harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
i\frac{d}{dt}\langle\tilde{\psi}^{(0)}|\tilde{x}^{m}\tilde{p}^{n}|\tilde{\psi}^{(0)}\rangle=\langle\tilde{\psi}^{(0)}|\left[\tilde{x}^{m}\tilde{p}^{n},\frac{1}{2\mu}\tilde{p}^{2}+V(\tilde{x})\right]|\tilde{\psi}^{(0)}\rangle=0.
{\calL}^{(0)}~=~\frac{\kappa}{2\pi}\epsilon^{mn}A_{n}\dot{A}_{m}+\pi_{a}\dot{z}_{a}+\pi_{a}^{*}\dot{z}_{a}^{*}+\Omega_{1}^{(0)}\dot{\alpha}-{\calH}^{(0)},~~~m,n=1,2,
1\simeq\frac{2\alpha}{\pi}T|eH|\int_{-\infty}^{\infty}dq_{3}\int_{0}^{\infty}d\hat{q}_{\perp}^{2}\mathrm{e}^{-\hat{q}_{\perp}^{2}}\sum_{l^{\prime}}\frac{1}{Q_{2}+4\pi^{2}T^{2}l^{2}}~\frac{1}{Q_{1}+\pi^{2}T^{2}(2l^{\prime}-1)^{2}}
\langleT_{ik}(x)\rangle_{\mathrm{reg}}=\langle0|T_{ik}(x)|0\rangle-\langle0_{M}|T_{ik}(x)|0_{M}\rangle=\hat{\theta}_{ik}\langle\varphi(x)\varphi(x^{\prime})\rangle_{\mathrm{reg}},
\dot{q}^{\lambda}\bar{\nabla}_{\lambda}(\dot{q}^{\delta}\bar{\nabla}_{\delta}\xi^{\mu})-\bar{R}_{\sigma\rho\lambda}^{\mu}\dot{q}^{\rho}\dot{q}^{\sigma}\xi^{\lambda}-\xi^{\mu}-2H_{\rho\sigma}^{\mu}\dot{q}^{\rho}(\dot{q}^{\delta}\bar{\nabla}_{\delta}\xi^{\sigma}-\xi^{\sigma})=U^{\mu}.
\phi_{ia,\overline{{k}}\overline{{b}}}(0)=-\phi_{ia,kb}(0)+\frac{4\pi^{2}}{|\beta^{2}|}\lambda_{i}^{\mathrm{v}}\cdot\left(\lambda_{k}^{\mathrm{v}}+\lambda_{\overline{{k}}}^{\mathrm{v}}\right)\,\,\,\,\,\,\,(\mathrm{mod}2\pi\hbar)
q^{\mu}\left\{\tilde{\Gamma}_{\mu}(q;p,p^{\prime})-ie\mu^{\frac{\epsilon}{2}}(p^{\prime}+p)_{\mu}\tilde{D}\right\}=-e\mu^{\frac{\epsilon}{2}}\left\{(\tilde{\Gamma_{2}}(p^{\prime})+ip^{\prime2}\tilde{D})-(\tilde{\Gamma_{2}}(p)+ip^{2}\tilde{D})\right\}\;.
\rho_{1}(L_{i_{1}}\cdotsL_{i_{n}})={\frac{1}{n}}\sum_{(i_{1},\ldots,i_{n})}\Bigl(L_{i_{1}}\rho_{1}(L_{i_{2}}\cdotsL_{i_{n}})+P(L_{i_{1}},L_{i_{2}}\cdotsL_{i_{n}})\Bigr).
\frac{1}{24\pi^{2}}\int_{S^{3}}d^{3}x\;\varepsilon_{ijk}\mathrm{Tr}\Big[(\partial_{i}S^{(m)})(S^{(m)})^{-1}(\partial_{j}S^{(m)})(S^{(m)})^{-1}(\partial_{k}S^{(m)})(S^{(m)})^{-1}\Big]\;=m\;.
\tilde{\zeta}(s)=\frac{V_{q}}{(4\pi)^{q/2}}\frac{\Gamma(s-q/2)}{\Gamma(s)}\Big\lbrace(\mu_{c}^{2}-\mu^{2})^{q/2-s}+\sum_{{\bfp}\ne{\bf0}}(\sigma_{\bfp}^{d}+m^{2}-\mu^{2})^{-s}\Big\rbrace\;,
cosh^{\left({2/\beta}\right)}({\frac{\betac}{2L}})sech^{\left({2/\beta}\right)}({\frac{\betar}{2L}}){\frac{d\theta}{dr}}={\frac{-1}{3}}\theta^{2}+{\frac{3\beta}{2L^{2}}}sech^{2}({\frac{\betar}{2L}}).
\mathrm{Tr}(\Phi\Phi)=\frac{\mathrm{Tr}({\tilde{F}}^{2})}{\mathrm{det}(E^{a}(q))^{2}}\\\\\mathrm{and}\\\\{\calT}=\frac{1}{2}\frac{\mathrm{Tr}({\tilde{F}}^{2})}{\mathrm{det}(E(q))}~.
\widehatT^{\mu\nu}(q)=e^{2}\left(\eta^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\widehatH(q^{2})-e^{2}\left(\eta^{\mu\nu}-{\frac{q^{\mu}q^{\nu}}{q^{2}}}\right)\left.\hat{h}\right|_{q^{2}=0},
\begin{array}{c}{\omega=\mathrm{d}{\bfp}\wedge\mathrm{d}{\bfx}+\mathrm{i}\sum_{i}\mathrm{d}{\bfz}_{i}\wedge\mathrm{d}{\bar{\bfz}}_{i},}\\{\;\;}\\{{\calH}=\frac{s}{2c}\left[\mathrm{i}{\sqrt{2}}{\bfp}({\bf{\bar{z}}}_{1}-{\bf{z}}_{1})+\mathrm{i}\sum_{i=1}^{N-1}{k}_{i}({\bfz}_{i}{\bf{\bar{z}}}_{i+1}-{\bfz}_{i+1}{\bf{\bar{z}}}_{i})+{k}_{N}({\bfz}_{N}{\bf{\bar{z}}}_{N}-c^{2})\right],}\\\end{array}
\begin{array}{c}{X_{2}X_{1}=q\X_{1}X_{2}\\\D_{2}D_{1}=q^{-1}\D_{1}D_{2}}\\{D_{1}X_{1}=q\X_{1}D_{1}+q^{-N}\\\D_{2}X_{2}=q^{-1}X_{2}D_{2}+q^{N}}\\{\left[D_{i},X_{j}\right]=0\\\i\neqj}\\\end{array}
{\calP}_{j}^{i}=2\,\pi_{gj}^{i}+\frac{1}{(p-1)!}\\pi_{A}^{is_{1}\ldotss_{p-1}}\,A_{js_{1}\ldotss_{p-1}}\,,\;\;{\calE}^{i_{1}\ldotsi_{p}}=\pi_{A}^{i_{1}\ldotsi_{p}}\,.
e^{2i\delta_{l,j}(k)}={\frac{-\kappa+i{\frac{M{\gamma}^{\prime}}{k}}}{|\lambda|-i{\frac{E\gamma}{k}}}}{\frac{\Gamma(1+|\lambda|-i{\frac{E\gamma}{k}})}{\Gamma(1+|\lambda|+i{\frac{E\gamma}{k}})}}e^{-i\pi\lambda}.
\left.\left\{\begin{cases}{}&{cB_{x}g_{0}+E_{y}g_{z}=(1-\beta^{2})E_{y}g_{z}=0}\\{}&{cB_{y}g_{0}-E_{x}g_{z}=-(1-\beta^{2})E_{x}g_{z}=0}\\\end{cases}\right.\right\}\qquad\Rightarrow\beta^{2}=1,
\check{R}(z,w)=\left[\begin{matrix}{1}&{}&{}&{}&{}&{}&{}&{}&{}\\{}&{e_{1}}&{}&{p}&{}&{}&{}&{}&{}\\{}&{}&{c_{1}}&{}&{g_{1}}&{}&{b}&{}&{}\\{}&{p}&{}&{e_{2}}&{}&{}&{}&{}&{}\\{}&{}&{h_{1}}&{}&{o}&{}&{h_{2}}&{}&{}\\{}&{}&{}&{}&{}&{e_{1}}&{}&{p}&{}\\{}&{}&{b}&{}&{g_{2}}&{}&{c_{2}}&{}&{}\\{}&{}&{}&{}&{}&{p}&{}&{e_{2}}&{}\\{}&{}&{}&{}&{}&{}&{}&{}&{1}\\\end{matrix}\right]
E_{M}{}^{A}=\left(\begin{matrix}{e^{-z}}&{ie^{-z/2}\lambda_{l}s_{j}{}^{l}}&{-ie^{-z/2}\bar{\lambda}^{l}(s^{-1})_{l}{}^{j}}\\{0}&{-e^{-z/2}s_{j}{}^{i}}&{0}\\{0}&{0}&{-e^{-z/2}(s^{-1})_{i}{}^{j}}\\\end{matrix}\right)\,.
\int\frac{d^{4}l}{(2\pi)^{4}}\frac{l\cdotq}{l^{2}(l+q)^{2}((l+k_{1})^{2}-m_{1}^{2})((l-k_{3})^{2}-m_{2}^{2})}\rightarrow\frac{-q^{2}}{2}\int\frac{d^{4}l}{(2\pi)^{4}}\frac{1}{l^{2}(l+q)^{2}((l+k_{1})^{2}-m_{1}^{2})((l-k_{3})^{2}-m_{2}^{2})}
\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)=\left(\begin{matrix}{A_{0}}&{0}\\{0}&{A_{0}}\\\end{matrix}\right)+\frac{1}{N^{2}}\left(\begin{matrix}{A_{1}}&{B_{1}}\\{B_{1}}&{A_{1}}\\\end{matrix}\right)+\mathcal{O}(N^{-4}).
\begin{array}{lcr}{\eta_{j}^{+}=u_{\alpha}^{+}\eta_{j}^{\alpha};\qquad\overline{{\eta}}_{j}^{+}=u_{\alpha}^{+}\overline{{\eta}}_{j}^{\alpha}}\\{\eta_{j}^{\alpha}\overline{{\eta}}_{\alphaj}=1;\qquad\eta_{j}^{\alpha}{\eta}_{\alphaj}=\overline{{\eta}}_{j}^{\alpha}\overline{{\eta}}_{\alphaj}=0}\\\end{array}
\left(\frac{2\pi\omega^{+j}R^{+j}+Y^{+j}}{2\pi\alpha^{\prime}}\right)\left(\frac{2\pi\omega_{-j}R_{-j}+Y_{-j}}{2\pi\alpha^{\prime}}\right)-\frac{1}{8}++N_{B}(DD)+N_{F}^{NS}(DD),
\begin{aligned}{}&{{}-\lambda_{1}=(a-1)(\lambda_{2}-\lambda_{1}),}&{}&{{}\qquad2\lambda_{1}-\lambda_{2}=-a(\lambda_{2}-\lambda_{1})}\\{}&{{}\lambda_{2}-\lambda_{1}=-(b-1)\lambda_{1},}&{}&{{}\qquad2\lambda_{1}-\lambda_{2}=b\lambda_{1}.}\\\end{aligned}
ds^{2}=\frac{\rho_{1}^{2}}{a(x_{1})}dx_{1}^{2}+\frac{\rho_{2}^{2}}{a(x_{2})}dx_{2}^{2}+\frac{(\rho_{1}\rho_{2})^{2}}{a(x)}dx^{2}+d\rho_{1}^{2}+d\rho_{2}^{2},
d^{D}\mathbf{x}^{\prime}\,d^{D}\mathbf{p}^{\prime}=d^{D}\mathbf{x}\,d^{D}\mathbf{p}\left[1-\left\{2(D-1)\beta\left(\frac{1+(\beta+\beta^{\prime})p^{2}}{1+\betap^{2}}\right)+(2\beta+\beta^{\prime})\right\}p_{j}\frac{\partialH}{\partialx_{j}}\deltat\right]\;.
\Gamma_{+}^{-1}C_{3}^{-2}B=\frac{1}{2}\frac{\alpha}{1+\mu\alphak}\left[(k_{2}\mu-k_{1})+(k-\mu^{2}k_{2})C_{3}^{-1}+(\frac{2}{\alpha}+k\mu+k_{1}\mu^{2})C_{3}^{-2}\right]Y.
V_{\mathrm{tot}}(Y_{1})=\int{\left[\lambda{\frac{(1+2g_{c}Y_{1})}{(1+4g_{c}Y_{1})}}-g_{c}{\frac{(1+2g_{c}Y_{1})}{(1+4g_{c}Y_{1})^{2}}}\right]\tilde{V}(Y_{1})}dY_{1}
\{\tilde{\Theta}_{\alpha}^{\mu},\tilde{\Theta}_{\beta}^{\nu}\}=\{\tilde{\Theta}_{\alpha}^{\mu},\tilde{\Psi}_{l}\}=\{\tilde{\Theta}_{\alpha}^{\mu},\tilde{H}\}=0~~;~~\{\tilde{\Psi}_{l},\tilde{\Psi}_{n}\}\approx0~;~\{\tilde{\Psi}_{l},\tilde{H}\}\approx0.
S[\varphi]=\frac12\int\!dx\,g^{1/2}\,\,\left(\nabla_{\mu}\varphi^{T}\nabla^{\mu}\varphi+\frac{R}{6}\varphi^{T}\varphi+\frac{\lambda^{2}}{4!}(\varphi^{T}\varphi)^{2}\right),
\begin{array}{l}{S=\intd^{d-1}\vec{z}\;\int{\frac{dz}{z^{d}}}\left[z^{2}\partial_{z}(z^{-s}h^{I_{1}\ldotsI_{s}})\partial_{z}(z^{-s}h^{I_{1}\ldotsI_{s}})+\right.}\\{+\left.[s^{2}+(d-5)s-2(d-3)]z^{-2s}h^{I_{1}\ldotsI_{s}}h^{I_{1}\ldotsI_{s}}\right]}\\\end{array}
S_{str}={\frac{1}{2}}\int_{{\calM}^{2}}d^{2}\xi\left(\partial_{m}\hat{X}^{a}\partial^{m}\hat{X}_{a}-\partial_{m}\hat{X}^{i}\partial^{m}\hat{X}^{i}\right)+\int_{\partial{\calM}^{2}}d\tau{\calP}_{i}\left(\hat{X}^{i}\left(\xi(\tau)\right)-\tilde{X}^{i}\right)
\delta\psi_{A}=D_{A}\epsilon+\frac{1}{6\sqrt{2}\eta}\,\gamma_{A}\Gamma_{45}\,\epsilon-\frac{1}{6\sqrt{2}}\,(\gamma_{A}^{~BC}-4\delta_{A}^{B}\gamma^{B})\left(\etaF_{BC}^{a}\Gamma_{a}+\frac{1}{\sqrt{2}\eta^{2}}\,f_{BC}\right)\epsilon\,,
{\Omega_{M}}^{N}=\left(\begin{array}{cc}{\gamma^{\mu}~{\Omega_{1\muM}}^{N}}&{\gamma_{5}\phi~{l_{M}}^{N}}\\{}&{}\\{\gamma_{5}\phi~{l_{N}}^{M}}&{\gamma^{\mu}~{\Omega_{2\muM}}^{N}}\\\end{array}\right)~~~,~~~~~N,M=1,\ldots,5
T_{0}=\Gamma_{11},\quadT_{2}={\frac{1}{2}}\psi^{2}+{\calF}\Gamma_{11},\quadT_{4}={\frac{1}{24}}\Gamma_{11}\psi^{4}+{\frac{1}{2}}{\calF}\psi^{2}+{\frac{1}{2}}{\calF}^{2}\Gamma_{11},\quad\mathrm{etc.}
j^{0}(x,t)\Bigm|_{x=0}=\frac{1}{2}(\psi_{1}^{\dagger}(0)\psi_{1}(0^{-})+\psi_{2}^{\dagger}(0)\psi_{2}(0^{-})+\psi_{1}^{\dagger}(0^{-})\psi_{1}(0)+\psi_{2}^{\dagger}(0^{-})\psi_{2}(0))=0
\begin{array}{ll}{\,\underline{{M^{2}=n}}}&{\,\quad\underline{{States\,\,\sum\,p(l^{\prime})\times(l^{\prime},l)}}}\\{1}&{(0,1)}\\{2}&{(0,2)+(1,1)}\\{3}&{(0,3)+(1,2)+2(2,1)}\\{4}&{(0,4)+(1,3)+2(2,2)+3(3,1)}\\{5}&{(0,5)+(1,4)+2(2,3)+3(3,2)+5(4,1)}\\\end{array}
\left[\frac{d^{2}}{dz^{2}}-\mathrm{sgn}\left(\beta\right)\lambda\,\alpha^{2}\kappa^{-2/\alpha}+\sigma\alpha^{2}z^{\widetilde{\beta}}-\frac{(\widetilde{l}+\widetilde{\nu})^{2}-1/4}{z^{2}}\right]w_{l}(z)=0\;,
2\left(\left(\bar{D}_{\mu}A_{\nu}\right)_{a}\left(\bar{D}_{\mu}A_{\nu}\right)_{a}-\left(\bar{D}_{\nu}A_{\mu}\right)_{a}\left(\bar{D}_{\mu}A_{\nu}\right)_{a}+\bar{F}_{\mu\nua}f_{abc}A_{\mub}A_{\nuc}\right)
{\calL}=-\,\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}-\,\frac{1}{12}\,H_{\mu\nu\rho}\,H^{\mu\nu\rho}\,+{\frac{m}{4}}\,\epsilon^{\mu\nu\rho\lambda}\,B_{\mu\nu}F_{\rho\lambda}\,,
k^{2}=-\left(1-\frac{{U_{T}}^{2}}{U^{2}}\right)\left[U^{2}\left(1-\frac{{U_{-}}^{2}}{U^{2}}\right)+\left(1-\frac{{U_{-}}^{2}}{{U_{T}}^{2}}\right)({U_{T}}^{2}+{U_{-}}^{2})\right],
e^{-i{t}^{a}{\theta}_{a}}\partial_{\mu}e^{i{t}^{b}{\theta}_{b}}=i{t}^{a}\left\{\bar{C}\partial_{\mu}S_{a}-\bar{S}_{a}\partial_{\mu}C+({f}_{abc}-i{d}_{abc})\bar{S}^{b}\partial_{\mu}{S}^{c}\right\},
\theta_{i}^{2}\equiv\stackrel{.}{\theta}_{i}=\left\{\theta_{i},H_{2}^{\left(1\right)}\right\}=0;\\theta_{i}^{\ast2}\equiv\stackrel{.}{\theta}_{i}^{\ast}=\left\{\theta_{i},H_{2}^{\left(1\right)}\right\}.
R_{\alpha}=-2G{\mathcalE}_{\alpha}=2G{\bar{D}}^{2}D_{\alpha}{\mathcalF}^{\prime}=-2{\bar{D}}^{\dot{\alpha}}(G{\bar{D}}_{\dot{\alpha}}D_{\alpha}{\mathcalF}^{\prime}-{\bar{D}}_{\dot{\alpha}}GD_{\alpha}{\mathcalF}^{\prime})\.
h_{d,s}(1\GeV)=F_{u}\left(\frac{\alpha_{1}(M_{Z})}{\alpha_{1}(M_{S})}\right)^{\frac{12}{168}}\left(\frac{\alpha_{1}(M_{S})}{\alpha_{1}(M_{Str})}\right)^{\frac{6}{198}}h_{d,s}(M_{Str})
\hat{\calA}=\hat{\calA}_{1}+\hat{\calA}_{2}=(-1)^{n+1}{\frac{1}{2^{2n-1}n!}}({\frac{g}{\pi}})^{n}\epsilon_{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}[\hat{F}_{\mu_{1}\nu_{1}}...\hat{F}_{\mu_{n}\nu_{n}}-g\theta^{\alpha\beta}\partial_{\beta}(\hat{A}_{\alpha}\hat{F}_{\mu_{1}\nu_{1}}...\hat{F}_{\mu_{n}\nu_{n}})].
\left(\begin{array}{cc}{-2\,m\,\Omega}&{i\wp_{_{S,A}}\left[\Omega+{\bbox{\sigma\cdotp}}\right]\Theta_{[1/2]}{\mit\Xi}\left[\Omega+{\bbox{\sigma\cdotp}}\right]}\\{i\wp_{_{S,A}}\left[\Omega-{\bbox{\sigma\cdotp}}\right]{\mit\Xi}^{-1}\Theta_{[1/2]}\left[\Omega-{\bbox{\sigma\cdotp}}\right]}&{-2\,m\,\Omega}\\\end{array}\right)\lambda(p^{\mu})=0\quad,
\langle\bar{P}^{n}\cdots\rangle={\frac{\partial^{n}}{\partial\bar{t}^{n}}}\,\langle\cdots\rangle=\left({\frac{\partialt_{s}}{\partial\bar{t}}}\,{\frac{\partial}{\partialt_{s}}}\right)^{n}=\left(-{\frac{1}{\langlePP\rangle_{0}}}\,{\frac{\partial}{\partialt_{s}}}\right)^{n}\\langle\cdots\rangle\,
{\calL}_{kin}=\intd^{4}\theta\left[Q^{+}e^{2qV_{X}+V_{N}}Q+{\tilde{Q}}e^{2{\tilde{q}}V_{X}-V_{N}}{\tilde{Q}}^{+}+\phi^{+}e^{-2V_{X}}\phi\right]+{\calL}_{S,V}\
\prod_{j=1}^{g}\int_{\Sigma_{g}}\omega_{\bar{z}}^{j}G_{z}d^{2}z\cdot\delta\left(\int_{\Sigma_{g}}\omega_{\bar{z}}^{j}J_{z}d^{2}z\right)\sim\tilde{\Omega}_{g}\delta({\calV}_{g})=c_{g}({\calE}_{hol}).
Y_{i}\left(\theta\right)=(1+x_{i}(\theta))\prod_{j=1}^{r}(1+x_{j}^{-1}(\theta))^{-g_{ij}}\quad\mathrm{and\quad~}\quadZ_{i}\left(\theta\right)=Y_{i}\left(\theta\right)(1+x_{i}^{-1}(\theta))\quad.
\lambda^{2}\left[e^{2i\varphi}\left(-{\frac{1}{2\epsilon^{2}}}+{\frac{1}{2\epsilon}}\right)+e^{i\varphi}\left(-{\frac{J-2}{\epsilon^{2}}}-{\frac{2}{\epsilon}}\right)+{\frac{A_{2,2}}{\epsilon^{2}}}+{\frac{A_{2,1}}{\epsilon}}+\mathrm{finite}\right]
<e^{ip\cdotX}>_{osc}=e^{-\frac12p_{\mu}p_{\nu}<(X+\bar{X})_{osc}^{\mu}(X+\bar{X})_{osc}^{\nu}>}=e^{-[(p_{0}^{2}+p_{1}^{2})A_{v}+\vec{p}_{T}^{2}(A+C_{v}-C)+2p_{0}p_{1}B_{v}]}
\Lambda^{(4)}={\frac{B}{A}}\left[-{\frac{3}{2}}{\frac{B^{\prime\prime}}{B}}+{\frac{3}{4}}{\frac{A^{\prime}B^{\prime}}{AB}}-3{\frac{B^{\prime}}{Br}}+{\frac{A^{\prime}}{Ar}}+{\frac{A-1}{r^{2}}}\right]-B\kappa^{2}\left[{\frac{\phi^{2}}{2A}}+{\frac{\phi^{2}}{r^{2}}}+V(\phi)\right]-B\Lambda\,.
\frac{1}{|\sqrt{2}|a|-K(r)|}\frac{|\sqrt{2a^{2}-b}-K(r)|^{\frac{|a|}{\sqrt{2(2a^{2}-b)}}+\frac{1}{2}}}{(\sqrt{2a^{2}-b}+K(r))^{\frac{|a|}{\sqrt{2(2a^{2}-b)}}-\frac{1}{2}}}=\frac{r}{r_{3}}
\gamma^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=\int_{0}^{\infty}d\taue^{-(D-4)\tau}I_{\tau}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;.
S_{22}=\left\langle\frac{2\pi}{H}\right\rangle\left\langle\frac{2\pi}{3}\right\rangle\left\langle\frac{2\pi}{H}+\frac{\pi}{3}\right\rangle\left\langle\frac{\pi}{3}-\frac{4\pi}{H}\right\rangle\left\langle\frac{\pi}{3}-\frac{6\pi}{H}\right\rangle\left\langle\frac{4\pi}{H}-\pi\right\rangle\left\langle\frac{6\pi}{H}-\pi\right\rangle\,
F(x^{-},x^{+})=\left|{\frac{L}{\pi}}\langle0|T^{++}(k)|n\rangle\right|^{2}\int{\frac{d\left({\frac{k\pi}{L}}\right)}{2\pi}}{\frac{1}{2L}}{\frac{\pi^{2}}{L^{2}}}e^{-i{\frac{M_{n}^{2}}{2({\frac{k\pi}{L}})}}x^{+}-i{\frac{k\pi}{L}}x^{-}}.
S_{p}=\frac12\intd\taud^{p}\sigma\\rho^{\mu}\left(2u^{\alpha}\omega_{\mu\alpha\dot{\alpha}}\bar{u}^{\dot{\alpha}}+u^{\alpha}\omega_{\mu\alpha\beta}u^{\beta}+\bar{u}^{\dot{\alpha}}\bar{\omega}_{\mu\dot{\alpha}\dot{\beta}}\bar{u}^{\dot{\beta}}\right),
{\bfU}_{i}=\frac1{\sqrt{2}}\left(\begin{array}{cc}{1}&{\Gamma_{i}}\\{1}&{-\Gamma_{i}}\\\end{array}\right),\;{\bfH}_{S}{}^{\prime}={\bfU}_{1}{\bfH}_{S}{\bfU}_{1}^{\dagger}=\left(\begin{array}{cc}{H_{-}}&{0}\\{0}&{H_{+}}\\\end{array}\right);
\Gamma_{(0)}=({{\widehaty}_{2}}^{2})^{2}+({{\widehaty}_{3}}^{2})^{2}+({\widehaty}_{2}\cdot{\widehaty}_{3})^{2}+({{\widehaty}_{2}}^{2}+{{\widehaty}_{3}}^{2})\({\widehaty}_{2}\cdot{\widehaty}_{3})-{{\widehaty}_{2}}^{2}{{\widehaty}_{3}}^{2}
\intd^{4}xh_{\mu\nu}f_{\mu\nu}^{V}=\intd^{4}xB_{\mu\nu}{}^{*}f_{\mu\nu}^{V}=\intd^{4}x\partial_{\mu}b_{\nu}{}^{*}f_{\mu\nu}^{V}=\intd^{4}xb_{\mu}k_{\mu},
e^{-A};\;\;\;\frac{\rho}{\chi},\;\frac{\rho}{\chi\Omega};\;\;\;\frac{\dot{\chi}}{\chi},\;\frac{\Omega\dot{\chi}^{2}}{\chi^{2}};\;\;\;\mathrm{and}\;\;\;\frac{\chi^{n}}{\Omega}\;\frac{d^{n}\Omega}{d\chi^{n}}\;\left(\frac{\dot{\chi}}{\chi}\right)^{n},\;\;\foralln\ge1\;.
(\alpha_{opt}^{2})^{(odd)}=\sqrt{\frac{2\mid\beta\mid^{2}coth\mid\beta\mid^{2}+2\mid\beta\mid^{2}cos~2\theta+1}{2\mid\beta\mid^{2}coth\mid\beta\mid^{2}-2\mid\beta\mid^{2}cos~2\theta+1}}(\alpha_{opt}^{2})^{(o)}\nonumber\,
\frac{d^{p-2}}{\partialx_{i_{1}}^{p-2}}\frac{\partial^{p-3}}{\partialx_{i_{2}}^{p-3}}\dots\frac{\partial}{\partialx_{i_{p-2}}}f(x_{1},x_{2},\dots)\biggr|_{x_{i_{1}}=x_{i_{2}}=\dots=x_{i_{p-2}}=x_{i_{p-1}}}=0.
\frac{d^{2}f}{dx^{2}}+\frac{a^{2}-6a+2-(a^{2}-8a+2)x^{a}}{2x(1-x^{a})}\frac{df}{dx}+\frac{\lambda^{2}}{x^{4-a}(1-x^{a})^{2}}f-\frac{l(l+a)}{x^{2}(1-x^{a})}f=0\,.
\hat{I}_{1}=\int_{0}^{r}\left[(E_{1}-E_{0})^{2}u_{1}^{2}+\frac{u_{0}^{2}W^{2}}{(I_{0}+\lambda)^{2}}+2(E_{1}-E_{0})\frac{u_{0}u_{1}}{(I_{0}+\lambda)}W\right]dr^{\prime}.
A=\frac{y_{2}}{W(y_{1},y_{2})}\frac{1}{R^{7}u^{\prime}\sqrt{{u^{\prime}}^{6}-(1+\theta){u^{\prime}}^{2}+\theta}},B=\frac{y_{1}}{W(y_{1},y_{2})}\frac{1}{R^{7}u^{\prime}\sqrt{{u^{\prime}}^{6}-(1+\theta){u^{\prime}}^{2}+\theta}}.
\left.h_{a}^{(I)}\right.^{2}=\left.h_{b}^{(I)}\right.^{2}=\sigma^{(I)},\qquad\mathrm{where}\qquad\sigma^{(I)}=\frac{1}{t^{(I)}d_{I}+\left.\vec{\alpha}^{(I)}\right.^{2}/4}.
u_{{m}}^{{\{f\}}}\left(\Gamma^{{m}}C^{-{1}}\right)_{{\alpha}{\beta}}=\left(v_{{\alpha}{A}}^{{\hat{a}}}v_{{\beta}{\hat{b}}{A}}+v_{{\alpha}{\dot{A}}}^{{\hat{a}}}v_{{\beta}{\hat{b}}{\dot{B}}}\right)\gamma_{{\hat{a}}}^{{\{f\}{\hat{b}}}},
\underline{{hs(2,2)}}:f(a,\bar{a})=\alpha+\alpha_{i\bar{j}}a^{i}\bar{a}^{\bar{j}}+\alpha_{i_{1}i_{2}\bar{j}_{1}\bar{j}_{2}}a^{i_{1}}a^{i_{2}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}+\ldots
G_{F}(x,x^{\prime})=-i{\frac{\pi}{(4\pii)^{\frac{d+1}{2}}}}\left({\frac{m^{2}}{\sigma+i\epsilon}}\right)^{\frac{d-1}{4}}H_{(d-1)/2}^{(2)}(m(-\sigma-i\epsilon)^{1/2}),
\left.\partial_{p_{1}^{\nu}}\Gamma_{A_{\mu}\varphi_{1}\varphi_{2}}(-p_{1}-p_{2},p_{1},p_{2})\right|_{\{p_{i}\}=p_{norm},\xi_{=}\xi_{0},\xi_{A}=\xi_{A_{0}}}=-ie\eta^{\mu\nu}\;\;\;,
\frac{k\nu}{2\pi}\widetilde{Q}_{\alpha}\equiv\widetilde{J}^{+}\widetilde{F}_{-\alpha}+\frac{\widetilde{J}^{3}}{2}\widetilde{F}_{+\alpha}+\frac{d-1}{2C_{\rho}}\widetilde{B}^{a}\eta^{\beta\gamma}\left(\lambda^{a}\right)_{\alpha\beta}\widetilde{F}_{+\gamma}-\frac{k}{2\pi}\partial_{\theta}\widetilde{F}_{+\alpha}(\theta)
\begin{array}{lclclcl}{x}&{=}&{x(z)+x(\bar{z}),}&{\qquad}&{\bar{x}}&{=}&{\bar{x}(z)+\bar{x}(\bar{z}),}\\{C}&{=}&{C(z),}&{\qquad}&{\bar{C}}&{=}&{\bar{C}(\bar{z}),}\\{B}&{=}&{B(z),}&{\qquad}&{\bar{B}}&{=}&{\bar{B}(\bar{z}).}\\\end{array}
(\tau_{1})_{\mathrm{vac}}\approx1.29\,\frac{\Lambda_{3}}{h^{1/7}}\,,\,\,\,(\tau_{2})_{\mathrm{vac}}\approx1.25\,\frac{\Lambda_{3}}{h^{1/7}}\,,\,\,\,{\calE}\approx3.59\,h^{10/7}\,\Lambda_{3}^{4}\,.
\alpha_{U}\;\sim\;{\frac{g_{\mathrm{I}}}{({\tilde{r}}M_{\mathrm{I}})^{6-n}}}\\,\\\M_{\mathrm{Planck}}^{2}\sim{\frac{r^{n}{\tilde{r}}^{\;6-n}M_{\mathrm{I}}^{8}}{g_{\mathrm{I}}^{2}}},
\psi_{m,\ell}(r,\theta)=<r,\theta|m,\ell>=(-1)^{m}\,\left(\frac{\omega}{\pi\hbar}\right)^{1/2}\,\left(\frac{m!}{(m+|\ell|)!}\right)^{1/2}\,e^{i\ell\theta}\,u^{|\ell|}\,e^{-u^{2}/2}\,L_{m}^{|\ell|}(u^{2})\\\,
=-\frac{1}{2}\sum_{i=1}^{N}\frac{\partial^{2}}{\partialx_{i}^{2}}+\frac{\lambda}{2}\sum_{i\neqj}^{N}\frac{1}{(x_{i}-x_{j})}\left(\frac{\partial}{\partialx_{i}}-\frac{\partial}{\partialx_{j}}\right).
q_{\mathrm{NS}}~\rightarrow~{\widetildeq}_{\mathrm{NS}}=\mathrm{e}^{-\varphi_{0}/2}\,q_{\mathrm{NS}}+\chi_{0}\,\mathrm{e}^{\varphi_{0}/2}\,q_{\mathrm{R}}~~~~,~~~~q_{\mathrm{R}}~\rightarrow~{\widetildeq}_{\mathrm{R}}=\mathrm{e}^{\varphi_{0}/2}\,q_{\mathrm{R}}~~.
{\calH}^{(2)}=\sum_{\sigma}\int\frac{d^{3}p}{(2\pi)^{3}}E_{p}\,\left[a_{\sigma}^{\dagger}(\vec{p})c_{\sigma}(\vec{p})+b_{\sigma}(\vec{p})d_{\sigma}^{\dagger}(\vec{p})+c_{\sigma}^{\dagger}(\vec{p})a_{\sigma}(\vec{p})+d_{\sigma}(\vec{p})b_{\sigma}^{\dagger}(\vec{p})\right]\,,
\begin{array}{l}{W_{k}(\sigma^{+},\sigma^{-})=W\left[\left(F(u_{k},V_{k}(\sigma^{+}))+F(u_{k},V_{k}(\sigma^{-}))-F(u_{k},v_{k-1})\right)\,,u_{k}\right]}\\{\bar{W}_{k}(\sigma^{+},\sigma^{-})=\bar{W}\left[\left(\bar{F}(U_{k}(\sigma^{+}),v_{k})+\bar{F}(U_{k}(\sigma^{-}),v_{k})-\bar{F}(u_{k-1},v_{k})\right)\,,v_{k}\right].}\\\end{array}
M_{pl}^{2}={\frac{M_{X}^{3}}{k}}e^{2\pik\rho-2c}\left(1-e^{-2ky_{2}}+e^{-2k(\pi\rho-y_{4}+y_{2})}-e^{-2k(2\pi\rho-y_{4})}\right)~.~\,
^{\star}{\calF}_{\alpha\beta}=\frac{1}{2}e^{-\sqrt{3}k}\epsilon_{\alpha\beta\gamma\delta}{\calF}^{\gamma\delta},\qquad^{\star}F_{I\alpha\beta}=e^{-k/\sqrt{3}}G_{IJ}\epsilon_{\alpha\beta\gamma\delta}F^{J\gamma\delta}.
\Phi(\theta^{a})=\sum_{i=0,1,..,3,5,..,d}\quad\sum_{\{a_{1}<a_{2}<...<a_{i}\}\in\{0,1,..,3,5,..,d\}}\alpha_{a_{1},a_{2},...,a_{i}}\theta^{a_{1}}\theta^{a_{2}}\cdots\theta^{a_{i}},
{\calL}_{0}=\int_{\Sigma}\left\{\partial\bar{X}^{\bar{J}}\bar{\partial}X^{I}G_{I\bar{J}}-\frac{i}{2}\left(\rho_{\bar{z}}^{I}D_{z}\bar{\psi}^{\bar{J}}+\rho_{z}^{\bar{J}}D_{\bar{z}}\psi^{I}\right)G_{I\bar{J}}-\mathrm{~\frac{1}{4}~}\psi^{I}\bar{\psi}^{\bar{I}}\rho_{\bar{z}}^{J}\rho_{z}^{\bar{J}}\;R_{I\bar{I}J\bar{J}}\right\},
\delta_{\eta}\Pi_{m}^{r}=i\partial_{m}X^{\mu}\left(2\bar{\theta}\Gamma^{r}\delta_{\eta}\psi_{\mu}+\frac{1}{\sqrt{2}}e_{\mu}^{~r}\bar{\theta}\Gamma^{11}\delta_{\eta}\lambda\right)-\Lambda_{~s}^{r}\Pi_{m}^{s}+{\calO}(\theta^{3})~,
{\calL}_{I}={\frac{1}{2}}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a}-V_{I}(\phi),\quadV_{I}(\phi)=-{\frac{m^{2}}{\beta^{2}}}\sum_{0}^{r}n_{j}\left(e^{i\beta\alpha_{j}\cdot\phi}-1\right),
:\phi\left(X\left(\tau\right)\right)::\psi\left(X\left(0\right)\right):\,=\,:e^{\frac{i}{2}\theta^{ij}\frac{\partial^{2}}{\partialX^{i}\left(\tau\right)\partialX^{j}\left(0\right)}}\phi\left(X\left(\tau\right)\right)\psi\left(X\left(0\right)\right):.
\frac{1}{N}Tr\rightarrow\int\frac{dwdz}{(2\piR)^{2}}=\int\frac{dw^{\prime}dz^{\prime}}{(2\piR^{\prime})^{2}}=\int\frac{d{x}_{2}^{\prime}d{x}_{4}^{\prime}}{(2\piR^{\prime})^{2}}
\chi_{2\ell}^{k}(q,z)\chi_{2\ell^{\prime}}^{1}(q,z)=\sum_{2\ell^{\prime\prime}=0}^{k+1}\chi_{2\ell^{\prime\prime}}^{k+1}(q,z){}~\chi_{2\ell+1,2\ell^{\prime\prime}+1}^{Vir~(k+2)}(q),
F_{\mu\nu}=F_{\mu\nu}^{i}\lambda_{i}+G_{\mu\nu}^{i}\rho_{i}+ig(2A_{\mu}^{j}A_{\nu}^{k}+2B_{\mu}^{j}B_{\nu}^{k}-A_{\mu}^{j}B_{\nu}^{k}-B_{\mu}^{j}A_{\nu}^{k})[\rho_{k},\lambda_{j}].
m=m_{b}={\frac{1}{10l^{2}}}[{r_{b}}^{7}+(l^{2}-28n^{2}){r_{b}}^{5}+5n^{2}(14n^{2}-l^{2}){r_{b}}^{3}+5(3l^{2}-28n^{2})r_{b}+5n^{6}(l^{2}-7n^{2})/r_{b}]
\frac{1}{M_{11}^{3}}\intd^{3}x^{\prime}\,(\partial_{0}^{\prime}X^{i}\partial_{0}^{\prime}X^{i})^{2}\sum_{n=-\infty}^{\infty}\left(X^{2}+[2\pinR_{11}-X^{11}]^{2}\gamma^{2}\right)^{-3}\.
\left.\langleT_{\mu\nu}(x)T_{\rho\sigma}(0)\rangle\right|_{D=2}=\frac{1}{8\pi}C_{T}\left.\left[g_{\mu\rho}(x)g_{\nu\sigma}(x)+g_{\mu\sigma}(x)g_{\nu\rho}(x)-\delta_{\mu\nu}\delta_{\rho\sigma}\right]\frac{1}{(x^{2})^{2+\epsilon}}\right|_{\epsilon=0},
{\calH}=\frac{1}{2}E^{2}-\frac{i\hbar}{2}\psi^{\dagger}\sigma_{3}\left(\partial_{1}+ieA\right)\psi+\frac{i\hbar}{2}\left[\left(\partial_{1}-ieA\right)\psi^{*}\right]\sigma_{3}\psi-A_{0}\,{\calG}.
ds_{10}^{2}=f_{p}(r)^{-1/2}\left[\left(1-k(r)\right)dt^{2}-d{\vec{y}}^{2}\right]-f_{p}(r)^{1/2}\left[\left(1-k(r)\right)^{-1}dr^{2}+r^{2}d\Omega_{8-p}^{2}\right]\quad,
\delta_{\Lambda}A_{\mu}^{a}=0,\;\;\;\delta_{\Lambda}B_{\mu_{1}\ldots\mu_{n-2}}^{a}=-D_{[\mu_{1}}\Lambda_{\mu_{2}\ldots\mu_{n-2}]}^{a},\;\;\;\delta_{\Lambda}\eta_{\mu_{1}\ldots\mu_{n-3}}=\Lambda_{\mu_{1}\ldots\mu_{n-3}}^{a}\,\,.
{\calI}_{s}^{+}=\left\{\begin{array}{cl}{1}&{\qquadq_{1}=0}\\{8|q_{1}|}&{\qquad0<|q_{1}|<|\tilde{a}_{1}|\quad\mathrm{and}\quadq_{1}a_{1}>0}\\{0}&{\qquad\mathrm{otherwise}}\\\end{array}\right\},
2\Sigma_{12}=\left(\begin{array}{cccc}{1}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{-1}&{\cdot}&{\cdot}\\{\cdot}&{\cdot}&{1}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{-1}\\\end{array}\right)\;\;,\;\;2\Sigma_{34}=\left(\begin{array}{cccc}{1}&{\cdot}&{\cdot}&{\cdot}\\{\cdot}&{1}&{\cdot}&{\cdot}\\{\cdot}&{\cdot}&{-1}&{\cdot}\\{\cdot}&{\cdot}&{\cdot}&{-1}\\\end{array}\right)\nonumber
y^{2}~t^{2}~+~t~(y^{2}-a_{1}^{2})~(y^{2}-a_{2}^{2})~\ldots~(y^{2}-a_{n}^{2})~+~~\Lambda_{SU}^{2b_{0}^{SU}}~\Lambda_{SO}^{b_{0}^{SO}}~y^{2}~=~0~.
\frac{d}{du}{\calV}^{(n)}(u)|_{u=0}=\sum_{i=1}^{n}t_{\beta_{1}\beta_{2}}(0)\otimest_{\beta_{2}\beta_{3}}(0)\otimes\cdots\otimes\dot{t}_{\beta_{i}\beta_{i+1}}(0)\otimes\cdots\otimest_{\beta_{n}\beta_{1}}(0)~.
\sum_{A}M_{A}\int_{x}^{\infty}d\thetae^{\theta}L_{A}^{>}(\theta)=\sum_{A}M_{A}\int_{x}^{\infty}d\thetaL_{A}^{>}(\theta)\frac{d\epsilon_{A}^{>}}{d\theta}+\sum_{A,B}\int_{x}^{\infty}L_{A}^{>}(\theta)\,\Phi_{AB}*(L_{B}^{>})^{\prime}(\theta),
\begin{aligned}{[m]H_{6}}&{{}=\star_{10}\left(H_{4}+A\wedgeH_{3}\right)}\\{H_{7}}&{{}=A\wedge\star_{10}H_{4}+\star_{10}\imath_{A^{\sharp}}(A\wedgeH_{3})-e^{-2\Phi}\star_{10}H_{3}~,}\\\end{aligned}
\int{\frac{dk}{2\pi}}\Bigl({\frac{1}{4\piR^{2}}}\sum_{l=0}^{\infty}{\frac{2l+1}{k^{2}+(l+1/2)^{2}/4\pi^{2}R^{2}+\lambda_{c}^{2}/4\pi^{2}}}-\int{\frac{d^{2}{\bfk}}{(2\pi)^{2}}}{\frac{1}{k^{2}+{\bfk}^{2}}}\Bigr)=0.
\ddot{\Phi}_{S}-\Phi_{S}^{\prime\prime}-\bar{R}_{\mu\sigma\rho\lambda}n_{S}^{\mu}n_{R}^{\lambda}\dot{q}^{\rho}\dot{q}^{\sigma}\Phi^{R}=2H_{\mu\rho\sigma}\dot{q}_{S}^{\mu}n_{R}^{\sigma}(\dot{\Phi}^{R}-\Phi^{R}),
C_{T}\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(c-a)\Gamma(b)}=\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(c-b)\Gamma(a)}\left(C_{R}+\frac{\Gamma(2-c)\Gamma(c-b)}{\Gamma(c)\Gamma(2-c-b)}\right)(\alpha-1)^{b-a}2^{a-b}
\hat{{\calV}}=-\,\hat{V}_{a}^{U(1)_{Y}}\cdot\frac{Y}{2}-\sum_{a=1}^{3}\hat{V}_{a}^{SU(2)_{L}}\frac{\sigma^{a}}{2}-\sum_{a=1}^{8}\hat{V}_{a}^{SU(3)_{C}}\frac{\lambda^{a}}{2}
S=\int\left[\Pi_{\alpha}\dot{z}_{\alpha}+\Pi_{\alpha}^{*}\dot{z}_{\alpha}^{*}+\Pi_{\phi}\dot{\phi}+C^{\alpha}\dot{\bar{P}}_{\alpha}+P^{\alpha}\dot{\bar{C}}_{\alpha}+p_{\alpha}\dot{q}^{\alpha}-H_{U}\right]
\begin{array}{lcl}{S[\phi]}&{=}&{\frac{1}{2}\int\phi(x_{1})D(x_{1},x_{2})\phi(x_{2})dx_{1}dx_{2}}\\{}&{+}&{\frac{1}{4!}\intV(x_{1},x_{2},x_{3},x_{4})\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})dx_{1}dx_{2}dx_{3}dx_{4}}\\\end{array}