harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi(x;y,\bar{y})=\sum_{\begin{array}{c}{\scriptstyle\ell+1,m,n\,\ge\,0}\\{\scriptstyle|m-n|=4(\ell+1)}\\\end{array}}\frac1{m!n!}\Phi_{\alpha_{1}\ldots\alpha_{m}\dot{\alpha}_{1}\ldots\dot{\alpha}_{n}}(x)y^{\alpha_{1}}\ldotsy^{\alpha_{m}}\bar{y}^{\alpha_{1}}\ldots\bar{y}^{\dot{\alpha}_{n}}.
\psi_{k}^{+}(x)=\left(\begin{matrix}{\eta_{k}^{S}}\\{\tilde{\eta}_{k}^{A}}\\\end{matrix}\right)\mathrm{~~~~and~~~~}\psi_{k}^{-}(x)=\left(\begin{matrix}{\eta_{k}^{A}}\\{\tilde{\eta}_{k}^{S}}\\\end{matrix}\right)
\deltaS={\frac{i}{2}}\int{dtd}\Theta\delta\{(D_{\Theta}\phi)\dot{\phi}\}\Rightarrow\deltaL={\frac{1}{2}}\epsilon{\frac{d}{dt}}(D_{\Theta}\phi)\dot{\phi}\mid_{\Theta=0}={\frac{i}{2}}\epsilon{\frac{d}{dt}}\{\psi\dot{q}\}\neq0.
\tau_{p,st}=2\times10^{36}\,\mathrm{years}\left(.037L(Q)^{2/3}Ig_{s}^{1/3}\right)^{-2}\left(\frac{.04}{\alpha_{GUT}}\right)^{4/3}\left(\frac{M_{GUT}}{2\times10^{16}\,\mathrm{GeV}}\right)^{4}.
\left|{\frac{dz}{d\tau}}\right|^{2}=G_{ab}{\frac{d\phi^{a}}{d\tau}}{\frac{d\phi^{b}}{d\tau}}={\frac{\partial^{2}K}{\partialz^{i}\partial\bar{z}^{n}}}{\frac{dz^{i}}{d\tau}}{\frac{d\bar{z}^{n}}{d\tau}}\.
S_{3}(k_{1},k_{2};k_{3})\equiv\langleV_{-1;\chi}V_{-1;\chi^{*}}V_{0,A_{\mu}}\rangle=i{\frac{C_{D2}g_{0}^{3}}{\sqrt{2\alpha^{\prime}}}}(2\pi)^{4}\delta^{(4)}(\sum_{i=1}^{3}k_{i})\alpha^{\prime}(k_{1}-k_{2})\cdote\,,
\mathrm{Re}\int_{1}^{\infty}dz\frac{\sqrt{-1+i\sqrt{z^{2}-1}}}{z(z^{2}+a^{2})}=-\frac{\pi}{\sqrt{2}a^{2}}\left[1-\frac{1}{\sqrt{2}}\sqrt{1+\sqrt{1+a^{2}}}\right]
\bigg(\langle\langleF_{\mu\nu}(z_{1})F_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle-\langle\langleF_{\mu\nu}(z_{1})\rangle\rangle\,\langle\langleF_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle\bigg)=0
\omega^{-1}(x)\tilde{F}_{\mu\nu}(x)\omega(x)=-\frac{2}{\bar{N}}\epsilon_{\mu\nu\rho\sigma}\int\delta\xidsE^{\rho}[\xi|s]\dot{\xi}^{\sigma}(s)\dot{\xi}^{-2}(s)\delta(x-\xi(s)).
\widetilde{W}=\left(\begin{array}{ll}{A-\widehat{\theta}\PhiA}&{A\theta-\widehat{\theta}(\PhiA\theta+(A^{t})^{-1})}\\{\PhiA}&{\PhiA\theta+(A^{t})^{-1}}\\\end{array}\right)=\left(\begin{array}{ll}{M}&{N}\\{R}&{S}\\\end{array}\right)
H_{C}=\int{d^{2}}x\left\{\begin{array}{r}{-\frac{1}{2}F_{i0}\left({1+e^{2}a\left[{1-12\alpha^{2}g^{2}\partial^{2}}\right]}\right)F^{i0}+\frac{1}{4}F_{ij}\left({1+e^{2}a\left[{1-12\alpha^{2}g^{2}\partial^{2}}\right]}\right)F^{ij}}\\{+\Pi^{i}\partial_{i}A_{0}-\frac{\chi}{2}\varepsilon^{ij}A_{0}\partial_{i}A_{j}+A_{0}J^{0}}\\\end{array}\right\}.
e^{-\beta\lambda_{j}\cdot\phi}\rightarrowe^{-2\pii\lambda_{i(2)}^{\mathrm{v}}\cdot\lambda_{j}}\frac{<\Lambda_{j}|e^{X_{i(1)i(2)}^{1/p_{i(1)}}(\theta_{12})A_{i(1)}\widehat{F}^{i(1)}(\theta_{1})}|\Lambda_{j}>}{<\Lambda_{0}|e^{X_{i(1)i(2)}^{1/p_{i(1)}}(\theta_{12})A_{i(1)}\widehat{F}^{i(1)}(\theta_{1})}|\Lambda_{0}>^{m_{j}}}\,.
S=\frac{N^{2}\Omega_{8-p}}{(2\pi)^{7}}\intd^{p+2}x\sqrt{-g}\left[R-\frac{1}{2}(\partial\Phi)^{2}+V(\Phi)\right]-\frac{2N^{2}\Omega_{8-p}}{(2\pi)^{7}}\intd^{p+1}x\sqrt{-h}K,
\intdt\dot{\xi}^{\mu}A_{\mu}(y+\xi)={\frac{1}{2}}F_{\mu\nu}(y)\intdt\dot{\xi}^{\mu}\xi^{\nu}+{\frac{1}{3}}\partial_{\lambda}F_{\mu\nu}(y)\intdt\dot{\xi}^{\mu}\xi^{\nu}\xi^{\lambda}+\cdots.
\frac{\partial\Gamma_{\Lambda}[\phi]}{\partial\Lambda}=\frac{1}{2}tr\left[\frac{\partialC^{-1}}{\partial\Lambda}\;_{.}\left(C^{-1}\delta^{ab}+\frac{\delta^{2}\Gamma_{\Lambda}}{\delta\phi^{a}\delta\phi^{b}}\right)^{-1}\right]
\Phi^{m_{1}m_{2}m_{3}}\equive_{\underline{{m}}_{1}}{}^{m_{1}}e_{\underline{{m}}_{2}}{}^{m_{2}}e_{\underline{{m}}_{3}}{}^{m_{3}}\Phi^{\underline{{m}}_{1}\underline{{m}}_{2}\underline{{m}}_{3}}\,,
f(T,U)=\int_{(T_{0},U_{0})}^{(T,U)}\{dT^{\prime}Q(U,U^{\prime})(T-T^{\prime})^{2}\partial_{T^{\prime}}^{3}f(T^{\prime},U^{\prime})+(T\leftrightarrowU,T^{\prime}\leftrightarrowU^{\prime})\}\,
\intd^{p+1}y\;\Phi(X,y)\;\partial_{\alpha}X^{i}\partial^{\alpha}X^{i}=\sum_{l}{\frac{1}{l!}}\intd^{p+1}y\;(\partial_{i_{1}}\dots\partial_{i_{l}}\Phi)X^{i_{1}}\dotsX^{i_{l}}\;\partial_{\alpha}X^{i}\partial^{\alpha}X^{i}.
\begin{array}{ll}{ds_{6}^{2}=}&{e^{2\lambda}\left(dy^{2}+dz^{2}\right)+e^{2\varphi}\left(-d\tau^{2}+\tau^{2}d\Omega_{3,-1}^{2}\right),}\\{H_{e}=}&{id\left(e^{2\lambda}dy\wedgedz\right).}\\\end{array}
+\frac{(1-\sigma_{r})(1-\sigma_{r^{\prime}})}{(t-t^{\prime})+i(r+r^{\prime})}(G+\bar{G^{\prime}})-\frac{(1+\sigma_{r})(1+\sigma_{r^{\prime}})}{(t-t^{\prime})-i(r+r^{\prime})}(\bar{G}+G^{\prime})
\left.\frac{H_{-4/5}^{(1)}(z)}{H_{-4/5}^{(2)}(z)}\right|_{z=2l/5}=\left.\frac{H_{-4/5}^{(1)}(z)}{H_{-4/5}^{(2)}(z)}\right|_{z=2l(\sqrt{2}\alpha\pi\rho/3+1)^{5/2}/5}.
\begin{array}{rcl}{D^{\,(\pm)}\:e_{\,j}\:(D^{\,(\pm)})^{-1}}&{=}&{e_{\,j+1}\qquad1\leqj\leq2M-2}\\{D^{\,(\pm)}\:e_{\,2M-1}\:(D^{\,(\pm)})^{-1}}&{=}&{e_{2M}^{\,(\pm)}}\\{D^{\,(\pm)}\:e_{\,2M}^{\,(\pm)}\:(D^{\,(\pm)})^{-1}}&{=}&{e_{\,1}}\\\end{array}
e^{ik\eta_{c}}c_{-}\simeq\frac{1}{2\sqrt{2}}[(\frac{1}{2}-\nu)\frac{\Gamma(\nu)}{\pi}(\frac{k\eta_{c}}{2})^{-\nu-\frac{1}{2}}-i(\frac{k\eta_{c}}{2})^{\nu-\frac{1}{2}}(\nu+\frac{1}{2})\frac{1}{\nu\Gamma(\nu)}],
\begin{array}{l}{\frac{2M}{(D-3-p)A_{D-2-p}}=\frac{D-2-p}{D-3-p}r_{H}^{D-3-p}+(D-2)\delta\alpha,}\\{\left(\frac{Q_{0}}{(D-3-p)A_{D-2-p}}\right)^{2}=(D-2)\delta\alpha\left(\alpha+r_{H}^{D-3-p}\right),}\\\end{array}
{\calH}=\zeta^{\,0}+\frac{1}{2}\left\{(\dot{\phi})^{2}+(\partial_{\,z}\phi+F)^{2}+i\psi_{2}(\partial_{\,z}-F^{\prime})\psi_{1}+i\psi_{1}(\partial_{\,z}+F^{\prime})\psi_{2}\right\}\,,
\hat{\varphi}_{s}^{[m]}(e^{xD})=\varphi_{s}^{[m]}(e^{xD})-\frac{e^{sx}-1}{e^{x}-1}-\sum_{j=1}^{m}\frac{x^{j}e^{sx}}{e^{x}-1}t^{j}/j!,\quad\hat{\varphi}_{s}^{[m]}(C)=1\inR_{m}
\displaystyle\frac{K_{l-1}(x)}{K_{l}(x)}=\displaystyle\frac{x}{2l}\left[1+\displaystyle\frac{1}{l}+\left(1-\displaystyle\frac{1}{4}x^{2}\right)\displaystyle\frac{1}{l^{2}}+0\left(\displaystyle\frac{1}{l^{3}}\right)\right],
\left.{\frac{\partial}{\partials}}{\calD}(s,\nu,{\frac{\theta}{2\pi}})\right|_{s=0}=\left.-\left[{\frac{\Gamma^{\prime}(s)}{\Gamma^{2}(s)}}G(s)\right]\right|_{s=0}+\left.\left[{\frac{1}{\Gamma(s)}}G^{\prime}(s)\right]\right|_{s=0}
{\mathbf\Phi}_{-}\ni\phi^{+}=\int_{m_{0}^{2}}^{\infty}d{\mathsfs}\,|[j,{\mathsfs}],\phi^{+}\rangle\phi^{+}({\mathsfs})=\int_{m_{0}^{2}}^{\infty}d{\mathsfs}\sum\!\!\!\!\!\!\!\!\int_{b}\|[j,{\mathsfs}],b^{+}\rangle\langle^{+}b,[j,{\mathsfs}]|\phi^{+}\rangle
\tilde{L}=-\frac{1}{8\pi}\left(\frac{4\pi}{g^{2}\bar{\tau}\tau}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{i}{\bar{\tau}\tau}\frac{\theta}{2\pi}\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\tilde{F}^{\mu\nu}\tilde{F}^{\rho\sigma}\right).
{\tilde{\sigma}}^{\muM{\dot{N}}}=-\epsilon^{MK}\,\overline{{{\sigma^{\mu}}_{{\dot{K}}L}}}\,\epsilon^{{\dot{L}}{\dot{N}}}=\epsilon^{MK}\,\epsilon^{{\dot{N}}{\dot{L}}}\,{{\bar{\sigma}}^{\mu}}_{K{\dot{L}}}=\epsilon^{MK}\,\epsilon^{{\dot{N}}{\dot{L}}}\,{\sigma^{\mu}}_{{\dot{L}}K}={\sigma^{\mu}}^{{\dot{N}}M}.
\begin{array}{ll}{\pi^{\kappa}(\lambda)|0>=0,}&{\pi^{\nu}(\lambda)|0>=0}\\{<0|q^{\kappa}(\lambda)=\mathrm{ln}a(\lambda)<0|,}&{<0|q^{\nu}(\lambda)=\mathrm{ln}d(\lambda)<0|}\\\end{array}<0|0>=1
j=\left(\begin{array}{cc}{j_{0}/2}&{j_{-}}\\{j_{+}}&{-j_{0}/2}\\\end{array}\right),\qquad\bar{\jmath}=\left(\begin{array}{cc}{\bar{\jmath}_{0}/2}&{\bar{\jmath}_{-}}\\{\bar{\jmath}_{+}}&{-\bar{\jmath}_{0}/2}\\\end{array}\right).
{\calL}_{\mathrm{hyp.kin.}}=-g^{MN}\partial_{M}\phi_{i}^{*}\partial_{N}\phi^{i}-\frac12\bar{\Psi}\gamma^{M}\partial_{M}\Psi+\frac12\partial_{M}\bar{\Psi}\gamma^{M}\Psi+F_{i}^{*}F^{i}+\frac{15}{4}k^{2}\phi_{i}^{*}\phi^{i},
\hat{K}=\frac{\left|\Phi_{w}\right|+\left|\Phi_{\bar{w}}\right|}{\left|\Phi_{w}\right|-\left|\Phi_{\bar{w}}\right|}\mathrm{~,~}\hat{K}>1\mathrm{~\quad~if~\quad}\left|\Phi_{\bar{w}}\right|\neq0.\tag{4.35b}
\{\chi_{\mu\nu}\}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{\chi_{11}}&{\chi_{12}}&{\chi_{13}}\\{0}&{\chi_{12}}&{\chi_{22}}&{\chi_{23}}\\{0}&{\chi_{13}}&{\chi_{23}}&{\chi_{33}}\\\end{array}\right);~~~\chi_{11}+\chi_{22}+\chi_{33}=0.
\begin{array}{lcr}{(|X_{a-1}|^{2}+|X_{a+1}|^{2}-2|X_{a}|^{2})-(|Y_{a-1}|^{2}+|Y_{a+1}|^{2}-2|Y_{a}|^{2})=R_{a}}&{(a)}\\{X_{a-1}\overline{{Y}}_{a-1}+X_{a+1}\overline{{Y}}_{a+1}-2X_{a}\overline{{Y}}_{a}=0}&{(b)}\\{\overline{{X}}_{a-1}{Y}_{a-1}+\overline{{X}}_{a+1}{Y}_{a+1}-2\overline{{X}}_{a}{Y}_{a}=0.}&{(c)}\\\end{array}
Z_{g=2}=\sum_{e}C(e)\int\prod_{i=1}^{6}d^{2}a_{i}\frac{1}{dv_{pr}^{2}}T^{-1}\overline{{\prod_{k<l}a_{kl}^{-3}}}\overline{{F(\Lambda_{24})}}\prod_{k<l}a_{kl}^{-2}[{\calP}^{X}+{\calP}_{e}^{gh}]{\calO}_{e}^{3};
u\equiv\frac{1}{4}\tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{2}Tr\tilde{G}_{\mu\nu}\tilde{G}^{\mu\nu}+\frac{m^{2}}{\sqrt{-g}}\varepsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu\alpha\beta}
\Gamma_{+}^{-1}A^{(r)}C_{r}-\alpha_{r}C_{3}\Gamma_{+}^{-1}A^{(r)}=C\delta_{r3}-\frac{1}{2}\frac{\alpha}{1+\mu\alphak}CU_{3}^{-1}YY^{\mathrm{T}}A^{(r)}CU_{r}^{-1}.
\{{\bar{\omega}}^{A^{\prime}},\{\bar{\omega}}^{B^{\prime}}\}=\{{\bar{\omega}}^{A^{\prime}},\{\bar{\pi}_{A}}\}=\{{\bar{\pi}_{A}},\{\bar{\pi}_{B}}\}=\{{\omega}^{A},\{\bar{\omega}}^{B^{\prime}}\}=0.
P_{rh}^{-}={\frac{g^{2}L}{4\pi^{2}}}\left[\sum_{N\neq0}{\frac{C_{N}^{3}C_{-N}^{3}}{N^{2}}}+\sum_{N}\left[{\frac{C_{N}^{+}C_{-N}^{-}}{(Z_{R}+N)^{2}}}+{\frac{C_{N}^{-}C_{-N}^{+}}{(Z_{R}-N)^{2}}}\right]+\Pi_{R}^{2}\right]
<\Phi(d_{l},t)\overline{{\Phi}}(d_{m},t^{\prime})>_{q\approx0}=\frac{q}{M_{1}}\sum_{n=1}^{l}q^{-n}sin\left(\sqrt{q};\kappa_{n}d_{l}\right)sin\left(\sqrt{q};\kappa_{n}d_{m}\right)e^{i\kappa_{n}^{2}(t^{\prime}-t)}.
\psi_{{\bfk}s}^{\pm}=\left[-\gamma^{0}\left({\frac{d}{d\tau}}+{\frac{1}{2\tau}}\right)-i\gamma_{\bf{{\perp}}}\cdot{\bf{k_{\perp}}}-i\gamma^{3}\pi_{\eta}+m\right]\chi_{s}{\frac{f_{{\bfk}s}^{\pm}}{\sqrt\tau}}\,.
{\calR}_{abc}^{d}=\partial_{b}\Omega_{ac}^{d}-\partial_{c}\Omega_{ab}^{d}+\Omega_{0b}^{d}\Omega_{ac}^{0}+\Omega_{eb}^{d}\Omega_{ac}^{e}-\Omega_{0c}^{d}\Omega_{ab}^{0}-\Omega_{ec}^{d}\Omega_{ab}^{e}.
V_{eff}=\frac{4}{\kappa^{4}}\left[(e^{\frac{\alphaG}{2}})_{\bar{A}}G^{\bar{A}D}(e^{\frac{\alphaG}{2}})_{D}-\frac{3}{4}e^{\alphaG}\right]=\frac{e^{\alphaG}}{\kappa^{4}}\left[\alpha^{2}G^{A}G_{A}-3\right].
\{D_{d_{1}d_{2}}^{d_{4}d_{5}}C_{d_{3}d_{4}}^{d_{6}}+\mathrm{symm}(d_{5},d_{6})\}+\,C_{d_{1}d_{2}}^{d_{4}}D_{d_{3}d_{4}}^{d_{5}d_{6}}+\mathrm{cyclic}(d_{1},d_{2},d_{3})=0
{\calL}_{eff}=\frac{1}{2}\partial_{\mu}{\vec{\pi}}\partial^{\mu}{\vec{\pi}}+\frac{1}{6F^{2}}\left[({\vec{\pi}}\cdot\partial_{\mu}{\vec{\pi}})^{2}-{\vec{\pi}}^{2}(\partial_{\mu}{\vec{\pi}}\cdot\partial^{\mu}{\vec{\pi}})\right]+.....
na_{n}(q^{\prime},q)+(q^{\prime}-q)\frac{\partiala_{n}(q^{\prime},q)}{\partialq^{\prime}}=\frac{1}{2}\frac{\partial^{2}a_{n-1}(q^{\prime},q)}{\partialq^{2}}-V(q^{\prime})a_{n-1}(q^{\prime},q).
j_{M}^{\mu}=\delta_{M}\phi^{K}\cdot\frac{\partial{\calL}}{\partial\partial_{\mu}\phi^{K}}+\frac1{\left(q-1\right)!}\delta_{M}A_{\nu_{2}\ldots\nu_{q}}\cdot\frac{\partial{\calL}}{\partial\partial_{\mu}A_{\nu_{2}\ldots\nu_{q}}}\quad,
E\geq\frac{2}{g^{4}\sum_{i}\frac{N_{i}}{g_{i}^{2}}}\left(\bigl(\sum_{i}r_{i}d_{i}\bigr)^{2}+\big|\sum_{i}r_{i}f_{i}\bigr|^{2}\right)=\frac{2\bigl|\sum_{i}r_{i}\vec{\zeta}_{i}\bigr|^{2}}{g^{4}\sum_{i}\frac{N_{i}}{g_{i}^{2}}}.
\int\!\frac{\mathrm{d^{3}}k}{(2\pi)^{3}}\,\frac{1}{8(\omega_{k}^{0})^{3}}=\frac{1}{2}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{-i}{(k^{2}-m^{2}+io)^{2}}\;.
\int_{0}^{x^{+}}du_{1}f(u_{1})\int_{0}^{u_{1}}du_{2}f(u_{2})\cdots\int_{0}^{u_{n-1}}du_{n}f(u_{n})={\frac{1}{n!}}\Biggl[\int_{0}^{x^{+}}dyf(y)\Biggr]^{n}\;,
\int_{-\frac{1}{2}}^{+\frac{1}{2}}d\tau_{1}\,\overline{{\rho}}=\frac{90}{\pi\tau_{2}}\,\\,int_{1}^{\infty}\frac{dx}{x^{2}}\vartheta_{3}^{2}\left(i\frac{x}{R^{2}}\right)=\frac{R^{2}}{2}+\frac{\kappa}{R^{2}}+{\calO}\left(e^{-\piR^{2}}\right)
\prod_{i=3}^{N-1}(\intd\tau_{i})\langle{\calO}(z_{1}){\calO}(z_{2})\ointb(z_{1}^{\prime})dz_{1}^{\prime}{\calO}(z_{3})\ointb(z_{2}^{\prime})dz_{2}^{\prime}\cdots{\calO}(z_{N})\rangle_{R},
[T_{\alpha},T_{\beta}^{\dagger}]=\bar{U}_{\alpha\beta}^{\gamma}T_{\gamma}+T_{\gamma}^{\dagger}\bar{U}_{\beta\alpha}^{\dagger\gamma}+\bar{U}_{\alpha\delta}^{\gamma}\bar{U}_{\beta\gamma}^{\dagger\delta}(-1)^{\varepsilon_{\gamma}\varepsilon_{\delta}}.
\mp\left[-e^{3A}\dot{\Psi}^{(0)}F_{ww}+2e^{A}\dot{\varphi}+2e^{2A}(e^{A}\dot{\Psi}^{(0)})^{\cdot}\phi_{\pmw}\right]={V_{\pm}^{\prime\prime}}^{(0)}(\varphi+e^{2A}\dot{\Psi}^{(0)}\phi_{\pmw})+2{\alpha_{\pm}^{\prime}}^{(0)}e^{2A}\bar{\tau}_{\pm(Y)},
\Big(\Psi_{k}^{a\tilde{a}},(\Psi\tilde{\Phi})_{k}^{a},(\Psi\tilde{\Pi})_{k}^{a},(\Phi\tilde{\Psi})_{k}^{\tilde{a}},(\Pi\tilde{\Psi})_{k}^{\tilde{a}},(\Phi\tilde{\Phi})_{k},(\Phi\tilde{\Pi})_{k},(\Pi\tilde{\Phi})_{k},(\Pi\tilde{\Pi})_{k}\Big).
\tau_{+}^{2}(r)\sim2(r_{+}-r)\left({\frac{r_{+}^{2}-|\Upsilon|^{2}}{r_{0}}}\right),\quad\tau_{-}^{2}(r)\sim2(r-r_{-})\left(\frac{r_{-}^{2}-|\Upsilon|^{2}}{r_{0}}\right).
{\calS}_{\mathrm{ghost}}=\int\mathrm{d}^{4}x~\Big[(\partial^{\mu}\overline{{C}}{}^{\nu})~\partial_{\mu}C_{\nu}+(\partial^{\mu}\overline{{D}})~\partial_{\mu}D+(\partial^{\mu}\overline{{E}})~\partial_{\mu}E\Big].
\frac{\deltaK_{m}\left(\xi_{1}^{\prime},\cdots,\xi_{m}^{\prime}\right)}{\deltaK_{n}\left(\xi_{1},\cdots,\xi_{n}\right)}=\delta_{m,n}\frac{1}{n}\left(2\pii\right)^{n}\sum_{c}\delta\left(\xi_{1}-{\xi^{\prime}}_{c(1)}\right)\cdots\delta\left(\xi_{n}-{\xi^{\prime}}_{c(n)}\right).
\mu^{2}\phi(z)=\left(m^{2}-\frac{g^{2}N}{2\pi}\right)\left(\frac1z+\frac{1}{1-z}\right)\phi(z)-\frac{Ng^{2}}{2\pi}\int_{0}^{1}dy\frac{{\calP}}{(y-z)^{2}}\phi(y),
H_{0}\left(q_{n,\alpha_{n}},p_{n,\alpha_{n}};\mu_{n,i_{n}}\right)=\overline{{H}}_{0}\left(q_{n,\alpha_{n}},p_{n,m_{n}-1}\right)-\sum_{n}\sum_{i_{n}=1}^{m_{n}-1}\mu_{n,i_{n}}q_{n,i_{n}}\.
\langle\partialX^{\mu}(z)\bar{\partial}X^{\nu}(\bar{w})\rangle=-\delta^{\mu\nu}\left[\pi\delta^{(2)}(z-w)-{\frac{\pi}{2}}\,\omega(z)\cdot(\mathrm{Im}\tau)^{-1}\cdot\bar{\omega}(\bar{w})\right],
{\calF}_{\epsilon}^{i}(\vec{x},t)=\frac{1}{2\pi}\intd^{2}\vec{p}\;\mathrm{e}^{i\vec{p}\vec{x}}\,\tilde{\calF}_{\epsilon}^{i}(\vec{p},t),\qquad\overline{{\tilde{\calF}}}{}_{\epsilon}^{i}(\vec{p},t)=\tilde{\calF}_{\epsilon}^{i}(-\vec{p},t),
J_{\nu-1}^{2}(ix)-e^{2\theta}J_{\nu}^{2}(ix)={J_{\nu}^{\prime}}^{2}(ix)-\left({\frac{\nu^{2}}{x^{2}}}+e^{2\theta}\right)J_{\nu}^{2}(ix)+{\frac{2\nu}{ix}}J_{\nu}(ix)J_{\nu}^{\prime}(ix).
V_{\left\{m\right\},\left\{m^{\prime}\right\}}^{n}=\sum_{k_{1},\ldotsk_{n}}^{\prime}\sum_{j_{1},\ldotsj_{n},}V_{\left\{m\right\},\left\{m^{\prime}\right\}}^{n}\left(k_{1},\ldotsk_{n}\right)\prod_{i=1}^{n}\left(a_{k_{i}}^{+}\left(j_{i}\right)\right)^{m_{i}}\left(a_{k_{i}}\left(j_{i}\right)\right)^{m_{i}^{\prime}},
<{\phi}|{\psi}>=-i{\int}d^{d}x{\int}_{0}^{\infty}d{p_{+}}{\sum}_{{\sigma}{\sigma^{\prime}}={\pm}}(-1)^{(1+{\sigma}^{\prime})/2}{\phi*}_{{\sigma}{\sigma^{\prime}}}(x^{{\mu}},p_{+},{\tau}){\psi}_{{-\sigma}{-\sigma^{\prime}}}(x^{{\mu}},p_{+},{\tau}).
P_{\mu}^{GG}=\alpha^{2}\left(e^{2\gamma\psi}N(K^{\prime})^{2}+\frac{1}{2}Nx^{2}(H^{\prime})^{2}+\frac{1}{2x^{2}}(K^{2}-1)^{2}e^{2\gamma\psi}+K^{2}H^{2}+\frac{1}{2}Nx^{2}(\psi^{\prime})^{2}\right)\,
_\mp\langle\{\lambda\},\pmM+{\tilde{q}}|\prod_{j=1}^{M}\Phi^{\pm}(z_{j})|{\tilde{q}}\rangle=\prod_{i<j}(z_{i}-z_{j})^{1/\beta}\prod_{j}^{M}(z_{j})^{\pm{\tilde{q}}/\beta}(-1)^{|\lambda|}J_{\{\lambda^{\prime}\}}^{(1/\beta)}(z_{j})\(j_{\lambda}^{\beta})^{1/2}
\chi_{1/2}^{1,2}|\Omega{\rangle}_{L}\otimes{\Psi}_{1/2}^{a}{\Psi}_{1/2}^{i*};\,\,{\Psi}_{1/2}^{a*}{\Psi}_{1/2}^{i}|\Omega{\rangle}_{R}\\\mathrm{and~exchange}\\Psi\longrightarrow\Phi,
\wp^{-1}=\sigma{_{2}F_{1}}\left({\frac{1}{2}},\,{\frac{1}{3}};\,{\frac{4}{3}};\,-4\sigma^{3}\right)-{\frac{4^{1/3}}{3}}B\left({\frac{1}{3}},\,{\frac{1}{6}}\right)
T^{MN}(x)=\sigma\,\eta^{\mu\nu}\,\delta_{\mu}^{\phantom{\mu}M}\,\delta_{\nu}^{\phantom{\nu}N}\,\delta(\eta)+\frac{3\,\Lambda\,L_{p}^{(3-d)}}{2\,\pi\,d\,(d-1)}\,g^{MN}\,[\Theta(\eta)+\Theta(-\eta)],
j_{q1}(\tilde{x},t)=\partial_{\tilde{x}}\mathcal{J}_{q1},\qquad\mathcal{J}_{q1}=-\frac{i}{12\mu}\bigl\{\psi^{*}(\tilde{x},t)\tilde{x}\partial_{\tilde{x}}\psi(\tilde{x},t)-\bigl[\partial_{\tilde{x}}\psi^{*}(\tilde{x},t)\bigr]\tilde{x}\psi(\tilde{x},t)\bigr\}.
\left(Q_{2}^{\dagger}\right)^{31}u_{B}^{+}({\bfr})\propto\sum_{n=0}^{\infty}{\frac{(-1)^{n}}{(2n)!!(2n-2\left(2\ell+1-\lambda\right))!!}}\left(\sqrt{E}r\right)^{2n-\left(2\ell+1-\lambda\right)}e^{-i\left(2\ell+1\right)\theta}
\begin{array}{ll}{V}&{=F_{\sigma}\bar{F}_{\sigma}+F_{A}\bar{F}_{A}+F_{\phi}\bar{F}_{\phi}}\\{}&{=4k^{2}\phi\bar{\phi}(\sigma_{3}\bar{\sigma}_{3}+\pi_{A}\bar{\pi}_{A})}\\{}&{+k^{2}[f_{\pi}^{2}-\sigma_{3}^{2}-\pi_{A}\pi_{A}][f_{\pi}^{2}-\bar{\sigma}_{3}^{2}-\bar{\pi}_{A}\bar{\pi}_{A}].}\\\end{array}
q^{-\beta}\frac{\prod_{k=1}^{2l}\phi(u+(k-l-\frac12)\eta)}{Q(u-\eta/2)Q(u+\eta/2)}=q^{\frac{2\beta}{\eta}u}\left(\frac{P(u+\eta/2)}{Q(u+\eta/2)}-\frac{P(u-\eta/2)}{Q(u-\eta/2)}\right).
S_{F}[\overline{{\psi}},\psi,A,h]=\sum_{b=1}^{N}\intd^{2}x\;\overline{{\psi}}^{(b)}(x)\gamma_{\mu}\Big(\partial_{\mu}-ieA_{\mu}(x)-i\sqrt{g}h_{\mu}(x)\Big)\psi^{(b)}(x)\;.
{\delta}A_{i}^{\gamma}({\bfr})=\left({\delta}_{i\,{\gamma}}-\frac{r_{i}\,r_{\gamma}}{r^{2}}\right)\frac{{\delta}{\calN}}{gr}+\frac{r_{i}\,r_{\gamma}}{gr^{2}}\frac{d\,({\delta}{\calN})}{dr}+{\delta}_{{i}\,{\gamma}}\,{\delta}{\calT}_{A}(r)+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,{\delta}{\calT}_{B}(r)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\delta}{\calT}_{C}(r)\,.
\deltah_{\mu\nu}=\frac{1}{2}\left(\bar{\epsilon}\gamma_{\mu}\Psi_{\nu}+\bar{\epsilon}\gamma_{\nu}\Psi_{\mu}\right),~~~~~\delta\Psi_{\mu}=\frac{1}{4}\hat{\omega}_{\mu}^{~~mn}\gamma_{mn}+\alphaM\,h_{\mu\nu}\gamma^{\nu}\epsilon+\betaM\,h\gamma_{\mu}\epsilon
F_{\eta_{0}\alpha_{0}}^{\delta_{0}}F_{\beta_{0}\gamma_{0}}^{\eta_{0}}-R_{\alpha_{0}}^{i}F_{\beta_{0}\gamma_{0},i}^{\delta_{0}}+\mathrm{cyclic~perm}(\alpha_{0},\beta_{0},\gamma_{0})=3Z_{\alpha_{1}}^{\delta_{0}}H_{\alpha_{0}\beta_{0}\gamma_{0}}^{\alpha_{1}},
<\!{{T}_{a}}^{b}(\gamma)\!>_{\mathrm{(FT)}}={\frac{{\rho}^{\gamma}}{3}}\left[4u_{a}u^{b}+{{\delta}_{a}}^{b}\right]\,\quad\mathrm{where}\quad\rho^{\gamma}={\frac{\pi^{2}N^{2}}{2{{\beta}_{\gamma}}^{4}}}\,
H=\intd^{3}x\left[{\textstyle{\frac{1}{2}}}E_{i}^{a}E_{i}^{a}+{\textstyle{\frac{1}{2}}}B_{i}^{a}B_{i}^{a}+\dot{\Phi}^{\dagger}\dot{\Phi}+(D_{i}\Phi)^{\dagger}(D_{i}\Phi)+\lambda(\Phi^{\dagger}\Phi)^{2}\right]\,.
\partial_{1}\left(\frac{h_{2}h_{3}}{h_{1}}\right)=\partial_{2}\left(\frac{A_{1}h_{2}h_{3}}{h_{1}}\right)~,~~~~\partial_{3}\left(\frac{h_{2}h_{1}}{h_{3}}\right)=\partial_{2}\left(\frac{A_{3}h_{2}h_{1}}{h_{3}}\right)~.
S_{N}^{(1)}(x,y)=K_{N}(x,y)-\psi_{1}(x)\sum_{l=1}^{4}[AC(I-BAC)^{-1}]_{l1}\\varepsilon\psi_{l}(y)-\psi_{2}(x)\sum_{k=1}^{4}[AC(I-BAC)^{-1}]_{k2}\\varepsilon\psi_{k}(y),
{\renewcommand{\arraystretch}{1.4}\begin{array}{rl}{(x-q^{4})(x+q^{6})\lambda^{3}\;+\;q^{6}(2+q^{2})(\lambda^{2}+x^{2}\lambda)}&{}\\{+\;x\,(q^{2}-1)(1-3q^{4}+q^{8})(\lambda^{2}+\lambda)}&{}\\{-\;q^{2}(1+2q^{2})(x^{2}\lambda^{2}+\lambda)\;+\;(1-q^{4}x)(1+q^{6}x)}&{=0\;.}\\\end{array}}
\begin{array}{rll}{\partial_{+}s_{0}s_{0}^{-1}}&{=}&{[t,A_{+}]\,}\\{\partial_{-}s_{0}s_{0}^{-1}}&{=}&{{\textstyle{\frac12}}(s_{0}Bs_{0}^{-1}-A_{-})\,}\\\end{array}
{\calL}=-\frac{1}{4g^{2}}F_{\mu\nu}^{k}F_{\mu\nu}^{k}+\frac{1}{g^{\prime}}F_{\mu\nu}^{k}\tilde{F}_{\mu\nu}^{k}+\frac{2}{g^{\prime}}\left(D_{\mu}\cal{B}_{\mu}\right)^{k}\!\left(D_{\nu}\cal{A}_{\nu}\right)^{k}+{\calL}_{FP}\,,
{\calS}_{H}=\intd^{4}x~\sum_{n=0}^{\infty}~[\eta^{\mu\alpha}\eta^{\nu\beta}\eta^{\lambda\gamma}H_{\mu\nu\lambda}^{n}H_{\alpha\beta\gamma}^{n}+3m_{n}^{2}\eta^{\mu\alpha}\eta^{\nu\beta}B_{\mu\nu}^{n}B_{\alpha\beta}^{n}]
\frac{1}{2}\epsilon^{ij}(\partial_{i}{\calA}_{j}^{a}-\partial_{j}{\calA}_{i}^{a}+f^{abc}{\calA}_{i}^{b}{\calA}_{j}^{c})=-\frac{i}{\kappa}\sum_{p=1}^{N-1}T_{(p)}^{a}\delta^{2}({\bfr}-{\bfr}_{p}).
\alpha_{AB}=\left(\begin{array}{cccccc}{2}&{-1}&{0}&{1}&{-1}&{1}\\{-1}&{2}&{-1}&{1}&{1}&{0}\\{0}&{-1}&{2}&{-1}&{1}&{1}\\{1}&{1}&{-1}&{2}&{0}&{1}\\{-1}&{1}&{1}&{0}&{2}&{1}\\{1}&{0}&{1}&{1}&{1}&{2}\\\end{array}\right)
(F,G)^{a}=(-1)^{\epsilon_{A}}\intd^{2}\theta\,\Bigr\{\frac{\partial^{2}\deltaF}{\partial\theta^{2}\delta\Phi^{A}(\theta)}\theta^{a}\frac{\deltaG}{\delta\bar{\Phi}_{A}(\theta)}-(-1)^{(\epsilon(F)+1)(\epsilon(G)+1)}(F\leftrightarrowG)\Bigr\}.
Z_{AB}=-\frac{1}{16}\,\left(q_{1}\,\left[\frac{H_{2}(x)}{H_{1}^{3}(x)}\right]^{1/4}\,\tau_{AB}^{+}\,+\,q_{2}\,\left[\frac{H_{1}(x)}{H_{2}^{3}(x)}\right]^{1/4}\,\tau_{AB}^{-}\right)\.
{\calR}=\sum_{p^{\prime}}\frac{1}{2m_{p^{\prime}}c^{2}}\left(\sum_{I}\frac{q_{p^{\prime}}^{I}q_{p}^{I}}{\kappa_{I}}\right)^{2}\int\!d^{2}\!{\bfr}^{\prime}\,\left(\frac{1}{4\pi^{2}}\frac{1}{|{\bfr}-{\bfr}^{\prime}|^{2}}\right)\rho_{p^{\prime}}({\bfr}^{\prime},t)\Psi_{p}({\bfr},t),
\begin{array}{rcl}{a^{\pm}(\zeta)+d^{\pm}(\zeta)}&{=}&{\rho(\zeta^{\pm2})^{\pm1}\displaystyle\frac{\bar{\alpha}(\zeta^{-1})}{\bar{\alpha}(\zeta)}~,}\\{b^{\pm}(\zeta)+c^{\pm}(\zeta)}&{=}&{\rho(\zeta^{\pm2})^{\pm1}q^{\pm1}\displaystyle\frac{1+(q^{-1}\zeta)^{\pm1}}{1+(q\zeta)^{\pm1}}\displaystyle\frac{\bar{\beta}(\zeta^{-1})}{\bar{\beta}(\zeta)}~.}\\\end{array}
\gamma^{0}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\,\,\gamma^{i}=\left(\begin{matrix}{0}&{\sigma^{i}}\\{-\sigma^{i}}&{0}\\\end{matrix}\right)\,\,,\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{matrix}{-1}&{0}\\{0}&{1}\\\end{matrix}\right)\.
S=\int_{M}E^{\mu\nu\alpha\beta}\left(c_{1}\,{^+G}_{\mu\nu}^{ab}\,{^+G}_{\alpha\beta}^{cd}+c_{2}\,{^-G}_{\mu\nu}^{ab}\,{^-G}_{\alpha\beta}^{cd}+c_{3}\,{^+{\calR}}_{\mu\nu}^{ab}\,{^+G}_{\alpha\beta}^{cd}+c_{4}\,{^-{\calR}}_{\mu\nu}^{ab}\,{^-G}_{\alpha\beta}^{cd}\right)\epsilon_{abcd}
\left(\begin{array}{cc}{t}&{0}\\{0}&{t^{-1}}\\\end{array}\right),\;\left(\begin{array}{cc}{-t}&{0}\\{0}&{-t^{-1}}\\\end{array}\right),\;\left(\begin{array}{cc}{0}&{t}\\{t^{-1}}&{0}\\\end{array}\right),\;\left(\begin{array}{cc}{0}&{-t}\\{-t^{-1}}&{0}\\\end{array}\right),
m_{0}^{2}={\frac{4\kappa}{3}}\nu_{2}^{2}\varphi_{(-)}^{2}e^{-2d/\ell}\left[{\frac{\sqrt{1+(M^{2}\ell^{2}/4)}}{1-e^{-4\sqrt{1+(M^{2}\ell^{2}/4)}d/\ell}}}\right].
\sum_{N=1}^{h_{k}+h_{l}-1}\left(\sum_{q,q^{\prime}}C^{ijq}C^{klq^{\prime}}D_{qq^{\prime}}\right)\,\,p_{h_{i}}^{h_{j}h_{q}}(m,n+p)p_{h_{q}}^{h_{k}h_{l}}(n,p)+\mathrm{cyclic}=0.
f^{\mathrm{{i}}}(r)=br^{2}-\textstyle{\frac{3+20b}{3+12b}}b^{2}r^{4}+\textstyle{\frac{9+153b+618b^{2}-464b^{3}-3584b^{4}}{9(1-2b)(1+4b)^{3}}}b^{3}r^{6}+\cdots
H^{2}=\frac{\rho_{m}^{2}}{36}+\frac{1}{12}U_{B}\rho_{m}-\frac{1}{16a^{4}}\intd\tau\frac{da^{4}}{d\tau}(\dot{\phi}^{2}-2V)-\frac{1}{12a^{4}}\intd\taua^{4}\rho_{m}\frac{dU_{B}}{d\tau}.
\left|\sum_{\alpha\beta\gamma}\langle\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\rangle_{c}\left.\right/\sum_{\alpha\beta\gamma}\langle\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}\rangle\right|=\left|\langle\Phi^{3}(x)\rangle_{c}\left.\right/\langle\Phi^{3}(x)\rangle\right|\;(n=3)
T_{0i}=-\frac{gM}{m}\gamma^{3}\varepsilon_{ijk}S_{j}\left[\frac{3(\vec{a}\vec{n})a_{k}}{1-(\vec{n}\vec{v})}+\frac{3v_{k}(\vec{a}\vec{n})^{2}}{(1-(\vec{n}\vec{v}))^{2}}-\frac{\omega^{2}{v}_{k}}{1-(\vec{n}\vec{v})}\right],
F(\Phi,\tilde{\Phi}^{*})=\Phi^{A}\tilde{\Phi}_{A}^{*}-\tilde{\bar{b}}{}_{\alpha}^{*\dot{\alpha}}\bar{D}_{\dot{\alpha}}\frac{1}{\Box}i\partial^{\alpha\dot{\beta}}\tilde{\bar{\sigma}}_{\dot{\beta}}^{*}-\tilde{\sigma}_{\beta}^{*}\frac{1}{\Box}i\partial^{\beta\dot{\alpha}}D_{\alpha}\tilde{b}_{\dot{\alpha}}^{*\alpha}\.
{\calL}_{(2+1)}=\frac1{4\bar{\varkappa}}\epsilon^{\alpha\beta\gamma}\hat{A}_{\alpha}\hat{F}_{\beta\gamma}+i\hbar\Psi^{*}(\partial_{t}+i\hat{A}_{0})\Psi-\frac{\hbar^{2}}{2m}\sum_{i=1}^{2}\left|(\partial_{i}+i\hat{A}_{i})\Psi\right|^{2}-V(\Psi^{*}\Psi)\,
{\mathcal{K}}(\hat{M}_{f},\hat{M}_{b})=\left\langle\frac{\prod_{k=1}^{n_{f}}{\mathcal{Z}}\left(\hat{J},m_{f}^{(k)}\right)}{\prod_{l=1}^{n_{b}}{\mathcal{Z}}\left(\hat{J},m_{b}^{(l)}\right)}\right\rangle_{J}
{\calH}=\frac{1}{2}\,\mathrm{e}^{-2k}\left(\frac{1}{8}\,p_{k}^{2}-\frac{1}{4}\,p_{h}^{2}-\mathrm{e}^{2h}\,p_{a}^{2}\right)+\mathrm{e}^{2k}\,{\calV}=0~~.
S_{c}=\int_{z=z_{c}}d^{4}x\sqrt{\bar{g}}e^{-4kz_{c}}\left[e^{2kz_{c}}\bar{g}^{\mu\nu}\nabla_{\mu}H^{\dagger}\nabla_{\nu}H-\lambda\left(|H|^{2}-m_{0}^{2}\right)^{2}\right]
4\pi\intd\rho\rho^{2}\Theta^{\mu\nu}K_{\nu}=-\frac{K^{\mu}}{2}\{\frac{1+K^{2}}{\rho}+2K^{2}\mathrm{\Large~a}.K\;ln\rho+K^{2}(\mathrm{\Large~a}^{2}+{\mathrm{\Large~a}_{K}}^{2})\rho\}.
V_{\mathrmeff}(B,m)=\frac{B^{2}}{2}-\frac{eB}{2\pi}\sum_{n=1}^{\infty}\sqrt{2neB+m^{2}}+\frac{m^{2}eB}{2\pi}\sum_{n=1}^{\infty}\frac{1}{\sqrt{2neB+m^{2}}}.