(z).
I_{IR}(\rho)=\frac{1}{2\pii}\int_{\tilde{C}}\frac{ds}{s}e^{\rho/s}F(s)=\frac{1}{2\pii}\int_{\tilde{C}}\frac{ds}{s}e^{\rho/s}\tilde{F}(s^{a})=\frac{k}{2\pii}\int_{\tilde{C}^{\prime}}\frac{ds}{s}e^{(\frac{\tilde{\rho}}{s})^{k}}\tilde{F}(s),
\left[\sum_{k=1}^{n_{c}}(-1)^{k}s_{k}(\phi^{2})x^{n_{c}-k+1}+2\Lambdam_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{2}\prod_{i=1}^{n_{f}}(x+m_{i}^{2})=-4\Lambda^{2}x\prod_{a=1}^{n_{c}}(x-\alpha_{a})^{2}.
\mathrm{d}\tilde{s}^{2}=\frac{r+2m}{r}\,\mathrm{d}r^{2}+r(r+2m)\,\left[(\Sigma^{1})^{2}+(\Sigma^{2})^{2}\right]+\frac{4rm^{2}}{r+2m}\,\left[\Sigma^{3}\right]^{2}\,
\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[a_{n}^{(\alpha)}~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+b_{n}^{(\alpha)}~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right].
\left[\hat{M}_{\hat{a}\hat{b}},\hat{M}_{\hat{c}\hat{d}}\right]=-\hat{\eta}_{\hat{a}\hat{c}}\hat{M}_{\hat{b}\hat{d}}-\hat{\eta}_{\hat{b}\hat{d}}\hat{M}_{\hat{a}\hat{c}}+\hat{\eta}_{\hat{a}\hat{d}}\hat{M}_{\hat{b}\hat{c}}+\hat{\eta}_{\hat{b}\hat{c}}\hat{M}_{\hat{a}\hat{d}}\,.
I=\int_{\cal{M}}\!d^{4}\!x\sqrt{\gamma}\left(\!R(\gamma)\!-\!{\frac{1}{2}}\!\left(\!DU\right)^{2}\!-\!{\frac{1}{4}}e^{2U}V_{ij}V^{ij}\!-\!{\frac{1}{2}}\!\left(\left(\!D\Phi\!\right)^{2}\!-\!e^{-2\Phi}\!\left(\!D\Psi\!\right)^{2}\right)\!-\!{\frac{1}{2}}e^{-2\left(\Phi+U\right)}(\!\partial_{0}\!\Psi\!)^{2}\right)\.
Z_{\mathrm{gr}}[\varphi,\mu,\bar{\mu}]=\int{\calD}T{\calD}\bar{T}\,\,e^{-\intd^{2}w\,\,T\mu-\intd^{2}w\,\,\bar{T}\bar{\mu}-K[\varphi,T,\bar{T},\mu,\bar{\mu}]}\left[e^{-W[\mu]}e^{-W[\bar{\mu}]}\right].
(H-\epsilon_{a})J_{a}\propto\left|\begin{array}{ccc}{(\epsilon_{c}-\epsilon_{a})m_{c}}&{(\epsilon_{b}-\epsilon_{a})m_{b}+v_{bc}m_{c}}&{(\epsilon_{a}-\epsilon_{a})m_{a}+v_{ab}m_{b}+v_{ac}m_{c}}\\{\epsilon_{c}-\epsilon_{a}}&{v_{bc}}&{v_{ac}}\\{0}&{\epsilon_{b}-\epsilon_{a}}&{v_{ab}}\\\end{array}\right|\,.
F_{A}=\left(\begin{matrix}{{\frac{1}{2}}F^{0}}&{0}\\{0}&{{-{\frac{1}{2}}}F^{0}}\\\end{matrix}\right)=\left(\begin{matrix}{{-1}}&{0}\\{0}&{1}\\\end{matrix}\right)h^{2}dz\wedged{\bar{z}}\quad.
T(z)=g_{ab}(z)\frac{\deltaS}{\deltag_{ab}(z)}=\frac{\varepsilon\mu^{-\varepsilon}}{8\pi\alpha^{\prime}}g^{ab}(z)\partial_{a}x^{\mu}\partial_{b}x^{\nu}G_{\mu\nu}-\frac{\mu^{-1-\varepsilon}}{4\pi\alpha^{\prime}}H_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}\delta_{\partialM}(z)
\Gamma\lbrackA\rbrack=\mathrm{tr}\int_{0}^{\infty}{\frac{dT}{T}}\,\mathrm{e}^{-m^{2}T}\int{\calD}x\,{\calP}\mathrm{exp}\left[-\int_{0}^{T}\!\!\!d\tau\left({\frac{1}{4}}{\dot{x}}^{2}+igA_{\mu}\dot{x}^{\mu}\right)\right]
T(\vec{k^{\prime}},n^{\prime};\vec{k},n)=\frac{-1}{8\pi^{2}}\intd\vec{x^{\prime}}\int_{o}^{2\pi}d\phi^{\prime}e^{-i\vec{k^{\prime}}.\vec{x^{\prime}}}e^{-in^{\prime}\phi^{\prime}}V(x^{\prime},\phi^{\prime})\Psi_{\vec{k},\vec{n}}(x^{\prime},\phi^{\prime}),
\langleN\rangle=\left\{\begin{array}{ll}{-{\frac{1}{2}}\mathrm{sgn}(m)F,}&{\Phi^{(0)}-\Upsilon>0}\\{{\frac{1}{2}}\mathrm{sgn}(m)(1-F),}&{\Phi^{(0)}-\Upsilon<0}\\\end{array}\right\},\quad0<F<1,
\sigma(x,\theta,\bar{\theta})=(1+\theta^{\alpha}{\sigma^{m}}_{\alpha\dot{\alpha}}\bar{\theta}^{\dot{\alpha}}\partial_{m}+\frac{1}{4}\theta^{2}\bar{\theta}^{2}\Box)A(x)+\sqrt{2}\theta\psi(x)-\frac{i}{\sqrt{2}}\theta^{2}\partial_{\alpha\dot{\alpha}}\psi^{\alpha}(x)\bar{\theta}^{\dot{\alpha}}+\theta^{2}F(x)
lA_{1}+nB_{1}=(U_{l}^{n})\sum_{q=1}^{l}Str\left(\ldots\phi^{i_{2k-1}^{\prime}}\ldots\underbrace{\partial_{a_{q}}\phi^{j_{q}}\phi^{i_{2k}^{\prime}}}\ldots\right)+(U_{l}^{n})\sum_{s=1}^{n}Str\left(\ldots\phi^{i_{2k-1}^{\prime}}\ldots\underbrace{\phi^{i_{s}}\phi^{i_{2k}^{\prime}}}\ldots\right)
\begin{matrix}{s_{1}}&{,}&{s_{1}+1}&{,}&{s_{1}+2}&{,}&{\ldots}&{,}&{s_{1}+n_{1}}\\{s_{2}}&{,}&{s_{2}+1}&{,}&{s_{2}+2}&{,}&{\ldots}&{,}&{s_{2}+n_{2}}\\{\vdots}&{}&{\vdots}&{}&{\vdots}&{}&{\ddots}&{}&{\vdots}\\{s_{i}}&{,}&{s_{i}+1}&{,}&{s_{i}+2}&{,}&{\ldots}&{,}&{s_{i}+n_{i}}\\\end{matrix}\,
A(x)=\frac{1}{2}\left\{-I_{0}(\frac{e^{x}}{2})\int_{x}^{\infty}dtK_{0}(\frac{e^{t}}{2})-K_{0}(\frac{e^{x}}{2})\int_{-\infty}^{x}dt\left[I_{0}(\frac{e^{x}}{2})-1\right]\right\}.
R_{a\bar{c}b\bar{d}}^{(0)}=G_{a\bar{c}}^{(0)}G_{b\bar{d}}^{(0)}+G_{a\bar{d}}^{(0)}G_{b\bar{c}}^{(0)}-e^{2G^{(0)}}W_{abe}\overline{{W}}_{\bar{c}\bar{d}\bar{f}}G^{(0)e\bar{f}},
g(x)={\frac{\theta\left({\frac{x}{2\omega}}+\alpha,\beta|{\frac{1}{N}}\left(\begin{array}{ll}{{\frac{\omega^{\prime}}{\omega}}}&{1}\\{1}&{\tau^{\prime}}\\\end{array}\right)\right)}{\vartheta_{3}\left({\frac{Nx}{2\omega}}+N\alpha|\tau\right)}},
\hat{L}^{(\phi^{N})}:=a^{2}\dot{\varphi}_{0}^{2}\partial_{y}{\frac{1}{a^{2}\dot{\varphi}_{0}^{2}}}\partial_{y}a^{2}-{\frac{2\kappa}{3}}a^{2}\dot{\varphi}_{0}^{2}.
V(y_{i})_{\mathrm{reg}}=\frac{-k\,\pi^{\frac{D-2}{2}}}{\Gamma(\frac{D-2}{2})V_{T}}\int_{0}^{\infty}dx\,\Big[\prod_{i=1}^{D}\theta_{3}\Big(\frac{y_{i}}{L_{i}},\frac{ix}{L_{i}^{2}}\Big)-1\Big]\,.
\frac{\partial{u_{3}}}{\partial{t}}=\frac{\partial}{\partial{t}}\Gamma_{k}^{(6)}[\varphi=0]=\frac{1}{2}{T}{r}\frac{\delta^{6}}{\delta\varphi_{1}(q=0)\cdots\delta\varphi_{6}(q=0)}\left\{(\Gamma_{k}^{(2)}[\varphi]+{R}_{k})^{-1}\frac{\partial}{\partial{t}}{R}_{k}\right\}\Bigg\vert_{\varphi=0}
{\calA}={\calA}^{0}+\intd^{4}x\lambda_{\mu}(x)\{\partial_{\nu}\mathrm{\mathrm{}~^*\!~}F^{\mu\nu}(x)+4\pi{\tilde{e}}\intd\tau\frac{dY^{\mu}(\tau)}{d\tau}\delta(x-Y(\tau))\}.
\frac{1}{q^{2}+i{\epsilon}}{\delta}^{\prime}(q_{-})=-\frac{1}{n^{2}q_{+}^{2}+q_{\bot}^{2}}{\delta}^{\prime}(q_{-})+\frac{2n_{+}q_{+}}{(n^{2}q_{+}^{2}+q_{\bot}^{2})^{2}}{\delta}(q_{-}).
F({\calA},G,q)=\sum_{k=1}^{\infty}F_{k}^{(\mathrm{cov)}}(G)\sum_{n=-\infty}^{+\infty}\mathrm{e}^{-k{\calA}v(n)}q^{k}=\sum_{n=-\infty}^{+\infty}F^{(\mathrm{cov)}}(G,\mathrm{e}^{-{\calA}v(n)}q)
c:=(ab)^{2}={\frac{(E_{p}-\sqrt{q_{3}^{2}+M^{2}})(E_{q}-\sqrt{q_{3}^{2}+M^{2}})}{(E_{p}+\sqrt{q_{3}^{2}+M^{2}})(E_{q}+\sqrt{q_{3}^{2}+M^{2}})}}
S={\frac{1}{2}}\int_{\partial\calM}[{\frac{1}{2}}\stackrel{(0)}g\strut^{ab}\partial_{a}\Phi\partial_{b}\Phi-\frac8{\ell^{2}}e^{\Phi}+\stackrel{(0)}{\calR}\Phi]\,\theta\,dt\,d\phi\qquad,
{\cal{L}}_{\Lambda}=\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}+\frac{1}{2}\deltaZ_{1}(\partial\phi)^{2}-\frac{1}{2}\deltam^{2}\phi^{2}-\frac{\delta\lambda}{4!}\phi^{4}.
\begin{array}{lcl}{a_{1}}&{=}&{\displaystyle{\frac{16(4\pi)^{3/2}}{7\mu^{3}}\biggl\{\Lambda+\frac{9\mu^{2}}{5(11-36~\xi)}\biggr\}}\,\nonumber}\\{}&{}&{\nonumber}\\{a_{2}}&{=}&{\displaystyle{\frac{72(4\pi)^{3/2}}{(11-36~\xi)\mu^{3}}}\.}\\\end{array}
{\calL}_{B}=D_{\mu}\phi_{i}D^{\mu}\bar{\phi}_{i}+D_{\mu}\sigma_{I}D^{\mu}\bar{\sigma}_{I}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}-\frac{\alpha}{2}F_{\mu\nu}H^{\mu\nu};
W({\bfx}_{n+1},{\bfx}_{n};\varepsilon)\simeq{\frac{m({\bfx}_{n+1}-{\bfx}_{n})^{2}}{2\varepsilon}}+i{\frac{3}{2}}\,ln\,\varepsilon+\Phi({\bfx}_{n+1},{\bfx}_{n})\,,
P_{0}{\Phi}_{0}=\{{\alpha}_{+}{\Pi}^{1}+{\alpha}_{-}{\pi}^{1}-{\phi}-e_{1}{\phi}_{(1)}^{\mathrm{self}}-e_{2}{\phi}_{(2)}^{\mathrm{self}}+{\beta}_{1}m_{1}+{\beta}_{2}m_{2}\}{\Phi},
\chi_{F}^{(3)}\left(e_{i},e^{-\varphi}\right)=\chi_{F}^{(3)}\left(e_{i},2\varphi_{\bar{z}}e^{-\varphi}\right)=\chi_{F}^{(3)}\left(e_{i},(2\varphi_{\bar{z}})^{2}e^{-\varphi}\right)=0,\qquad\foralli.
S=\intd\tau\{p_{m}(\dot{x}^{m}+i\psi\sigma^{m}\bar{\rho}-i\rho\sigma^{m}\bar{\psi})-\frac{ep^{2}}{2}-{\rho^{\alpha}}{\dot{\theta}}_{\alpha}-{\bar{\rho}}_{\dot{\alpha}}{\dot{\bar{\theta}}}^{\dot{\alpha}}\},
\Theta_{\mu\nu,\kappa\lambda}^{-1}=\left[XP^{(2)}+YP_{m}^{(1)}+ZP_{s}^{(0)}+WP_{\omega}^{(0)}+RP_{sw}^{(0)}+SP_{\omegas}^{(0)}\right]_{\mu\nu,\kappa\lambda}\.
\delta_{L}V_{k}^{R}=i\epsilon_{i}[K_{i}^{R},V_{k}^{R}]=\epsilon_{i}(x_{i}\partial_{t}+t\partial_{i})V_{k}^{R}-\partial_{k}{\frac{1}{\vec{\partial}^{2}-M^{2}}}\epsilon_{i}\dot{V}_{i}^{R}~,
X_{i}\left(x\right)=C_{ij}\int\frac{d^{3}\mathbf{x}}{\left(2\pi\right)^{3/2}}e^{i\mathbf{k\cdotx}}\Big[U_{r}^{jk}\left(k,\,\eta\right)a_{r}^{k}\left(k\right)+V_{r}^{jk}\left(k,\,\eta\right)b_{r}^{+k}\left(-k\right)\Big]\,,
N_{k}=a_{k}^{\dagger}a_{k}+\sum_{n=1}^{\infty}\sum_{i_{1},\dots,i_{n}}\sum_{\pi\inS_{n}}[Y_{k,\pi(i_{1},\dots,i_{n})}]^{\dagger}Y_{k,i_{1},\dots,i_{n}}(A^{(k,i_{1},\dots,i_{n})})_{k,i_{1},\dots,i_{n};k,\pi(i_{1},\dots,i_{n})}^{-1},
v^{2}=\beta^{2}+(1-\beta^{2})\,{v^{\prime}}^{2}=\frac{{\bfE}_{0}^{2}+({\bfE}_{0}\cdot{\bfB}_{0})^{2}}{1+{\bfB}_{0}^{2}}=1-\dot{T}_{\infty}^{2}.
{\calL}=-{\textstyle{\frac{1}{2}}}(\partial\phi)^{2}-\bar{\lambda}^{i}\partial\llap/\lambda_{i}+{\textstyle{\frac{1}{2}}}H^{2}-{\textstyle{\frac{1}{4}}}F^{2}+{\textstyle{\frac{1}{2}}}D^{2}\,.
Z_{R}=\frac{\theta_{3}^{4}-\theta_{4}^{4}}{2\eta^{12}}\Theta_{\Gamma_{8}^{+}}-\frac{\theta_{2}^{4}}{2\eta^{12}}\Theta_{\Gamma_{8}^{-}}+\frac{\theta_{3}^{4}+\theta_{4}^{4}}{2\eta^{12}}\Theta_{\Gamma_{8}^{+}+\delta}-\frac{\theta_{2}^{4}}{2\eta^{12}}\Theta_{\Gamma_{8}^{-}+\delta}.
\begin{array}{lcr}{ds^{2}={\frac{1}{2}}K_{\rho\sigma}[d\phi_{\rho}d\phi_{\sigma}+a_{\rho}a_{\sigma}\sum_{i=1}^{3}d\Omega_{\rho,i}d\Omega_{\sigma,i}];}\\{H=-2a_{\rho}d\Omega_{\rho},}\\\end{array}
\hat{\mathrm{P}}_{\mathrm{R}}^{s}=\frac{1}{2}\hbar\intdx({\psi}_{\mathrm{R}}^{\daggers}(-i\partial_{x}){\psi}_{\mathrm{R}}^{s}-{\psi}_{\mathrm{R}}^{s}(i\partial_{x}){\psi}_{\mathrm{R}}^{\daggers})-\intdx\hat{\mathrm{E}}{\partial_{1}}A_{1}.
\left(\frac{l}{r^{2}}\frac{dr}{d\phi}\right)^{2}+\frac{l^{2}}{r^{2}}=c^{2}(k^{2}-1)+\frac{\alphac^{2}}{r}e^{-\betar}+\frac{\alphal^{2}}{r^{3}}e^{-\betar}.
{\calL}_{b}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}+\frac{1}{2}(D^{a})^{2}-\frac{1}{2}(D_{\mu}C^{a})^{2}-\frac{1}{2}(v_{\mu}^{a})^{2}+\frac{k}{2}C^{a}D^{a}-\frac{k}{2}A_{\mu}^{a}v^{a\mu},
E_{D}=\frac{1}{2}\sum_{n_{1},..,n_{D-1}}\left[\left(\frac{n_{1}\pi}{L_{1}}\right)^{2}+\dots+\left(\frac{n_{p}\pi}{L_{p}}\right)^{2}+\sum_{j=p+1}^{D-1}\nu_{n_{j}}^{2}(a_{j})\right]^{1/2},
\begin{array}{ll}{(\:{\partial_{\tau}}^{2}+{\partial_{\sigma}}^{2}\:)\:\phi_{\omega}^{E}(\sigma,\tau)=0}\\{(\:{\partial_{\tau}}^{2}-{\partial_{\sigma}}^{2}\:)\:\phi_{\omega}^{L}(\sigma,\tau)=0}\\\end{array}
\langle0|T^{+}A_{\mu}({x})A_{\nu}({y})|0\rangle=i\int\frac{d^{d+2}k}{(2\pi)^{d+2}}\frac{e^{-i{k}\cdot({x}-{y})}}{k^{2}+i\epsilon}\left[-g_{\mu\nu}+\frac{(k_{\mu}n_{\nu}^{Weyl}+k_{\nu}n_{\mu}^{Weyl})}{k_{+}+i\\epsilon^{\prime}\\mathrm{sgn}(k_{-})}\right].
\begin{matrix}{A=\left(\begin{matrix}{0}&{a}&{0}&{b}\\{c}&{0}&{d}&{0}\\{0}&{e}&{0}&{f}\\{g}&{0}&{h}&{0}\\\end{matrix}\right),}&{B=i\left(\begin{matrix}{0}&{-f}&{0}&{e}\\{-h}&{0}&{-g}&{0}\\{0}&{-b}&{0}&{a}\\{d}&{0}&{c}&{0}\\\end{matrix}\right)}\\\end{matrix}.
4\pi\sqrtn\,(n-2)\,{\frac{|\xi|}{\sqrt\lambda}}\,\left({\frac{6|\xi|}{1+6|\xi|}}\right)^{1/2}<\left({\frac{\phi}{M_{\mathrm{p}}}}\right)^{n-2}<8\pi\sqrt{2n}\,{\frac{|\xi|}{\sqrt\lambda}}\,
\frac{de}{d\omega}=\frac{\hbar}{\pi^{2}c^{3}}\Big[\frac{\omega^{3}}{2}+\gamma\omega_{c}^{3}x^{2}\sum_{n=0}^{\infty}\frac{\beta^{2n}}{2n+1}+\sum_{k=0}^{n}(-1)^{k}\frac{n-k-x}{k!(2n-k)}\Theta[n-k-x]\Big]
F(x^{+},x^{-},0)=\sum_{n,m,s,t}\,\,\left(\frac{\pi}{2L^{2}l}\right)^{2}\langle0|\frac{L}{\pi}T(n,m)e^{-iP_{op}^{-}x^{+}-iP^{+}x^{-}}\frac{L}{\pi}T(s,t)|0\rangle\,,
ds^{2}=H^{-\frac{1}{3}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+2H^{-\frac{1}{3}}g_{m\overline{{n}}}dz^{m}dz^{\overline{{n}}}+H^{\frac{2}{3}}\delta_{\alpha\beta}dx^{\alpha}dx^{\beta}
\frac{2}{\bar{N}}\{\delta_{\rho}^{\alpha}\delta_{\sigma}^{\beta}-\delta_{\rho}^{\beta}\delta_{\sigma}^{\alpha}\}\dot{\zeta}_{\beta}(u)\int\delta\xidsE^{\rho}[\xi|s]\dot{\xi}^{\sigma}(s)\dot{\xi}^{-2}\delta(\xi(s)-\eta(t)).
E={\frac{1}{2}}\int_{-\infty}^{+\infty}dra^{4}\left\{[f_{i}^{a}(\phi^{i})^{\prime}\mp3f^{ai}\partial_{i}W]^{2}-12[{\frac{a^{\prime}}{a}}\pmW]^{2}\right\}\pm3\int_{-\infty}^{+\infty}dr{\frac{\partial}{\partialr}}[a^{4}W].
\zeta(s\midD_{0}D_{0}^{\dagger})=\sum_{m}d_{m}\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{dt}{t}t^{s}e^{-t\lambda_{m}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{dt}{t}t^{s}\mathrm{Tr}^{\prime}(e^{-t(D_{0}D_{0}^{\dagger})}).
\langleR_{\mu}(x)R_{\nu}(y)K_{\rho}(z)\rangle=\left[\frac{1}{9}\mathrm{dim}\,T+\frac{g^{2}}{32\pi^{2}}\,\mathrm{Tr}\,T^{a}T^{a}\left(\frac{8}{3}+\frac{1}{3}-3\right)\right]A_{\mu\nu\rho}(x,y,z)\,,
\frac{1}{2}\mathrm{Tr}\left[O+\left(\partial\!\!\!/B\!\!\!\!/\right)\right]^{2}=2\left(O^{2}+O(\partial\cdotB)+(\partial\cdotB)O\right)+\frac{1}{2}\mathrm{Tr}\left[\left(\partial\!\!\!/B\!\!\!\!/\right)\left(\partial\!\!\!/B\!\!\!\!/\right)\right].
\intd^{R}x\left[D_{i}^{\delta}\frac{\delta}{\deltaA_{i}}+gf^{\alpha\delta\beta}c^{\beta}\frac{\delta}{\deltac^{\alpha}}+gf^{\alpha\delta\beta}\overline{{c}}^{\beta}\frac{\delta}{\delta\overline{{c}}^{\alpha}}\right]S_{eff}=0
\Sigma_{gf}=s\int_{\cal{M}}d^{4}x\sqrt{g}\left\{g^{\mu\alpha}g^{\nu\beta}\chi_{\alpha\beta}^{a}F_{\mu\nu}^{a+}-g^{\mu\nu}(\partial_{\mu}\bar{\varphi}^{a})\psi_{\nu}^{a}-g^{\mu\nu}(\partial_{\mu}\bar{c}^{a})A_{\nu}^{a}\right\},
\Omega_{\partial_{0}^{\prime},\overline{{\partial}}{}_{0}^{\prime}}^{\,^{\bullet,\bullet}}({\mathcalQ})\=\\Omega_{\partial_{0}^{\prime},\overline{{\partial}}{}_{0}^{\prime}}^{\,^{\bullet,\bullet}}({{\mathcalF}}_{\mathcalQ}^{(0)})\.
\begin{array}{c}{\left(\Phi,\left\{a(f),a^{*}(f)\right\}\Phi\right)=\left(f,f\right)\left(\Phi,\Phi\right)}\\{i.e.\quad\left|\left|a(f)\Phi\right|\right|^{2}\leq\left(f,f\right)\left(\Phi,\Phi\right)}\\\end{array}
E_{(p,q)}(r)=\frac{1}{2\pi\alpha^{\prime}}\int_{-\frac{r}{2}}^{\frac{r}{2}}dx\,e^{-2y}T_{\mathrm{eff}}^{(p,q)}(z)\,\sqrt{1+\left(\frac{dz}{dx}\right)^{2}}\quad.
\left[\frac{d^{2}}{dv^{2}}+a(v)\frac{d}{dv}+b(v)\right]\left[\frac{d}{dv}+\frac1{2+v}+\frac1{1-v}+\frac{3L^{2}p^{2}}{8v(2+v)}\right]\varphi(v)=0~.
{\calL}_{\mathrm{matter}}=\frac{1}{4}\sum_{i}Z_{i}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\,\bar{\theta}\bar{Q}_{i}e^{V}Q_{i}+\frac{1}{2}\left\{\int\!\mathrm{d}^{2}\theta\,{\calW}(Q_{i})+\mathrm{H.c.}\right\}\,.