harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0=\Im\left\{5\,\mathrm{Li}_{2}(\lambda)-6\,\mathrm{Li}_{2}\left(\lambda/2\right)+12\,\mathrm{Li}_{2}\left(i/2\right)-16\,\mathrm{Li}_{2}\left(\frac{1+i}{2}\right)+8\,\mathrm{Li}_{2}\left(\frac{1+i}{4}\right)\right\}
{\tilde{\cal{E}}}_{m>0}={\cal{E}}_{m>0}(B)-{\cal{E}}(0)=\frac{B^{2}}{2}+\frac{(eB)^{\frac{3}{2}}}{2\pi}g\left(\frac{eB}{m^{2}}\right)+\frac{eB}{2\pi}|m|\;.
H_{SB}(\ell)=\left(\begin{array}{cccc}{h_{L}}&{h_{L\bullet}}&{h_{L\circ}}&{h_{LH}}\\{h_{L\bullet}}&{h_{\bullet}}&{h_{\bullet\circ}}&{h_{H\bullet}}\\{h_{L\circ}}&{h_{\bullet\circ}}&{h_{\circ}}&{h_{H\circ}}\\{h_{LH}}&{h_{H\bullet}}&{h_{H\circ}}&{h_{H}}\\\end{array}\right)
\begin{array}{ll}{{[}J_{i},J_{j}]=i\epsilon_{ij}^{k}J_{k}}&{(i,j,k=1,2,3)}\\{{[}J_{i},S_{j}]=i\epsilon_{ij}^{k}S_{k}}&{}\\{{[}S_{i},S_{j}]=-i\epsilon_{ij}^{k}J_{k}(2\vec{J}^{2}-C_{2}-4)}&{}\\{{[}C_{2},J_{i}]=[C_{2},S_{i}]=0}&{\mathrm{with~}\vec{J}^{2}=J_{1}^{2}+J_{2}^{2}+J_{3}^{2}}\\\end{array}
\begin{array}{rcl}{{\calK}^{(1)}}&{=}&{\partial\omega^{(0)}-\frac{1}{2\pi\alpha^{\prime}}(i_{k}B^{(2)})\,,}\\{}&{}&{}\\{{\calK}^{(2)}}&{=}&{2\partial\omega^{(1)}+\frac{1}{2\pi\alpha^{\prime}}(i_{k}C^{(3)})-2{\calK}^{(1)}(DXC^{(1)})\,,}\\\end{array}
\sum_{r=0}^{k-1}\frac{(-1)^{r}\Gamma(\mu)}{2^{r+\lambda}r!\Gamma(\mu-r)(r+\lambda)}+\sum_{r=0}^{s-1}\frac{(-1)^{r}\Gamma(\lambda)}{2^{r+\mu}r!\Gamma(\lambda-r)(r+\mu)}
\frac{d^{2}\tilde{F}_{0}}{d\tilde{z}^{2}}+\tilde{z}\frac{d\tilde{F}_{0}}{d\tilde{z}}+(1-\frac{1}{4\tilde{z}^{2}}+\frac{\mu^{2}}{\tilde{z}^{4}})\tilde{F}_{0}=0
G_{\mu\nu}^{i}\,\equiv\,g^{i{{j}^{\star}}}\,{\bar{f}}_{j^{\star}}^{I}\,\mathrm{Im}{\calN}_{IJ}F_{\mu\nu}^{J}\quad;\quad{\bar{f}}_{j^{\star}}^{I}\equiv\nabla_{j^{\star}}{\bar{Y}}^{I}
{\vec{r}}={\bar{z}}^{\alpha}{\vec{\sigma}}_{\alpha\beta}z^{\beta},\quad{\hat{\vec{p}}}=\frac{i\hbar}{2(z\bar{z})}(z^{\alpha}{\hat{\sigma}}_{\beta\alpha}\frac{\partial}{\partialz^{\beta}}+{\bar{z}}^{\alpha}{\hat{\sigma}}_{\alpha\beta}\frac{\partial}{\partial{\bar{z}}^{\beta}}),
ds^{2}=-{\frac{(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})}{C^{2d}}}dt^{2}+{\frac{r^{2d-2}C^{2}}{(r^{d}-r_{1}^{d})(r^{d}-r_{2}^{d})}}dr^{2}+C^{2}d\Omega_{D-2}^{2}
\frac{(\eta^{\omega}+\eta^{-\omega})^{3/2}}{1-\eta^{2}}\left(\frac{3\omega}2\frac{\eta^{\omega}-\eta^{-\omega}}{\eta^{\omega}+\eta^{-\omega}}+\frac{1+\eta^{2}}{1-\eta^{2}}\right)=0.
ds^{2}=\frac{r^{2}}{l^{2}}\left[-\left(1-\frac{r_{0}^{4}}{r^{4}}\right)\dt^{2}+dx_{1}^{2}+dx_{2}^{2}+d\theta^{2}\right]+\frac{l^{2}}{r^{2}}\left(1-\frac{r_{0}^{4}}{r^{4}}\right)^{-1}dr^{2}
\hbar^{2}{\frac{1}{K}}{\frac{\partial^{2}K}{\partialt^{2}}}-\left({\frac{\partial^{2}S}{\partialt^{2}}}\right)(\Omega+i\eta\hbar+S^{2})+(\mu^{2}+\phi^{2})\pmi\hbar\phi^{\prime}=0
Q_{0}=-\sum_{\kappa=-\infty}^{\infty}\vert\kappa\vert\left\{N_{\kappa}^{+}-N_{\kappa}^{-}+\int_{0+}^{\infty}\frac{\mathrm{d}k}{2\pi}\int_{0}^{\infty}\mathrm{d}r\left[u_{1\varepsilon\kappa}^{2}\left(r\right)+u_{2\varepsilon\kappa}^{2}\left(r\right)-u_{1,-\varepsilon\kappa}^{2}\left(r\right)-u_{2,-\varepsilon\kappa}^{2}\left(r\right)\right]\right\}\,.
n_{k}=\frac{1}{2}\,\langle\,\frac{p_{k}^{2}}{\omega_{k}}+\omega_{k}q_{k}^{2}\,\rangle\approx\frac{{\calB}_{k}}{\omega_{k}}+\frac{\omega_{k}}{16{\calA}_{k}}-\frac{1}{2}\,,
\left|\sum_{\alpha\beta}\langle\Phi_{\alpha}\Phi_{\beta}\rangle_{c}\left.\right/\sum_{\alpha\beta}\langle\Phi_{\alpha}\Phi_{\beta}\rangle\right|=\left|\langle\Phi^{2}(x)\rangle_{c}\left.\right/\langle\Phi^{2}(x)\rangle\right|\;(n=2)\;,
H_{I}=\intd^{2}z\left[-ie\pi_{\psi}\cdot\gamma^{0}\gamma^{j}A_{T}^{j}\psi\,+\,ie\theta\pi_{\psi}\cdot\psi\epsilon^{ij}\frac{\partial^{i}}{\nabla^{2}}A^{j}\,+\,\frac{e^{2}}{2}(\pi_{\psi}\cdot\psi)\frac{1}{\nabla^{2}}(\pi_{\psi}\cdot\psi)\right]\,.
\psi_{12}(p_{30})=\,{\frac{-1}{2i\pi}}\,{\frac{\Lambda^{+++}}{(P_{0}-p_{30})-(S_{12}+\Lambda_{12}^{++}V_{12}\Lambda_{12}^{++})+i\epsilon}}\,\left[\,V_{23}\,+\,V_{31}\,\right]\,\Lambda^{+++}\,\psi
<A_{1}|n>=\left(\frac{k\tau_{2}}{2\pi^{2}}\right)^{1/4}\,\frac{1}{\sqrt{n!}}\,\left(\frac{i\bar{\tau}}{2\tau_{2}}\right)^{n/2}\,\left[u-\frac{d}{du}\right]^{n}\,e^{\frac{1}{2}\frac{\tau}{\bar{\tau}}u^{2}}\\\,\\\u=\sqrt{\frac{ik\bar{\tau}}{2\pi}}\,A_{1}\\\.
a_{4,4}=ra_{4,4}^{\mathrm{\tiny{scal}}}+{\frac{1}{4!}}\int_{\Sigma_{r}}\bar{h}^{1/2}d^{3}x\left[{\frac{1}{16}}\mathrm{Tr}(\bar{\gamma}^{i}\bar{\gamma}^{j}\bar{F}_{ij})^{4}-{\frac{r}{2}}(\bar{F}^{ij}\bar{F}_{ij})^{2}\right]~~~,
{\tilde{Z}}_{m}^{T}~=~\int\prod_{\mu;<{ij}>}dP_{ij}^{\mu}e^{-\sum_{\mu;<{ij}>}(P_{ij}^{\mu})^{2}}Det\left(-\sum_{\mu}(P^{\mu})^{2}\right)\prod_{\mu;i}\delta(\sum_{j}P_{ij}^{\mu})~.
|\Phi_{\vec{x},\vec{y}}\rangle\;=\;\sum_{i,\alpha,\beta}\psi_{i,\alpha}(\vec{x})\;U_{\alpha\beta}\Big({\calC}_{\vec{x},\vec{y}}\Big)\;\overline{{\psi}}_{i,\beta}(\vec{y})\;|0\rangle\;.
\frac{1}{2}\nabla^{R}h^{MN}\nabla_{M}h_{RN}=\frac{1}{2}\nabla_{M}h^{MN}\nabla^{R}h_{RN}+\frac{1}{2}h^{MN}h^{PQ}R_{MNPQ}-\frac{1}{2}h^{MN}h_{NP}R_{M}^{P},
\left(\hat{\Gamma}_{0}\hat{\Gamma}_{i}\hat{\Gamma}_{j}\right)^{\dagger}=\hat{\Gamma}_{j}^{\dagger}\hat{\Gamma}_{i}^{\dagger}\hat{\Gamma}_{0}^{\dagger}=\hat{\Gamma}_{j}\hat{\Gamma}_{i}(-\hat{\Gamma}_{0})=\hat{\Gamma}_{0}\hat{\Gamma}_{i}\hat{\Gamma}_{j}
I_{N}(s)={\frac{\sqrt\pi\beta_{N}}{2\Gamma(s)}}\sum_{n=1}^{\rho_{N}}{\frac{(2n-1)!!}{2^{n}}}c_{2n}^{N}(a^{2}b)^{n+\frac12-s}\Gamma\left(s-n-\frac12\right),
fromQw3a=\frac{2\left[6\!-\!2wNB\!-\!12Q^{2}B\!+\!3(w\!-\!2)NQ^{2}B^{2}\!+\!(w\!+\!2)N\Lambda\right](6\!-\!wNB)B}{\left[36\!-\!12(w\!+\!1)NB\!+\!w^{2}N^{2}B^{2}\right](\Lambda\!-\!Q^{2}B^{2})}.
Z_{K}(V)=\frac{C^{K}}{K!}\int\prod_{i=1}^{K}d{\bfa}_{i}d\rho_{i}\prod_{j=1}^{K}\rho_{j}^{b-d-1}\mathrm{e}^{-U(\{{\bfa}_{k}\},\{\rho_{k}\})}\,,
\Gamma^{\alpha}{}_{\!\mu\beta}=\big[\big\{{}^{\alpha}{}_{\!\mu\beta}\big\}+\frac{1}{6}(\delta^{\alpha}{}_{\!\beta}T_{\mu}-T^{\alpha}g_{\mu\beta})\big]+\big[\frac{1}{6}(\delta^{\alpha}{}_{\!\beta}T_{\mu}+T^{\alpha}g_{\mu\beta})\big]\;.
\Gamma[\phi^{2}]=\intd^{D}x\left\{V(\phi^{2},t)+\frac{1}{2}K(\phi^{2},t)\left(\partial_{\mu}\phi^{a}\right)^{2}+\frac{1}{2}Z(\phi^{2},t)\left(\phi^{a}\partial_{\mu}\phi^{a}\right)^{2}\right\}
i\Pi_{odd}^{\mu\nu}(p)=-\frac{mg^{2}}{2\pi^{\frac{3}{2}}}p^{\alpha}\epsilon^{\mu\nu\alpha}\intdx\int\frac{d\alpha}{\sqrt{\alpha}}~[e^{-\alphaM^{2}}-e^{-\alphaM^{2}-\frac{\tilde{p}^{2}}{4\alpha}}]~~~.
L=\intd^{3}{x}\left[-\frac14(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})(\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}V_{\mu}-M)\psi\right]\;.
\left(\omega_{\alpha}^{g_{1},g_{2},g_{3}g_{4}}\right)\,\left(\omega_{\alpha}^{g_{1}g_{2},g_{3},g_{4}}\right)\:=\:\left(\omega_{\alpha}^{g_{1},g_{2}g_{3},g_{4}}\right)\,\left(\omega_{\alpha}^{g_{2},g_{3},g_{4}}\right)\,\left(g_{4}^{*}\omega_{\alpha}^{g_{1},g_{2},g_{3}}\right)
\begin{array}{rclrcl}{\left[L_{1},L_{-2}\right]}&{=}&{L_{-1},}&{\quad\left[L_{1},L_{2}\right]}&{=}&{0,}\\{\left[L_{2},L_{-2}\right]}&{=}&{N+\frac{D+1}2,}&{\quad\left[L_{0},L_{n}\right]}&{=}&{0,}\\{\left[L_{1},L_{-1}\right]}&{=}&{L_{0},}&{\quad\left[N,L_{n}\right]}&{=}&{{}-nL_{n},\quadn=0,\pm1,\pm2.}\\\end{array}
V_{ang}=V_{+}\:\left[\bfr\times\left(v_{\mathrm{1}}-v_{\mathrm{2}}\right)\right]^{\mathrm{2}}/4\mathrm{r^{2}\;+\;V_{-}\:\left[\bfr\times\left(v_{\mathrm{1}}+v_{\mathrm{2}}\right)\right]^{\mathrm{2}}/4\mathrm{r^{2},}}
S=\frac{1}{2\kappa_{n+2}^{2}}\int{d^{n+2}x\,\sqrt{g_{n+2}}\,\left(R-\frac{1}{2}(\nabla\phi)^{2}-2\kappa_{n+2}^{2}e^{-\sigma\phi}F_{\mu\nu}F^{\mu\nu}\right)}+S_{\mathrm{br}}+S_{\mathrm{G.H.}}\,,
Q=\left(\begin{array}{cccc}{0}&{}&{}&{}\\{}&{\fbox{\ensuremath{\begin{array}{rr}{-1}&{i}\\{\phantom{-}i}&{1}\\\end{array}}}}&{}&{}\\{}&{}&{\fbox{\ensuremath{\begin{array}{rr}{-2}&{2i}\\{\phantom{-}2i}&{2}\\\end{array}}}}&{}\\{}&{}&{}&{\ddots}\\\end{array}\right).
\Psi_{-\nu}^{\overline{{4}}}|0\rangle^{\prime}=\Psi_{-\nu}^{\overline{{4}}}\Psi_{\nu}^{4}|0\rangle=\{\Psi_{-\nu}^{\overline{{4}}},\Psi_{\nu}^{4}\}|0\rangle=|0\rangle\neq0\,.
{\bfA}=-{\frac{i}{2}}\intdt\left[v^{\dagger}{\bf\nabla}v-{\bf\nabla}v^{\dagger}\,v\right]-{\frac{i}{2}}\sum_{j}\left[S_{j}^{\dagger}{\bf\nabla}S_{j}-{\bf\nabla}S_{j}^{\dagger}S_{j}\right]\,.
[\ell_{j}^{\infty},\ell_{k}^{\infty}]=[\ell_{j}^{\infty},t_{k}]+[t_{j},\ell_{k}^{\infty}]=2ag[\ell_{j}^{\infty},\check{s}_{jk}]=-2ag[\ell_{k}^{\infty},\check{s}_{jk}].
\delta\langle\tau^{\prime\prime}|\tau^{\prime}\rangle=i\langle\tau^{\prime\prime}|{p_{\mu}}^{\prime}\delta{x^{\mu}}^{\prime\prime}+({x^{\mu}}^{\prime\prime}-2\lambdap_{\mu}^{\prime}(\tau^{\prime\prime}-\tau^{\prime}))\delta{p_{\mu}}^{\prime}|\tau^{\prime}\rangle,
T_{\phi\mu\nu}=B_{\mu\rho}B_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}g^{\rho\sigma}g^{\lambda\tau}B_{\rho\lambda}B_{\sigma\tau}-2\frac{\partialV(\phi)}{\partialg^{\mu\nu}}+g_{\mu\nu}V(\phi).
E_{Cas}(v)=\frac{1}{2\pi}\int_{0}^{\infty}\mathrm{ln}\Bigl(1-e^{-2\pix}\left(\frac{x-v}{x+v}\right)^{2}\Bigr)dx-\frac{v}{\pi}\mathrm{ln}(vT)-\frac{v}{\pi}\gamma\,.
\int_{\Lambda}\left[\prod_{i=1}^{n_{a}}d^{2}x_{i}^{(a)}d^{2}y_{i}^{(a)}\right]\Big|A(a,a)\Big|^{\frac{\pi}{\pi+gN}p}\;\leq\;(2n_{a})!\bigg[\Xi\Big(\frac{\pi}{\pi+gN}p\Big)\bigg]^{2n_{a}}\;,
\hat{U}\left[\hat{\triangle}+\frac{i}{k}\hat{J}_{-}^{a}(\sigma_{n})M^{a}\right]\hat{U}^{-1}=U(\sigma)\left[\hat{\triangle}+\frac{i}{k}\hat{J}_{-}^{a}(\sigma_{n})M^{a}\right]U(\sigma)^{-1}~,
\left.\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}}&{}&{h^{0,1}}&{}\\{h^{2,0}}&{}&{h^{1,1}}&{}&{h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}\right.=\left.\begin{array}{ccccc}{}&{}&{1}&{}&{}\\{}&{2}&{}&{2}&{}\\{1}&{}&{4}&{}&{1}\\{}&{2}&{}&{2}&{}\\{}&{}&{1}&{}&{}\\\end{array}\right.
P_{L}^{-}=\sum_{N>0}\left({\frac{N\pi}{L}}\right)\alpha{_N^{\dagger}}\alpha_{N}+\sum_{n>0}\left({\frac{n\pi}{L}}\right)\left[\beta{_n^{\dagger}}\beta_{n}+\delta{_n^{\dagger}}\delta_{n}\right]-{\frac{g^{2}}{2\pi}}\int_{0}^{2L}dx^{+}\mathrm{Tr}(A^{2})\;.
S_{0}=\intd^{2}x\,\Big(i\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-g_{0}\overline{\psi}\gamma^{\mu}\gamma_{5}\psi\partial_{\mu}\phi+{\frac{1}{2}}\partial_{\mu}\phi\partial^{\mu}\phi-{\frac{m^{2}}{2}}\phi^{2}\Big)
V(\Phi^{(1)},\Phi^{(2)})=\frac{g_{1}\eta^{4}}{8}\sum_{A=1}^{2}\left(\frac{1}{2}\mathrm{Tr}[\Phi^{(A)}]^{2}-1\right)^{2}+\frac{g_{2}\eta^{4}}{4}\left(\frac{1}{2}\mathrm{Tr}[\Phi^{(1)}\Phi^{(2)}]-c\right)^{2}.
{\mathcal{L}_{SD}}=\beta^{2}g\left(m^{2}A^{2}/2\beta^{2}\right)-\frac{\beta^{2}}2h\left(m^{2}B^{2}/2\beta^{3}\right)-\frac{\chi\theta}{2}\varepsilon^{\mu\nu\rho\lambda}\partial_{\mu}B_{\nu\rho}A_{\lambda}\;.
\lambda_{(0)}^{\alphaaA}{\lambda_{(0)\alphaa}}^{B}=\frac{3\cdot2^{6}}{g^{2}}\frac{\rho_{0}^{4}}{\left(\rho_{0}^{2}+(x-x_{0})^{2}\right)^{4}}\:\zeta^{\alphaA}{\zeta_{\alpha}}^{B}\,.
={\frac{1}{Z}}\sum_{n=0}^{\infty}\big(-{\frac{1}{4\pi\alpha^{\prime}}}\big){\frac{1}{n!}}\int_{{\calH}_{n}^{\varepsilon}}d^{d}zd^{d}\bar{z}\langle\Phi_{1}\Phi_{2}\dots\Phi_{N}\prod_{a=1}^{n}B_{i_{a}j_{a}}J^{i_{a}}(z_{a})\bar{J}^{i_{a}}(\bar{z}_{a})\rangle
S_{HG}^{eff}=-\frac{1}{2k^{2}}e^{\frac{3}{4}\Phi_{0}}\alpha_{E}^{3/2}\frac{9y^{2}}{4}\Bigg|_{0}^{\infty}\,\quadS_{br}^{eff}=\frac{1}{2k^{2}}e^{\frac{3}{4}\Phi_{0}}\alpha_{E}^{3/2}\frac{8y^{2}}{4}\Bigg|_{0}^{\infty}\.
\frac{d}{dt}\langleF(x)\rangle_{t}\sim\frac{1}{2\gamma}\Bigl(\dot{x}_{+}F^{\prime}(x_{+})+\dot{x}_{-}F^{\prime}(x_{-})\Bigr)=\frac{1}{2\gamma}\left(\frac{d}{dt}F(x_{+})+\frac{d}{dt}F(x_{-})\right).
{\hat{D}}_{\mu}^{(A)}L^{\mu}-g^{2}{\hat{T}}^{a}(L_{\psi}T^{a}\psi+\bar{\psi}T^{a}L_{\bar{\psi}}+L_{\phi\,i}{\tilde{T}}^{a}\phi_{i}-\phi_{i}^{*}{\tilde{T}}^{a}L_{\phi^{\dagger}\,i})\equiv0.
\left(\lambda^{\alpha,i}\right)^{\dagger}=i\lambda^{\beta,j}\epsilon_{ji}\epsilon_{\beta\alpha},\qquad\left(\bar{\lambda}_{\alpha^{\prime}}^{i}\right)^{\dagger}=-i\bar{\lambda}_{\beta^{\prime}}^{j}\epsilon_{ji}\epsilon^{\beta^{\prime}\alpha^{\prime}},
\alpha:=\left(\begin{array}{cccccccc}{\gamma_{1}}&{0}&{\gamma_{3}}&{\gamma_{4}}&{\ldots}&{\gamma_{N-2}}&{\gamma_{N-1}}&{\gamma_{N}}\\{0}&{\beta_{2}}&{\gamma_{3}}&{c_{4}\gamma_{4}}&{\ldots}&{c_{N-2}\gamma_{N-2}}&{\beta_{N-1}}&{\beta_{N}}\\\end{array}\right)
A=\left(\begin{matrix}{D}&{E}\\{-E^{\ast}}&{-D^{\ast}}\\\end{matrix}\right)^{2}=\left(\begin{matrix}{D^{2}-EE^{\ast}}&{(D-D^{\ast})E}\\{-(D-D^{\ast})E^{\ast}}&{(D^{\ast})^{2}-EE^{\ast}}\\\end{matrix}\right),
\begin{array}{ccrccc}{\underline{{System}}}&{}&{}&{\underline{{Dual\;System}}}&{}&{}\\{Gauge\;Group}&{\#\mathrm{flavors}}&{}&{Gauge\,Group}&{\#flavor}&{\#\mathrm{singlets}}\\{SU(N_{C})}&{N_{F}}&{}&{SU(N_{F}-N_{C})}&{N_{F}}&{N_{F}^{2}}\\{SO(N_{C})}&{N_{F}}&{}&{SO(N_{F}-N_{C}+4)}&{N_{F}}&{N_{F}^{2}}\\{Sp(N_{C})}&{2N_{F}}&{}&{Sp(N_{F}-N_{C}-2)}&{2N_{F}}&{N_{F}^{2}}\\\end{array}
\begin{matrix}{M2}&{0}&{1}&{2}&{}&{}&{}&{}&{}\\{M2}&{0}&{}&{}&{3}&{4}&{}&{}&{}\\{M5}&{0}&{1}&{}&{3}&{}&{5}&{6}&{7}\\{M5}&{0}&{}&{2}&{}&{4}&{5}&{6}&{7}\\\end{matrix}
\Lambda^{3}=-{\frac{1}{16\pi^{3}}}{\frac{\Gamma^{2}(\beta^{2}/4)\Gamma(\beta^{2})}{\Gamma^{2}(1-\beta^{2}/4)\Gamma(1-\beta^{2})}}\left[{\frac{\piM}{\sqrt{3}\Gamma(1/3)}}{\frac{\Gamma\left({\frac{2}{3(2-\beta^{2})}}\right)}{\Gamma\left({\frac{\beta^{2}}{3(2-\beta^{2})}}\right)}}\right]^{3(2-\beta^{2})}
E_{s}^{\varepsilon}\;=\;E_{s}\:+\:s\:\frac{\varepsilon}{2\mu_{k}}\left(E_{s}\:{\cal{B}}_{0}\:E_{\bar{s}}+E_{\bar{s}}\:{\cal{B}}_{0}\:E_{s}\right)\:+\:{\cal{O}}(\varepsilon^{2})
\int_{0}^{1}{\mathrmd}x{\frac{x}{\sqrt{1-x^{2}}}}K_{0}(\frac{nz~x}{2})={\frac{1}{nz}}\Big[\,{\mathrmEi}\big(\frac{nz}{2}\big)\,{\mathrme}^{-{nz}/{2}}-{\mathrmEi}\big(-\frac{nz}{2}\big)\,{\mathrme}^{{nz}/{2}}\,\Big].
\eta_{0}={\frac{C^{2(D-2)}}{(D-3)!}}\left({\frac{A^{\prime\prime}}{AB^{2}}}-{\frac{A^{\prime}B^{\prime}}{AB^{3}}}-{\frac{C^{\prime\prime}}{CB^{2}}}+{\frac{B^{\prime}C^{\prime}}{B^{3}C}}\right)
\left.\hat{F}_{\alpha\beta}^{I}\right|_{\mathrm{sing}}=-\frac{\partialx^{\mu}}{\partialq^{\alpha}}\frac{\partialx^{\nu}}{\partialq^{\beta}}\,\Omega_{-}^{-1}\,\hat{N}_{+}^{-1}\frac{2\hat{\eta}_{\mu\nu}^{+}}{\left(r^{2}+\rho^{2}\right)^{2}}\hat{N}_{+}\,\Omega_{-}.
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\frac{\dot{a}}{a}}(1+w)+2{\frac{\dot{c}}{c}}{\frac{1+\epsilon_{\delta\sigma}}{1+\epsilon}}+\left({\frac{\dot{G}_{4}}{G_{4}}}-4{\frac{\dot{c}}{c}}\right)(1+\epsilon_{\delta\sigma})\right].
K_{np}^{(2)}=m^{2}\frac{\lambda^{2}}{24}\frac{1}{a}\frac{1}{(16\pi^{2})^{2}}\int_{0}^{1}dx\int_{0}^{\infty}d\alpha^{\prime}\int_{0}^{\infty}dz^{\prime}\frac{e^{-a(\alpha^{\prime}+z^{\prime})}}{({(\alpha^{\prime})}^{2}x(1-x)+\alpha^{\prime}z^{\prime}+1)(\alpha^{\prime}x(1-x)+z^{\prime})},
Z_{S}^{T}(1)~=~\int\prod_{\mu;<{ij}>}dP_{ij}^{\mu}e^{-\sum_{\mu;<{ij}>}(P_{ij}^{\mu})^{2}}Det\left(\sum_{\mu}(\sigma^{\mu}P^{\mu})\right)\prod_{\mu;i}\delta(\sum_{j}P_{ij}^{\mu})~~,
W=\tilde{d}_{1}\frac{e^{\frac{-3S}{2\tilde{\beta}_{1}}}}{\eta(T)^{6-\frac{9\delta_{GS}}{4\pi^{2}\tilde{\beta}_{1}}}}+\tilde{d}_{2}\frac{e^{\frac{-3S}{2\tilde{\beta}_{2}}}}{\eta(T)^{6-\frac{9\delta_{GS}}{4\pi^{2}\tilde{\beta}_{2}}}}\;.
\left[-\left(1-\frac{2M}{\rho}\right)\frac{d^{2}}{d\rho^{2}}-\frac{2(\rho-4M)(2\rho-3M)}{\rho^{2}(\rho-3M)}\frac{d}{d\rho}+\frac{8M}{\rho^{2}(\rho-3M)}\right]H_{1}(\rho)=\lambdaH_{1}(\rho),
B_{i}\frac{\prod_{j<i}C_{ji}(\zeta-a_{ji})}{\prod_{j>i}C_{ij}(\zeta-a_{ij})}e^{x_{i}-\bar{z}_{i}\zeta}=E_{i}\frac{\prod_{j<i}(\zeta-a_{ji})}{\prod_{j>i}(\zeta-a_{ij})}e^{x_{i}-\bar{z}_{i}\zeta},
S=\frac{1}{2\alpha_{0}}\intd^{2}\sigma\,\Bigl\{\rho^{-1}\partial^{2}X^{\mu}\partial^{2}X_{\mu}+\lambda^{ab}(\partial_{a}X^{\mu}\partial_{b}X_{\nu}-\rho\delta_{ab})\Bigr\}+\mu_{0}\intd^{2}\sigma\,\rho.
\hat{\tilde{\phi}}\Psi=0,\quad2\hat{\tilde{\phi}}=\hat{\calP}^{2}+m^{2}+\frac{1}{2}\epsilon_{\mu\nu\lambda}\gamma^{\lambda}\otimes1\cdot\left(\frac{1}{2}{\calQF}^{\mu\nu}\cdot1\otimes\sigma_{3}+qF^{\mu\nu}\cdot1\otimes1\right),
{\calF}(\beta)=-\frac{2^{-14}\pi^{-26}}{(2\alpha^{\prime})^{13}}\sum_{r=1}^{\infty}\sum_{k=-1}^{\infty}a_{k}^{2}\int_{0}^{\infty}d\tau_{2}\tau_{2}^{-14}e^{-\frac{\beta^{2}r^{2}}{4\pi\tau_{2}\alpha^{\prime}}}e^{-4\pik\tau_{2}}
H[\varphi,\pi]=\frac{1}{2}\sum_{p}\left[\pi_{p}\pi_{-p}+p^{2}\varphi_{p}\varphi_{-p}\right]+{\frac{\lambda}{4!V}}\sum_{p_{1}p_{2}p_{3}p_{4}}\varphi_{p_{1}}\varphi_{p_{2}}\varphi_{p_{3}}\varphi_{p_{4}}~\delta_{p_{1}+p_{2}+p_{3}+p_{4},0}~,
U_{\{(n)\}}^{\{r,r\}}(z)U_{\{(m)\}}^{\{s,s\}}=\frac{1}{n!m!}\frac{\partial^{n}}{\partialz^{n}}\frac{\partial^{m}}{\partial\xi^{m}}\left(:U^{r}(z)U^{s}(\xi):(z-\xi)^{r\cdots}\right)
\begin{array}{cccc}{p_{+}=\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)~~~}&{~~~p_{-}=\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)~~~}&{~~~\tau_{+}=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right)~~~}&{~~~\tau_{-}=\left(\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right)}\\\end{array}
q=-{\frac{a_{0}\ddot{a}_{0}}{\dot{a}_{0}^{2}}}={\frac{\dot{I}}{2HI}}+{\frac{1}{2}}(I^{-3/2}\Omega_{\varrho}+I^{-1}\Omega_{\Phi})+{\frac{16\pi^{2}G_{5}^{2}\sigma}{3c^{6}H^{2}}}(I^{-1/2}\wp+I^{-1}\wp_{\Phi}+2I^{-1/2}\wp_{\Lambda_{4}})+I^{-1}\Omega_{\calC},
{\bar{\theta}}^{\dot{2}}=Y(x){\bar{\theta}}^{\dot{1}},\quad{\bar{\theta}}^{\dot{\alpha}}={\bar{\theta}}^{\dot{1}}{\bar{\psi}}^{\alpha},\quad{\bar{\zeta}}^{\dot{2}}=Y(x){\bar{\zeta}}^{\dot{1}},\quad{\bar{\zeta}}^{\dot{\alpha}}={\bar{\zeta}}^{\dot{1}}{\bar{\psi}}^{\alpha}.
\frac{1}{g_{YM}^{3}}=\frac{\tau_{5}(2\pi\alpha^{\prime})^{2}}{2}\int_{{\cal{C}}_{2}}{e}^{-\phi}B_{2}=\frac{1}{4\pig_{s}(2\pi\sqrt{\alpha^{\prime}})^{2}}\int_{{\cal{C}}_{2}}{e}^{-\phi}B_{2}
\overline{{P_{\{N_{\nu_{\alpha_{1}}}^{(\alpha_{1})},\dots,\,N_{\nu_{\alpha_{n}}}^{(\alpha_{n})}\}}^{\Lambda,\lambda_{\Lambda}}}}={\calA}^{\{{\calN}\}}d(\Lambda)\int_{G}\,d\mu(g)\bar{\chi}^{(\Lambda)}(g)[D_{\nu_{1}\nu_{1}}^{(\alpha_{1})}]^{N_{\nu_{\alpha_{1}}}^{(\alpha_{1})}}\cdots[D_{\nu_{n}\nu_{n}}^{(\alpha_{n})}]^{N_{\nu_{\alpha_{n}}}^{(\alpha_{n})}}\,;
F_{1}=\left(\begin{array}{c}{P^{2}}\\{P^{1}}\\\end{array}\right|\left|\begin{array}{c}{1}\\{1}\\\end{array}\right)_{4},\;\;\left(\begin{array}{c}{P^{2}}\\{P^{1}}\\\end{array}\right|\left|\begin{array}{c}{3}\\{1}\\\end{array}\right)_{12}etc.
p_{1,th,\mathrm{q}}=\frac{(m_{2}+m_{3})^{2}-m_{1}^{2}}{4\epsilon}+\frac{p_{1,th,\mathrm{q}}^{4}}{12\epsilon\omega^{2}}\left(\frac{m_{2}^{3}+m_{3}^{3}}{(m_{2}+m_{3})^{3}}-1\right)(q-1)^{2}.
\left(1|\mathbb{E}_{\left({\frac{dim}{2}}\right)}\right)_{\alpha\beta}=\delta_{0\alpha}\delta_{\beta\left({\frac{dim}{2}}\right)}-\delta_{0\beta}\delta_{\alpha\left({\frac{dim}{2}}\right)}-\epsilon_{\alpha\beta\left({\frac{dim}{2}}\right)}.
\nabla_{\nu}^{\varphi}=-(\nabla_{\nu}\Phi)\,\qquad\nabla_{\nu}^{\bar{h}_{\rho\sigma}}=g_{\rho\sigma}\nabla_{\nu}-g_{\nu\rho}\nabla_{\sigma}-g_{\nu\sigma}\nabla_{\rho}\,\qquad\nabla_{\nu}^{\eta}=-(\nabla_{\nu}\psi)+{\frac{1}{8}}\left[\gamma_{\nu},\gamma_{\lambda}\right]\psi\nabla^{\lambda}\.
\begin{array}{c}{\sum_{0\lep\le1-a_{ij}}(-1)^{p}\left[\begin{array}{clcr}{}&{1-a_{ij}}\\{}&{p}\\\end{array}\right]_{Q_{i}}e_{i}^{1-a_{ij}-p}e_{j}e_{i}^{p}=0}\\{\sum_{0\lep\le1-a_{ij}}(-1)^{p}\left[\begin{array}{clcr}{}&{1-a_{ij}}\\{}&{p}\\\end{array}\right]_{Q_{i}}f_{i}^{1-a_{ij}-p}f_{j}f_{i}^{p}=0}\\\end{array}
{\calL}=-\partial_{\mu}\overline{\psi}_{1}\gamma_{\mu\nu}\partial_{\nu}\psi_{1}-\partial_{\mu}\overline{\psi}_{2}\gamma_{\mu\nu}\partial_{\nu}\psi_{2}-m^{2}\overline{\psi}_{1}\psi_{1}+m^{2}\overline{\psi}_{2}\psi_{2}\quad.
\tilde{W}_{tree}=\frac{1}{2\mu}S^{ij}\tilde{Q}_{i}\tilde{Q}_{j}+mS^{2N_{f}-1,2N_{f}}+\frac{1}{2\mu}\Phi^{st}\tilde{q}_{s}\tilde{q}_{t}+\frac{1}{\mu}N^{is}\tilde{Q}_{i}\tilde{q}_{s}.
H_{\perp}=l_{P}^{N-1}G_{ab,cd}\pi^{ab}\pi^{cd}+\frac{1}{l_{P}^{N-1}}C^{ab,cd,ef}g_{ab}g_{cd,ef}+\frac{1}{l_{P}^{N-1}}D^{ab,e,cd,f}g_{ab,e}g_{cd,f}.
\frac{1}{2}i\psi_{\pm}^{\dagger}\left(\partial_{1}-ieA^{1}\right)\psi_{\pm}-\frac{1}{2}i\left(\partial_{1}+ieA^{1}\right)\psi_{\pm}^{\dagger}\psi_{\pm}=\pm\frac{1}{4\pi}\left(\partial_{1}\phi_{\pm}\mpe\lambdaA^{1}\right)^{2}\mp\frac{\pi}{12L^{2}}\.
\left(\begin{matrix}{N_{22}}&{-N_{21}}\\{-N_{12}}&{N_{11}}\\\end{matrix}\right)\left(\begin{matrix}{N_{11}}&{N_{12}}\\{N_{21}}&{N_{22}}\\\end{matrix}\right)=\left(\begin{matrix}{\Omega}&{0}\\{0}&{\Omega}\\\end{matrix}\right)=\left(\begin{matrix}{N_{11}}&{N_{12}}\\{N_{21}}&{N_{22}}\\\end{matrix}\right)\left(\begin{matrix}{N_{22}}&{-N_{21}}\\{-N_{12}}&{N_{11}}\\\end{matrix}\right)
\triangle=\int_{{\tilde{\calF}}}\frac{d^{2}\tau}{\tau_{2}}\sum_{(g_{0},h_{0})\in{\calO}}b_{a}^{(h_{0},g_{0})}{\calZ}_{(h_{0},g_{0})}(\tau,{\bar{\tau}})-b_{a}^{N=2}\int_{\calF}\frac{d^{2}\tau}{\tau_{2}}.
{\mathfrakJ}^{\mu}(x,\kappa)\equiv\frac1{2\kappa}~\varepsilon^{\mu\nu}\partial_{\nu}\;\Bigl(\chi(x,\kappa)-\chi^{T}(x,\kappa)\Bigr)\equiv\sum_{n=0}^{\infty}\kappa^{n}~J_{(n)}^{\mu}(x)\;.
\begin{array}{l}{\Phi_{j}\bar{\Phi}_{i}=\delta_{ij}+qR_{kijl}\bar{\Phi}_{l}\Phi_{k},}\\{\Phi_{l}\Phi_{k}=q^{-1}R_{jikl}\Phi_{j}\Phi_{i},\qquadi,j=1,2,...,N,}\\\end{array}
(*)\,\,\,\,\,{\vec{k}_{d}}\,=\,{(k_{1},...,k_{2})}\,\,\stackrel{{\pi}^{-1}}{\Longrightarrow}\,\,{\vec{k}_{d+1}^{ex(i)}}\,=\,{(k_{1},...,0^{i},...,k_{d})}.
\mathbf{O}_{T}=\mathbf{S}_{T}+\mathrm{tr}\Bigr\{\gamma\mathbf{G}+\tau\mathbf{R}+[\tau,\gamma]\frac{\partial}{\partial\gamma}-\gamma\frac{\partial}{\partial\tau}+\tau^{2}\frac{\partial}{\partial\tau}\Bigr\}.
{\calL}_{g}=-{\frac{1}{2}}\sqrt{-g}g^{\alpha\beta}G_{\alpha\beta}+{\frac{\tilde{Y}M\tilde{Y}}{2\sqrt{-g}(\partiala)^{2}}}-{\frac{\tilde{Y}L\,\partialY\cdot\partiala}{2(\partiala)^{2}}}.
{\calI}_{\mathrm{1-loop}}^{I}=\frac{1}{2}V^{(8)}\sum_{\sigma}\rho_{\sigma}\int_{0}^{\infty}\frac{dt}{t}\frac{1}{(2\pi^{2}t)^{4}}\Big(\sum_{p\in\Gamma_{2}}e^{-\pit|p|^{2}/2}\Big)\Z(\tau_{\sigma})\int\prod_{i=1}^{4}d^{2}z_{i}\langle\prod_{i=1}^{4}{V}_{\mathrm{gauge},i}\rangle
\begin{array}{c}{\rho_{\zeta}(x_{1})=\rho(x_{1}),\;x=e,f}\\{\rho_{\zeta}(e_{0})=\zeta\rho(f_{1})}\\{\rho_{\zeta}(f_{0})={\frac{1}{\zeta}}\rho(e_{1})}\\{\rho_{\zeta}(t_{0})=(\rho_{\zeta}(t_{1}))^{-1}=(\rho(t_{1}))^{-1}}\\\end{array}
<BR>{\cal=L}_{St}=3D-{\frac{1}{=204}}A_{\mu\nu}A^{\mu\nu}+{\frac{\mu}{4}}\epsilon_{\mu\nu\lambda=}A^{\mu\nu=20}A^{\lambda}+{\frac{m^{2}}{2}}(A_{\mu}-\partial_{\mu}\theta)(A^{\mu}-\partial=^{\mu}=20\theta),<BR><BR>
\left[G_{a}^{(1)},G_{b}^{(1)}\right]=0,\;\left[G_{a}^{(1)},G_{b}^{(2)}\right]=0,\;\left[G_{a}^{(2)},G_{b}^{(2)}\right]=f_{\;\;ab}^{c}G_{c}^{(2)},
b_{\beta\delta}^{ik}=\frac{1}{2}\kappa^{2}\left(\epsilon^{ij}\epsilon^{kl}e_{j}^{\alpha}e_{l}^{\gamma}+\frac{1}{2}\eta^{jk}\eta^{il}e_{l}^{\alpha}e_{j}^{\gamma}\right)R_{\alpha\beta}R_{\gamma\delta}
\langle0|T\{\psi_{\sigma_{1},\sigma_{2},m_{I}}^{j_{1},j_{2},I}(x_{1})\bar{\psi}_{\sigma_{4},\sigma_{3},m_{I}^{\prime}}^{j_{4},j_{3},I^{\prime}}(x_{2})\phi_{a}(y_{1})\}|0\rangle,
\Psi_{n,m}(x,y)=y^{p_{2}}H_{n}(\sqrt{2\omega}x)L_{m}^{(-1/2+p_{2})}(\omegay^{2})e^{-\omegax^{2}-\frac{\omegay^{2}}{2}}\,\E_{n,m}=\\omega[2(n+m)+p_{2}+
S_{cl}(A_{\mu}^{p},H_{\mu\nu}^{p})=\int{d^{4}}x\left(-\frac{1}{4}{\varepsilon}^{\mu\nu\rho\sigma}F_{\mu\nu}^{p}H_{\rho\sigma}^{p}+\frac{1}{2}A_{\mu}^{p}A^{p\mu}\right),
S_{\mathrm{YM}}=-\,\frac{1}{g^{2}}\intd^{4}x\left\{\frac{1}{4}F_{\alpha\beta}^{a}F_{a}^{\alpha\beta}+\frac{1}{2}\partial_{\alpha}{\overline{\phi}}\,\partial^{\alpha}\phi\right\}+\frac{\theta_{\mathrm{YM}}}{32\pi^{2}}\intd^{4}xF_{\alpha\beta}^{a}{\widetilde{F}}_{a}^{\alpha\beta}
J_{1}\left(m_{eff},\beta\right)=-\frac{\mu_{eff}^{2}\beta}{\left(2\pi\right)^{2}r}\,\sum_{n=1}^{\infty}\frac{1}{n^{2}}K_{2}\left(2\pin\mu_{eff}\right)+\frac{\mu_{eff}^{3}\beta}{2\pir}\sum_{n=1}^{\infty}\frac{1}{n}K_{3}\left(2\pin\mu_{eff}\right),
C_{m_{d}}(\xi)\;=\;\frac{1}{2\pi}\intd^{2}p\;\hat{C}_{m_{d}}(p)\;e^{ip\xi}\;=\;\frac{1}{(2\pi)^{2}}\intd^{2}p\;\frac{1}{p^{2}+m_{d}^{2}}\;e^{ip\xi}\;,
h\left(t,\frac{D_{B}}{\mu^{2}}\right)=\sum_{n=-\infty}^{-t\left(\left(\frac{2n\pi}{L\mu}\right)^{2}+\left(\fracm\mu\right)^{2}\right)^{\frac12}}=\sum_{p=-\infty}^{\infty}c_{p}\left(t\right)
\begin{array}{c}{\bar{\Phi}(x,\theta,\bar{\theta})=\bar{\alpha}(x)+\bar{\theta}\bar{\psi}(x)-i\theta\sigma^{n}\bar{\theta}\partial_{n}\bar{\alpha}(x),}\\{\Box\bar{\alpha}(x)=0,\qquad\tilde{\sigma}^{n}\partial_{n}\bar{\psi}(x)=0.}\\\end{array}
\intDA_{\mu}^{a}e^{S[A_{\mu}^{a}]}=\intD(A_{\mu}^{a})_{dev}\left[\intD(A_{\mu}^{a})_{cl}e^{iS[{(A_{\mu}^{a})}_{cl}+{(A_{\mu}^{a})}_{dev}]}\right]
J^{-1}=\left(\begin{array}{cccc}{0}&{0}&{\delta^{\,2}\,\left(x-y\right)}&{0}\\{0}&{0}&{0}&{\nabla^{-2}}\\{-\,\delta^{\,2}\,\left(x-y\right)}&{0}&{0}&{0}\\{0}&{-\,\nabla^{-2}}&{0}&{0}\\\end{array}\right).