harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C_{NUT}=-\frac{(4\pi)^{(n-2)/2}\beta}{16\pi^{3/2}\ell^{2}}\Gamma\left(\frac{3-n}{2}\right)\Gamma\left(\frac{n}{2}\right)N^{n-3}\left[n(n-1)(n-2)N^{2}-(n-2)(n-3)\ell^{2}\right]
\begin{array}{c}{[\,H_{j}\otimesH_{j}\,,\,E_{\alpha}\otimes1\,]~=~\alpha_{j}\;E_{\alpha}\otimesH_{j}~,}\\{[\,H_{j}\otimesH_{j}\,,\,1\otimesE_{\alpha}\,]~=~\alpha_{j}\;H_{j}\otimesE_{\alpha}~,}\\\end{array}
d{\hat{s}}^{2}=(1+\Omega^{n}\,H(\vec{x}))d{\hat{s}}_{\circ}^{2}-\frac{2}{n(n-2)}\,\,\Omega^{n}\,E_{ab}(\vec{x})\,dx^{a}dx^{b}-\Omega^{n}\,H(\vec{x})\,d\Omega^{2}+{\calO}(\Omega^{n+1}),
|<\nu|\mu>|^{2}=\frac{I_{2j+k-1}(2\sqrt{\nu^{*}\mu})I_{2j+k-1}(2\sqrt{\mu^{*}\nu})}{I_{2j+k-1}(2\sqrt{\nu^{*}\nu})I_{2j+k-1}(2\sqrt{\mu^{*}\mu})}
\tilde{\calO}_{\;\;B}^{A}=\left(\frac{\epsilon^{2}-1}{{\calR}}\right)_{\;\;B}^{A}=\int_{0}^{1}\frac{dt}{\Lambda^{2}}\,exp\left(\,t\,\frac{{\calR}_{B}^{A}}{\Lambda^{2}}\right)\;\;.
C_{\mu\nu}^{-1}=-\frac{1}{m^{2}}\left(\begin{array}{cccc}{0}&{\partial_{1}^{x}}&{\partial_{2}^{x}}&{-1}\\{\partial_{1}^{x}}&{0}&{-m}&{0}\\{\partial_{2}^{x}}&{m}&{0}&{0}\\{1}&{0}&{0}&{0}\\\end{array}\right)\delta^{2}(x-y).
\oint_{C}\frac{d\zeta}{\zeta^{n}}\hat{f}=\left(\oint_{0}\frac{d\zeta}{\zeta^{n-2}}\chi-\oint_{\infty}\frac{d\zeta}{\zeta^{n-2}}\chi\right)+\left(\oint_{\infty}\frac{d\zeta}{\zeta^{n}}\left(\zeta^{2}\chi-\tilde{\chi}\right)\right).
e^{-\alpha{\bf\tilde{p}}_{l}{\bf\tilde{p}}_{n-l}}e^{-\beta{\bf\tilde{x}}_{l}{\bf\tilde{x}}_{n-l}}={\frac{1}{(1+\alpha\beta)^{2}}}e^{-{\frac{\beta}{1+\alpha\beta}}{\bf\tilde{x}}_{l}{\bf\tilde{x}}_{n-l}},
+G_{mn,nm}(q_{1}))+i\frac{e^{2}{\hbar}c}{4}\sum_{(m<n)}\frac{1}{{\omega}_{nm}}\mathrm{Re}\left[G_{nm,mn}(\frac{1}{c}{\omega}_{nm})+G_{nm,mn}(\frac{1}{c}{\omega}_{mn})\right],
L_{D}=\int_{\calM}\varepsilon^{ijk}\bigg(\dots+{\frac{k}{8\pi}}f_{abc}A_{i}^{a}A_{j}^{b}B_{k}^{c}+{\frac{1}{2\pi}}\chi_{i}^{a}\partial_{j}B_{ka}+{\frac{1}{4\pi}}f_{abc}\chi_{i}^{a}B_{j}^{b}B_{k}^{c}+\dots\bigg).
\left[{\frac{dn_{d}(\omega|D)}{d\omega}}\right]_{\tiny\mathrm{div}}=r_{d}\,{\frac{\Gamma\left(1-\fracD2\right)}{(4\pi)^{D/2}}}\,\,{\frac{m^{D-4}}{\kappa}}\,\,\int_{\Sigma}\left[2\left(m^{2}+{\frac{R}{12}}-{\frac{\calQ}{8}}\right)-{\frac{\omega^{2}}{\kappa^{2}}}{\calP}\right]~~~.
\begin{array}{ll}{D_{\mu}\phi=(\nabla_{\mu}-i\frac{q}{2\sqrt{\beta}}H_{\mu})\phi,}\\{D_{\mu}\sigma=(\nabla_{\mu}+i\frac{e}{2}A_{\mu}-i\frac{\alphae}{2\sqrt{\beta}}H_{\mu})\sigma,}\\\end{array}
\frac{S}{V}=-\frac{\mu_{5}}{2g}2\pi\alpha^{\prime}\intd^{2}\xi\,G_{\perp}^{1/2}(F_{ab}F^{ab})^{1/2}\eta^{\mu\nu}\partial_{\mu}x^{m}\partial_{\nu}x^{n}=-\frac{n}{2\pig}\eta^{\mu\nu}\partial_{\mu}\bar{\phi}\partial_{\nu}\phi\.
\intdp^{+}\intd^{2}p_{\perp}f(p^{+},{\bfp}_{\perp})\simeq\frac{\pi}{L_{\|}}\left(\frac{2\pi}{L_{\perp}}\right)^{2}\sum_{n,n_{x},n_{y}}f(nP^{+}/K,2{\bfn}_{\perp}\pi/L_{\perp})\,.
\nabla^{[\mu}\nabla^{<\alpha}R^{\nu]\beta>}\biggl|_{{\calI}^{-}}=\frac{\nabla^{[\mu}v\nabla^{<\alpha}v}{(\nablau,\nablav)^{2}}(\nabla_{\gamma}u)(\nabla_{\sigma}u)\nabla^{\gamma}\nabla^{\sigma}R^{\nu]\beta>}+O\left(\frac{1}{r^{3}}\right)\;.
\frac{1}{2}\Big(\partial_{\mu}\Phi^{(1)}\Big)^{2}\;+\;\frac{1}{2}\Big(\partial_{\mu}\Phi^{(2)}\Big)^{2}\;+\;\frac{1}{2}\Big(\Phi^{(1)}\Big)^{2}\frac{2e^{2}}{\pi+2g}\;.
e^{2U}=(1-{\frac{r_{+}}{r}})(1-{\frac{r_{-}}{r}})^{{\frac{1-a^{2}}{1+a^{2}}}},\qquadR=r\;(1-{\frac{r_{-}}{r}})^{{\frac{a^{2}}{1+a^{2}}}}\,
T_{\mu\nu}^{P}=\frac{\pi}{\sqrt{-g}}\sum_{a}\intd\tau_{a}\delta^{2}(x-z_{a})e_{a}^{-1}g_{\mu\alpha}g_{\nu\beta}\frac{dz_{a}^{\alpha}}{d\tau_{a}}\frac{dz_{a}^{\beta}}{d\tau_{a}}.
h_{0}^{(1)}(k)=-k^{2}-m^{2}-\int{\frac{d^{4}l}{(2\pi)^{4}}}{\frac{R_{00}(k,l)\,(l^{2}+m^{2})}{(l^{2}+m^{2})^{2}+[h_{1}^{(0)}(l)]^{2}+[h_{2}^{(0)}(l)]^{2}+[h_{3}^{(0)}(l)]^{2}}}\,.
\frac{W_{\xi^{\prime},\eta}(i)}{W_{\xi^{\prime},\eta}(i-1)}=\frac{M_{\xi^{\prime}}}{M_{\eta}}\frac{Y_{\eta}-X_{\xi^{\prime}}\omega^{i}}{Y_{\xi^{\prime}}-X_{\eta}\omega^{i}}\qquad\frac{W_{\xi,\eta^{\prime}}(i)}{W_{\xi,\eta^{\prime}}(i-1)}=\frac{M_{\xi}}{M_{\eta^{\prime}}}\frac{Y_{\eta^{\prime}}-X_{\xi}\omega^{i}}{Y_{\xi}-X_{\eta^{\prime}}\omega^{i}}
\tilde{E}_{i}^{2}=E_{i}^{b}R_{2b}=E_{i}^{b}\hat{E}_{3}^{b}\hat{E}_{3}^{a}\hat{\lambda}^{a}-E_{i}^{b}\hat{\lambda}^{b}\hat{E}_{3}^{a}\hat{E}_{3}^{a}\,.
L_{fk}=\frac{i}{2}\left[\overline{{L_{L}}}\not\!\!D^{(L)}L_{L}+\,\overline{{E}}_{R}\not\!\!D^{(R)}E_{R}+\,\overline{{Q}}_{L}\not\!\!D^{(L)}Q_{L}+\,\overline{{U}}_{R}\not\!\!D^{(R)}U_{R}+\,\overline{{D}}_{R}\not\!\!D^{(R)}D_{R}\right]
S=\intd^{4}x\sqrt{g}\left({\frac{m^{8}v_{1}v_{2}v_{3}}{(2\pi)^{7}\lambda^{2}}}R+{\frac{1}{4}}{\frac{m_{s}^{2}v_{2}}{(2\pi)^{3}\lambda}}F^{2}+{\frac{1}{4}}{\frac{m_{s}^{2}v_{1}}{(2\pi)^{3}\lambda}}\tilde{F}^{2}+\dots\right)
\sumq^{\frac{1}{2}p^{2}}=\sum_{n}q^{\frac{6}{2}\left(2n+\frac{4m_{1}+3m_{2}}{12}\right)^{2}}=\sum_{n}q^{12\left(n+\frac{4m_{1}+3m_{2}}{24}\right)^{2}}=\theta_{4m_{1}+3m_{2},12}\:(\tau).
|\alpha\rangle\rangle_{\Omega}\equiv|S_{0};(L_{j},M_{j},S_{j})_{j=1}^{r}\rangle_{\Omega}={\frac{1}{\kappa_{\alpha}^{\Omega}}}\sum_{\lambda+1\in\mathrm{Exp}(G),\mu}^{\beta}\delta_{\Omega}\;B_{\alpha}^{\lambda,\mu}\,|\lambda,\mu\rangle\!\rangle_{\Omega},
2\chi\;\frac{dA}{d\chi}=-1+\epsilon\;\mathrm{sign}(\dot{\chi})\;\sqrt{K}\;,\;\;\;\;K\equiv\frac{\Omega}{3}\;\left(1+\frac{2\rho_{0}\chie^{3(1-\gamma)A}}{(\sigma(t)+c)^{2}}\right)\;.
i{\calM}~=~\frac{ip_{1}^{2}p_{2}^{2}}{-t}~\frac{\Gamma\left(1-ip_{1}^{2}p_{2}^{2}/32\piEp\right)}{\Gamma\left(1+ip_{1}^{2}p_{2}^{2}/32\piEp\right)}~\left(\frac{4\mu^{2}}{-t}\right)^{-i\frac{p_{1}^{2}p_{2}^{2}}{32\piEp}}~,
|\Psi\rangle\equiv\prod_{l=1}^{\nu}\prod_{i_{l}=1}^{m_{l}^{\prime}}a_{l}^{A_{l}^{i_{l}}}\Psi_{\phantom{1}}^{A_{1}^{1},\ldots,A_{1}^{m_{1}^{\prime}},\ldots,A_{\nu}^{1},\ldots,A_{\nu}^{m_{_\nu}^{\prime}}}|0\rangle\,.
\begin{aligned}{\deltax^{0}}&{{}=-x^{0}(\lambda+2a_{i}x^{i}),}\\{\deltax^{i}}&{{}=-x^{i}(\lambda+2a_{j}x^{j})+a^{i}x^{2}-b^{i}+\omega^{ij}x^{j},}\\\end{aligned}
2\Bigl[J_{0}\left(\frac{m_{n}}{k}e^{Rk\pi}\right)-\frac{J_{0}\left(\frac{m_{n}}{k}\right)}{Y_{0}\left(\frac{m_{n}}{k}\right)}Y_{0}\left(\frac{m_{n}}{k}e^{Rk\pi}\right)\Bigr]=\frac{W}{2M_{5}^{2}R}\Bigl[J_{1}\left(\frac{m_{n}}{k}e^{Rk\pi}\right)-\frac{J_{0}\left(\frac{m_{n}}{k}\right)}{Y_{0}\left(\frac{m_{n}}{k}\right)}Y_{1}\left(\frac{m_{n}}{k}e^{Rk\pi}\right)\Bigr]\,,
C_{L}\stackrel{I}{\longrightarrow}-C_{R}\stackrel{h\circf^{(m)}}{\longrightarrow}\left\{z;\mathrm{arg}z=\frac{\pi}{m}\right\}\stackrel{h\circ\widehat{G}_{2}\circh^{-1}}{\longrightarrow}\left\{z;\mathrm{arg}z=2n^{\prime}\pi+\frac{3m-1}{2m-1}\pi\right\},
S_{\mathrm{m}}=-\intd^{4}xd^{2}\thetad^{2}{\overline{\theta}}\,\left[\frac{1}{2}m\delta_{IJ}\Phi_{a}^{I}\Phi^{Ja}\delta({\overline{\theta}})+\frac{1}{2}m^{*}\delta^{IJ}{\Phi^{\dagger}}_{I}^{a}{\Phi^{\dagger}}_{Ja}\delta(\theta)\right].
\varphi({\bfr}_{1},t)\varphi({\bfr}_{2},t)=-\varphi({\bfr}_{2},t)\varphi({\bfr}_{1},t)~e^{i\Gamma({\bfr}_{1},~{\bfr}_{2})},~~~~{\bfr}_{1}\neq{\bfr}_{2},
{\frac{1}{\beta}}\sum_{n}{\frac{e^{i\omega_{n}\tau}}{\omega_{n}^{2}+E_{p}^{2}}}={\frac{e^{-E_{p}|\tau|}}{2E_{p}}}+{\frac{e^{E_{p}\tau}+e^{-E_{p}\tau}}{2E_{p}}}{\frac{1}{e^{\betaE_{p}}-1}}\,\;.
\frac{1}{\prod_{i\geq1}(1-q^{i})}\Bigl(\sum_{n\geq0}\sum_{m=0}^{n}z^{n-2m}\bigl(q^{\frac{n^{2}-n}{2}}(1-q^{2n+1})+q^{\frac{n^{2}+n}{2}}(1-q^{2n+3})\bigr)\Bigr)\,.
\begin{array}{c}{Q\;|\;q,z,w>^{^{\prime}}=q|\;q,z,w>^{^{\prime}},}\\{\;a^{\dagger}b^{\dagger}|\;q,z,w>^{^{\prime}}=w^{*}|\;q,z,w>^{^{\prime}},\;b^{\dagger}a^{\dagger}|\;q,z,w>^{^{\prime}}=z^{*}|\;q,z,w>^{^{\prime}}.}\\\end{array}
S[x^{\mu},P_{\mu};e,\pi_{e};\tilde{\lambda},u]=\int_{\tau_{i}}^{\tau_{f}}d\tau\left[\dot{x}^{\mu}P_{\mu}+\dot{e}\pi_{e}-\frac{1}{2}\left[|e|+\tilde{\lambda}\right]\left[P^{2}+m^{2}\right]-u\pi_{e}\right]\.
{\frac{3}{c^{2}n^{2}}}{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}+{\frac{\dot{b}}{b}}\right)-{\frac{3}{b^{2}}}\left[{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}-{\frac{b^{\prime}}{b}}\right)+{\frac{a^{\prime\prime}}{a}}\right]+{\frac{3k}{a^{2}}}={\frac{8\piG_{5}}{c^{4}}}\left[\Lambda+(\sigma+\varrhoc^{2}){\frac{\delta(y)}{b}}\right],
\hat{Z}_{888}^{10,10}(\Omega)=\frac{(det\Im\\Omega)^{2}}{2^{3}}\left|\Theta\left[\begin{array}{lll}{1/2}&{1/2}&{1/2}\\{1/2}&{0}&{1/2}\\\end{array}\right](0\vert\Omega)\right|^{8}
\left(H+{\frac{1}{2}}{\frac{\dot{\Phi}}{\Phi}}\right)^{2}={\frac{1}{12}}\left({\frac{\dot{\Phi}}{\Phi}}\right)^{2}+{\frac{8\pi}{3}}{\frac{\rho}{\Phi}}-{\frac{k}{a^{2}}}\,,
{\calL}_{g}=-\frac{\zeta^{2}}{2}(\partial_{\mu}A^{\mu})(1+\frac{\Box}{M^{2}})(\partial_{\nu}A^{\nu})-\bar{c}(1+\frac{\Box}{M^{2}})\Boxc+\frac{1}{2\zeta^{2}}B(1+\frac{\Box}{M^{2}})B
D^{\dagger}=\left(\begin{array}{ccc}{-B_{2}+\hat{z}_{2}}&{B_{1}-\hat{z}_{1}}&{I}\\{B_{1}^{\dagger}-\widehat{\bar{z}_{1}}}&{B_{2}^{\dagger}-\widehat{\bar{z}_{2}}}&{J^{\dagger}}\\\end{array}\right)
G_{A}(x,y)=\Sigma\!\!\!\!\!\!\int\frac{d^{4}p}{(2\pi)^{4}}E_{p}(x)\frac{1}{\gamma\cdot\bar{p}+{\Sigma}_{A}(\bar{p})}\bar{E}_{p}(y),~~~\Sigma\!\!\!\!\!\!\intd^{4}p\equiv\sum_{k}\intdp_{0}dp_{2}dp_{3}.
W^{CS}[g_{\mu\nu},\beta]={\frac{\beta}{\beta_{H}}}W[g_{\mu\nu}]+U(\beta)-{\frac{\beta}{\beta_{H}}}U(\beta_{H})+\sum_{s}2\pi\xi_{s}(1-\beta/\beta_{H})\int_{\Sigma}d\sigma<\hat{\phi}_{s}^{2}>_{\beta}
\left({\frac{\dot{a}_{R}}{a_{R}}}\right)^{2}+2m_{0}\dot{R}{\frac{\dot{a}_{R}}{a_{R}}}+m_{0}^{2}\dot{R}^{2}+{\frac{kc^{2}\Omega_{R}^{2}}{a_{R}^{2}}}={\frac{8\pim_{0}G_{5}}{3(1-\Omega_{R}^{2})}}\left[{\frac{\tilde{\varrho}_{1}\Omega_{R}+\varrho_{2}\Omega_{R}^{5}}{\sqrt{\Omega_{R}^{2}-\dot{R}^{2}/c^{2}}}}+V_{r}/c^{2}\right].
F^{10}=\frac{B_{0}}{2a}\Bigl(\frac{\theta}{2bB_{0}}\Bigr)^{\frac{2}{3}}\,\biggl\{\biggl[1-\Bigl(1-\frac{32E_{0}^{3}}{27\theta^{2}B_{0}b^{4}}\Bigr)^{\frac{1}{2}}\biggr]^{\frac{1}{3}}+\biggl[1+\Bigl(1-\frac{32E_{0}^{3}}{27\theta^{2}B_{0}b^{4}}\Bigr)^{\frac{1}{2}}\biggr]^{\frac{1}{3}}\biggr\}^{2}
\frac{1}{g^{2}}\sum_{i<j}(\bar{\xi^{i}}-\bar{\xi^{j}})\Gamma^{\mu\alpha\nu}(\xi^{i}-\xi^{j})\frac{(x_{\alpha}^{i}-x_{\alpha}^{j})}{(x^{i}-x^{j})^{2}}\tilde{A}_{\mu}^{ij*}\tilde{A}_{\nu}^{ij},
V_{3}=\sum_{J_{1},J_{2},J_{3}}{\frac{1}{4}}\sqrt{\frac{J_{3}!}{NJ_{1}!J_{2}!}}(J_{1}\omega+J_{2}\omega-J_{3}\omega)(a_{J_{1}}^{\dagger}a_{J_{2}}^{\dagger}a_{J_{3}}+a_{J_{3}}^{\dagger}a_{J_{1}}a_{J_{2}})\delta_{J_{1}+J_{2},J_{3}}.
E_{1}=\frac{1}{\sqrt{6}}\left(\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{2}\\\end{array}\right)~;~~E_{2}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right)~;~~E_{3}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}{0}&{0}&{1}\\{0}&{0}&{0}\\{1}&{0}&{0}\\\end{array}\right)
ds^{2}={\frac{\sqrt{am}}{[1+{\frac{\lambda}{2}}(x^{+}-x_{0}^{+})]}}{\frac{dx^{+}dx^{-}}{[1-{\frac{am}{4e}}({\frac{x^{+}+{\frac{\lambdax_{0}^{+}}{2}}(x^{+}-x_{0}^{+})}{1+{\frac{\lambda}{2}}(x^{+}-x_{0}^{+})}})x^{-}]}}
\begin{array}{lll}{d_{v}}&{=}&{4\sum_{n=1}^{k-1}n+(2N-4k+1)k=2Nk-k(2k+1),}\\{d_{H}}&{=}&{[\frac{1}{2}\times2\sum_{n=1}^{k-1}((2n)^{2}+2n(2n+2))+\frac{1}{2}(2(2k)^{2}+(2N-4k-2)2k(2k+1))+2k]}\\{}&{-}&{[2\sum_{n=1}^{k-1}(n(2n-1)+n(2n+1))+(2N-4k-1)k(2k+1)+2k(2k-1)]}\\{}&{=}&{k.}\\\end{array}
\psi_{\alpha}^{II}(q_{3})=\sum_{i=1,2}\left(a_{i}e^{\frac{i}{\hbar}p_{3}^{(i)}q_{3}}+b_{i}e^{-\frac{i}{\hbar}p_{3}^{(i)}q_{3}}\right)\chi_{\alpha}^{i}
\begin{array}{rcl}{T_{B}(z_{1},z_{2})}&{=}&{\displaystyle\sum_{j,k}\phi_{j}(z_{1}+z_{2})K_{k}^{j}(z_{1})\phi^{k}(z_{1}-z_{2})}\\{}&{=}&{\displaystyle\sum_{j,k}\phi^{*j}(-z_{1}-\frac{n}{2}w-z_{2})K_{k}^{j}(z_{1})\phi^{k}(z_{1}-z_{2}).}\\\end{array}
V_{i}\cdotV_{j}=\left(\begin{array}{cccc}{-1}&{-{\frac{1}{2}}}&{0}&{0}\\{-{\frac{1}{2}}}&{(v^{I})^{2}}&{v^{I}A^{I}}&{0}\\{0}&{v^{I}A^{I}}&{(A^{I})^{2}+{\frac{2}{3}}}&{0}\\{0}&{0}&{0}&{({\tilde{A}}^{I})^{2}+1}\\\end{array}\right)~.
\langle\good\,^{\prime}\midA_{j_{good}}B_{k_{good}}\midgood\\rangle=\sum_{good\,^{\prime\prime}}\langle\good\,^{\prime}\midA_{j_{good}}\midgood\,^{\prime\prime}\\rangle\langle\good\,^{\prime\prime}\midB_{k_{good}}\midgood\\rangle.
P_{I}=\frac12\delta_{j}^{i}(1+\Gamma)=\frac12\delta_{j}^{i}\left(\begin{array}{cc}{\delta_{\alpha}^{\beta}}&{{\displaystyle\frac{im^{*}\omega_{\alpha\dot{\beta}}}{\sqrt{-\omega^{2}}(m+ie\bar{W}/2)}}}\\{{}}&{{}}\\{{\displaystyle\frac{-im^{*}\tilde{\omega}^{\dot{\alpha}\beta}}{\sqrt{-\omega^{2}}(m-ieW/2)}}}&{\delta_{\dot{\beta}}^{\dot{\alpha}}}\\\end{array}\right),
D_{j_{1},m_{1},\overline{{m}}_{1};j_{2},m_{2},\overline{{m}}_{2}}^{j_{3},m_{3},\overline{{m}}_{3}}=\sqrt{\frac{(2j_{1}+1)(2j_{2}+1)}{(2j_{3}+1)}}\;\langlej_{1}j_{2}m_{1}m_{2}|j_{3},m_{3}\rangle\;\langlej_{1}j_{2}\overline{{m}}_{1}\overline{{m}}_{2}|j_{3},\overline{{m}}_{3}\rangle\;d_{j_{1},j_{2}}^{j_{3}},
S[\,H\,B\,]=\intd^{4}x\left[\,{\frac{1}{12}}H_{\lambda\mu\nu}\,H^{\lambda\mu\nu}-{\frac{1}{6}}H^{\lambda\mu\nu}\partial_{[\,\lambda}B_{\mu\nu\,]}+{\frac{m^{2}}{4}}B_{\mu\nu}\,B^{\mu\nu}\,\right]
\begin{array}{l}{\rho^{\mu}\partial_{\mu}x^{AB^{\prime}}o_{A}=0,}\\{\partial_{\mu}x^{AB^{\prime}}o_{A}\bar{o}_{B^{\prime}}=0,}\\{\partial_{\mu}(\rho^{\mu}o_{A}\bar{o}_{B^{\prime}})+3\kappa\varepsilon^{\mu_{1}\mu_{2}}\partial_{\mu_{1}}x^{A_{1}{B^{\prime}}_{1}}\partial_{\mu_{2}}x^{A_{2}{B^{\prime}}_{2}}\partial_{[AB^{\prime}}T_{A_{1}A_{2}{B^{\prime}}_{1}{B^{\prime}}_{2}]}=0,}\\\end{array}
P_{Y}^{(20)I_{1}I_{2}...I_{n},J_{1}J_{2}...J_{n}}=\sum_{i}Q_{Y,i_{1}i_{2}...i_{2n}}^{T,I_{1}I_{2}...I_{n}}Q_{Y,i_{1}i_{2}...i_{2n}}^{J_{1}J_{2}...J_{n}}
\begin{array}{ll}{A:\qquadu=U(\sigma^{+}),}&{\quadv=\bar{V}(\sigma^{-})}\\{B:\qquadu=\bar{U}(\sigma^{-}),}&{\quadv=V(\sigma^{+})}\\{C:\qquadu=u_{0},}&{\quadv=W[\alpha(\sigma^{+})+\bar{\beta}(\sigma^{-}),\,\,u_{0}]}\\{D:\qquadu=\bar{W}[\bar{\alpha}(\sigma^{-})+\beta(\sigma^{+}),\,\,v_{0}],}&{\quadv=v_{0}\,\,\,,}\\\end{array}
\hat{H}=\hat{H}_{0}+\lambda(t)\hat{\phi}\\,\\\hat{H}_{0}=\frac{1}{2}g^{2}\hat{p}_{i}^{a}\hat{p}_{i}^{a}+\frac{1}{2}\omega^{2}\hat{q}_{i}^{a}\hat{q}_{i}^{a}\\,\\\hat{\phi}=\epsilon^{ab}\hat{p}_{i}^{a}\hat{q}_{i}^{b}\\,
\mu^{\prime\prime}+\mu\biggl[n^{2}+n^{2}\sum_{q=1}^{m}\frac{b_{q}}{(2\pi)^{2q}}\biggl(\frac{\epsilon\,n}{\vert\eta\vert^{1+\beta}}\biggr)^{2q}-\frac{a^{\prime\prime}}{a}\biggr]=0.
\frac{\partial\overline{{H}}_{0}}{\partialq_{n,\alpha_{n}}}\left(q_{n,\alpha_{n}},p_{n,m_{n}-1}\right)=-\frac{\partialL_{0}}{\partialq_{n,\alpha_{n}}}\left(q_{n,\alpha_{n}},\dot{q}_{n,m_{n}-1}(q_{n,\alpha_{n}},p_{n,m_{n}-1})\right)\,
\left(\begin{matrix}{E_{a\mu}}&{E_{a\underline{\mu}}}\\{E_{\underline{a}\mu}}&{E_{\underline{{a\mu}}}}\\\end{matrix}\right)=\left(\begin{matrix}{e_{a\mu}-x^{\underline{b}}b_{\underline{b}a\mu}}&{0}\\{-x^{\underline{b}}A_{\underline{{ba}}\mu}}&{\eta_{\underline{{a\mu}}}}\\\end{matrix}\right).
\partial_{r}\varphi={\frac{1}{3(r-r_{H})}}[\eta_{P}e^{3\varphi}{\bfP}_{ij\,\infty}+\eta_{Q}e^{-{3\varphi}}{\bfQ}_{ik\,\infty}]^{2}{\frac{1}{\eta_{P}{\bfP}_{ij\,\infty}+\eta_{Q}{\bfQ}_{ik\,\infty}}}.
H^{\ast}C^{2}\dot{\tilde{x}}^{0}=\psi^{0}(\sigma^{1},\sigma^{2},..,\sigma^{p}),~~~~~~~H^{\ast}C^{2}\dot{\tilde{x}}^{i}=\psi^{i}(\sigma^{1},\sigma^{2},..,\sigma^{p}),
\bar{\Psi}^{\prime\rhor}(z^{\prime})=\Omega(z;g)^{-\eta}\Upsilon(z;g)^{\kappa}\bar{D}_{~\sigma}^{\rho}(\hat{\bar{L}}(z_{-};g))D_{~s}^{r}(\hat{u}(z;g)^{-1})\bar{\Psi}^{\sigmas}(z)\,.
\phi_{2}^{4}=\frac{\sqrt{g}}{\alpha}\left[p_{1}^{2}-m^{2}+\sqrt{g}~\frac{\alpham+\beta\sigma\sqrt{m^{2}-p_{1}^{2}}}{(q_{2}^{2})^{3/2}}\right],~~~~~~\sigma=\mathrm{sign}~\varepsilon(p_{1}q_{2}q_{3}),
H=-\frac{1}{2}\sum_{j=1}^{L}\{(\sigma_{x}^{j}\sigma_{x}^{j+1}+\sigma_{y}^{j}\sigma_{y}^{j+1})(1-U\tau_{z}^{j+1})+(\tau_{x}^{j}\tau_{x}^{j+1}+\tau_{y}^{j}\tau_{y}^{j+1})(1-U\sigma_{z}^{j})\}
\beta_{R}(\alpha_{R})=q^{2}\frac{\partial\alpha_{R}(q)}{\partialq^{2}}=-\frac{\beta_{0}}{4\pi}\alpha_{R}^{2}+\frac{\beta_{0}}{4\pi}\frac{\left[\frac{\beta_{0}}{4\pi}-\alpha_{R}\right]^{2}}{1+q^{2}/\Lambda_{R}^{2}}
{\calL}=-\frac{i}{2}\epsilon_{\mu\nu\lambda}B_{\mu}^{*}(\partial_{\nu}-iga_{\nu})B_{\lambda}+\frac{M}{2}B_{\mu}^{*}B_{\mu}-\frac{i}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+(\partial_{\mu}\bar{c})(\partial_{\mu}c)+\frac{1}{2\rho}(\partial_{\mu}a_{\mu})^{2}\;.
{\widetilde\xi}|\Psi\rangle=\frac{i}{cos^{2}(\theta)-sin^{2}(\theta)e^{2i\phi}}(\sqrt{2}sin(\theta)cos(\theta)e^{i\phi}-(cos^{2}(\theta)+sin^{2}(\theta)e^{2i\phi})\xi)|\Psi\rangle,
f={\frac{1}{2}}g_{s}^{-1}\left[\sqrt{m^{2}+g_{s}^{2}\left({\frac{\mu}{r}}\right)^{7}\left(P_{r}^{2}+{\frac{\vec{L}^{2}}{r^{2}}}\right)}+m\right],\\\\g={\frac{\vec{L}^{2}}{r^{2}}},
\{A,C\}_{(1,2)}=\beta({\partial}_{1}a{\partial}_{2}c-{\partial}_{2}a{\partial}_{1}c)+\beta\cdot\sum_{k=1}^{\infty}{\gamma}^{2k}({\partial}_{2}c{\partial}_{1}a_{2k}-{\partial}_{1}c{\partial}_{2}a_{2k}),
\textsf{P}_{0}=VP_{0}V^{-1}=P_{0}-l[P_{0}(\overrightarrow{P}.\overrightarrow{\bigtriangledown}),P_{0}]+l^{2}[P_{0}(\overrightarrow{P}.\overrightarrow{\bigtriangledown})[P_{0}(\overrightarrow{P}.\overrightarrow{\bigtriangledown}),P_{0}]]-...
\sum_{A,B}\int_{x}^{\infty}d\theta\frac{L_{A}^{>}(\theta)}{d\theta}(\Phi_{AB}*L_{A}^{>})(\theta)\simeq\sum_{A}\int_{\epsilon_{A}^{>}(x)}^{\epsilon_{A}^{>}(\infty)}d\epsilon_{A}^{>}\frac{\epsilon_{A}^{>}}{(1+e^{\epsilon_{A}^{>}})}-\sum_{A}\int_{x}^{\infty}d\thetaM_{A}e^{\theta}L_{A}^{>}(\theta).
\alpha^{\prime}M_{n}^{2}\sim4\alpha^{\prime}\lambdan^{2}+\frac{32\sqrt{\pi\alpha^{\prime}\lambda}}{\Gamma\left(\frac{1}{4}\right)^{2}}n^{3/2}+\left(4\alpha^{\prime}\lambda+\frac{128\pi}{\Gamma\left(\frac{1}{4}\right)^{4}}\right)n+{O}(\sqrtn),
A_{\mu}^{(1)}=\mp{\frac{2i}{g}}\,(H_{1}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+H_{2}\,{\frac{v_{\mu}^{2}}{\rho_{2}}})+\kappa\,(E_{13}^{\pm}\,R_{\mu}^{1}+E_{23}^{\pm}\,R_{\mu}^{2})\,\delta(R^{1}\cdotR^{2}).
\underline{{\nabla}}\,\cdot\,\underline{{E}}\;=\;\langle{\mathcalJ}^{0}\rangle_{\beta,\underline{{\mu}}}~,~~\underline{{\nabla}}\wedge\underline{{B}}-\partial_{t}\,\underline{{E}}\;=\;\langle\underline{{{\mathcalJ}}}\rangle_{\beta,\underline{{\mu}}}~.
I^{a}(w;\eta^{\prime},n^{\prime};\eta,n)=\int_{-Y_{0}}^{Y_{0}}\,dy\int_{0}^{\infty}\,dk^{+}\int_{-\infty}^{\infty}\,dk^{-}\delta(k^{+}k^{-}+w)E_{w}^{a}(y;\eta^{\prime},n^{\prime},\eta,n),
\dot{\hat{\varrho}}+3\left(\hat{\varrho}+{\frac{\hat{\wp}}{\hat{c}^{2}}}\right){\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}=\left(\hat{\varrho}-3{\frac{\hat{\wp}}{\hat{c}^{2}}}\right){\frac{\dot{\hat{c}}}{\hat{c}}}.
\delta\widetilde{G}_{\mu\nu}=\partial^{\rho}\partial_{\left(\nu\right.}h_{\left.\mu\right)\rho}-\frac{1}{2}\partial^{\rho}\partial_{\rho}\,h_{\mu\nu}-\frac{1}{2}\partial_{\mu}\partial_{\nu}h_{\rho}^{\rho}-\frac{1}{2}\eta_{\mu\nu}\left(\partial^{\rho}\partial^{\kappa}\,h_{\rho\kappa}-\partial^{\rho}\partial_{\rho}\,h_{\kappa}^{\kappa}\right),
\prod_{i=1}^{L}\intd^{d}k_{i}\prod_{j=1}^{N}P_{\overline{{k}}_{j},m_{j}}^{\nu_{j}}\prod_{r=1}^{n_{1}}\overline{{k}}_{1\mu_{r}}\ldots\prod_{s=1}^{n_{N}}\overline{{k}}_{N\lambda_{s}}=T(q,\partial,{\bfd^{+}})\prod_{i=1}^{L}\intd^{d}k_{i}\prod_{j=1}^{N}P_{\overline{{k}}_{j},m_{j}}^{\nu_{j}},
f(\omega,\epsilon)={\frac{i}{4\pi\omega}}{\frac{\Gamma(1+i\omega/\kappa)}{\Gamma(1-i\omega/\kappa)}}\left({\frac{m\epsilon}{2}}\right)^{-2i\omega/\kappa}-{\frac{1}{2\pi\kappa}}\psi(1-i\omega/\kappa)~,
(\,\Lambda\,\vert\Lambda\,)=1\,,\,\,(\,\Phi\vertX\Psi\,)=(\,X^{*}\Phi\vert\Psi\,)\,\,\mathrm{for}\,\,\,\Phi\,,\Psi\,\in{\calV}_{\Lambda}\,,\,\,X\inU_{q}^{(n)}\,.
j_{p}=d\zeta_{\tilde{m}}^{\wedgep}\int_{\partial{\calM}^{p+1}}d{\hat{\zeta}}^{\tilde{m}}(\tau)\delta^{p+1}\left(\zeta-{\hat{\zeta}}(\tau)\right)~=(-)^{p}d\zeta_{\tilde{n}}^{\wedgep}j^{\tilde{n}}.\qquad
<\rho^{\mathrm{vort}}({\bfq})\rho^{\mathrm{vort}}({\bf-q})>=\sum_{a,a^{\prime}}e^{{\bfiq.(x}_{a}-{\bfx}_{a^{\prime}})}<m({\bfx}_{a})m({\bfx}_{a^{\prime}})>\,\,\,\,.
S^{\dagger}=-\;\intd\tau\;d\sigma\;[\mu\sqrt{-\mathrm{det}G}+\frac{1}{2}\sqrt{-\mathrm{det}G}\;G^{\alpha\beta}\Psi_{,\alpha}\Psi_{,\beta}+\epsilon^{\alpha\beta}A_{\mu}X_{,\alpha}^{\mu}\Psi_{,\beta}],
\begin{array}{c}{\{Q^{a\alpha},\bar{Q}_{b\beta}\}=2\delta^{a}{}_{b}\gamma^{\mu}{}^{\alpha}{}_{\beta}P_{\mu}\,,}\\{{}}\\{{}[{P}_{\mu},{P}_{\nu}]=[{P}_{\mu},Q^{a\alpha}]=0\,.}\\\end{array}
\vec{\calQ}={\frac{1}{\sqrt{2}}}{\calU}^{T}\left(\begin{matrix}{U_{5}({\bfe}_{u}-{\bfe}_{d})}\\{U_{21}\left(\begin{matrix}{{\bfe}_{u}+{\bfe}_{d}}\\{0_{16}}\\\end{matrix}\right)}\\\end{matrix}\right),
<W_{1}W_{2}W_{3}>=\frac{<[W_{1},[W_{2},W_{3}]]>}{(p_{1}-2m_{1})(p_{3}+2m_{1}+2m_{2})}+\frac{<[W_{2},[W_{1},W_{3}]]>}{(p_{2}-2m_{2})(p_{3}+2m_{1}+2m_{2})}.
\frac{1}{2}\intdz\,\left(-\hbar^{2}\frac{\delta^{2}}{\delta\phi\left(z\right)^{2}}+\phi\left(z\right)\left(m^{2}-\nabla_{z}^{2}\right)\phi\left(z\right)\right)\Psi[\phi]=E_{0}~\Psi[\phi].
0<i\left[e_{\alpha}^{A}\bar{h}_{A\bar{\beta}}-h_{A\alpha}\bar{e}_{\bar{\beta}}^{A}\right]=e_{\alpha}^{A}\left[i\bar{h}_{A\bar{\gamma}}(\bar{e}^{-1})_{B}^{\bar{\gamma}}-i(e^{-1})_{A}^{\gamma}h_{B\gamma}\right]\bar{e}_{\bar{\beta}}^{B}=e_{\alpha}^{A}\left[i\bar{\Pi}_{AB}-i\Pi_{BA}\right]\bar{e}_{\bar{\beta}}^{B}\,
{\calL}=\intd^{2}z(-|\partialx_{A}|^{2}+\psi{\bar{\partial}}\rho_{A}+{\bar{\psi}}_{A}\partial{\bar{\rho}}_{A}-|\partial_{A}W|^{2}+\partial_{A}\partial_{B}W\psi_{A}{\bar{\psi}}_{B}+{\bar{\partial}}_{A}{\bar{\partial}}_{B}{\bar{W}}\rho_{A}{\bar{\rho}}_{B})
\frac{\partial^{2}}{\partialt^{2}}\phi_{\alphaj}(t)+3\frac{\dot{R}(t)}{R(t)}\frac{\partial}{\partialt}\phi_{\alphaj}(t)+\left(m^{2}+\xi{\bfR}+\frac{{\bfk}^{2}}{R^{2}(t)}\right)\phi_{\alphaj}(t)=0,
\tilde{T}(z)=-{\frac{1}{2}}(\partial\varphi)^{2}-{\frac{i}{2\alpha_{+}}}(\alpha_{1}+\alpha_{3})\cdot\partial^{2}\varphi+\tilde{\beta}_{\alpha_{1}}\partial\gamma_{\alpha_{1}}-\tilde{\eta}_{\alpha_{2}}\partial\tilde{\xi}_{\alpha_{2}}-\tilde{\eta}_{\alpha_{1}+\alpha_{2}}\partial\tilde{\xi}_{\alpha_{1}+\alpha_{2}}.
\pi_{1}^{\prime\prime}+\pi_{1}^{\prime}\Bigl({\frac{6}{\lambda\phi^{5}}}+{\frac{9}{\phi}}\Bigr)+\pi_{1}\Bigl({\frac{6}{\lambda\phi^{6}}}+{\frac{15}{\phi^{2}}}+{\frac{36\pi}{\lambda\phi^{4}}}-{\frac{\lambda_{1}}{\pi\phi^{6}}}\Bigl({\frac{6\pi}{\lambda}}\Bigr)^{3/2}\Bigr)=0\.
+\frac{ie(\phi_{1}\phi_{0}^{\dagger}+\phi_{0}\phi_{1}^{\dagger})}{(\lambda_{1}^{2}-{\widehat\partial}^{2}){\widehat\partial^{2}}}[(2\partial_{0}^{2}-{\widehat\partial}^{2})V_{0,1}]+\frac{ie\phi_{0}\phi_{0}^{\dagger}}{({\widehat\partial}^{2})^{2}}[(2\partial_{0}^{2}-{\widehat\partial}^{2})V_{0,0}].
\left[{\frac{m_{r}^{2}}{2p^{+}}}+{\frac{m_{r}^{2}}{2q^{+}}}-(p^{-}+q^{-})\right]\phi(p^{+},q^{+})\approx{\frac{g^{2}N_{c}}{2\pi^{2}}}\intdk^{+}K_{0}(\sqrt{2}k^{+}R)\phi(p^{+}-k^{+},q^{+}+k^{+}),
ds_{\mathrm{brane}}^{2}=H_{\mathrm{brane}}^{-{\frac{2}{p+1}}}dx^{m}\eta_{mn}dx^{n}+H_{\mathrm{brane}}^{\frac{2w}{p+1}}dX^{m^{\prime}}\delta_{m^{\prime}n^{\prime}}dX^{n^{\prime}}\.
{\calG}=\frac{1}{2}e^{-4\phi}\omega_{\kappa}^{2}+2d\phi^{2}+\frac{1}{2}\left(\frac{df^{2}}{f^{2}}+f^{2}\omega_{\chi}^{2}\right)+f\left\{e^{2\phi}\omega_{v}^{2}+e^{-2\phi}\omega_{a}^{2}\right\},
{}_{\mathrm{in}}\big\langle\,{\bfB}_{\sigma_{n}^{\prime}}(k_{n}^{\prime})\ldots{\bfB}_{\sigma_{1}^{\prime}}(k_{1}^{\prime})\,|\,{\bfB}_{\sigma_{1}}(k_{1})\ldots{\bfB}_{\sigma_{n}}(k_{n})\,\big\rangle_{\mathrm{in}}=(2\pi)^{n}\\prod_{j=1}^{n}\delta_{\sigma_{j}\sigma_{j}^{\prime}}\\delta(\theta_{j}-\theta_{j}^{\prime})\,,
\gamma^{0}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right),\quad\gamma^{1}=\left(\begin{matrix}{0}&{-1}\\{1}&{0}\\\end{matrix}\right),\quad\gamma^{5}=\gamma^{0}\gamma^{1}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right).
K(P^{\prime\prime},P^{\prime})=i\int_{0}^{\infty}d\lambda^{\prime}\;se^{i\lambda^{\prime}(\pi^{\prime\prime}-\pi^{\prime})}e^{-i[s\lambda^{\prime}(P^{2}-m^{2})+{\frac{i}{s}}({\bar{\eta}}^{\prime\prime}-{\bar{\eta}}^{\prime})(\eta^{\prime\prime}-\eta^{\prime})]}\delta^{4}(P^{\prime\prime}-P^{\prime})\,.
S\approx(1/2\pi)\intd^{2}\sigma\left\{U^{2}/R^{2}+(1/2)\left[(U_{T}^{2}/R^{2})\left((\partial_{i}Z)^{2}+(\partial_{i}T)^{2}+(\partial_{i}x_{3})^{2}\right)+R^{2}(\partial_{i}y^{M})^{2}\right]\right\},
(1-\delta_{IJ}\delta_{bb^{\prime}})\left\{{\frac{\vec{\alpha}^{(I)}\vec{\alpha}^{(J)}}{2}}-{\frac{d_{I}d_{J}}{D-2}}+\sum_{L=0}^{D-1}\Lambda_{bL}^{(I)}\Lambda_{b^{\prime}L}^{(J)}-2\right\}=0,
{\Gamma}_{\mathrm{vac}}^{(3)}(D)=m^{3D-12}(4\pi)^{-{\frac{3}{2}}D}\Gamma(6-{\frac{3}{2}}D)\,\biggl\lbrace\Bigl[{\frac{2B(2-{\frac{D}{2}},2-{\frac{D}{2}})}{D-2}}\Bigr]^{2}+12\zeta(3)-4\zeta(2)-4\biggr\rbrace
\begin{array}{l}{E[((\phi_{\tau},f)-E[(\phi_{\tau},f)])((\phi_{\tau^{\prime}},f^{\prime})-E[(\phi_{\tau^{\prime}},f^{\prime})])]=}\\{\frac{\hbar^{2}}{2i}\Big(f,{\calA}^{-1}\Big(exp(-i{\calA}|\tau-\tau^{\prime}|)-exp(-i(\tau+\tau^{\prime}-2\tau_{0}){\calA})\Big)f^{\prime}\Big)}\\\end{array}