harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\deltaS=-\int_{\mathcalM}\left[\delta\bar{\phi}\left(\nabla^{a}\nabla_{a}+m^{2}\right)\phi+\delta\phi\left(\bar{\nabla}^{a}\bar{\nabla}_{a}+m^{2}\right)\bar{\phi}\right]+\sum_{s}\int_{\mathcalB}\left(\delta\bar{\phi}_{s}\nabla_{a}\phi_{s}+\delta\phi_{s}\bar{\nabla}_{a}\bar{\phi}_{s}\right)n_{s}^{a}\,.
S_{0}\left(\,\sigma\;\Omega\,\right)\equiv{\frac{B}{2\Omega_{p+1}}}\,{\frac{1}{(p+1)!}}\,\left(\,\sigma^{\mu_{1}\dots\mu_{p+1}}-\sigma_{0}^{\mu_{1}\dots\mu_{p+1}}\,\right)^{2}-\frac{m_{p+1}}{2}\,p\,V_{p+1}\,
\Psi^{\prime}=-\intd^{4}x\left(\overline{{\eta}}_{1}^{a}\left(M^{2}A_{a}^{0}-\frac1\lambdaf_{\;ac}^{b}\pi_{b}^{i}\tilde{A}_{i}^{c}\right)+\lambdaM^{2}\overline{{\eta}}_{2}^{a}\varphi^{a}\right),
B=-{\frac{\sqrt{-2\nabla^{2}}}{2}}(Q_{1}+Q_{2});~~E_{i}=-{\frac{1}{\sqrt{-2\nabla^{2}}}}[\epsilon_{ij}\partial_{j}(P_{1}+P_{2})+(\mu+m)\partial_{i}Q_{1}+(\mu-m)\partial_{i}Q_{2}],
\{\varepsilon^{\mu\nu}\}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{\varepsilon^{12}}&{\varepsilon^{13}}\\{0}&{-\varepsilon^{12}}&{0}&{\varepsilon^{23}}\\{0}&{-\varepsilon^{13}}&{-\varepsilon^{23}}&{0}\\\end{array}\right)
\alpha^{a}=\sqrt{\frac{m}{2}}\,\left[\hat{A}^{a}+\frac{i}{m}\hat{\pi}^{a}\right]\\,\\\{\alpha^{a}}^{\dagger}=\sqrt{\frac{m}{2}}\,\left[\hat{A}^{a}-\frac{i}{m}\hat{\pi}^{a}\right]\\,
\vec{a}_{DES}^{\,\prime}=R_{x}(-\gamma)R_{z}(\phi-{\frac{\pi}{2}})R_{x}({\frac{\pi}{2}}-\theta)R_{z}(2u)R_{z}(-\theta_{2})R_{x}(\phi_{2})R_{z}(\theta_{2})R_{z}(u)\hat{x},
h\,\left(\,t;{\bfx},{\bfx}\,|\,M\,\right)=\frac{1}{\left(4\,\pi\,t\right)^{d/m}}\:\left[\,a_{0}\,(\,x\,|\,M\,)+a_{1}\,(\,x\,|\,M\,)\:t+\dots\,\right]\.
S_{\mathrm{q}}^{(B,e)}=\int_{M}d^{3}x\,tr\left((B_{\mu}^{i}\,\,e^{i})\left\{\begin{array}{cc}{-\epsilon^{\mu\nu\gamma}D_{\nu}}&{-D^{\alpha}}\\{D^{\gamma}}&{0}\\\end{array}\right\}^{ij}\left(\begin{array}{c}{B_{\nu}^{j}}\\{e^{j}}\\\end{array}\right)\,\,\right).
\left(\begin{array}{c|c}{-\bar{D}^{2}\delta_{\mu\alpha}}&{0}\\{0}&{\frac{1}{\theta}}\\\end{array}\right)\left(\begin{array}{c|c}{-\frac{1}{\bar{D}^{2}}\delta_{\alpha\gamma}}&{0}\\{0}&{\theta}\\\end{array}\right)\left(\begin{array}{c|c}{-\bar{D}^{2}\delta_{\gamma\sigma}+\bar{D}_{\gamma}\bar{D}_{\sigma}}&{-i\bar{D}_{\gamma}}\\{i\bar{D}_{\sigma}}&{0}\\\end{array}\right)
S_{E}=-\frac{1}{16\piG_{4}}\int_{M}d^{4}x\sqrt{\hat{g}}[R-\frac{1}{2}(\partial\Phi)^{2}-\frac{1}{8}e^{-\Phi}\bar{F}^{2}-\frac{1}{12}e^{-2\Phi}H^{2}],
S_{lm}(\theta)={\bfP}\,s_{lm}(\theta)\sum_{p=0}^{\mathrm{Min}(n-l,m)}\sum_{q=0}^{m-p}\tau_{\lambda_{l+p-q}+\lambda_{m-p-q}}(\theta)P_{\lambda_{l+p-q}+\lambda_{m-p-q}}
\int(x_{k})^{4}\mathcal{V}_{J}\,d^{2}x=6\int(x_{k})^{2}\partial_{j}\partial_{j}h\,d^{2}x=6\int\left(\partial_{j}((x_{k})^{2}\partial_{j}h)-2\partial_{k}(x_{k}h)+2h\right)d^{2}x,
\begin{array}{rcl}{\left((q-q^{-1})f\right)^{p}\left((q-q^{-1})e\right)^{p}}&{=}&{\displaystyle\prod_{i=0}^{p-1}\left(u+u^{-1}-q^{i}t-q^{-i}t^{-1}\right)}\\{}&{=}&{\displaystyle\prod_{i=0}^{p-1}\left(u-q^{i}t\right)\left(1-q^{-i}t^{-1}u^{-1}\right)~.}\\\end{array}
\frac{\partialS_{n,l}^{*}}{\partialt_{1}}=S_{n+1,l}^{*}\frac{h_{n+1,l}}{h_{n,l}},\;\;\;\frac{\partialS_{n,l}^{*}}{\partialt_{-1}}=-S_{n-1,l}^{*}\frac{h_{n+1,l}}{h_{n,l}},
\sigma_{1}+\frac{1}{\alpha^{2}}\mathrm{ln}cosh(F_{1}t)-\frac{1}{\alpha^{2}}e^{-2(q_{0}+F_{1})t}\intdt^{\prime}e^{2(q_{0}+F_{1})t^{\prime}}\mathrm{ln}cosh(F_{1}t^{\prime})=\frac{c_{1}}{2(q_{0}+F_{1})}+e^{-2(q_{0}+F_{1})t}c_{2}~,
Z={\frac{H(t_{f})a(t_{f})}{H(t_{i})a(t_{i})}}=\left({\frac{\Phi(t_{i})}{\Phi(t_{f})}}\right)^{1/\sqrt{3}}=\left({\frac{t_{\mathrm{sing}}-t_{i}}{t_{\mathrm{sing}}-t_{f}}}\right)^{(1+\sqrt{3})/\sqrt{3}}\,.
f_{\pi}^{2}=\frac{N_{c}}{\left(2\pi\right)^{2}}\,\int\,xdx\,\frac{\Sigma(x)\Bigl[\Sigma(x)-x\Sigma^{\prime}(x)/2\Bigr]}{\Bigl[x+\Sigma^{2}(x)\Bigr]^{2}},
S=\int\,d^{2}x\;\left[{\dot{\varphi}}\varphi^{\prime}+{\frac{1}{2}}({\dot{A}}_{-}-{A_{+}^{\prime}})^{2}+ae^{2}[A_{+}+{\frac{2}{ae}}(\dot{\omega}+\dot{\varphi})]A_{-}\right]
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}+\tilde{\omega}^{2}\left(1+{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right)-{\frac{s\tilde{R}_{\frac{\pi}{2}}^{8}}{r^{10}\tilde{f}^{2}}}\right\}\psi_{s}=0.
\begin{array}{rcl}{}&{}&{V_{j}(z)V_{j^{\prime}}(w)=q^{jj^{\prime}}V_{j}(w)V_{j}(z)~,}\\{}&{}&{J(z)V_{j}(w)=q^{-j}V_{j}(w)J(z)~,}\\{}&{}&{J(z)J(w)=qJ(w)J(z)~.}\\\end{array}
\nabla_{\\\dot{A}}^{C}\overline{{R}}=6\alpha\left(\nabla_{\\\dot{A}}^{C}W^{2}\right)\overline{{\nabla}}^{2}\overline{{W}}^{2}+6\alphaW^{2}\overline{{\nabla}}^{2}\nabla_{\\\dot{A}}^{C}\overline{{W}}^{2}+{\calO}\left(\alpha^{2}\right)
w(r)=r-\sum_{p=2}^{\infty}\left(\frac{t_{p}t_{0}^{p-1}}{(1-t_{1})^{p}}\frac{p!(p-1)!}{(2p)!}\right)\frac{(2p)!}{p!(p-1)!}r^{p}=r-\sum_{p=2}^{\infty}\frac{t_{p}t_{0}^{p-1}}{(1-t_{1})^{p}}r^{p}.
E_{n}^{A_{1}...A_{n}}(x_{1},...,x_{n};\Gamma)=\sum_{m_{1}...m_{n}}\intdxE_{n|m_{1}...m_{n}}^{A_{1}...A_{n}}(x,\Gamma(x),\partial_{\mu}\Gamma(x),...)\prod_{k=1}^{n}(\partial_{\mu})^{m_{k}}\delta(x-x_{k}),
I_{\epsilon}^{(1)}(a^{2})={\frac{\pi}{2}}{\frac{1}{cos{\frac{\pi}{2}}\epsilon}}\left\{{\frac{1}{a^{1+\epsilon}}}+2\sum_{n=1}^{\infty}{\frac{1}{(2\pin)^{1+\epsilon}}}+2\sum_{n=1}^{\infty}{\frac{1}{(2\pin)^{1+\epsilon}}}\;[{\frac{1}{(1+{\frac{a^{2}}{4\pi^{2}n^{2}}})^{\frac{1+\epsilon}{2}}}}-1]\right\}\;.
ds^{2}=\prod_{R=1}^{B}H_{R}^{\frac{p_{R}+1}{D-2}\sigma^{(R)}}\left\{\sum_{L=0}^{D-s-3}\left(\prod_{R=1}^{B}H_{R}^{\Upsilon_{RL}}\right)^{-\sigma^{(R)}}\eta_{KL}dy^{K}dy^{L}+\sum_{\gamma}dx^{\gamma}dx^{\gamma}\right\}.
\begin{array}{ccccc}{}&{SO(2n-1)}&{\subset}&{SO(2n)}&{R_{M}={\bf2n}}\\{}&{Sp(2n)}&{\subset}&{SU(2n)}&{R_{M}={\bfn(2n-1)}}\\{}&{G_{2}}&{\subset}&{SO(7)}&{R_{M}={\bf7}\mathrm{\or\}{\bf8}}\\{}&{F_{4}}&{\subset}&{E_{6}}&{{\bf27}}\\\end{array}
\begin{array}{r}{\overline{{M}}^{n}:~~1+\displaystyle\sum_{i,j=0}^{n}\eta_{jk}x^{j}x^{k}=0~,}\\{1+\displaystyle\sum_{i,j=0}^{n}\eta_{jk}y^{j}y^{k}=0~.}\\\end{array}
G_{\alpha_{1}\ldots\alpha_{n}}(p_{1},\ldots,p_{n})\;\prod_{i=1}^{n}\;J_{\alpha_{i}}^{i}(p_{i})=\hat{G}_{\alpha_{1}\ldots\alpha_{n}}(p_{1},\ldots,p_{n})\;\prod_{i=1}^{n}\;\hat{J}_{\alpha_{i}}^{i}(p_{i})\\\\\.
E_{D}(L_{i},a)=\frac{1}{2}\left(\prod_{i=1}^{D-2}L_{i}\right)\frac{1}{(2\pi)^{D-2}}\int_{0}^{\infty}dk_{1}\dots\int_{0}^{\infty}dk_{D-2}\sum_{n=1}^{\infty}\left[(k_{1})^{2}+\dots+(k_{D-2})^{2}+\left(\frac{n\pi}{a}\right)^{2}\right]^{\frac{1}{2}}.
E=\int_{\calC}d^{2}x\left[\frac{1}{4}W_{ij}^{a}W_{ij}^{a}+\frac{1}{4}B_{ij}B_{ij}+(D_{i}\Phi)^{\dagger}D_{i}\Phi+\lambda(\Phi^{\dagger}\Phi-\phi_{0}^{2})^{2}\right]~~,
{\calL}_{pYM}^{(1)}(D\rightarrow10,5)=-\frac{\hbarg^{5}f^{abf}f^{bcg}f^{cdh}f^{dei}f^{eaj}}{1024\pi^{5}(10-D)}\frac{4}{15}F_{\mu\nu}^{f}F_{\nu\rho}^{g}F_{\rho\sigma}^{h}F_{\sigma\lambda}^{i}F_{\lambda\mu}^{j}
\begin{array}{ll}{D_{a}=\partial_{a}-i\sigma_{a\dot{a}}^{\mu}\bar{\theta}^{\dot{a}}\partial_{\mu},}&{}\\{\bar{D}_{\dot{a}}=-\partial_{\dot{a}}+i\theta^{a}\sigma_{\dot{a}a}^{\mu}\partial_{\mu},}&{}\\\end{array}
\alpha_{IJ}=\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right)\;\qquad\beta_{IJ}=\left(\begin{matrix}{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right)\.
\alpha_{r}^{(N_{f})}(\phi_{3},\ldots,\phi_{N_{f}},\theta)=\sum_{j=1}^{N_{f}-r}e^{-i(j+r/2)(\phi_{3}+\ldots+\phi_{N_{f}}-\theta)}\beta_{j,j+r}^{(N_{f})}(\phi_{3},\ldots,\phi_{N_{f}})\;,
\mathrm{{\mathfrak~S}}_{\mu}=i/\!\!\!\partial\Phi^{a}\gamma_{\mu}G_{ab}\Psi^{b}+i\gamma_{p}\gamma^{\nu}\gamma_{\mu}(N\cdotK)_{\nu}^{~j}~(V_{(aj)}+\gamma_{p}V_{[aj]})\Psi^{a}.
f^{2}g^{2}{\frac{\partial^{2}}{\partialr^{*2}}}t+f\{fg^{\prime\prime}+f^{\prime}g^{\prime}\}{\frac{\partial}{\partialr^{*}}}t-[\sqrt2ff^{\prime}-2f(1-f)]t-{\frac{\partial^{2}}{\partialt^{2}}}t=0.
(\xi,\eta)=\frac{c}{2\pi}\int(\xi^{a}\nabla_{a}\nabla_{b}\nabla_{c}\eta^{c})\,dx^{b}+\frac{h}{2\pi}\int(\xi^{a}D_{ab}\nabla_{c}\eta^{c})\,dx^{b}-(\xi\leftrightarrow\eta).
X_{8}=\frac{1}{2}\frac{1}{\left(2\pi\right)^{6}}\frac{1}{48}\left(\frac{5}{4}trF^{4}-\frac{1}{8}\left(trF^{2}\right)^{2}-\frac{1}{8}trF^{2}trR^{2}+\frac{1}{8}trR^{4}+\frac{1}{32}\left(trR^{2}\right)^{2}\right)
\left(\begin{array}{cc}{\partial_{r}-{\frac{a+j-1/2}{r}}}&{-\sqrt{2}evf}\\{-\sqrt{2}evf}&{\partial_{r}+{\frac{j+1/2}{r}}}\\\end{array}\right)\left(\begin{array}{c}{h_{2}(r)}\\{h_{3}(r)}\\\end{array}\right)=0,
\hat{H}_{ph}^{F}\Phi_{n}^{F}=\left(\frac12\langlep_{a},\hat{g}^{11}p_{a}\rangle+V_{q}[a]+\frac12\langle{\bfB},{\bfB}\rangle\right)\Phi_{n}^{F}=E_{n}\Phi_{n}^{F}\.
H_{R}=\sum_{i}\sum_{n}\frac{1}{2}\left[e^{-3Ht}L^{3}\pi_{\eta_{i}}(k_{n},t)\pi_{\eta_{i}}(-{\bfk}_{n},t)+\frac{e^{3Ht}}{L^{3}}(e^{-2Ht}{\bfk}_{n}^{2}+\mu_{i}^{2})\eta_{i}({\bfk}_{n},t)\eta_{i}(-{\bfk}_{n},t)\right],
ds^{2}=e^{\pm2\eta}\left(e^{2\varphi_{0}}+2\alpha^{\prime}v\frac{2\rho^{2}+e^{\pm2\eta}}{(\rho^{2}+e^{\pm2\eta})^{2}}\right)\left(-d\eta^{2}+d\Omega_{-1}^{2}\right).
{\bar{\partial}}^{-1}\lambda_{n}(a_{1})={\frac{(-1)^{{\frac{n(n-1)}{2}}}}{2^{n}}}\int_{ST_{2}}\prod_{j=2}^{n+1}d\tau_{j}[\prod_{i=1}^{n-1}({\hat{\zeta}_{i,i+1}}\partial_{i+1}-D_{i}{\hat{\zeta}_{i,i+1}}D_{i+1})\partial_{n}{\hat{\zeta}_{n,n+1}}\prod_{l=2}^{n+1}(a_{l})]
\begin{array}{ccccccc}{{\mathcal{H}}_{+}}&{=}&{2\,d\gamma_{I}(z,{\bar{z}})\,\wedge\,\omega^{I}}&{;}&{{\mathcal{H}}_{-}}&{=}&{-2\,d{\bar{\gamma}_{I}}(z,{\bar{z}})\,\wedge\,\omega^{I}\}\\\end{array}
\langle\psi_{{\xi}}^{ml}\vert\frac{{\bf{p}}^{2}}{2m}\vert\psi_{{\xi}}^{ml}\rangle=\frac{2\sqrt{\beta}}{\pi}\int_{-\infty}^{+\infty}\frac{dp}{(1+\betap^{2})^{2}}\frac{p^{2}}{2m}=\frac{1}{2m\beta}\quad.
[~t_{0}^{ii}~,~t_{n}^{ij}~]~=~-~t_{n}^{ij}~,~~~[~t_{0}^{ii}~,~t_{n}^{ji}~]~=~t_{n}^{ji}~,~~~[~t_{0}^{ii}~,~t_{n}^{jk}~]~=~0~,
{\calL}_{g.f.}=-i\bar{\calZ}^{I}\partial_{\tau}{\calZ}^{I}-u^{0}(\bar{\calZ}^{I}\delta^{IJ}{\calZ}^{J}-sR)+\tilde{\pi}_{i}u^{i}+\pi_{0}(u^{0}-1)+b_{a}\partial_{\tau}c^{a}\,.
\Omega_{4-}{}^{ij}=\Omega_{i+}{}^{4j}=\pm\epsilon_{ijk}\partial_{\underline{k}}e^{-2\phi}\,,\qquad\Omega_{k-}{}^{im}=\Omega_{k+}{}^{im}=2\delta_{k[i}\partial_{\underline{m}]}e^{-2\phi}\,.
(F^{2})_{\mu\nu}{}_{ij}=\frac{1}{4}F_{\mu\lambda}^{a}F_{\lambda\nu}^{b}\delta_{ij}+\frac{1}{8}F_{\mu\lambda}^{a}F_{\lambda\nu}^{b}[\sigma^{a},\sigma^{b}]_{ij}
\sigma^{Ti}(\tau^{i})={\frac{1}{2}}(1+(-1)^{L})k^{\tau^{i}}|\tau^{i}|+\sum_{j_{i}=1}^{L}(-1)^{j_{i}+L}k^{\tau^{i}}|\tau^{i}-\tau_{j_{i}}^{i}|~.~\,
\left.{\calF}^{(N_{F})}(\mathrm{v};\{m_{i}\},\Lambda_{N_{F}})\right|_{n=1}=-\frac{i}{\pi}\frac{\Lambda_{N_{F}}^{4-N_{F}}}{\mathrm{v}^{2}}{\calF}_{1}^{(0)}\,\prod_{j=1}^{N_{F}}m_{j}
\begin{array}{l}{x_{i}x_{j}=q_{ij}x_{j}x_{i},}\\{\partial_{i}\partial_{j}=p_{ij}^{-1}\partial_{j}\partial_{i},}\\{\partial_{i}x_{i}=1+p_{ij}q_{ij}x_{i}\partial_{i}+(p_{ij}q_{ij}-1)\sum_{j=i+1}^{N}x_{j}\partial_{j},}\\{\partial_{i}x_{j}=p_{ij}x_{j}\partial_{i},}\\{\partial_{j}x_{i}=q_{ij}x_{i}\partial_{j},}\\\end{array}
\begin{array}{c}{{\bfz}_{i}{\bf{\bar{z}}}_{j}-|c|\delta_{ij}\approx0,}\\{{\bfz}_{i}{\bfz}_{j}\approx0,}\\{{\bfp}{\bfz}_{i}\approx0,}\\{{\bfp}^{2}\approx0.}\\\end{array}
\Gamma^{(2)}=\frac{209}{2880(4\pi)^{4}}\frac{1}{\epsilon}\intd^{4}x\sqrt{-g}R_{\;\;\;\;\gamma\delta}^{\alpha\beta}R_{\;\;\;\;\rho\sigma}^{\gamma\delta}R_{\;\;\;\;\alpha\beta}^{\rho\sigma}
[|\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}-\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}|^{2}]^{\nu}=\sum_{N=0}^{N_{0}}\sum_{\alpha=0}^{N}\frac{(2a)^{2\nu+n}{\pi}^{\frac{n}{2}}\Gamma(\nu+\frac{n}{2}+\frac{1}{N_{0}})\Gamma(N-\nu+\frac{1}{N_{0}})}{\Gamma(N+n+\nu+\frac{1}{N_{0}})\Gamma(-\nu)}Y_{\alpha}^{N}({\eta})Y_{\alpha}^{N}({\eta}^{\prime})
R\sim\left(\begin{matrix}{0}&{\lambda_{1}}&{}&{}&{}&{}&{}&{}\\{-\lambda_{1}}&{0}&{}&{}&{}&{}&{}&{}\\{}&{}&{0}&{\lambda_{2}}&{}&{}&{}&{}\\{}&{}&{-\lambda_{2}}&{0}&{}&{}&{}&{}\\{}&{}&{}&{}&{.}&{}&{}&{}\\{}&{}&{}&{}&{}&{.}&{}&{}\\{}&{}&{}&{}&{}&{}&{0}&{\lambda_{n}}\\{}&{}&{}&{}&{}&{}&{-\lambda_{n}}&{0}\\\end{matrix}\right).
\omega\widetilde{T}_{-}(q,p)\rightarrow\left(\mathrm{e}^{-i\pi\widetilde{T}_{1}}\ast\widetilde{T}_{-}\right)(q,p)=\omega\widetilde{T}_{+}(q,p)=\sum_{i=1}^{N}\frac{p_{i}^{2}}{2}.
^{+}S_{2}=\frac{i}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ab}\,{\calR}_{\mu\nu}^{ab}\,{\calR}_{\alpha\beta}^{cd}\,\epsilon_{abcd}-\frac{1}{2}{^+\tau}\int_{M}eF^{\mu\nui}F^{\alpha\betaj}g_{ij}\,{\calR}_{\mu\nu}^{ab}\,{\calR}_{\alpha\beta}^{cd}\delta_{ab,cd}\,.
G^{B=0}=\left(\begin{array}{cc}{e^{\lambda^{(0)}+\psi^{(0)}}}&{0}\\{0}&{e^{\lambda^{(0)}-\psi^{(0)}}}\\\end{array}\right)=\left(\begin{array}{cc}{\xi_{1}}&{0}\\{0}&{\xi_{2}}\\\end{array}\right)
\begin{array}{rl}{(z_{0};z_{1},z_{2})=}&{{\frac{1}{h}}\left({\frac{1+x_{1}^{2}+h^{2}}{2}}+i\alpha_{0}(x_{1})h^{3};\right.}\\{}&{\left.x_{1}+i\alpha_{1}(x_{1})h^{3},\;\;{\frac{i-ix_{1}^{2}-ih^{2}}{2}}+\alpha_{2}(x_{1})h^{3}\right)}\\\end{array}
(F^{2}-G^{2}p^{2})_{h}\;\left(\frac{h_{_{k_{0}}}}{k_{0}}+\frac{h_{_k}}{k}\right)+(F^{2}-G^{2}p^{2})_{p}\;\left(\frac{p_{_{k_{0}}}}{k_{0}}+\frac{p_{_k}}{k}\right)=0\;.
(e^{-2\phi})_{\mathrm{fix}}={\frac{|n_{2}m_{1}-n_{1}m_{2}|}{m_{1}^{2}+m_{2}^{2}}}\,\qquad(a)_{\mathrm{fix}}={\frac{n_{2}m_{2}+n_{1}m_{1}}{m_{1}^{2}+m_{2}^{2}}}\.
\frac{l}{2}G_{\\nu}^{\mu}+\frac{\chi_{\\nu}^{\mu}}{\Omega^{2}}+\frac{l^{2}}{4}\left(\frac{d}{\Omega^{2}}+\frac{l}{2}\right){\calS}_{\\nu}^{\mu}-\frac{l^{2}}{8}\left(\frac{1}{\Omega^{4}}+1\right)\Box\chi_{\\nu}^{\mu}+l\left(\delta\phi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Box\delta\phi\right)+\frac{C_{\\nu}^{\mu}}{\Omega^{2}}=-\frac{\kappa^{2}}{2}\stackrel{B}{T_{\\nu}^{\mu}}\.
\begin{array}{lclclcl}{\delta_{B}x}&{=}&{-\bar{C}}&{\qquad}&{\delta_{B}\bar{x}}&{=}&{-C}\\{\delta_{B}C}&{=}&{0}&{\qquad}&{\delta_{B}\bar{C}}&{=}&{0}\\{\delta_{B}B}&{=}&{\lambda}&{\qquad}&{\delta_{B}\bar{B}}&{=}&{\bar{\lambda}}\\{\delta_{B}\lambda}&{=}&{0}&{\qquad}&{\delta_{B}\bar{\lambda}}&{=}&{0}\\\end{array}
V^{0}(\vert\phi\vert,\chi,\sigma)~=~N\int{\frac{d^{3}p}{(2\pi)^{3}}}\mathrm{ln}(p^{2}+\sigma)+N(m^{2}\chi+{\frac{\lambda}{2}}\chi^{2}+{\frac{\eta}{3}}\chi^{3})+\sigma(\vert\phi\vert^{2}-N\chi).
S=-\intd^{4}\sigma\sqrt{-\mathrm{det}\,(G_{ij}+e^{-\frac{\phi}{2}}F_{ij}-b_{ij})}+\int\left(C_{4}+C_{2}\wedge(e^{-\frac{\phi}{2}}F-b_{2})+\frac{1}{2}C_{0}F\wedgeF\right)\,.
|1\rangle_{44}=\Bigl((\vec{b}\cdot\vec{c})^{2}-\frac{1}{9}\vec{b}^{2}\vec{c}^{2}-\frac{2}{9}\vec{b}\cdot\vec{c}(b^{2}+c^{2})+\frac{2}{63}(b^{4}+c^{4})\Bigr)|0\rangle
\begin{array}{c}{g(\vartheta|\vartheta_{k})=\displaystyle\sum_{k=1}^{N_{H}}\chi(\vartheta-h_{k})-\displaystyle\sum_{k=1}^{N_{S}}\left(\chi(\vartheta-\hat{y}_{k}+i\eta)+\chi(\vartheta-\hat{y}_{k}-i\eta)\right)+}\\{-\displaystyle\sum_{k=1}^{M_{C}}\chi(\vartheta-c_{k})-\displaystyle\sum_{k=1}^{M_{W}}\chi_{II}(\vartheta-w_{k})}\\\end{array}
\Omega_{h}^{ij}(X)=\sum_{w,{w^{\prime}}\inh}\mathrm{sign}(w,{w^{\prime}})\left(X_{w}^{i}X_{w^{\prime}}^{j}-{\frac{1}{3}}\delta^{ij}X_{w}^{k}X_{w^{\prime}\,k}\right)
I(p;m_{0},r)=\left(\prod_{\mu=1}^{d}c_{\mu}\right)\left[m_{0}+r\sum_{\rho}(c_{\rho}-1)+r\sum_{\rho}\frac{s_{\rho}^{2}}{c_{\rho}}\right](H_{0}^{2})^{-n-\frac{1}{2}}.
u=e^{i\gamma(\phi_{2}+\bar{\phi}_{2})}\left[e^{\gamma(\phi_{1}+\bar{\phi}_{1})}(1+\Phi\bar{\Phi})-\frac{1}{4}e^{-\gamma(\phi_{1}+\bar{\phi}_{1})}+\frac{i}{2}\left(\Phie^{\gamma(\phi_{1}-\bar{\phi}_{1})}+\bar{\Phi}e^{-\gamma(\phi_{1}-\bar{\phi}_{1})}\right)\right],
\{\Theta_{1},\Theta_{2}\}^{\pm}=\mp\frac{2}{N}(\Theta_{1}\Theta_{1}+\Theta_{2}\Theta_{2})+r_{+}^{12}\Theta_{1}\Theta_{2}+\Theta_{1}\Theta_{2}r_{+}^{12}-\Theta_{1}r_{\mp}^{12}\Theta_{2}+\Theta_{2}r_{\pm}^{12}\Theta_{1}.
{\calL}_{\mathrm{daughter}}=\frac{1}{2}\sum_{i=1}^{N}\dot{x}_{i}^{2}-\frac{1}{2}\left\{\sum_{i=1}^{N}x_{i}^{2}+g\sum_{\stackrel{i,j=1}{i\neqj}}^{N}\frac{1}{(x_{i}-x_{j})^{2}}\right\}
b_{k}^{\mathrm{s}}=\epsilon_{kij}\partial_{i}a_{j}^{\mathrm{s}}=\epsilon_{kij}\partial_{i}\left(\frac{1}{2ig}\mathrm{tr}\,\tau_{3}\,\Omega_{\mathrm{D}}\partial_{j}\Omega_{\mathrm{D}}^{\dagger}\right),
X\equivg_{2}^{4/3}(1+3g_{A}^{2})^{1/3}\left({\frac{m_{n}-m_{p}}{m_{W}}}\right)\left({\frac{m_{\mathrm{Planck}}}{m_{W}}}\right)^{1/3}\left({\frac{g_{*}}{10.75}}\right)^{-1/6}\.
Z=\sqrt{\pi}\int{\calD}G\sqrt{detM^{-1}}\int{\calD}\bar{A}exp\{-\int[\alphaG_{\mu\nu}^{a}G_{a}^{\mu\nu}-G_{a}^{\mu\nu}F_{\mu\nu}^{a}(\bar{A})]dx\}\delta(2\bar{A}_{\mu}^{a}+2{M^{-1}}_{\nu\mu}^{ab}\partial_{\rho}G_{b}^{\rho\nu}),
z=\frac{3\left(1+\frac{2{\bfQ}^{2}}{a^{2}}\right)^{2}}{\left[\rho^{2}\left(1+\frac{2{\bfQ}^{2}}{a^{2}}\right)^{2}+6\rho\left(1+\frac{2{\bfQ}^{2}}{a^{2}}\right)+27\right]}.
{\calL}_{E}(\phi_{3})=\frac{1}{2}\left(\partial_{\tau}\phi_{3}\right)^{2}+\frac{1}{2}\left(\nabla\phi_{3}\right)^{2}-\frac{1}{2}m^{2}(0)\phi_{3}^{2}+\frac{\lambda}{4!}\phi_{3}^{4}\,.
\langle\kappa_{\gamma}^{(p)},\kappa_{\gamma^{\prime}}^{(p^{\prime})}\rangle=e^{-{\frac{1}{2}}(\gamma^{*}{}^{i}\gamma_{i}+\gamma^{*}{}^{i}\gamma_{i}^{\prime}-2\gamma^{*}{}^{i}\gamma_{i}^{\prime})^{2}}\delta_{p_{1}p_{1}^{\prime}}\delta_{p_{2}p_{2}^{\prime}}...\delta_{p_{D}p_{D}^{\prime}}.
\left[\tilde{M}_{\tilde{a}\tilde{b}},\tilde{M}_{\tilde{c}\tilde{d}}\right]=\delta_{\tilde{a}\tilde{c}}\tilde{M}_{\tilde{b}\tilde{d}}+\delta_{\tilde{b}\tilde{d}}\tilde{M}_{\tilde{a}\tilde{c}}-\delta_{\tilde{a}\tilde{d}}\tilde{M}_{\tilde{b}\tilde{c}}-\delta_{\tilde{b}\tilde{c}}\tilde{M}_{\tilde{a}\tilde{d}}\,.
h_{1}=-f_{\;bc}^{a}\left(\frac{1}{2}A_{j}^{b}A_{k}^{c}B_{a}^{jk}+\left(\partial^{l}B_{0l}^{b}\right)\left(B_{a}^{0i}A_{i}^{c}+\eta_{a}^{\left(2\right)i}{\calP}_{i}^{\left(2\right)c}+\eta_{a}{\calP}^{c}\right)\right),
T_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,\qquadT_{+}=\left(\begin{array}{cc}{0}&{1}\\{0}&{0}\\\end{array}\right)\,\qquadT_{-}=\left(\begin{array}{cc}{0}&{0}\\{1}&{0}\\\end{array}\right)\.
R_{\mu\nu}=Tr[\frac{1}{2}(J_{\mu}^{p}J_{\nu}^{p}+J_{\mu}^{q}J_{\nu}^{q})-2p(F_{\mu\lambda}{F^{T}}_{\nu}^{\lambda}-\frac{1}{4}g_{\mu\nu}FF^{T})]
\lambda_{0}=-\frac{1}{m}\left(D_{i}D^{i}\varphi+m^{2}\varphi\right)+ie\varphi_{0}A_{0};\\lambda_{0}^{\ast}=-\frac{1}{m}\left(D_{i}^{\ast}D^{\asti}\varphi^{\ast}+m^{2}\varphi^{\ast}\right)-ie\varphi_{0}^{\ast}A_{0};
\sum_{c=1}^{N}\sum_{d=1}^{N}\left\{\left(\frac{\partial}{\partial\beta}-\frac{\partial^{2}}{\partialx^{2}}+\bar{v}_{a}^{2}\right)\delta_{ac}-\bar{V}_{ac}(x)\right\}[A]_{cd}(x,x^{\prime})[K_{\bar{\calV}}]_{db}(x,x^{\prime};\beta)=0\quad,
{\calR}_{z\doteqz_{n}=\cdots=z_{0}}F=\mathrm{Res}_{z=z_{0}}{\calR}_{z_{n}=\cdots=z_{0}}\partial_{\bar{z}}^{-1}F-\sum_{i=0}^{N}{\calR}_{z_{n}=\cdots=z_{0}}\mathrm{Res}_{z=z_{i}}\partial_{\bar{z}}^{-1}F.
J_{1lin}=\frac{m(T_{o})T_{o}}{8\pi^{2}}\left(He^{Ht_{o}}\right)^{4}\int_{1}^{\infty}\frac{dz}{z}\frac{\sqrt{z^{2}-1}}{\left[\frac{\lambda_{R}T_{c}^{2}}{24}+m^{2}(T)z\right]^{2}}
\begin{array}{c}{{\calE}=-\int_{M_{3}}d(\mathrm{tr}(\Phi_{A}H))-\int_{M_{3}}d(\mathrm{tr}(\Phi_{B}K))}\\{=-\int_{S^{2}}\mathrm{tr}(\Phi_{A}H)-\int_{S^{2}}\mathrm{tr}(\Phi_{B}K)}\\\end{array}
\delta_{h}S=\int_{{\calM}^{1+1}}\left({\frac{1}{2}}\hat{E}^{I}\wedge\hat{E}^{--}i_{\delta}f^{++I}-{\frac{1}{2}}\hat{E}^{I}\wedge\hat{E}^{--}i_{\delta}f^{++I}+mj_{1}\wedge\hat{E}^{i}i_{\delta}f^{i}\right)
{\calG}_{B}^{(1)}(\tau_{1},\tau_{2})={\calG}_{B}(\tau_{1},\tau_{2})+{\frac{1}{2}}{\frac{[{\calG}_{B}(\tau_{1},\tau_{a})-{\calG}_{B}(\tau_{1},\tau_{b})][{\calG}_{B}(\tau_{a},\tau_{2})-{\calG}_{B}(\tau_{b},\tau_{2})]}{{\bar{T}}-{\frac{1}{2}}{\calC}_{ab}}}\,
\left(\begin{matrix}{X^{I}}\\{F_{I}}\\\end{matrix}\right)\longrightarrow\left(\begin{matrix}{\tilde{X}^{I}}\\{\tilde{F}_{I}}\\\end{matrix}\right)=\left(\begin{matrix}{U}&{Z}\\{W}&{V}\\\end{matrix}\right)\left(\begin{matrix}{X^{I}}\\{F_{I}}\\\end{matrix}\right)\,.
g\equivG/H=e^{x^{\pm}L_{-1}^{\pm}}e^{\psi_{1}^{\pm}J_{1}^{\pm}}e^{\xi_{1}^{\pm}L_{1}^{\pm}}e^{\psi_{2}^{\pm}J_{2}^{\pm}}\ldots\;e^{u(L_{0}^{+}+L_{0}^{-})}e^{\phi(J_{0}^{+}+J_{0}^{-})}\quad.
\begin{array}{rcl}{\mathrm{integral}}&{\sim}&{\displaystyle\intd\ell_{E}\ell_{E}^{n-1}v^{\prime-1}\tau(k^{\prime2}-4(\ell_{E}^{2}+m^{2}))}\\{}&{\sim}&{\displaystyle\int_{0}^{\eta}dyy^{(n-2)/2}(\eta-y)^{-1/2}}\\{}&{\propto}&{\displaystyle\eta^{(n-1)/2},}\\\end{array}
\frac{1}{2}p^{i}p_{i}+\sum_{n=1}^{\infty}n(N_{bn}+N_{cn}+\sum_{i=d+1}^{25}N_{in})+\sum_{r=1/2}^{\infty}\sum_{a=0}^{d}rN_{ar}={\frac{15-d}{16}}.
\begin{array}{ll}{sA_{\mu}^{a}=-(D_{\mu}c)^{a},}&{}\\{s\varphi^{a}=f^{abc}c^{b}\varphi^{c},}&{}\\{sc^{a}=\frac{1}{2}f^{abc}c^{b}c^{c},}&{}\\{s\bar{c}^{a}=b^{a},}&{sb^{a}=0,}\\{s^{2}=0.}\\\end{array}
\left(\mu_{3}^{A},\bar{\Phi}_{B}^{(2)}\right)_{1}=\delta_{B}^{A},\;\left(\bar{\Phi}_{A}^{(3)},\mu_{2}^{B}\right)_{1}=\delta_{A}^{B},\;\left(\nu_{1}^{A},\tilde{\Phi}_{B}\right)_{1}=\delta_{B}^{A}.
\widehat{\calM}^{(B)}(\omega)=\frac{1}{2}\left(\begin{array}{cccc}{{\calM}_{1}+{\calM}_{2}^{-1}}&{0}&{0}&{{\calM}_{1}-{\calM}_{2}^{-1}}\\{0}&{{\calM}_{2}+{\calM}_{1}^{-1}}&{-{\calM}_{2}+{\calM}_{1}^{-1}}&{0}\\{0}&{{\calM}_{2}+{\calM}_{1}^{-1}}&{{\calM}_{2}+{\calM}_{1}^{-1}}&{0}\\{{\calM}_{1}-{\calM}_{2}^{-1}}&{0}&{0}&{{\calM}_{1}+{\calM}_{2}^{-1}}\\\end{array}\right)
F_{mn}^{a}=-4\rho^{2}\frac{\overline{\eta}_{mn}^{a}}{(x^{2}+\rho^{2})^{2}}+8\rho^{2}\frac{(\overline{\eta}_{mk}^{a}x_{n}-\overline{\eta}_{nk}^{a}x_{m})x_{k}}{x^{2}(x^{2}+\rho^{2})^{2}}~.
\overline{{\Gamma}}_{5}\left(\Gamma_{\mu}\partial_{\mu}+mP_{1}+\frac{m_{0}^{2}}{m}P_{0}\right)=\left(\Gamma_{\mu}\partial_{\mu}+mP_{1}+\frac{m_{0}^{2}}{m}P_{0}\right)\overline{{\Gamma}}_{5}.
S_{0}\left\vert\begin{array}{c}{N}\\{(p,q)}\\\end{array},k\right>=\left(k+\displaystyle\frac{1}{2}\left(\frac{2p-1}{2m}-\frac{2q-1}{2n}-N\right)\right)\left\vert\begin{array}{c}{N}\\{(p,q)}\\\end{array},k\right>,
\mu^{\prime\prime}(T_{c}^{+})\simeq-\frac{16\pi^{2}}{9M}\:,\qquad\qquad\qquad\qquad\left.T\frac{\partial\varrho}{\partialT}\right|_{T=T_{c}^{+}}\simeq2\varrho\:,\nonumber
D_{{\vec{c}},n-b}=D_{{\vec{c}},n}e^{-i2\pik_{2}/N}e^{-i2\pi\left(pw-qNb\right)\midc\mid/\ellpN}e^{\frac{i\pi\tau}{\omega\kappa^{2}}b^{2}}e^{\frac{-i2\pi\tauN_{A}}{\omega\kappa^{2}}\left(n+\beta/2\pi\right)b}e^{\frac{i\alphaN_{A}}{\omega\kappa^{2}}b}
Q_{\mu\rho,\nu\sigma}=P_{\mu\rho,\nu\sigma}^{\mathrm{{(0-s)}}}+\sqrt{2}(P_{\mu\rho,\nu\sigma}^{\mathrm{(0-sw)}}+P_{\mu\rho,\nu\sigma}^{\mathrm{{(0-ws)}}})+2P_{\mu\rho,\nu\sigma}^{\mathrm{(0-w)}}.
\begin{array}{l}{N\ge6,\quad\alpha=6,2,1,-1,}\\{\displaystyleT(f)=f\partial_{t}+\frac{1}{N}f^{\prime}(x\partial_{x}+2y\partial_{y}-w\partial_{w}),}\\{\displaystyleY(g)=g\partial_{y},\quadX(h)=h\partial_{x},}\\{\displaystyleW(k)=k\partial_{w}.}\\\end{array}
\frac{\partial^{a-1}}{\partialx_{i_{a}}^{a-1}}\frac{\partial^{a-2}}{\partialx_{i_{a-1}}^{a-2}}\dots\frac{\partial}{\partialx_{i_{2}}}F_{r_{1},r_{2},\dots,r_{n};\theta_{1},\theta_{2},\dots,\theta_{n}}^{p,g;\iota_{1},\dots,\iota_{n}}
\frac12\int_{\partial\Sigma_{t}}\nabla_{\mu}K_{\nu}d\Sigma^{\mu\nu}=\frac23g^{2}\int_{\Sigma_{t}}V(X)K_{\nu}d\Sigma^{\nu}+\int_{\Sigma_{t}}(F_{\nu\rho}^{2}-\frac16g_{\nu\rho}F^{2}+G_{IJ}\partial_{\nu}X^{I}\partial_{\rho}X^{J})K^{\rho}d\Sigma^{\nu},
\begin{array}{rcl}{e\vertj_{k},j_{k-(2i+1)}\rangle}&{=}&{[j_{k}+j_{k-(2i+1)}+1]\vertj_{k},j_{k-(2i+1)}+1\rangle}\\{}&{=}&{[n^{\prime}p]\vertj_{k},j_{k-(2i+1)}+1\rangle=0~.}\\\end{array}
{\hat{\calR}}=\sqrt{q}P_{12}{\calR}^{\frac{1}{2}\frac{1}{2}}=\left[\begin{matrix}{q}&{0}&{0}&{0}\\{0}&{q-1/q}&{1}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{q}\\\end{matrix}\right]
V=4h_{uv}k_{\Lambda}^{u}k_{\Sigma}^{v}\bar{X}^{\Lambda}X^{\Sigma}\mathrm{e}^{\mathcalK}+U^{\Lambda\Sigma}P_{\Lambda}^{i}P_{\Sigma}^{i}-3P_{\Lambda}^{i}\bar{X}^{\Lambda}P_{\Sigma}^{i}X^{\Sigma}\mathrm{e}^{\mathcalK}\,,\quad(i=1,2,3).
R_{ab}\left(\frac{\lambda}{\lambda^{\prime}}\right){\calZ}_{ba}^{-1}\Pi_{a}(\lambda){\hat{\calZ}_{ab}}^{-1}\Pi_{b}(\lambda^{\prime})={\calZ}_{ab}^{-1}\Pi_{b}(\lambda^{\prime}){\hat{\calZ}_{ba}}^{-1}\Pi_{a}(\lambda)R_{ab}\left(\frac{\lambda}{\lambda^{\prime}}\right)\,,
\Xi^{\mu}=\bar{\Xi}^{\mu}+{\frac{{l_{B}}^{2}}{4}}\varepsilon^{mn}{n_{m}}^{\mu}{\rho_{j}}^{l},_{n}\varepsilon^{jk}(\bar{\Pi}_{k}\bar{\Pi}_{l}+\bar{\Pi}_{l}\bar{\Pi}_{k})+{\calO}({l_{B}}^{4})
W_{\mu}^{T}=W_{\mu}^{F}+W_{\mu}^{G}=\intd^{4}\!x\,tr\big(F_{\mu\nu}(x){\frac{\delta}{\deltaA_{\nu}(x)}}+D_{\nu}A_{\mu}(x){\frac{\delta}{\deltaA_{\nu}(x)}}\big),
\sum_{n=0}^{\infty}\sum_{k=0}^{nd}(-1)^{k}y^{k}b_{k}(S^{n}X)q^{n}=\prod_{n=1}^{\infty}\prod_{k=0}^{d}(1-y^{k+(n-1)\fracd2}q^{n})^{-(-1)^{k}b_{k}(X)},
A_{\mu}^{(1)}(x)\;\longrightarrow\;\tilde{A}_{\mu}^{(1)}(x)\;=\;A_{\mu}^{(1)}(x)\;-\;\frac{i}{e}\Big(\partial_{\mu}\tilde{S}(x)\Big)\Big(\tilde{S}(x)\Big)^{-1}\;,