harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{E_{7}{\circ_{1,Z}}E_{1}=E_{5},\;\;\;\;E_{7}{\circ_{1,Z}}E_{2}=E_{3},\;\;\;\;E_{7}{\circ_{1,Z}}E_{4}=E_{6},}\\{E_{5}{\circ_{1,Z}}E_{7}=E_{1},\;\;\;\;E_{3}{\circ_{1,Z}}E_{7}=E_{2},\;\;\;\;E_{6}{\circ_{1,Z}}E_{7}=E_{4}.}\\\end{array}
\{\varepsilon_{\mu\nu}\}=\left(\begin{array}{cccc}{\varepsilon^{00}}&{\varepsilon^{01}}&{\varepsilon^{02}}&{\varepsilon^{00}}\\{\varepsilon^{01}}&{\varepsilon^{11}}&{\varepsilon^{12}}&{\varepsilon^{01}}\\{\varepsilon^{02}}&{\varepsilon^{12}}&{-\varepsilon^{11}}&{\varepsilon^{02}}\\{\varepsilon^{00}}&{\varepsilon^{01}}&{\varepsilon^{02}}&{\varepsilon^{00}}\\\end{array}\right).
P=\sum_{i_{1},i_{2},i_{3},i_{4},i_{5}}\,a_{i_{1}i_{2}i_{3}i_{4}i_{5}}\,z_{1}^{i_{1}}\,z_{2}^{i_{2}}\,z_{3}^{i_{3}}\,z_{4}^{i_{4}}\,z_{5}^{i_{5}}~,
\int|\Phi_{0}(w)^{2}|dw=\mathrm{K}^{2}\int\frac{dw}{z^{2}}\equiv\mathrm{K}^{2}\int_{-\infty}^{+\infty}\frac{{\calH}^{2}}{a^{3}\,{\varphi^{\prime}}^{2}}=1,
F=-W/\beta=V\rho,\quadP=-\left({\frac{\partialF}{\partialV}}\right)_{T},\quadU=T^{2}\left({\frac{\partialW}{\partialT}}\right)_{V},\quadS=-\left({\frac{\partialF}{\partialT}}\right)_{V},\quadC_{V}=T\left({\frac{\partialS}{\partialT}}\right)_{V}\quad.
i\Pi_{\mu\nu}^{(d)}(p)=\frac{ie^{2}N_{f}}{6\pi^{2}\epsilon}(p_{\mu}p_{\nu}-\eta_{\mu\nu}p^{2})=\frac{ie^{2}}{(4\pi)^{2}\epsilon}\frac{-8N_{f}}{3}(\eta_{\mu\nu}p^{2}-p_{\mu}p_{\nu})
h_{\mu\nu}=\left(\begin{array}{llll}{A_{00}}&{0}&{0}&{0}\\{0}&{A_{11}}&{A_{12}}&{0}\\{0}&{A_{12}}&{-A_{11}}&{0}\\{0}&{0}&{0}&{A_{00}}\\\end{array}\right)e^{i\tilde{k}z}cos\\omegat\,
\Gamma_{11}\equiv\Gamma^{0}\ldots\Gamma^{9}=\left(\begin{array}{cc}{1_{16}}&{0}\\{0}&{-1_{16}}\\\end{array}\right)\,\\\gamma^{0}\bar{\gamma}^{1}\ldots\gamma^{8}\bar{\gamma}^{9}=1_{16},\\gamma^{+}={\bar{\gamma}}^{-}=\sqrt2\left(\begin{array}{cc}{1_{8}}&{0}\\{0}&{0_{8}}\\\end{array}\right)\
|\mathrm{initial}\rangle_{\mathrm{anc}}=\int\prod_{\ell=1}^{n}\left(d\tilde{u}_{\ell}\right)|\tilde{u}_{1},\tilde{u}_{2},\dots,\tilde{u}_{n}\rangle\delta(\prod_{\ell=1}^{n}\tilde{u}_{\ell}-I)~.
\intdx\Bigl[J^{\mu}\partial_{\mu}\frac{\delta\mathcal{Z}_{J}}{\delta\bar{D}}+\frac{1}{\xie^{2}}\partial_{\mu}\frac{\delta\mathcal{Z}_{J}}{\deltaJ^{\mu}}D+J\frac{\delta\mathcal{Z}_{J}}{\deltaU}\Bigr]=0.
\int_{\mathrm{gauge\atopcoset}}dw\,=\,c_{k,N}\int\left(\mathrm{det}_{2k^{\prime}}W\right)^{N^{\prime}/2-{k^{\prime}}^{2}/2\pm1/4}\,dW^{0}\,\prod_{c=1,2,3}dW^{c}\.
G_{\mu\nu\rho\sigma}(p)=\int\!dp_{1}\int\!dp_{2}\sum_{s_{1}}\sum_{s_{2}}\\delta(p-p_{1}-p_{2})\langle0\vertT_{\mu\nu}^{V}(0)\vertn\rangle\langlen\vertT_{\rho\sigma}^{V}(0)\vert0\rangle,
{\displaystyle\int}{\calD}y\,\mathrm{exp}\Bigl[-\int_{0}^{T}d\tau{\frac{1}{4}}{\dot{y}}^{2}\Bigr]=\mathrm{Det^{\prime}}_{P}^{-{\frac{1}{2}}}\bigl[-{\frac{d^{2}}{d\tau^{2}}}\bigr]=(4\piT)^{-{\frac{D}{2}}}\,
\begin{array}{l}{3(3,2)_{(1,-1,0,0,0^{3})}+3(\bar{3},1)_{(-1,0,1,0,0^{3})}+3(\bar{3},1)_{(-1,0,0,1,0^{3})}}\\{+3(1,2)_{(0,1,0,0,\underline{{-1,0,0}})}+3(1,1)_{(0,0,-1,0,\underline{{1,0,0}})}+3(1,1)_{(0,0,0,-1,\underline{{1,0,0}})}}\\{+6(1,2)_{(0,1,-1,0,0^{3})}+6(1,2)_{(0,-1,0,1,0^{3})}+12(1,1)_{(0,0,1,-1,0^{3})}.}\\\end{array}
S=\intd^{4}x\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left(B_{\mu\nu}^{a}(x)R_{\lambda\rho}^{a}(x)+\phi^{ab}(x)(B_{\mu\nu}^{a}B_{\lambda\rho}^{b}-\frac{1}{3}\delta^{ab}B_{\mu\nu}^{c}B_{\lambda\rho}^{c})\right)\.
F(s)=\frac{1}{48}\int_{0}^{\infty}\!\!\!d\mu\,\tilde{c}(\mu)\mu\sqrt{s}\left(({\mu}^{3}{s}^{3/2}\!+\!24\mu\sqrt{s})K_{0}(\mu\sqrt{s})\!+\!(8\mu^{2}s\!+\!48)K_{1}(\mu\sqrt{s})\right)
F^{{\mu}{\nu}{\sigma}}=-G^{{\mu}{\nu}{\sigma}}+\frac{1}{({\partial}_{\lambda}a)({\partial}^{\lambda}a)}\left[{\partial}^{[{\mu}}a{\calG}^{{\nu}{\sigma}]{\rho}}{\partial}_{\rho}a+\frac{1}{3!}{\epsilon}^{{\mu}{\nu}{\sigma}{\rho}{\eta}{\delta}}{\partial}_{[{\rho}}a{\calG}_{{\eta}{\delta}]{\theta}}{\partial}^{\theta}a\right].
\hat{\eta}_{\mu\nu}\rightarrow\hat{\eta}_{\alpha\beta}=\hat{\eta}_{\mu\nu}\frac{\partialx^{\mu}}{\partialq^{\alpha}}\frac{\partialx^{\nu}}{\partialq^{\beta}}=\hat{\xi}_{ab}\,c_{\alpha}^{a}(q)\,c_{\beta}^{b}(q).
\Gamma_{3(T)}^{(1)\mu}=\frac{e^{3}}{8\pi\Theta}\int_{0}^{1}dx\int_{0}^{1}dy\left[\frac{A^{\mu}}{\left(Q_{0}^{2}-y{\bfQ}^{2}\right)^{3/2}}+\frac{B^{\mu}}{\left(Q_{0}^{2}-y{\bfQ}^{2}\right)^{1/2}}\right]\;,
L={\frac{1}{2}}\,\sum_{i=1}^{N}\Bigl(\partial_{\mu}\Phi_{i}\partial^{\mu}\Phi_{i}+h_{i}\Phi_{i}^{2}\Bigr)-{\frac{\alpha}{2}}\,\left(\sum_{i=1}^{N}\Phi_{i}^{2}-{\frac{1}{g^{2}}}\right),
E\equiv\frac{1}{\alpha}{\frac{\left({r_{+}}^{2}+{r_{-}}^{2}\right)^{2}+2\left(r_{+}+r_{-}\right)^{2}\left({r_{+}}^{2}+{r_{-}}^{2}+r_{+}r_{-}\right)}{\sqrt{\left(r_{+}+r_{-}\right)^{2}+2\left({r_{+}}^{2}+{r_{-}}^{2}\right)}}},
{\vec{A}}_{3}=\alpha{\vec{B}}_{3}-\frac{1}{|{\vec{B}}_{3}|^{2}}{\vec{B}}_{3}\times\left[({\vec{A}}_{2}\times{\vec{B}}_{2})+({\vec{A}}_{1}\times{\vec{B}}_{1})+(\partial_{i}{\vec{B}}_{i})\right].
\int_{{\calM}_{g,N}}\!\![dm][dz]\intdw\;\langle\;\oint_{C_{w}}\!\!G^{+}\bar{G}^{+}\bar{\phi}_{\bar{i}}(w)\:\prod_{i=1}^{N}\:\oint_{C_{z_{i}}}\!\!G^{-}\bar{G}^{-}\phi_{i}(z_{i})\:\prod_{a,{\bar{a}}=1}^{3g-3}G^{-}(\chi_{a})\bar{G}^{-}(\bar{\chi}_{\bar{a}})\;\rangle
O_{\omega\omega^{\prime}}^{0}=\frac{1}{4\pi}\{\int_{0}^{\theta}\!(\tilde{f}_{\omega}^{E}(\sigma)\:\tilde{f}_{\omega^{\prime}}^{E}(\sigma)+\tilde{f}_{-\omega}^{E}(\sigma)\:\tilde{f}_{-\omega^{\prime}}^{E}(\sigma))d\sigma+\int_{\theta}^{2\pi}\!(\tilde{f}_{\omega}^{L}(\sigma)\:\tilde{f}_{\omega^{\prime}}^{L}(\sigma)+\tilde{f}_{-\omega}^{L}(\sigma)\:\tilde{f}_{-\omega^{\prime}}^{L}(\sigma))d\sigma\}
M_{\alpha\beta}^{i^{\prime}}=\left(\partial_{[\alpha}e_{\beta]}^{c}-e_{[\alpha}^{a}e_{\beta]}^{b}\tilde{e}_{a}^{j^{\prime}}\tilde{e}_{b}^{k^{\prime}}\partial_{j^{\prime}}\tilde{e}_{k^{\prime}}^{c}\right)\tilde{e}_{c}^{i^{\prime}}.
{\calS}={\displaystyle\intdt\sqrt{-g_{oo}}\e^{-\varphi}\left[g^{oo}\left(-\sum_{i=1}^{N}\dot{\lambda}_{i}^{2}+\dot{\varphi}^{2}-\frac{3}{2}\left(1-\frac{c}{Y}\right)\dot{\sigma}^{2}\right)-2YV(Y,\sigma)\right]+{\calS}_{m}}.
\frac{\partial^{2}h_{n}}{\partialu\,\partialu}=\frac{\partial^{2}h_{n}}{\partialv\,\partialv}\;,\qquad\frac{\partial^{2}\tilde{h}_{n}}{\partialu\,\partialu}=\frac{\partial^{2}\tilde{h}_{n}}{\partialv\,\partialv}\;.
\frac{g^{2}}{8\pi}\,\left(C_{AB}d{\vec{x}}_{A}\cdotd{\vec{x}}_{B}+(C^{-1})_{AB}(d\psi_{A}+\vec{\omega}_{AC}\cdotd\vec{x}_{C})(d\psi_{B}+\vec{\omega}_{BD}\cdotd\vec{x}_{D})\right).
\epsilon_{\alpha\beta\gamma\delta}\epsilon_{\bar{\alpha}\bar{\beta}\bar{\gamma}\bar{\delta}}(\Sigma_{\alpha\bar{\alpha}}\cdot\zeta)(\Sigma_{\beta\bar{\beta}}\cdot\zeta)(\Sigma_{\gamma\bar{\gamma}}\cdot\zeta)(\Sigma_{\delta\bar{\delta}}\cdot\zeta)=t_{8}\zeta^{4}\.
V(g)\;\equiv\;-\sum_{n=1}^{\infty}\left(\kappa_{n}N\mathrm{Tr}(g^{n})+\bar{\kappa}_{n}N\mathrm{Tr}((g^{\dagger})^{n})+\lambda_{n}\vert\mathrm{Tr}g^{n}\vert^{2}\right)\;,
\Gamma^{*}[\phi]=\frac{1}{60}\frac{1}{(4\pi)^{2}}\int\mathrm{d}^{4}x\left\{a_{*}(\Box\phi)^{2}-(a_{*}-a_{*}^{\prime})\left[\Box\phi+(\partial_{\mu}\phi)^{2}\right]^{2}\right\}.
\longrightarrowH^{\pi^{+},\pi^{-}}(\bar{F})\stackrel{pr}{\longrightarrow}H^{\pi^{+}-1,\pi^{-}}(F_{-})\stackrel{X_{0}^{+}}{\longrightarrow}H^{\pi^{+},\pi^{-}}(F_{-})\stackrel{i}\longrightarrowH^{\pi^{+},\pi^{-}}(\bar{F})\longrightarrow.
y^{2}=x^{3}+\sum_{k=-4}^{4}\sum_{l=0}^{12-nk}f_{kl}z^{\primel}z^{4+k}\ldotsa^{4-6k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\primel}z^{6+k}\ldotsa^{6-6k}
ds^{2}=-(1-\frac{M}{r^{\prime}})d\tau^{2}+(1-\frac{Q^{2}}{Mr^{\prime}})^{-1}{df}^{2}+(1-\frac{M}{r^{\prime}})^{-1}(1-\frac{Q^{2}}{Mr^{\prime}})^{-1}\frac{kdr^{2}}{8r^{2}},
\deltaS_{\calR}\,=\,{\frac{1}{2}}\int\mathrm{d}^{4}{\bfx}\biggl[(v_{\bfk}^{\prime})^{2}-\delta^{ij}v_{{\bfk},i}v_{{\bfk},j}+{\frac{z^{\prime\prime}}{z}}v_{\bfk}^{2}\biggr]\,,
{\frac{1}{2}}\,e^{-i\{\nu\}\pi/2}I_{1}^{(1)}+{\frac{1}{2}}\,e^{i\{\nu\}\pi/2}I_{1}^{\:(2)}=e^{i\{\nu\}\varphi}e^{-2\pii\{\nu\}d}\mathrm{~.}
T(0)=\prod_{j=3}^{P}\;\prod_{i=2}^{j-1}R_{ij}^{\iota^{P-j+i-1}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})\prod_{j=2,..,k}^{\longleftarrow}R_{1j}^{\iota^{P+j-1}}\;\;\prod_{j=1}^{P-1}\;\prod_{i=j+1}^{P}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})
{\calA}[\Sigma]=\int_{\Sigma}\omega=\int_{\Sigma}{\calL}\left(z,\bar{z},\varphi(z,\bar{z}),v=\partial_{z}\varphi,\bar{v}=\partial_{\bar{z}}\varphi\right)\,{\mathrm{d}}z\wedge\mathrm{d}\bar{z}\,.
\varepsilon\approx(\xi-\xi_{c})\frac{\tilde{A}_{a}(D-2)}{\tilde{A}_{a}-B_{a}n}\frac{\Gamma(D-1/2)\Gamma\left(\frac{D+1}{2}\right)}{2^{D-1}\pi^{D/2}\Gamma^{2}(D/2)}\frac{a^{D-2}}{r^{2D-1}},\quadD>2.
\frac{1}{N}\frac{g_{0}^{2}}{4\pim_{0}^{2}}\leq\frac{g^{2}}{4\pim_{0}^{2}}\leq\frac{1}{N}\frac{g_{1}^{2}}{4\pim_{0}^{2}}\,,\quad\sigma_{1}\leq\sigma\leq\sigma_{0}\,,
m_{1R}^{2}=m_{1}^{2}+\frac{\sigma{}m_{2}^{d-2}}{(4\pi)^{\frac{d}{2}}}\left[\frac{(-1)^{\frac{d}{2}-1}}{(\frac{d}{2}-1)!}\frac{2}{\epsilon}+\Omega(\frac{\epsilon}{2})\right].
\int\!dz\!\wedge\!d\bar{z}\,\partialX\bar{\partial}X=\oint_{\mathrm{a}}~dz~\partialX\oint_{\mathrm{b}}~d\bar{z}~\bar{\partial}X-\oint_{\mathrm{a}}~d\bar{z}~\bar{\partial}X\oint_{\mathrm{b}}~dz~\partialX~,
Q(p^{\prime},p)=\frac{1}{N}\sum_{i}\left(\begin{array}{cc}{a_{i}^{\dagger}(p)a_{i}(p^{\prime})}&{b_{i}^{\dagger}(p)a_{i}(p^{\prime})}\\{a_{i}^{\dagger}(p)b_{i}(p^{\prime})}&{b_{i}^{\dagger}(p)b_{i}(p^{\prime})}\\\end{array}\right)\.
\kappa\theta_{a}^{k}\displaystyle\frac{\delta^{f}L_{B_{f}}}{\delta^{f}B_{a}^{k}}=\kappa\theta_{a}^{k}J_{k}^{a}\approx-\displaystyle\frac{\kappa^{2}}{2}\displaystyle\frac{\theta_{a}^{k}\partialg^{00}}{\partial(\kappaB_{a}^{k})}T_{00}=\kappa^{2}(\theta_{a}^{k}\theta_{k}^{a})T_{00}.
U_{M,N}(a)={\frac{2\pi}{L}}\left(M-{\frac{1}{2}}-{\frac{gLa}{2\pi}}\right)^{2}-{\frac{2\pi}{L}}\left(N-{\frac{gLa}{2\pi}}\right)^{2}+{\frac{\pi}{4L}}+V^{[N=2]}.
F=-\frac{4}{3}\intd\phid\theta\int_{r_{+}+h}^{L}dr\int_{\mu\sqrt{-{g^{\prime}}_{tt}}}^{\infty}dE\frac{1}{e^{\betaE}-1}\frac{\sqrt{g_{4}}}{\sqrt{-{g^{\prime}}_{tt}}}\left(\frac{E^{2}}{-{g^{\prime}}_{tt}}-\mu^{2}\right)^{3/2},
\mathrm{(3a)}+\mathrm{(3b)}=\frac{N}{96\pim}\int_{x}\left[\mathrm{tr}\left\{\partial_{[\mu}\tilde{A}_{\nu]}\partial^{[\mu}\tilde{A}^{\nu]}\right\}+O(\partial^{2}\frac{\partial^{2}}{m^{2}})\right].
\begin{aligned}{\bar{\Phi}_{R}}&{{}=\bar{A}_{L}+\thetaR\bar{\psi}+(\thetaR\theta)\bar{F}_{L}+\cdots,}\\{\bar{\Phi}_{L}}&{{}=\bar{A}_{R}+\thetaL\bar{\psi}+(\thetaL\theta)\bar{F}_{R}+\cdots,}\\\end{aligned}
\partial_{z}\partial_{\bar{z}}\psi=\frac{1}{4}\left(2m^{2}+\partial_{\varphi}^{2}V(\varphi_{0})\right)\,,\quad(\partial_{z}\psi)(\partial_{\bar{z}}\psi)=\frac{1}{4}\left(2m^{2}-\partial_{\varphi}^{2}V(\varphi_{0})\right)\,.
\intd^{\nu-1}q_{1}\;d^{\nu-1}q_{2}\frac{1}{\sqrt{2}}a^{+}\left({\vec{q}}_{1}\right)a{+}\left({\vec{q}}_{2}\right)|0><0|a\left({\vec{q}}_{1}\right)a\left({\vec{q}}_{2}\right)\frac{1}{\sqrt{2}}+....
J_{-}=\frac{1}{4\sqrt{d}}X_{10}\bar{Y}_{10}=\frac{2\sqrt{d}}{2N+1}\left(\begin{matrix}{0}&{\cdots}&{\cdots}&{\cdots}&{0}\\{\sqrt{N-1}}&{0}&{\cdots}&{\cdots}&{\vdots}\\{0}&{\sqrt{2(N-2)}}&{0}&{\cdots}&{\vdots}\\{\vdots}&{0}&{\ddots}&{0}&{\vdots}\\{0}&{\cdots}&{0}&{\sqrt{N-1}}&{0}\\\end{matrix}\right).
\Omega_{\pi^{+},\pi^{-},m,n}^{(1)}=-{\calP}_{\pi^{+},\pi^{-},m}^{(1)}(z)\tilde{\calO}_{\pi^{+},\pi^{-},n}(\bar{z})dz+{\calP}_{\pi^{+},\pi^{-},m}(z)\tilde{\calO}_{\pi^{+},\pi^{-},n}^{(1)}(\bar{z})d\bar{z}
=\frac{1}{S_{R}T_{R}^{3}|\eta(T)|^{12}}\left(|S_{R}{\calK}_{S}-{\calK}|^{2}|{\tilde{\Sigma}}|^{2}+\frac{T_{R}^{2}}{3}|{\tilde{\Sigma}}_{T}+\frac{3}{2\pi}{\tilde{\Sigma}}{\tilde{G}}_{2}|^{2}|{\calK}|^{2}-3|{\calK}|^{2}|{\tilde{\Sigma}}|^{2}\right),
\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{-\frac{\beta^{2}}{4s}n^{2}}=\frac{\sqrt{4\pis}}{\beta}\sum_{k=-\infty}^{\infty}{\mathrm{e}}^{-\frac{4\pi^{2}s}{\beta^{2}}k^{2}}.
\left(2\pi^{2}\mu_{0}M_{n}\right)^{1/2}={\frac{4\pi}{\beta(1/4,3/2)}}\left({\frac{1}{8\sqrt2V_{H}g_{\mathrm{YM}}^{2}}}\right)^{1/3}\left(n+{\frac{3}{4}}\right)^{2/3}\.
E_{jk}(\lambda,\mu):=\frac{1}{2}((-i)^{\delta_{j0}}S^{j_{-}k_{-}}+i\lambdaS^{j_{+}k_{-}}-i^{1+\delta_{j0}}\muS^{j_{-}k_{+}}-\lambda\muS^{j_{+}k_{+}}),
\ddot{U}=\dot{U}^{2}\Bigl[{\frac{1}{2U}}+{\frac{1}{U-1}}\Bigr]+{2A^{2}}g^{2}{\frac{(1+U)(1-U)^{3}}{U}}+\dot{U}{\frac{\dot{g}}{g}}+2(h^{2}-l){\frac{U(1+U)}{(U-1)}}.
\nabla_{-}x^{i}-\frac12\Omega_{-}^{+2i}x^{-2}-\frac12\Omega_{-}^{-2i}x^{+2}-ih_{-q}^{1+}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{1-}-i\chi_{\dot{q}}^{1}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{1+}+i\chi_{q}^{2}\gamma_{q\dot{q}}^{i}\theta_{\dot{q}}^{2+}+ih_{-\dot{q}}^{2+}\tilde{\gamma}_{\dot{q}q}^{i}\theta_{q}^{2-}=0.
\mathrm{Ext}_{[{\bfC}/{\bfZ}_{n}]}^{n}\left(i_{*}{\calE},i_{*}{\calF}\right)\:\cong\:\mathrm{Ext}_{[{\bfC}/{\bfZ}_{n}]}^{1-n}\left(i_{*}{\calF},i_{*}{\calE}\otimesK\right)^{\vee}
\psi_{j_{R}{\mathsf{s}}_{R}}^{\mathrm{G}}=\sum_{j_{3}}\int\frac{d^{3}\hat{p}}{2\hat{p}^{0}}|\hat{\bbox{p}}j_{3}({\mathsf{s}}_{R},j_{R})^{-}\rangle\phi_{j_{R}}(\hat{\bbox{p}}),
V_{i}\cdotP_{\overline{{\alphaV}}}^{\parallel}+{\frac{\delta_{i1}}{m_{1}}}D\equivV_{i}\cdotP_{\overline{{\alphaV}}}^{\parallel}+\sum_{r}T_{i}^{r}N_{\overline{{\alphaV}}}^{r}+{\frac{\delta_{i1}}{m_{1}}}\ell=s_{i}+\sum_{j}k_{ij}\alpha_{j}~({\mathrm{mod}}~1)~.
S^{ii}(\theta)=\left(\begin{array}{ccc}{(2)_{\theta}}&{(3)_{\theta}(1)_{\theta}}&{-(2)_{\theta}}\\{(3)_{\theta}(1)_{\theta}}&{(2)_{\theta}^{2}}&{(3)_{\theta}(1)_{\theta}}\\{-(2)_{\theta}}&{(3)_{\theta}(1)_{\theta}}&{(2)_{\theta}}\\\end{array}\right)
b=-{\frac{i}{2}}dx^{m}(\bar{\theta}_{1}\Gamma_{m}d\theta_{1}-\bar{\theta}_{2}\Gamma_{m}d\theta_{2})+{\frac{1}{2}}(\bar{\theta}_{1}\Gamma^{m}d\theta_{1})(\bar{\theta}_{2}\Gamma_{m}d\theta_{2})\.
3H_{0}^{2}=-\frac{k}{M^{D+2}\tilde{b}_{0}^{D+4}}-\frac{D-2}{D+2}\frac{V\left(\tilde{b}_{0}\right)}{M^{D-2}\tilde{b}_{0}^{D}}+\frac{V^{\prime}\left(\tilde{b}_{0}\right)}{(D+2)M^{D-2}\tilde{b}_{0}^{D-1}}
{\mathcalL}_{D3}={\calO}(\Theta^{0})+Z^{-1}\left(-\frac{i}{2}\overline{{\Theta}}\Gamma^{i}D_{i}\Theta-\frac{i}{48}e^{\phi}\overline{{\Theta}}\Gamma^{mnp}\ThetaRe\left[(*_{6}G)_{mnp}-iG_{mnp}\right]\right)
K(a,\bar{a})=\alpha\pi\left|{\frac{a}{\Lambda}}\right|^{2}|F|^{2}e^{\int_{\tau_{0}}^{\tau}\beta^{-1}+\overline{{\int_{\tau_{0}}^{\tau}\beta^{-1}}}}\left(\Im\,\tau\right)^{2}\\,
\widetilde{\tau}(M,\chi,g)=\prod_{q=0}^{m-1}det(\widetilde{\nabla}_{q}^{*}\widetilde{\nabla}_{q})^{(-1)^{q}}det(\widetilde{\nabla}_{m}^{*}\widetilde{\nabla}_{m})^{\frac{1}{2}(-1)^{m}}
K_{2-2}=\mathrm{Tr}_{\theta}\Bigl(\frac{1}{2}(\partial_{t}\deltau_{i})^{2}+\frac{1}{2}(\partial_{t}\deltab^{I})^{2}+\frac{1}{4}([\partial_{i},\deltau_{j}]-[\partial_{i},\deltau_{j}])^{2}+\frac{1}{2}[\partial_{i},\deltab^{I}]^{2}\Bigr)
\Pi_{\mu\nu}^{odd}(p)\equiv{\cal{\pi}}_{\mu\nu}(p)=\frac{8v^{2}}{\kappa}T\sum_{n}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{\epsilon_{\mu\nu\lambda}k^{\lambda}}{(k^{2}+M^{2})[(p-k)^{2}+m_{\sigma}^{2}]}.
{\calA}_{fi}\left(m_{2}^{2}\right)=-\frac{Q_{\mu\nu}V_{f}^{\mu}\left(m_{2}^{2}\right)V_{i}^{\nu}\left(m_{2}^{2}\right)}{m_{2}\Gamma_{2}-i\left[A(\bar{s})-A\left(m_{2}^{2}\right)\right]}+\tilde{N},
A_{v}(k_{1},\cdots,k_{N})=g_{v}^{N-2}\prod_{j=1}^{N}\intd^{2}\sigma_{j}\int\chi_{v}\left(-\frac{1}{h}\int{\calL}(X^{\mu},\partial_{\alpha}X^{\mu})d^{2}\sigma\right){\calD}_{v}X,
\{\Phi_{1}^{2},\Psi_{2}^{4}\}\stackrel{\Sigma}{=}0,~~~~\{\Phi_{1}^{3},\Psi_{3}^{2}\}\stackrel{\Sigma}{=}0,~~~~\{\Phi_{1}^{3},\Psi_{2}^{3}\}\stackrel{\Sigma}{=}0.
H\,=\,M_{0}\sum_{n=1}^{\infty}\,\sum_{j=1}^{D-2}\,\omega_{n}\,a_{n}^{j+}a_{n}^{j}\,+\,\frac{D-2}{2}\,M_{0}\,\sum_{n=1}^{\infty}\,\omega_{n}\,{,}
G({\vec{r}},{\vec{y}};{\vec{r}}_{0},{\vec{y}}_{0})={\frac{1}{V}}\sum_{\vec{q},j,m,n}G_{j\vec{q}}(r;r_{0})D_{mn}^{*j}(\psi_{0},\theta_{0},\phi_{0})D_{mn}^{j}(\psi,\theta,\phi)e^{i{\vec{q}}.({\vec{y}}-{\vec{y}}_{0})},
6G_{ij}\to{\frac{\alpha^{2}B^{2}}{\epsilon^{2}}}\delta_{ij},\\\G_{00}=-1+{\frac{\alpha^{2}B^{2}v_{j}v^{j}}{\epsilon^{2}c^{2}}},\\\G_{0i}=G_{i0}={\frac{\alpha^{2}B^{2}v^{i}}{\epsilon^{2}c}},
\frac{F}{V}=\left(\frac{1}{4\pi^{2}\alpha^{\prime}}\right)^{5}\int_{-1/2}^{1/2}d\tau_{1}\int_{0}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{6}}\left|\frac{1}{\eta(\tau)}\right|^{24}2\sum_{\stackrel{n=1}{n\odd}}^{+\infty}e^{-\frac{\beta^{2}n^{2}}{4\pi\alpha^{\prime}\tau_{2}}}\left|\theta_{2}(0,\tau)\right|^{8}
H=\frac{1}{2}(J_{1}^{2}+J_{2}^{2}+2J_{3}^{2})+c_{1}x_{1}+c_{2}x_{2}+c_{3}(x_{1}^{2}-x_{2}^{2})+c_{4}x_{1}x_{2}+c_{5}/x_{3}^{2},
\displaystyle{\left.\Pi_{22}(p)\right|_{\mathrm{G}}}=\displaystyle{\frac{1}{4\pi}\left[-\frac{p^{2}}{8}\mathrm{ln}\left(\frac{p^{2}}{\mu^{2}}\right)-\frac{53}{6}\frac{(p_{1})^{2}(p_{2})^{2}}{p^{2}}\right]\,.}
4\pi\mui\delta({\bfx},{\bfx^{\prime}})=\intd^{2}x^{\prime\prime}d^{2}x^{\prime\prime\prime}[\rho(x^{\prime\prime}),\rho(x^{\prime\prime\prime})]\epsilon_{lj}\frac{({\bfx}-{\bfx^{\prime\prime}})^{l}}{|{\bfx}-{\bfx^{\prime\prime}}|^{2}}\frac{({\bfx^{\prime}}-{\bfx^{\prime\prime\prime}})^{j}}{|{\bfx^{\prime}}-{\bfx^{\prime\prime\prime}}|^{2}}\Big|_{t^{\prime}=t}
\frac{1}{2}\left<P^{0}[h]_{\mu\nu}(x)\,P^{\infty}[h]_{\rho\sigma}(y)\right>_{0}+\frac{1}{2}\left<P^{\infty}[h]_{\mu\nu}(x)\,P^{0}[h]_{\rho\sigma}(y)\right>_{0}
\eta\sim-4\beta\left(\frac{2}{\sqrt{d_{\perp}}}\right)^{d_{\perp}}\frac{y^{2}}{\left(1-y\right)^{d_{\perp}}}\left[\left(3+\left(d_{\perp}-4\right)y\right)\left(1+\left(\frac{d_{\perp}}{4}-\frac{3}{2}\right)y\right)+y\left(1-y\right)\left(\frac{d_{\perp}}{4}-\frac{3}{2}\right)\right]
{\calH}_{class}=\frac{1}{2}\sum_{i=1}^{N_{f}}(\dot{\chi}_{i}^{2}+(\chi_{i}^{\prime})^{2})+\frac{e^{2}}{2\pi}(\sum_{i}\chi_{i}-\frac{\theta}{2\sqrt{\pi}})^{2}+\sum_{i}m_{i}^{2}[1-cos(2\sqrt{\pi}\chi_{i})],
|\phi(\tau)\rangle_{I}=V(\tau,\tau^{\prime})|\phi(\tau^{\prime})\rangle_{I}=\mathrm{T}e^{-i\int_{\tau^{\prime}}^{\tau}d\tau^{\prime\prime}\mathrm{K}_{I}(\tau^{\prime\prime})}|\phi(\tau^{\prime})\rangle_{I}
c_{l,m}\;=\;q^{m-l}\,i\tau_{l}^{2}\tau_{l}^{1}\;+\;q^{l-m}\,i\tau_{m}^{2}\tau_{m}^{1}\;-\;\eta^{m-l}\,i\tau_{l}^{2}\tau_{m}^{1}\;-\;\eta^{l-m}\,i\tau_{m}^{2}\tau_{l}^{1}\,
\begin{array}{cc}{\left\{\pi_{\mathrm{ab}}\left(\sigma,\tau\right),\chi_{1}\left(\sigma^{\prime},\tau\right)\right\}=0,}&{\left\{\pi_{\mathrm{ab}}\left(\sigma,\tau\right),\chi_{2}\left(\sigma^{\prime},\tau\right)\right\}=0.}\\\end{array}
\sum_{j}q_{a}^{j}\varphi_{j}^{+}\overline{{\varphi}}_{j}^{+}=\zeta_{a}^{+}\overline{{\zeta}}_{a}^{+}=u_{(\alpha}^{+}u_{\beta)}^{+}\zeta_{a}^{(\alpha}\overline{{\zeta}}_{a}^{\beta)}.
ds^{2}=-(N^{0})^{2}dt^{2}+\frac{dr^{2}}{f^{2}}+H^{2}{\biggl[}d\tilde{\varphi}+{\biggl(}N^{\varphi}(r)-N^{\varphi}(r_{+}){\biggr)}dt{\biggr]}^{2}\:,
V=-Ng^{2}K\int_{-R/2}^{R/2}\mathrm{d}x\,\langle\bar{\psi}\psi\rangle^{2}\,\qquad\langle\bar{\psi}\psi\rangle=\sum_{n}^{occ}\sum_{i=1}^{2}\frac{1}{{\calN}_{n,i}}\bar{\psi}_{n,i}\psi_{n,i}\.
\frac{\delta_{L}\Phi^{\prime}{}^{n}}{\delta\Phi^{m}}=\delta_{m}^{n}-\frac{\delta_{L}}{\delta\Phi^{m}}\left(\frac{\deltaF}{\delta\Phi_{n}^{*}}\right)_{\Phi^{*}=\delta\Psi/\delta\Phi+K}\;.
s_{\Gamma}=\int\left\{\partial_{\mu}c\frac{\delta}{\deltaA_{\mu}}+B\frac{\delta}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\underline{{Y}}}\frac{\delta}{\delta\underline{{\varphi}}}+\frac{\delta\Gamma}{\delta\underline{{\varphi}}}\frac{\delta}{\delta\underline{{Y}}}+\underline{{q}}\frac{\delta}{\delta\underline{{\hat{\varphi}}}}\right\}+\chi\partial_{\xi}
S_{F}(\cdot)=\intd^{4}x\left[\partial^{\mu}c\,{\frac{\delta(\cdot)}{\delta{A^{\mu}}}}+\left(\partial^{\mu}A_{\mu}+{\frac{ev}{\alpha}}\phi_{2}\right){\frac{\delta(\cdot)}{\delta{\bar{c}}}}\right]+({F},\cdot)+(\cdot,{F}).
\frac{\int_{S}\sum_{n_{1}=1}^{\infty}\left|\sum_{\gamma\in\Gamma_{\infty}^{2}\backslash\Gamma_{1}^{2}}\nabla_{\tau}\phi_{n_{1}}^{0}(\tau)\big|_{\gamma\circ\tau}\right|^{2}dV}{\int_{S}\sum_{n=1}^{\infty}\left|\sum_{\gamma\in\Gamma_{\infty}^{2}\backslash\Gamma_{1}^{2}}\phi_{n_{1}}^{0}(\tau)\big|_{\gamma\circ\tau}\right|^{2}dV}<10\lambda.
S(g)=\frac{k}{32\pi}\int_{\partialB}Tr\left(\partial_{\mu}g^{-1}\bar{\partial}^{\mu}g\right)+\frac{k}{48\pi}\int_{B}\varepsilon^{ijk}Tr\left(\partial_{i}gg^{-1}\partial_{j}gg^{-1}\partial_{k}gg^{-1}\right)
\begin{aligned}{H=}&{{}\int_{\Sigma}\frac{1}{2\sqrt{W}}[(P^{m})^{2}+(\Pi_{r})^{2}+1/2W\{X^{m},X^{n}\}^{2}+W(\mathcal{D}_{r}X^{m})^{2}+1/2W(\mathcal{F}_{rs})^{2}]}\\{}&{{}+\int_{\Sigma}[1/8\sqrt{W}n^{2}-\Lambda(\mathcal{D}_{r}\Pi^{r}+\{X^{m},P_{m}\})]-\frac{1}{4}\int_{\Sigma}\sqrt{W}n^{*}\mathcal{F}}\\\end{aligned}
\frac{\ddot{a_{0}}}{a_{0}}+\left(\frac{\dot{a_{0}}}{a_{0}}\right)^{2}=-\frac{\kappa^{4}}{36}\rho(\rho+3p)+\frac{\kappa^{4}}{72}\dot{\varphi}^{4}+\frac{\kappa^{4}}{72}\dot{\varphi}^{2}(\rho+3p),
\int_{-\infty}^{\infty}\frac{dk}{2\pi\omega_{k}}\left(\frac{\omega_{k}+k}{\omega_{k}-k}\right)^{\!i\left(\sigma\Omega-\sigma^{\prime}\Omega^{\prime}\right)/2}\!\!=\int_{-\infty}^{\infty}\!\frac{dy}{2\pi}\,e^{i\left(\sigma\Omega-\sigma^{\prime}\Omega^{\prime}\right)y}=\delta\left(\sigma\Omega-\sigma^{\prime}\Omega^{\prime}\right)\.
\left[m-\sigma^{i}(p_{i}+\frac{1}{2}i\partial_{i})\right]\hat{W}=-\frac{1}{4}i\kappa\sigma^{i}e_{i}^{\alpha}\hat{J}R_{\alpha\beta}\left\{\sigma_{3},\frac{\partial}{\partialp_{\beta}}\hat{W}\right\}
C(g,q)=-{\frac{g\phi_{0}}{(g+q)}}-{\frac{g^{2}q}{2(g+q)}}\int_{0}^{\infty}dkC(g,k)[{\frac{1}{(k^{2}+q^{2}+kq)}}-{\frac{1}{2}}{\frac{1}{(k^{2}+q^{2}-kq)}}].
\begin{array}{ll}{<\chi^{0}\parallel\hat{\Phi}(\zeta)\parallel\chi>=\S_{\lambda\mu=1}^{2}\S_{r^{\lambda\mu}=1}^{N_{\lambda\mu}}\bar{c}(r^{\lambda\mu})F_{r^{\lambda\mu}}(\zeta),}\\{<\chi\parallel\bar{\hat{\Phi}}(\zeta)\parallel\chi^{0}>=\S_{\lambda\mu=1}^{2}\S_{r^{\lambda\mu}=1}^{N_{\lambda\mu}}{\bar{c}}^{*}(r^{\lambda\mu})F^{r^{\lambda\mu}}(\zeta).}\\\end{array}
ds^{2}=-Z_{\phi}Z_{1}^{-1/2}Z_{5}^{-1/2}f_{3}(r)\beta^{2}dt^{2}+Z_{\phi}Z_{1}^{1/2}Z_{5}^{1/2}\left(Z_{0}^{-1}dr^{2}+r^{2}d\Omega^{(3)}\right),
I=\frac{1}{4}\intd^{d+1}x\,x_{0}^{1-d}\left(h_{j,i}^{k}h_{k}^{j,i}-h_{,i}h^{,i}-2h_{j,i}^{k}h_{k}^{i,j}+2h_{,i}h_{j}^{i,j}+h_{j,0}^{i}h_{i,0}^{j}-h_{,0}h_{,0}\right).
\tilde{\mathcal{H}}_{i}=-2(\tilde{\gamma}^{1/3}\tilde{h}_{ik}\tilde{\pi}^{kj})_{,j}+\tilde{\gamma}^{1/3}\tilde{\pi}^{kl}\tilde{h}_{kl,i}+\tilde{\pi}^{\phi}\tilde{\phi}_{,i}=0~~~.
g_{ab}\mathrm{d}X^{a}\mathrm{d}X^{b}=U(\vec{r})\mathrm{d}\vec{r}\cdot\mathrm{d}\vec{r}+4U^{-1}(\vec{r})(\mathrm{d}X^{4}+\vec{w}\cdot\mathrm{d}\vec{r})^{2}\,
\left|u_{k}\right|\rightarrow{\frac{2^{\nu-3/2}}{\sqrt{2k}}}{\frac{\Gamma\left(\nu\right)}{\Gamma\left(3/2\right)}}\left|C_{+}\left(k\right)+C_{-}\left(k\right)\right|\left({\frac{y}{1-\epsilon}}\right)^{1/2-\nu}.
\frac{1}{2}\left(\frac{\partialg_{\gamma\alpha}}{\partialx^{\beta}}+\frac{\partialg_{\beta\alpha}}{\partialx^{\gamma}}-\frac{\partialg_{\beta\gamma}}{\partialx^{\alpha}}\right)=\sum_{\sigma=0}^{3}g_{\alpha\sigma}\Gamma_{\beta\gamma}^{\sigma}.
\begin{array}{rcl}{H_{q-\mathrm{vib-rot}}}&{=}&{H_{q(J)-\mathrm{vib}}+H_{\mathrm{rot}}}\\{}&{=}&{\displaystyle{\frac{1}{2}}\left(a_{q(J)}^{\dagger}a_{q(J)}+a_{q(J)}a_{q(J)}^{\dagger}\right)hc\nu_{\mathrm{vib}}+\displaystyle\frac{h^{2}}{8\pi^{2}I}C_{I}~,}\\\end{array}
\frac{\partial\phi}{\partialx_{0}}=R-\xi_{1}\frac{\partialR}{\partial\xi_{1}}-\xi_{2}\frac{\partialR}{\partial\xi_{2}},\quad\frac{\partial\phi}{\partialx_{1}}=\frac{\partialR}{\partial\xi_{1}},\quad\frac{\partial\phi}{\partialx_{2}}=\frac{\partialR}{\partial\xi_{2}}
\begin{array}{ccc}{\varepsilon^{i}=\left(\begin{array}{cc}{\bar{\varepsilon}_{a}}&{0}\\{0}&{\varepsilon^{a}{}^{t}}\\\end{array}\right)~~~~}&{~~~~\rho^{i}=\left(\begin{array}{cc}{0}&{\bar{\rho}^{a}{}^{t}}\\{\rho_{a}}&{0}\\\end{array}\right)~~~~}&{~~~~a=1,2,\cdots,N}\\\end{array}
S={\frac{ge^{2}}{4\pi^{2}}}\intd^{4}x\,\epsilon^{\mu\nu\alpha\beta}\,\left[\,F_{\mu\nu}\,F_{\alpha\beta}\,{\frac{1}{\Box}}\,\left(\,\partial^{\rho}A_{\rho}^{5}\,\right)-2{\bfa}\,A_{\mu}\,A_{\nu}^{5}\,F_{\alpha\beta}\,\right]
\mathrm{e}^{\lambda^{a}\cdot\varphi}=\frac{\mathrm{e}^{\lambda^{a}\cdot\psi}}{1+\displaystyle{\frac{\mu^{2}}{4}A_{a}B_{a}+\biggl(\frac{\mu^{2}}{4}\biggr)^{2}}A_{a}\circA_{\bar{a}}B_{a}\circB_{\bar{a}}}~,
\Pi_{i}\left(r,K,\xi\right)=\frac{\kappa^{2}}{30}\left[\pi^{2}T^{4}\,l_{i}\left(r,K\right)+\left|\vec{k}\right|^{2}T^{2}\,n_{i}\left(r,K,\xi\right)+m^{2}T^{2}\,s_{i}\left(r,K,\xi\right)\right]+\cdots.
R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}~^{(4)}R=-\ell_{P}^{2}\left(\frac{\delta\stackrel{\sim}{\calL}_{m}^{\prime}}{\deltag^{\mu\nu}}-{\frac{1}{2}}g_{\mu\nu}\stackrel{\sim}{\calL}_{m}^{\prime}\right)-\ell_{P}^{2}e^{-1}\frac{\delta{\calL}_{g}}{\deltag^{\mu\nu}}.