harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega\equiv\frac{p}{\rho}=-\frac{1}{3}-\frac{2}{3}\frac{\Gamma\left[\alpha+7/2,K_{+}^{2}/(a^{2}k_{\mathrm{c}}^{2})\right]}{\Gamma\left[\alpha+5/2,K_{+}^{2}/(a^{2}k_{\mathrm{c}}^{2})\right]},
{{\calL}_{Q}}^{\prime}=\mathrm{Tr}[s{\bar{s}}(-{\frac{1}{2}}A_{\mu}A^{\mu}+a_{1}\,{\bar{\alpha}}\alpha+{\frac{1}{4}}B_{\mu\nu}B^{\mu\nu}+a_{2}\,{\bar{\beta}}_{\mu}{\beta}^{\mu}+{a_{2}}^{\prime}\,{\bar{\phi}}\phi+{a_{2}}^{\prime\prime}\,{\frac{1}{2}}{\rho}^{2})],
A_{2}=\intdy\left\langlec\bar{c}V_{-2,0}(\zeta^{(1)},k_{1})(c+\bar{c})V_{0,0}(\zeta^{(2)},k_{2})(iy)\right\rangle={\frac{1}{k_{1}.k_{2}}}{H_{\mu\nu\rho}^{(1)}H^{(2)\mu\nu\rho}}.
\frac{\chi_{i}^{\dagger}\chi_{i}}{(2\pi{\alpha^{\prime}})^{2}}\sum_{a=1}^{5}(X_{a}-Y_{a})^{2}+\frac{1}{4g_{0}^{2}}\sum_{I=1}^{3}(\chi_{i}^{\dagger}\tau^{I}\chi_{i})^{2}\.
S_{\mathrm{free}}=\frac{1}{2}\int\!\!d^{D}\!x[(\partial_{x}\Phi)^{2}-m^{2}\Phi^{2}]=\sum_{k}\frac{1}{2}\int\!\!d^{D-1}\!{\bfx}[(\partial_{\bfx}\phi_{k}^{*})(\partial_{\bfx}\phi_{k})-m_{k}^{2}\phi_{k}^{*}\phi_{k}],
{\calZ}_{BF}={\displaystyle\int}{\calD}(A^{\mu},\theta)~exp~{i{\displaystyle\int}d^{4}x\left[-\frac{1}{4}F_{\mu\nu}^{2}+\frac{m^{2}}{2}A_{\mu}^{2}+\frac{m^{2}}{2}\partial_{\mu}\theta\partial^{\mu}\theta-m^{2}\theta\partial_{\mu}A^{\mu}\right].}
I^{ANEC}=\int_{-\infty}^{\infty}e^{-2\phi}G_{{\hat{\mu}}{\hat{\nu}}}k^{\hat{\mu}}k^{\hat{\nu}}d\lambda=\int_{r_{H}}^{\infty}\left(\rho+\tau\right)e^{-2\psi-2\phi}\frac{d\lambda}{dr}dr
\mathcal{N}^{-1}\int(d\tilde{f}^{a}(x))\frac{\deltaS_{eff}}{\deltat_{\mu}^{a}(x)}e^{-\frac{1}{\ell}\int_{0}^{\infty}ds\tilde{f}^{a}(s)\tilde{f}^{a}(s)}=0.
a_{\nu}^{\alpha}=g^{-1}\left[{\calD}_{\nu}\,U_{\dot{\beta}}\,\bar{\omega}^{\dot{\beta}}\right]^{\alpha}\,,\qquad\left\|\frac{\partiala_{\nu}^{\alpha}}{\partial\bar{\omega}^{\dot{\beta}}}\right\|=\frac{\sqrt{2}\,\pi\rho}{g}\,.
{\calA}=1-\left|{\frac{A_{\mathrm{out}}}{A_{\mathrm{in}}}}\right|^{2}={\frac{(e^{2\sqrt{s-1}\pi}-1)(e^{\omega^{\prime}/T_{H}}-1)}{\left(e^{{\omega_{R}^{\prime}/2T_{R}}+\sqrt{s-1}\pi}+1\right)\left(e^{{\omega_{L}^{\prime}/2T_{L}}+\sqrt{s-1}\pi}+1\right)}},
\lambda^{\alpha}=-\frac{i}{4}{\bar{D}}^{2}L^{\alpha}\,\qquad\Omega_{\alpha\beta}=\frac{i}{8}{\bar{D}}^{2}D_{(\alpha}L_{\beta)}\,\qquad\sigma=-\frac{i}{24}{\bar{D}}^{2}D^{\alpha}L_{\alpha}\.
\Lambda_{LR}\left(\vec{\theta},\vec{\varphi}\right)\Psi_{LR}=\left(\begin{array}{cc}{e^{i\vec{J}\cdot(\vec{\theta}+i\vec{\varphi})}}&{0}\\{0}&{e^{i\vec{J}\cdot(\vec{\theta}-i\vec{\varphi})}}\\\end{array}\right)\left(\begin{array}{c}{\Phi_{L}}\\{\Phi_{R}}\\\end{array}\right)\,.
\begin{array}{l}{e(+q_{1}+q_{2}+q_{3}+\cdots+q_{k}+n/2)\beta_{1}+}\\{e(-q_{1}+q_{2}+q_{3}+\cdots+q_{k}+n/2)\beta_{2}+}\\{e(-q_{1}-q_{2}+q_{3}+\cdots+q_{k}+n/2)\beta_{3}+}\\{e(-q_{1}-q_{2}-q_{3}+\cdots+q_{k}+n/2)\beta_{4}+}\\{\vdots}\\{e(-q_{1}-q_{2}-q_{3}-\cdots-q_{k}+n/2)\beta_{k+1}.}\\\end{array}
a_{1}=\tilde{a}(1)+\frac{1}{e}F,\;\;\;b_{1}=\tilde{b}_{1}(1)-\frac{1}{\sqrt\betae}F,\;\;\;b_{2}=\tilde{b}_{2}(2)-\sqrt\betaF,\;\;\;c_{2}=\tilde{c}(2)+F.
X=-\sqrt{\frac{2}{|\kappa|}}u+{\lambda}^{2}\sqrt{\frac{2}{|\kappa|}}\int^{x^{+}}\mathrm{e}^{g_{+}}\int^{x^{-}}\mathrm{e}^{g_{-}}=-Y+\sqrt{\frac{|\kappa|}{2}}(g_{+}+g_{-}).
\begin{array}{l}{\Psi^{\prime}(u)=U_{R}\Psi(u),}\\{\left(g(u)\nabla\Psi(u)\right)^{\prime}=U_{R}\left(g(u)\nabla\Psi(u)\right),\quadU_{R}=R({\bfB}_{f}^{\prime})U_{f}R^{-1}({\bfB}_{f}),}\\\end{array}
(\frac{d^{2}}{d\sigma_{c}^{2}}+\omega^{2})D_{\omega}^{R}(\sigma_{c})=m\frac{2\sigma_{c}^{2}-b^{2}}{(\sigma_{c}^{2}+b^{2})^{5/2}}\left(a_{\omega}^{R}e^{i\omega\sigma_{c}}+b_{\omega}^{R}e^{-i\omega\sigma_{c}}\right),
H_{D}^{F}=\int{d^{3}\sigma\Big[\lambda^{\mu}(\tau,\vec{\sigma}){\calH}_{\mu}^{\ast}(\tau,\vec{\sigma})-A_{\tau}(\tau,\vec{\sigma})\Gamma(\tau,\vec{\sigma})+\mu_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})\Big]}.
W(a|efg|bcd|h)=R_{\alphae+\betac+\gammag+\deltaa,\alphad+\betae+\gammaa+\deltaf,\alphaf+\betaa+\gammag+\deltab}^{\alphad+\betah+\gammab+\deltaf,\alphah+\betac+\gammag+\deltab,\alphad+\betae+\gammac+\deltah}
J^{i^{\prime}j^{\prime}}=\widehat{J}^{i^{\prime}j^{\prime}}-\frac{i}{4}\intd\sigma(\psi_{+}^{1}\gamma^{i^{\prime}j^{\prime}}\psi_{+}^{1}+\psi_{-}^{2}\gamma^{i^{\prime}j^{\prime}}\psi_{-}^{2})~.
\begin{array}{l}{E_{1}{\circ_{1,Z}}E_{2}=E_{6},\;\;\;\;E_{3}{\circ_{1,Z}}E_{5}=E_{6},}\\{E_{5}{\circ_{1,Z}}E_{2}=E_{4},\;\;\;\;E_{1}{\circ_{1,Z}}E_{3}=E_{4}.}\\\end{array}
|m_{A_{5}}^{2}|=\frac{c_{5}g^{2}}{\pi^{4}R_{5}^{2}}<g^{2}A_{6}^{2}\,(\mathrm{initial})\lesssim\frac{1}{4R_{6}^{2}}\quad\Rightarrow\quadR_{5}>\frac{2\,\sqrt{c_{5}}\,g}{\pi^{2}}\,R_{6}\;.
Y_{J_{2},n_{2}}^{+}(1)W_{J_{3},n_{3}}^{+}(x)=\frac{1}{1-x}f_{J_{2}n_{2},J_{3}n_{3}}^{J_{2}+J_{3}-1,n_{2}+n_{3}}Y_{J_{2}+J_{3}-1,n_{2}+n_{3}}^{+}(1)+regular~terms
S=S_{a}+S_{vis}=\int\frac{d^{4}x}{z_{v}^{4}}\left(\left(\lambda_{v1}+\lambda_{h1}\left(\frac{z_{v}}{z_{h}}\right)^{2-\nu}\right)\chi+\left(\lambda_{v2}^{(0)}+\lambda_{h2}^{*(0)}+(\lambda_{h2}^{(0)}-\lambda_{2}^{*(0)})\left(\frac{z_{v}}{z_{h}}\right)^{-2\nu}\right)\frac{\chi^{2}}{2}\right).
\partial_{y}\bigg[e^{-2ky}\bigg(\partial_{y}(e^{2ky}h)+\frac{2}{3k}M^{-3}\bigg(1-\frac{4\alphak^{2}}{M^{2}}\bigg)^{-1}T_{55}\bigg)\bigg]=-\frac{2}{3k}M^{-3}\bigg(1-\frac{4\alphak^{2}}{M^{2}}\bigg)^{-1}\partial^{\mu}T_{\mu5}.
j_{n,a}^{\mu}(k)=-igf^{abc}\intd\Omega{\frac{1}{v\cdotk+i0^{+}}}\sum_{k^{\prime}+k^{\prime\prime}=k}(v\cdotA(k^{\prime}))_{b}j_{n-1,c}^{\mu,\Omega}(k^{\prime\prime})\\\\\\\\\\n>2,
J_{4}=\sum_{i=1}^{4}|\lambda_{i}|^{4}-2\sum_{j>i}^{4}|\lambda_{i}|^{2}|\lambda_{j}|^{2}+4(\bar{\lambda}_{1}\bar{\lambda}_{2}\bar{\lambda}_{3}\bar{\lambda}_{4}+\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}).
I(m_{0},r)=a^{d}\int_{\calB}d^{d}p\I(p;m_{0},r)=\sum_{n_{\pi}=0}^{[m_{0}/2r]}(-1)^{n_{\pi}}{\binom{d}{n_{\pi}}}\frac{2^{d+1}\pi^{n}n!}{d!}.
g_{1}:{\bff}\rightarrow{\frac{1}{T^{2}}}(M_{1}{\bff}+A(T,U)),~~g_{2}:{\bff}\rightarrow{\frac{1}{(T+1)^{2}}}(M_{2}{\bff}+B(T,U)),~~\sigma:{\bff}\rightarrowM_{\sigma}({\bff}+K(T,U))
\int_{0}^{r_{0}}rdr(\tilde{T}_{_{(0)}\theta}^{\theta}+\tilde{T}_{_{(0)}r}^{r})=r_{0}^{2}\tilde{T}_{_{(0)}r}^{r}(r_{0})=r_{0}^{2}\left[\frac{A^{2}(r_{0})}{2e^{2}}+\frac{\alpha}{2}\Lambda^{2}(r_{0})\right].
\rho_{j}=\rho_{j}(p,q):=\frac{1}{2\pi\hbar}\\int_{-\infty}^{+\infty}\<q-\frac{\xi}{2}|\psi_{j}><\psi_{j}|q+\frac{\xi}{2}>\\\mathrm{exp}\\\bigg(\frac{ip\xi}{\hbar}\bigg)d\xi
{\calL}_{\mathrm{eff}}/N={\frac{1}{2}}\tilde{A}_{\mu}^{(1)}\left\{(k^{2}-{\frac{e^{2}}{\pi}})\left(-g^{\mu\nu}+{\frac{k^{\mu}k^{\nu}}{k^{2}}}\right)+2e^{2}\xi_{0}^{2}U(k^{2}){\frac{k^{\mu}k^{\nu}}{k^{2}}}\right\}\tilde{A}_{\nu}^{(1)}~.
(A^{(V)})_{\mathrm{~}\hat{c}}^{\hat{b}}\equiv(\omega^{a}J_{a}^{(V)}+E^{a}P_{a}^{(V)})_{\mathrm{~}\hat{c}}^{\hat{b}}=\left(\begin{array}{cccc}{0}&{\omega^{2}}&{-\omega^{1}}&{-E^{0}}\\{\omega^{2}}&{0}&{-\omega^{0}}&{-E^{1}}\\{-\omega^{1}}&{\omega^{0}}&{0}&{-E^{2}}\\{-E^{0}}&{E^{1}}&{E^{2}}&{0}\\\end{array}\right).
\sum_{n}\frac{\partial\beta_{n}}{\partialM}+\sum_{B}\frac{\partial\beta_{B}}{\partialM}=0;~~~~\frac{1}{4\pi}(1-\frac{M}{4\pi})=\sum_{n}\frac{\partial\beta_{n}}{\partialM}z_{n}+\sum_{B}\frac{\partial\beta_{B}}{\partialM}z_{B}.
K^{\prime}(\zeta)=\frac{1}{f^{\prime}(z;r)}\overline{{K^{\prime}}}(\zeta),~~~~f^{\prime}(z;r)=x^{-2}z\frac{\varphi_{1}(z^{-1};r)}{\varphi_{1}(x^{-4}z;r)}\frac{\varphi_{2}(x^{2}z^{-1};r)}{\varphi_{2}(x^{-2}z;r)},
W^{A}[N]=\int{\calD}\phie^{iS_{\mathrm{eff}}^{L}[\phi]}e^{i(\epsilon[O_{1}^{\prime}+\sum\delta\phi_{i}[\phi]{\frac{\deltaO_{1}^{\prime}}{\delta\phi_{i}}}]+N[O+\sum_{i}\delta\phi_{i}[\phi]{\frac{\deltaO}{\delta\phi_{i}}}])}.
\simeq\int\mathrm{d}^{4}w\frac{(w-x)_{\mu}}{2\sigma}\frac{e^{-\frac{(w-x)^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}W_{Ax,Ew}\int\mathrm{d}^{4}z\frac{(z-w)_{\nu}}{2\sigma}\frac{e^{-\frac{(z-w)^{2}}{4\sigma}}}{(4\pi\sigma)^{2}}W_{Ew,Cz}\left(\frac{-1}{\bar{D}^{2}}\right)_{Cz,By}
<0|T_{\mu\nu}|0>={\frac{\hbark}{\delta^{4}}}\left[\begin{array}{cccc}{4(T/\delta)^{2}+1\quad}&{0\quad}&{0\quad}&{4T\ell/\delta^{2}}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{4T\ell/\delta^{2}}&{0}&{0}&{4(\ell/\delta)^{2}-1}\\\end{array}\right]
p^{\mu}(\Sigma)=2\beta\int_{\Sigma}\sqrt{h}\;\left\{\eta^{\mu}\left[\kappa_{AB}\kappa^{AB}-\kappa^{2}+\eta^{a}\eta^{b}{\calR}_{ab}\right]+\eta^{a}{\calR}_{ab}e^{\mu\,b}+K_{AB}^{i}(h^{AB}\kappa-\kappa^{AB})n^{\mu}{}_{i}\right\}\,.
M=\left(\begin{matrix}{1}&{\tau_{1}}&{w_{2}}\\{0}&{1}&{w_{1}}\\{0}&{0}&{1}\\\end{matrix}\right)\left(\begin{matrix}{\sqrt{\tau_{2}}e^{\phi/3}}&{0}&{0}\\{0}&{\frac{1}{\sqrt{\tau_{2}}}e^{\phi/3}}&{0}\\{0}&{0}&{e^{-2\phi/3}}\\\end{matrix}\right),
\xi_{A}(p,q;t)=\sum_{\alpha=1}^{n}\left[(C_{\alpha}^{-1}p_{\alpha})x_{\alphaA}+(D_{\alpha}^{-1}q_{\alpha})y_{\alphaA}\right]+\sum_{\beta=1}^{N-2n}(r_{\beta}t+K_{\beta})z_{\betaA}
\partial^{\mu}\partial_{\mu}\left[\;\;a_{H}^{ij}({\bfx},t)-\frac{1}{3}\delta^{ij}\delta_{kl}a_{H}^{kl}({\bfx},t)\;\;\right]=-\intd^{3}y\;D^{ij,kl}({\bfx}-{\bfy})\theta_{H,kl}({\bfy},t).
(2-2h)F^{h}(g)=r^{h}(g)+\sum_{n=1}^{\infty}\beta_{n}(g)\sum_{\stackrel{\scriptstyleh_{1}+\cdots+h_{n}=h,}{h_{i}\geq0}}\prod_{i=1}^{n}\frac{dF^{h_{i}}}{dg}.
(\xi)_{\mathrm{B}}=\xi,~~(f)_{\mathrm{B}}=f=\partial^{\mu}G_{\mu}+\frac{igv\xi}{\sqrt{2}}\left(\phi^{\dagger}t^{A}\hat{\phi}_{0}-\hat{\phi}_{0}^{\dagger}t^{A}\phi\right)t^{A},
j^{\mu}=\varepsilon^{\mu\nu\rho\sigma}\varepsilon_{abcd}[D_{\nu}\phi^{a}D_{\rho}\phi^{b}D_{\sigma}\phi^{c}\phi^{d}-\frac{3}{4}F_{\rho\sigma}^{cd}D_{\nu}\phi^{a}\phi^{b}].
\begin{array}{l}{P=\left(\begin{array}{cc}{\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}}&{iT}\\{0}&{\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}}\\\end{array}\right)\times\left(\begin{array}{cc}{L}&{0}\\{0}&{L^{\left(+-1\right)}}\\\end{array}\right)}\\\end{array}
\begin{array}{ccc}{\int_{M_{3}}K_{[\mu\nu}^{a}(D_{\rho]}^{B}\Phi_{B})^{b}\\mathrm{tr}(T^{a}T^{b})}&{-}&{K_{[\mu\nu}^{a}(D_{\rho]}^{A}\Phi_{A})^{b}}\\{mathrm{tr}(T^{a})\mathrm{tr}(T^{b})}\\{-H_{[\mu\nu}^{a}(D_{\rho]}^{B}\Phi_{B})^{b}}\\{mathrm{tr}(T^{a})\mathrm{tr}(T^{b})}&{+}&{(A\leftrightarrowB,\Phi_{A}\leftrightarrow\Phi_{B}).}\\\end{array}
\left[L_{j}^{ch},L_{k}^{ch}\right]=(k-j)L_{j+k}^{ch},\qquad\left[\overline{L}_{j}^{ch},\overline{L}_{k}^{ch}\right]=(k-j)\overline{L}_{j+k}^{ch},\qquad\left[L_{j}^{ch},\overline{L}_{k}^{ch}\right]=0.
\left.\underbrace{\frac{\partial^{n}\Omega}{\partialz^{\alpha}\partialz^{\beta}\cdots\partialz^{\rho}}}_{\mathrm{~m~times}}\right|_{(n-m,m)form\;part}=\chi_{\alpha}^{i_{1}}\wedge\chi_{\beta}^{i_{2}}\wedge\cdots\wedge\chi_{\beta}^{i_{m}}\Omega_{{i_{1}}{i_{2}}\cdots{i_{m}}.},
Z=\left(\begin{matrix}{0}&{-x_{1}}&{x_{2}}&{-x_{4}}&{-x_{6}}&{x_{9}}\\{}&{0}&{x_{3}}&{-x_{5}}&{-x_{8}}&{x_{12}}\\{}&{}&{0}&{x_{7}}&{x_{11}}&{-x_{15}}\\{}&{}&{}&{0}&{-x_{14}}&{x_{13}}\\{}&{a/s}&{}&{}&{0}&{x_{10}}\\{}&{}&{}&{}&{}&{0}\\\end{matrix}\right)
t^{(N)}(\lambda,\{\lambda_{i}\},\omega)=\displaystyle\sum_{a_{1}}T_{a_{1}a_{1}}^{(N)}=\displaystyle\sum_{a_{1},...a_{N}}t_{a_{1}a_{2}}(\lambda-\lambda_{1})\otimest_{a_{2}a_{3}}(\lambda-\lambda_{2})\otimes...\otimest_{a_{N}a_{1}}(\lambda-\lambda_{N})e^{i\omegab}
J^{(N)}(N-1;1,\ldots,1)=\frac{1}{\Lambda^{(N)}}\;\left(\prodm_{i}^{2}\right)\;\sum_{i=1}^{N}\frac{1}{m_{i}^{2}}\;F_{i}^{(N)}\;J_{i}^{(N)}(N-1;1,\ldots,1).
X={\frac{1}{\sqrtg}}\left[\left({\dot{D}}_{\mu}{\sqrtg}g^{\mu\nu}A_{\nu}\right)+{\sqrtg}A_{\mu}A^{\mu}+{\frac{1}{4}}{\sqrtg}\phi_{\mu}\phi^{\mu}-{\frac{1}{2}}\left({\sqrtg}g^{\mu\nu}\phi_{\nu}\right)_{,\mu}\right]-{\frac{1}{2}}\sigma^{\mu\nu}\left(F_{\mu\nu}-\sigma^{ab}R_{\mu\nuab}\right).
W_{loop}=\sum_{i=1}^{n}{X_{i}^{a_{i}}X_{i+1}}\bar{W}_{loop}=\bar{X_{1}}\bar{X_{2}}^{a_{1}}+\cdots+{\bar{X}}_{n-1}\bar{X_{n}}^{a_{n-1}}+\bar{X_{n}}\bar{X_{1}}^{a_{n}}.
\frac{8\pi^{2}}{e^{2}}\left(\sum_{A}\left(\frac{1}{y_{A}}d\vec{y}_{A}^{2}+y_{A}\,(D\psi_{A})^{2}\right)+\frac{\left(\,\sum_{A}d\vec{y}_{A}\,\right)^{2}}{|\vec{y}_{N}+2\pi\vec{\zeta}|}-\frac{\left(\,\sum_{A}y_{A}\,D\psi_{A}\,\right)^{2}}{|\vec{y}_{N}+2\pi\vec{\zeta}|+\sum_{A}|\vec{y}_{A}|}\right),
R_{112}^{2}=R_{113}^{3}=\frac{1}{g^{2}}\left(\frac{h^{\prime\prime}}{h}-\frac{h^{\prime}}{h}\frac{g^{\prime}}{g}\right)\quad;\quadR_{232}^{3}=\frac{1}{h^{2}}\left(1-\left(\frac{h^{\prime}}{g}\right)^{2}\right)
\int\prod_{x\in\Sigma_{g}}\!\!dc^{\alpha}(x)\,\,c(x_{i})\left(1+S_{\mathrm{q}}(c)\right)=\int\prod_{x\in\Sigma_{g}\setminusx_{i}}\!\!dc^{\alpha}(x)\,e^{-S_{\mathrm{q}}},
f(x,y)=\left(\begin{array}{cc}{-2\partial_{x^{-}}}&{\alpha\partial_{x^{-}}^{2}-2\partial_{x^{-}}^{2}\phi-2\partial_{x^{-}}\phi\partial_{x^{-}}}\\{-\alpha\partial_{x^{-}}^{2}-2\partial_{x^{-}}\phi\partial_{x^{-}}}&{0}\\\end{array}\right)\delta(x-y)
\omega(x,y)=\frac{1}{2}+\frac{x^{2}y^{2}}{4\cdot5!}-\frac{5}{12\cdot7!}\,(x^{4}y^{2}+x^{2}y^{4})+\frac{1}{34500\,}(x^{2}y^{6}+x^{6}y^{2})+\frac{1}{86400}\,x^{4}y^{4}+\ldots
\left[1+\frac{\,g^{2}}{2m^{2}}\,V^{(2)}(\phi)\right]\frac{dp^{\mu}}{d\tau}=-\frac{\,g^{2}}{2m}\left[\partial^{\mu}V^{(2)}(\phi)-\frac{p^{\mu}}{p^{2}}\,p\cdot\partial\,V^{(2)}(\phi)\right]\,,
S_{GB}^{volume}=2(e^{2U})^{\prime}|_{r=r_{h}})^{-1}\left\{\left[(e^{2U})^{\prime}\left(1-(e^{U}R^{\prime})^{2}\right)\right]_{r=r_{h}}-\left[(e^{2U})^{\prime}\left(1-(e^{U}R^{\prime})^{2}\right)\right]_{r=r_{0}}\right\}\.
H^{2}=\left(\begin{matrix}{-\nabla^{2}+\phi_{1}^{2}+\phi_{2}^{2}}&{0}\\{0}&{-\nabla^{2}+\phi_{1}^{2}+\phi_{2}^{2}}\\\end{matrix}\right)+\left(\begin{matrix}{\phi_{2}^{\prime}}&{i\phi_{1}^{\prime}}\\{-i\phi_{1}^{\prime}}&{-\phi_{2}^{\prime}}\\\end{matrix}\right)
\Lambda^{\pm\pm}=\Lambda_{1}^{\pm}\Lambda_{2}^{\pm},\quad\Lambda_{i}^{\pm}={\frac{E_{i}\pmh_{i}}{2E_{i}}},\qquadE_{i}=\sqrt{h_{i}^{2}}=(\vec{p}_{i}^{2}+m_{i}^{2})^{\frac{1}{2}}.
\begin{array}{l}{\{\Omega_{n}^{a},\phi_{n^{\prime}}^{b}\}\approx0\\\\\a\neb}\\{\{\Omega_{n}^{a},\phi_{n^{\prime}}^{a}\}\approx0\\\\\n^{\prime}<n}\\{\{\Omega_{n}^{a},\phi_{n}^{a}\}\approx(-1)^{N_{a}-n}\eta^{a}(q,p)}\\\end{array}
I=\intdx(\Omega^{\prime}\rho^{\prime}+\frac{3}{4}e^{2\rho}\Omega^{-1/3}-\frac{1}{4}\Omegaf^{\prime2}-\frac{1}{4}\Omegaf_{1}^{\prime2}-\frac{1}{4}\Omegaf_{2}^{\prime2}-\frac{1}{4}\Omegaf_{1}^{\prime}f_{2}^{\prime}
\left(\begin{array}{cccc}{\thetaV^{\prime\prime}(c_{0})+2}&{-2}&{}&{}\\{-2}&{\thetaV^{\prime\prime}(c_{1})+6}&{-4}&{}\\{}&{-4}&{\thetaV^{\prime\prime}(c_{2})+10}&{-6}\\{}&{}&{-6}&{\ddots}\\\end{array}\right).
\left[-\frac{1}{2L}\left(\frac{\partial^{2}}{\partial(a^{3})^{2}}+\frac{\partial^{2}}{\partial(a^{8})^{2}}\right)+U_{eff}(a^{3},a^{8})\right]\tilde{\Phi}(a^{3},a^{8})=E\tilde{\Phi}(a^{3},a^{8})
\mathrm{det}\left(\begin{matrix}{\alpha_{1}+\beta}&{\beta}&{\ldots}&{\beta}\\{\beta}&{\alpha_{2}+\beta}&{\ldots}&{\beta}\\{\ldots}&{\ldots}&{\ldots}&{\ldots}\\{\beta}&{\beta}&{\ldots}&{\alpha_{n}+\beta}\\\end{matrix}\right)=\left(1+\sum_{j}{\frac{\beta}{\alpha_{j}}}\right)\prod_{i=1}^{n}\alpha_{i}\;\;\;.
{\phi}^{\mathrm{self}}(x,t)\equiv\int_{0}^{\mathrm{L}}dy\int_{0}^{\mathrm{L}}dz(e_{1}D(x,z|\mathrm{L})+e_{2}D(x,y|\mathrm{L}))\overline{{\Phi}}(z,t|y,t)({\gamma}^{0}\otimes{\gamma}^{0}){\Phi}(z,t|y.t).
\tau_{1}=\left(\begin{array}{cc}{i}&{0}\\{0}&{-i}\\\end{array}\right),\,\,\tau_{2}=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right),\,\,\tau_{3}=\left(\begin{array}{cc}{0}&{i}\\{i}&{0}\\\end{array}\right),
|a_{j}({\mathsfs})|^{2}=\left|B_{j}({\mathsfs})+\frac{\tilde{R}^{(i)}}{{\mathsfs}-{\mathsfs}_{R_{i}}}\right|^{2}=|B_{j}({\mathsfs})+a_{j}^{\mathrm{resonance}}({\mathsfs})|^{2}\,.
S_{\mathrm{F}}=\intdt\sum_{\bfx}\left\{\psi^{*}({\bfx},t)[iD_{0}+\mu]\psi({\bfx},t)-{\frac{J}{2}}\sum_{j=1,2}\left[\psi^{*}({\bfx}+e_{j},t)e^{-i{\calA}_{j}({\bfx},t),t)}\psi({\bfx},t)+\mathrm{c.c.}\right]\right\}
S=\frac1{2k_{d+1}^{2}}\intd^{d}x\int_{\rho\geq\varepsilon}d\rho\frac{d}{\rho^{{\frac{d}{2}}+1}}\sqrt{g(x,\rho)}-\frac{1}{k_{d+1}^{2}}\int_{\partialM_{d}^{\varepsilon}}d^{d}\sigma\sqrt{\gamma}K.
\frac{1}{\sqrt{-i}}\left(\frac{x^{-}}{x^{+}}\right)^{2}F(x^{+},x^{-},0)=\sum_{\alpha}\frac{1}{2(2\pi)^{5/2}}\frac{M_{\alpha}^{9/2}}{\sqrt{r}}K_{9/2}(M_{\alpha}r)\frac{|\langleu|\alpha\rangle|^{2}}{lK^{3}|N_{u}|^{2}}.
\left<\Psi\right|{\frac{T_{ij}(\vec{y},\tau)}{N}}\left|\Psi\right>=-\xi_{mn}{\frac{2}{N^{2}}}\int{dtf(t)\epsilon(\tau-t)Im\intd^{3}\vec{x}\left<T_{ij}(\vec{y},\tau)T_{mn}(\vec{x},t)\right>}+O({\frac{1}{N}}).
\Omega_{15}^{(4)}\equiv\left(\begin{array}{cccc}{\frac{1+\mathrm{ch}}{2}}&{\frac{1-\mathrm{ch}}{2}}&{\frac{\mathrm{sh}}{\sqrt{2}}}&{0}\\{}&{}&{}&{}\\{\frac{1-\mathrm{ch}}{2}}&{\frac{1+\mathrm{ch}}{2}}&{-\frac{\mathrm{sh}}{\sqrt{2}}}&{0}\\{}&{}&{}&{}\\{\frac{\mathrm{sh}}{\sqrt{2}}}&{-\frac{\mathrm{sh}}{\sqrt{2}}}&{\mathrm{ch}}&{0}\\{}&{}&{}&{}\\{0}&{0}&{0}&{1}\\\end{array}\right)\,,
[\,A_{{\sfGI}\,i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]\,V_{\cal{C}}^{-1}({\bf{r}})+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\;,
\widetildeD_{\mu\nu}(x^{T})\=\\frac{i}{4\pi}\left(g_{\mu\nu}^{LL}+g_{\mu\nu}^{TT}(1-\frac{\xi}{2})+\frac{x_{\mu}^{T}x_{\nu}^{T}}{x^{T2}}\frac{\xi}{2}\right)\frac{1}{r}\;
\tilde{M}_{2}R_{12}^{-1}M_{1}^{-1}=M_{1}^{-1}R_{12}^{-1}\tilde{M}_{2}\;,\qquad\tilde{M}_{1}R_{21}^{-1}M_{2}^{-1}=M_{2}^{-1}R_{21}^{-1}\tilde{M}_{1}\quad.
\Gamma^{\mu\nu}=(p^{2}+A)\left(\eta^{\mu\nu}-{\frac{p^{\mu}p^{\nu}}{p^{2}}}\right)+B\,{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}+{\frac{p^{\mu}p^{\nu}}{\xi}}
\begin{aligned}{\mathrm{{\bf~AAA}:}}&{{}\qquad\left[M-32\right]^{2}=0\;,}\\{\mathrm{{\bf~AAB}:}}&{{}\qquad\left[M-16\right]^{2}=0\;,}\\{\mathrm{{\bf~ABB}:}}&{{}\qquad\left[M-8\right]^{2}=0\;,}\\{\mathrm{{\bf~BBB}:}}&{{}\qquad\left[M-4\right]^{2}=0\;,}\\\end{aligned}
{\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}}\Bigl(\bar{\calG}_{ab}+\bar{\calG}_{ba}}\Bigr)}}\,
K:=UMU^{T}=\left(\begin{array}{ccccc}{\frac{1}{-\triangle+e^{2}\frac{N}{\pi}}}&{}&{}&{}&{}\\{}&{\frac{1}{-\triangle}}&{}&{}&{}\\{}&{}&{.}&{}&{}\\{}&{}&{}&{.}&{}\\{}&{}&{}&{}&{\frac{1}{-\triangle}}\\\end{array}\right)\;.
C^{A}\rightarrow\pm\bar{C}^{A},\quad\bar{C}^{A}\rightarrow\mpC^{A},\quadB^{A}\rightarrow-\bar{B}^{A},\quad\bar{B}^{A}\rightarrow-B^{A},\quad({\calA}_{\mu}^{A}\rightarrow{\calA}_{\mu}^{A}).
\partial_{t^{\prime}}^{2}\phi(\theta^{\prime},z^{\prime})-l^{2}\rho^{2}\left({\frac{1}{\rho^{2}}}\partial_{\theta^{\prime}}^{2}+\partial_{z^{\prime}}^{2}\right)\phi(\theta^{\prime},z^{\prime})+O(l^{3})=0\,.
M_{-+}^{2}=g^{2}v^{2}\left(\begin{array}{ccccccc}{2}&{-1}&{0}&{\cdots}&{0}&{0}&{0}\\{-1}&{2}&{-1}&{\cdots}&{0}&{0}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}&{0}&{0}\\{}&{}&{}&{\ddots}&{}&{}&{}\\{0}&{0}&{0}&{\cdots}&{-1}&{2}&{-1}\\{0}&{0}&{0}&{\cdots}&{0}&{-1}&{1}\\\end{array}\right).
\Gamma_{fin}^{1}=\frac{(-i\lambda)^{2n}(i)^{3n-1}}{2}\prod_{j=1}^{n}\int\frac{d^{6}k_{j}}{(2\pi)^{6}}\Theta^{(n)}(k_{1},k_{2},...k_{n};p,m^{2})
\sum_{n_{1}=-\infty}^{\infty}f(n_{1})\equiv\sum_{n_{1}=-\infty}^{\infty}{\frac{e^{-2\piin_{1}\sigma_{-}^{1}}}{n_{1}^{2}+n_{2}^{2}l^{2}/4}}=\sum_{n_{1}=-\infty}^{\infty}{\frac{\mathrm{Cos}(2\piin_{1}\sigma_{-}^{1})}{n_{1}^{2}+n_{2}^{2}l^{2}/4}}\quad.
\Psi_{E}\left(r,\theta\right)=\left(\begin{array}{l}{\sum_{n=-\infty}^{\infty}\left(A_{n}\,J_{n-\kappa}\left(|E|r\right)+B_{n}\,J_{\kappa-n}\left(|E|r\right)\right)e^{in\theta}}\\{\sum_{n=-\infty}^{\infty}-i\frac{|E|}{E}\left(A_{n}\,J_{n+1-\kappa}\left(|E|r\right)-B_{n}\,J_{\kappa-n-1}\left(|E|r\right)\right)e^{i\left(n+1\right)\theta}}\\\end{array}\right)
\left(\partial_{\mu}1\otimes1+\frac{1}{4}\epsilon_{\alpha\beta\delta}\omega_{\mu}^{\alpha\beta}1\otimes\gamma^{\delta}+\frac{i}{2}A_{\mu}^{c}\sigma_{c}\otimes1+\frac{1}{2\ell}e_{\mu\alpha}1\otimes\gamma^{\alpha}\right)\varepsilon_{r}^{\prime}=0\,.
Q^{ab}=(\delta^{ab}+\nu^{ab})^{-1}+(\delta^{ac}+\nu^{ac})^{-1}\left\{\nu^{cd}Q^{cb}-\mathrm{e}^{-i\mathbf{q.x}}C(-\Box)\frac{\delta^{2}\Gamma}{\delta\phi^{c}\delta\phi^{d}}\mathrm{e}^{i\mathbf{q.x}}Q^{db}\right\}
{\calS}[g_{\mu\nu},\Phi]=\intd^{4}x\,\sqrt{-g}\;{\calL}(g_{\mu\nu},\Phi)=\intd^{4}x\,\sqrt{-g}\;\left\lbrace{\frac{R}{16\pi}}+{\frac{1}{2}}\,g_{\mu\nu}\partial^{\mu}\Phi\partial^{\nu}\Phi-{\frac{1}{2}}\,m^{2}\Phi^{2}\right\rbrace,
S_{F}^{reg}[{\bar{\psi}},\psi,{\bar{\phi}},\phi;A]\;=\;\intd^{2}x\,\left[{\bar{\psi}}(i\not\!\partial-e\not\!\!A)\psi+{\bar{\phi}}(i\not\!\partial-e\not\!\!A-M)\phi\right]\;.
{}_{\nu\bar{\nu}}\langleO_{2}|(\deltaH)^{k-2}{|O_{k}\rangle}_{\nu\bar{\nu}}={}_{\nu\bar{\nu}}\langleO_{2}|{\left(\Sigma_{i,j}\Lambda_{j}^{i}(\nu_{i}^{3}{\bar{\nu}}^{4j}-\nu_{i}^{4}{\bar{\nu}}^{3j})\right)^{k-2}}{|O_{k}\rangle}_{\nu\bar{\nu}}
\Gamma_{\mu}(p,p-k)=Z_{2}\gamma_{\mu}-ie^{2}Z_{2}\int\frac{d^{n}p^{\prime}}{(2\pi)^{n}}\gamma_{\lambda}S(p^{\prime})\Gamma_{\nu\mu}(p^{\prime}k^{\prime};pk)D^{\lambda\nu}(k^{\prime}).
\begin{array}{l}{\gamma_{_{\!1}i}\otimes\mathbf{1}_{2}\;\longmapsto\;\gamma_{i}=\vartheta(\mathbf{a}_{_{\!1}i},\mathbf{0}),\;i=1,2,...,n_{1}=dim(L_{1})}\\{\omega_{1}\otimes\gamma_{_{\!2}j},\longmapsto\;\gamma_{n_{1}+j}=\vartheta(\mathbf{0},\mathbf{a}_{_{\!2}j}),\;j=1,2,...,n_{2}=dim(L_{2})\;.\;}\\\end{array}
\begin{array}{lll}{{{\omega}^{0}}_{3}=J^{0}{\omega}^{0}\nonumber}\\{{{\omega}^{1}}_{3}=c\frac{H^{\prime}}{2}{\omega}^{1}\nonumber}\\{{{\omega}^{2}}_{3}=-c\frac{H^{\prime}}{2}{\omega}^{2}\nonumber}\\\end{array}
p^{+}=\frac{1}{2}(1+K),\quadp^{-}=\frac{1}{2}(1-K),\quad{K_{\alpha}}^{\beta}=\frac{1}{\sqrt{-\pi^{2}}}\pi_{m}(X\tilde{\Gamma}^{m})_{\alpha}{}^{\beta},\quadK^{2}=1.
[\psi_{\dot{\alpha}},\tilde{\psi}_{\dot{\beta}}^{\dagger}]_{+}=[\tilde{\psi}_{\dot{\alpha}},\psi_{\dot{\beta}}^{\dagger}]_{+}=\delta_{\dot{\alpha}\dot{\beta}}\mid0\rangle\langle0\mid,\;\;\;\;\;\;[\psi_{\dot{\alpha}},\tilde{\psi}_{\dot{\beta}}]_{+}=[\psi_{\dot{\alpha}}^{\dagger},\tilde{\psi}_{\dot{\beta}}^{\dagger}]_{+}=0
S_{\mathrm{ct}}=-\frac{1}{8\piG}\intd^{3}x\sqrt{-h}\frac{c_{0}}{l_{\mathrm{eff}}},\\\c_{0}=2N\sqrt{\frac{3}{1-N}},\\\\frac{1}{l_{\mathrm{eff}}}=e^{b\phi}\sqrt{\frac{\Lambda}{3}}.
[\hat{x}^{\mu},\hat{\pi}_{\nu}]=i\{x^{\mu},\pi_{\nu}\}_{D(\Phi_{4}^{(1)})}=i\delta_{\nu}^{\mu}\;,\;\;[\hat{\psi}^{n},\hat{\psi}^{m}]_{+}=i\{\psi^{n},\psi^{m}\}_{D(\Phi_{4}^{(1)})}=-\frac{1}{2}\eta^{nm}\;.
g~=~\left(\begin{matrix}{e^{i\alpha_{1}}}&{0}&{0}&{0}&{\ldots}&{0}\\{0}&{e^{i\alpha_{2}}}&{0}&{0}&{\ldots}&{0}\\{0}&{0}&{e^{i\alpha_{3}}}&{0}&{\ldots}&{0}\\{0}&{0}&{\ldots}&{}&{}&{0}\\{0}&{0}&{\ldots}&{}&{}&{e^{i\alpha_{N}}}\\\end{matrix}\right)\;,
\begin{array}{rcl}{U_{d}/m^{2}}&{=}&{\displaystyle+2D_{m}^{2}(k_{1})D(k_{1}+p)D_{m}(k_{2})(2+K_{12})}\\{}&{}&{\displaystyle+2D_{m}^{2}(k_{1})D(k_{1}+p)D(k_{2})(2-K_{12})}\\{}&{}&{\displaystyle+2D^{2}(k_{1})D_{m}(k_{1}+p)[D_{m}(k_{2})+D(k_{2})]}\\{}&{}&{\displaystyle+4(N-1)D_{m}^{2}(k_{1})D(k_{1}+p)D(k_{2})}\\{}&{}&{\displaystyle+4ND^{2}(k_{1})D_{m}(k_{1}+p)D(k_{2}),}\\\end{array}
\sum_{\alpha=0}^{N}Y_{\alpha}^{N}(\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta})Y_{\alpha}^{N}(\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime})=\sum_{\alpha=0}^{N}Y_{\alpha}^{N}({\eta})Y_{\alpha}^{N}({\eta}^{\prime})
D\!\!\!\!/^{~2}=\left[{\frac{1}{\sqrtg}}\left({\sqrtg}g^{\nu\lambda}\right)_{,\lambda}+(1-{\calD})\phi^{\nu}-2\sigma^{qp}e_{~q}^{\mu}e_{~p}^{\nu}\phi_{\mu}+\left(g^{\mu\nu}+2\sigma^{\mu\nu}\right)D_{\mu}\right]D_{\nu}.
A_{M}=\left(\begin{array}{c}{a}\\{\psi}\\\end{array}\right),\;\;\;\;\;A_{M}^{\dagger}=\left(\begin{array}{c}{a^{\dagger}}\\{\psi^{\dagger}}\\\end{array}\right),\;\;\tilde{A}_{M}=\left(\begin{array}{c}{\tilde{a}}\\{\tilde{\psi}}\\\end{array}\right),\;\;\;\;\;\tilde{A}_{M}^{\dagger}=\left(\begin{array}{c}{\tilde{a}^{\dagger}}\\{\tilde{\psi}^{\dagger}}\\\end{array}\right)
\Gamma[\varphi]=-{\frac{l}{4\piG_{3}}}\intd\taudye^{-2\varphi}\left[{\frac{1}{l^{2}}}+\frac12(\varphi^{\prime})^{2}e^{2\varphi}+{\frac{l^{2}}{24}}(\varphi^{\prime})^{4}e^{4\varphi}+...\right]~~.
B_{\widetilde{S}}=tr\intd^{4}x\left(\frac{\delta\widetilde{S}}{\delta\Phi_{i}^{*}}\frac\delta{\delta\Phi_{i}}+\frac{\delta\widetilde{S}}{\delta\Phi_{i}}\frac\delta{\delta\Phi_{i}^{*}}\right)\;,
\mu_{\mathbf{R}}^{1}=\mu_{\mathbf{C}}^{1}=\mu_{\mathbf{C}}^{2}=\mu_{\mathbf{C}}^{2\,\,\,\prime}=\mu_{\mathbf{R}}^{3}=\mu_{\mathbf{C}}^{3}=0\,,
(c_{mn}^{pr}+c_{nm}^{pr})(a_{pt}^{i}a_{ry}^{j}+a_{py}^{i}a_{rt}^{j})+(c_{ty}^{qs}+c_{yt}^{qs})(a_{mq}^{i}a_{ns}^{j}+a_{nq}^{i}a_{ms}^{j}).