harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{m}=\left(\begin{array}{cccc}{\frac{k_{3}^{2}}{m^{2}}}&{0}&{0}&{\frac{\left|k_{3}\right|\left(m^{2}+k_{3}^{2}\right)^{1/2}}{m^{2}}}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{\frac{\left|k_{3}\right|\left(m^{2}+k_{3}^{2}\right)^{1/2}}{m^{2}}}&{0}&{0}&{\frac{\left(m^{2}+k_{3}^{2}\right)}{m^{2}},}\\\end{array}\right)
\int_{0}^{2\pi}d\sigma\left(-\frac{1}{2\pi}p^{+}\partial_{\sigma}X^{-}+{\calP}^{I}\partial_{\sigma}X^{I}+\frac{i}{2}\psi_{+}^{A}\partial_{\sigma}\psi_{+}^{A}+\frac{i}{2}\psi_{-}^{A}\partial_{\sigma}\psi_{-}^{A}\right)=0~.
R_{lij}^{(+)k}=\partial_{i}\Gamma_{jl}^{(+)k}-\partial_{j}\Gamma_{il}^{(+)k}+\Gamma_{im}^{(+)k}\Gamma_{jl}^{(+)m}-\Gamma_{jm}^{(+)k}\Gamma_{il}^{(+)m},\\R_{ij}^{(+)}=R_{ikj}^{(+)k}.
L=-{\frac{1}{4\pi}}{\epsilon}^{\mu\nu\lambda}[K_{ij}{a^{i}}_{\mu}\partial_{\nu}{a^{j}}_{\lambda}+2eA_{\mu}\partial_{\nu}(t_{i}{a^{i}}_{\lambda})]+{\ell}_{i}{a^{i}}_{\mu}J^{\mu},
\begin{array}{l}{\delta\theta_{\alpha}=\epsilon^{\alpha},\qquad\delta\bar{\theta}_{\alpha}=\bar{\epsilon}^{\alpha},}\\{\deltax^{\bar{\mu}}=-i\bar{\epsilon}_{\alpha}\tilde{\Gamma}^{\bar{\mu}\alpha\beta}\bar{\theta}_{\beta}-i\epsilon^{\alpha}\Gamma_{\alpha\beta}^{\bar{\mu}}\theta^{\beta},\qquad\deltax^{11}=0,}\\\end{array}
\vertF(z)\vert<\vertz\vert\varepsilone^{(c-2t)\verty\vert}\quad\textrm{or}\quad\vertf(z)\vert<\frac{Me^{2(1-t)\verty\vert}}{\vertz\vert^{\alpha-1}},\quadz=x+iy,\quad\vertz\vert\to\infty
G\approx\frac{1}{2\pi(1+g^{2}/\pi)}\bigg[1-2^{(5\pi+3g^{2})/(\pi+g^{2})}\frac{(1+g^{2}/\pi)^{2}}{16\pi^{2}}\bigg(\frac{m^{2}}{T^{2}}\bigg)^{(\pi+2g^{2})/(\pi+g^{2})}\bigg].
c_{2}^{D}={\mathrm{Tr}}\biggr[-{\frac{1}{360}}q+{\frac{2}{35}}p-{\frac{1}{3}}\Bigr(X-{\frac{1}{6}}R\Bigr)({\mathrm{Tr}}K)-{\frac{1}{2}}n^{\sigma}\nabla_{\sigma}\Bigr(X-{\frac{1}{6}}R\Bigr)+{\frac{1}{15}}C_{\mu\nu\alpha\beta}K^{\mu\alpha}n^{\nu}n^{\beta}\biggr]\;,
\intdx\,\phi_{n}^{*}\left(x\right)A\left(x,\partial_{x}\right)\psi_{n}\left(x\right)=\intdx\,\left[\tilde{A}^{*}\left(x,\partial_{x}\right)\phi_{n}^{*}\left(x\right)\right]\psi_{n}\left(x\right).
\begin{array}{l}{H=\displaystyle\frac{e}2p^{2}+\xi(pp_{z})+\phi(p_{z}^{2}+1)+\lambda_{e}\pi_{e}+\lambda_{\xi}p_{\xi}+\lambda_{\phi}\pi_{\phi}+\lambda_{B}^{\mu}p_{B\mu}+}\\{\qquad\lambda_{n\mu}p_{n}^{\mu}+\lambda_{z\mu}(p_{z}^{\mu}-n^{\nu})+L_{\alpha}\lambda_{\theta}^{\alpha},}\\\end{array}
p_{N{\mathrm{(int)}}}^{(2)}(\xi_{1},\xi_{2})=\frac{A_{d}}{2\xi_{2}^{2}}\int_{0}^{\infty}dkk^{d-2}\int_{0}^{\infty}d\omega\frac{I_{\omega}^{\prime}(k\xi_{1})(1+\omega^{2}/k^{2}\xi_{2}^{2})}{I_{\omega}^{\prime}(k\xi_{2})N_{\omega}^{\prime}(k\xi_{1},k\xi_{2})}.
E=L_{\perp}^{2}\int\frac{d^{2}k}{\left(2\pi\right)^{2}}\sum_{n=-\infty}^{\infty}\left[\left(k^{2}+\frac{\pi^{2}}{L^{2}}\left(2n\right)^{2}\right)^{1/2}+2\left(k^{2}+M^{2}+\frac{\pi^{2}}{L^{2}}\left(2n+1\right)^{2}\right)^{1/2}\right],
V(\overline{{\rho}})=-\frac{1}{12\pi}[|M_{+}(\overline{{\rho}}^{2})|^{3}+|M_{-}(\overline{{\rho}}^{2})|^{3}]+\frac{\lambda}{4!}\overline{{\rho}}^{4}+\frac{\eta}{6!}\overline{{\rho}}^{6}.
\tilde{F}^{\prime\prime}+\frac{1}{r}\tilde{F}^{\prime}-(\frac{1}{r^{2}}\tilde{\chi}^{2}+\frac{q^{2}}{m_{H}^{2}}\tilde{A}^{2})\tilde{F}+\frac{1}{2}(\tilde{F}-\tilde{F}^{3})=0,
B(q^{2})=b_{0}(q^{2})g^{3}+g^{5}{\frac{B_{1}(q^{2},Q^{2})}{\epsilon}}+g^{5}B_{2}(q^{2},Q^{2})+g^{7}\left[{\frac{B_{3}(q^{2},Q^{2})}{\epsilon^{2}}}+{\frac{B_{4}(q^{2},Q^{2})}{\epsilon}}+B_{5}(q^{2},Q^{2})\right]+...
\int_{0}^{\infty}dy^{\prime}\\chi(y,y^{\prime})e^{-ty^{\prime}}=\frac{1}{2\pii}\int_{\cal{C}}\frac{dw}{\sqrt{1+w}\sqrt{w}}\left[1+e^{twy}\sqrt{\pitwy}(\Phi(\sqrt{twy})-1)\right],
A_{parent}^{\mathrm{IV}}=-\frac{T_{p}}{2}\intd^{p+1}{\xi}\left[\sqrt{-g}\left(-{\Psi}^{(p+3)/2}g_{ij}h^{ij}+(p+3){\Psi}^{(p+1)/2}\right)+{\Lambda}\left({\Psi}-{\Phi}\right)\right].
\prod_{i=1}^{5}\biggl({\frac{\partial}{\partial\mu_{i}}}\biggr)^{w_{i}}\tilde{\Pi}(t,\{\mu_{i}\})=e^{-t}\biggl({\frac{\partial}{\partial\mu_{6}}}\biggr)^{-w_{6}=24}\tilde{\Pi}(t,\{\mu_{i}\}).
\langle\varphi_{i}^{(0)}(X_{0})\varphi_{j}^{(0)}(X_{0})\rangle_{\zetac}\simeq-\frac{\lambda}{192\pi}(N+2)\frac{T}{\omega^{2}}\frac{\delta_{ij}}{V}e^{2\omegaX_{0}}.
v_{2}(s_{0})={\frac{\sqrt{2}\,r}{\sqrt{r+a}}}\left(\begin{array}{c}{0}\\{0}\\{1}\\{0}\\\end{array}\right)+{\frac{\sqrt{2}\,a}{\sqrt{r+a}}}\left(\begin{array}{c}{0}\\{-1}\\{1}\\{0}\\\end{array}\right)\,.
P_{eq}^{DK}=\frac{8}{5\sqrt{3}}\frac{<\mid{\rho}\mid^{-3}\hat{b}(\hat{n}-F_{DK})>}{<\mid{\rho}\mid^{-3}(1-2/9(\hat{n}\cdot\hat{v})^{2}+11/18\midF_{DK}\mid^{2})>}~,
{\pi}^{+}=\frac{{\delta}L}{{\delta}{\partial}_{+}A_{+}}=0,\;{\pi}^{-}=\frac{{\delta}L}{{\delta}{\partial}_{+}A_{-}}=F_{+-},\;{\pi}^{i}=\frac{{\delta}L}{{\delta}{\partial}_{+}A_{i}}=F_{\;\;i}^{+},\;{\pi}_{B}=\frac{{\delta}L}{{\delta}{\partial}_{+}B}=0,
a_{2}=(Q-R/6)^{2}-\frac{1}{3}\BoxQ-\frac{1}{90}R_{\mu\nu}R^{\mu\nu}+\frac{1}{90}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}+\frac{1}{15}\BoxR+\frac{1}{6}\tilde{R_{\mu\nu}}\tilde{R^{\mu\nu}},
\langle0|T_{i}^{k}|0\rangle^{(1b)}(\xi_{2},\xi)=\langle0_{R}|T_{i}^{k}|0_{R}\rangle-\frac{1}{2}\delta_{i}^{k}A_{d}\int_{0}^{\infty}dkk^{d}\int_{0}^{\infty}d\omega\sum_{\sigma=0,1}\beta_{\sigma}\frac{K_{\omega}^{(\sigma)}(k\xi_{2})}{I_{\omega}^{(\sigma)}(k\xi_{2})}F_{{\mathrm{em}}}^{(i)}[I_{\omega}(k\xi)],
A=(N-1)!\int_{0}^{1}...\int_{0}^{1}d\alpha_{1}...d\alpha_{n}\delta(1-\sum_{i=1}^{n}\alpha_{i})\int...\times\int\prod_{i}d^{4}k_{i}\prod_{j}\frac{W_{j}(k_{j}^{2})}{\sum_{j}\alpha_{j}(k_{j}^{2}+m_{j}^{2})},
\begin{array}{l}{e_{1}^{\sharp}(x)=P_{11}(x)P_{22}(x)-P_{21}(x)P_{12}(x)=0}\\{e_{2}^{\sharp}(x)=P_{31}(x)P_{42}(x)-P_{41}(x)P_{32}(x)=0\;}\\\end{array}
{\calL\mit}\,=\,-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\mu^{2}A_{\mu}A^{\mu}-\frac{1}{2}\lambda(\partial_{\mu}A^{\mu})^{2}+\bar{\psi}(i\not\!{\partial}-e\not\!\!{A}-m)\psi,
{\frac{H(t_{f})a(t_{f})}{H(t_{i})a(t_{i})}}=\left({\frac{t_{\mathrm{sing}}-t_{i}}{t_{\mathrm{sing}}-t_{f}}}\right)^{(1+\sqrt{3})/\sqrt{3}}=\left({\frac{\tilde{t}_{\mathrm{sing}}-\tilde{t}_{i}}{\tilde{t}_{\mathrm{sing}}-\tilde{t}_{f}}}\right)^{2/3}={\frac{\tilde{H}(t_{f})\tilde{a}(t_{f})}{\tilde{H}(t_{i})\tilde{a}(t_{i})}}\,.
{\frac{\delta}{\deltaA_{\mu}^{a}(x)}}U_{\gamma_{u}}^{R}(s,t)=\int_{s}^{t}dw\,\dot{\gamma}_{u}^{\mu}(w)\delta^{(3)}(x,\gamma_{u}(w))U_{\gamma_{u}}^{R}(s,w)T^{a}U_{\gamma_{u}}^{R}(w,t).
{{\calG}^{-1}}_{\mu\nu}^{\alpha\beta}\eta_{\alpha\beta}^{\mathrm{LTT}}=\left(k^{2}-\Lambda_{l}+\frac{1}{2r^{2}}(N-1)-\frac{N+d}{N+d-2}\Lambda\right)\eta_{\mu\nu}^{\mathrm{LTT}}.
\left({\frac{x^{-}}{x^{+}}}\right)^{2}F(x^{-},x^{+})={\frac{N^{2}}{k}}\sum_{n}\left({\frac{3(k-2n)^{2}n_{f}}{4\pi^{2}k^{2}r^{4}}}+{\frac{3n(k-n)n_{b}}{\pi^{2}k^{2}r^{4}}}\right)={\frac{N^{2}(2n_{b}+n_{f})}{4\pi^{2}r^{4}}}\left(1-{\frac{1}{k}}\right)
{\bfp}_{j}={\bfp}(c_{j}),\quad{\bfp}(c)=\left(\begin{array}{c}{p^{(0)}(c)}\\{p^{(1)}(c)}\\\end{array}\right)=\left(\begin{array}{c}{{[}\theta_{2}\theta_{3}{]}(c\eta)}\\{-{[}\theta_{1}\theta_{4}{]}(c\eta)}\\\end{array}\right),
E_{00}^{(i)}\approx\frac{1}{2}\left\{\frac{\ddot{a}_{0}}{a_{0}}+\frac{\dot{a}_{0}}{a_{0}}\frac{\dot{d}_{0}}{d_{0}}-\frac{\dot{a}_{0}^{2}}{a_{0}^{2}}-\frac{\ddot{d}_{0}}{d_{0}}-\sum_{i=1}^{2}\frac{(\rho_{i}+P_{i})}{2d_{0}}\right\}\,.
\frac{d^{2}}{dt^{2}}\eta_{\vec{p}}\left(t\right)+\omega^{2}\left(\vec{p},t\right)\,\eta_{\vec{p}}\left(t\right)=0;\,\,\eta_{\vec{p}}\left(t\right)=e^{-i\,\omega_{0}\left(\vec{p}\right)\,t}\,\mathrm{~at~}t\rightarrow0\,\,
\hat{H}=\frac{1}{2}\hat{p}_{z}^{2}+\hbar\omega\left[a_{+}^{\dagger}a_{+}+a_{-}^{\dagger}a_{-}\right]\\\,\\\\hat{\phi}=\hat{p}_{z}+\hbarg\left[a_{+}^{\dagger}a_{+}-a_{-}^{\dagger}a_{-}\right]\\\.
(\Gamma)_{\\beta}^{\alpha}={\frac{\zeta}{(p+1)!\varphi\sqrt{-g}}}\varepsilon^{i_{1}\cdotsi_{p+1}}E_{i_{1}}^{a_{1}}\cdotsE_{i_{p+1}}^{a_{p+1}}(\Gamma_{a_{1}\cdotsa_{p+1}})_{\\beta}^{\alpha},
D_{\mu}^{ij}[A]\frac{\delta\Gamma}{\deltaA_{\mu}^{j}}+gf^{ijk}(\langleQ_{\mu}^{j}\rangle\frac{\delta\Gamma}{\delta\langleQ_{\mu}^{k}\rangle}+\langleC^{j}\rangle\frac{\delta\Gamma}{\delta\langleC^{k}\rangle}+\langle\overline{{C}}^{j}\rangle\frac{\delta\Gamma}{\delta\langle\overline{{C}}^{k}\rangle})=0
\langle0|{\hat{f}}_{1}^{(0)}(0)|\epsilon,\epsilon^{\prime}\rangle=\pi^{2}{\frac{\hbar}{a^{3}}}{\frac{ll^{\prime}}{\sqrt{\omega_{\epsilon}\omega_{\epsilon^{\prime}}}}}\delta_{n_{2},-n_{2}^{\prime}}...\delta_{n_{D},-n_{D}^{\prime}},
\eta^{(s_{1},s_{2},s_{3},s_{4})}=\left[1+\sum_{k=1}^{5}u_{k}^{(s_{1},s_{2},s_{3},s_{4})}\Gamma^{0}\left(\Gamma^{2k}x^{2k-1}-\Gamma^{2k-1}x^{2k}\right)\right]\eta_{0}^{(s_{1},s_{2},s_{3},s_{4})},
{\calT}_{\alpha}=[\bar{C}_{\alpha},\Omega]=T_{\alpha}+C^{\dagger\beta}U_{\beta\alpha}^{\gamma}\bar{C}_{\gamma}(-1)^{\varepsilon_{\alpha}}+\bar{U}_{\alpha\beta}^{\gamma}\bar{C}_{\gamma}C^{\beta}(-1)^{\varepsilon_{\beta}}+\bar{C}_{\gamma}^{\dagger}\bar{U}_{\beta\alpha}^{\dagger\gamma}C^{\beta}(-1)^{\varepsilon_{\alpha}},
E=+E_{n}\,,\;n\geq1\;\;\psi_{2}^{(+)}=\sqrt{\frac{E_{n}+m}{2E_{n}}}{e^{ipx_{2}-\frac{1}{2}{\zeta}^{2}}}\left(\begin{array}{c}{0}\\{0}\\{-\frac{\sqrt{E_{n}^{2}-m^{2}}}{E_{n}+m}N_{n}H_{n}(\zeta)}\\{N_{n-1}H_{n-1}(\zeta)}\\\end{array}\right)
W({\bfx}_{n+1},{\bfx}_{n};\varepsilon\vert{\hat{\bfA}})\simeq\sum_{a=1}^{N}{\frac{m({\bfx}_{a}^{n+1}-{\bfx}_{a}^{n})^{2}}{2\varepsilon}}+iN\,ln\,\varepsilon+\Phi({\bfx}_{n+1},{\bfx}_{n}\vert{\hat{\bfA}})\,,
Z^{(\mathrm{cov)}}(G,q)=\sum_{h_{1}>h_{2}>....}\left[\prod_{\{h_{i}>k_{j}\}}(h_{i}-k_{j})\right]^{2G-2}q^{|h|}=\sum_{\{h\}}\left[\frac{d_{\{h\}}}{|h|!}\right]^{2-2G}q^{|h|}~.
\langle\,X\,[\,\phi\,,\,\phi^{\ast}\,]\,\rangle_{_J}\,\,=\,\,\int\,{\calD}\phi\,X\,\,\,exp{\frac{i}{\hbar}}\,\{S[\phi\,,\,\phi^{\ast}\,]\,+\,J^{A}\,\phi^{A}\,\,\}
g_{\mu\nu}=\eta_{\mu\nu}+\frac{Kx_{\mu}x_{\nu}}{1-Kx_{\lambda}x^{\lambda}}\;,\;\;\;\;\eta_{=}\mathrm{diag}(\pm1,\pm1,\cdots)\;,\;\;\;K\mathrm{~is~constant~and~}x_{\mu}:=\eta_{\mu\lambda}x^{\lambda}\;.
\sum_{r\geq0}\nu^{r}\sum_{u+v=r\atopu,v\geq0}\rho_{u}(\rho_{v}(FG)H)=\sum_{r\geq0}\nu^{r}\sum_{u+v=r\atopu,v\geq0}\rho_{u}(F\rho_{v}(GH)),\quad\forallF,G,H\inN[\nu].
C:\quad\left(\gamma_{\mu\nu}\,D_{(+)}^{\mu}\,D_{(+)}^{\nu}\,+\,m^{2}\right)\,u(x)\,=\,0\,\,\,\longrightarrow\,\,\,\left(\gamma_{\mu\nu}\,D_{(-)}^{\mu}\,D_{(-)}^{\nu}\,-\,m^{2}\right)\,v(x)\,=\,0\quad,
f_{\bfk}({\bfx},t)={\frac{n_{T}^{(m-2)/2}}{\left[2\omega(2\pi)^{m}\right]^{1/2}}}\left\{\Theta(-x)T_{lg}e^{-i\beta^{\prime}x_{1}}+\Theta(x)\left[e^{in_{T}k_{1}x_{1}}+R_{lg}e^{-in_{T}k_{1}x_{1}}\right]\right\}e^{in_{T}(k_{2}x_{2}+\cdots+k_{m}x_{m})}e^{-i\omegat},
\sum_{j}^{n}\frac{\partialX^{\mu}}{\partialx_{i}}\left(\frac{\partial}{\partialx_{j}}\frac{\partialL}{\partial\frac{\partialX^{\mu}}{\partialx_{j}}}\right)\,+\,\sum_{j}\sum_{k\neqi}F_{ik}\left(\frac{\partial}{\partialx_{j}}\frac{\partialL}{\partial\frac{\partialA^{k}}{\partialx_{j}}}\right)\,=\,0
\begin{array}{c}{L=-\left(A_{x}+\frac\lambda2\left(B_{x}+B_{t}\right)+\frac1{2\lambda}\left(B_{x}-B_{t}\right)\right),}\\{T=-\left(+\frac\lambda2\left(B_{x}+B_{t}\right)-\frac1{2\lambda}\left(B_{x}-B_{t}\right)\right),}\\\end{array}
D=\frac{2ng_{s}}{r^{2}}\frac{\sqrt{\pi}\Gamma(2-\frac{\gamma}{2})}{\Gamma(\frac{3-\gamma}{2})},\quadB=\frac{4\pim}{l}\frac{\sqrt{\pi}\Gamma(\frac{\gamma+1}{2})}{\Gamma(\frac{\gamma}{2})}
\Psi^{(S)}~=~\sum_{\{\kappa\}}~\prod_{i}^{i_{\scriptstyle\kappa}}~\bar{\psi}^{(a)(+)}({\bfx}_{i})~\prod_{j}^{j_{\scriptstyle\kappa}}~\psi^{(a)(+)}({\bfx}_{j})~~\prod_{k}^{i_{\scriptstyle\kappa}}~\bar{\psi}^{(b)(+)}({\bfx}_{k})~\prod_{l}^{l_{\scriptstyle\kappa}}~\psi^{(b)(+)}({\bfx}_{l})~~\prod_{m}^{m_{\scriptstyle\kappa}}~\varphi^{(+)}({\bfx}_{m})\quad.
\Gamma_{1}^{k}=\left(\begin{array}{cc}{\gamma^{k}}&{0}\\{0}&{0}\\\end{array}\right)\quad\Gamma_{2}^{k}=\left(\begin{array}{cc}{0}&{0}\\{0}&{\sigma^{k}}\\\end{array}\right)\quad\Gamma_{3}^{k}=\left(\begin{array}{cc}{0}&{{\xi^{k}}^{T}}\\{-{\xi^{k}}}&{0}\\\end{array}\right)
\nabla_{\nu}\hat{E}^{\mu\nu}=16\piG_{N}\,T_{\\\\\\\\\rho}^{\mathrm{matter}\,\,\mu}U^{\rho}+16\pi\sqrt{G_{N}}\bar{\epsilon}(J^{\mu}+i\gamma_{5}\tilde{J}^{\mu})\epsilon+4\overline{{\hat{\nabla}_{\nu}\epsilon}}\Gamma^{\mu\nu\rho}\hat{\nabla}_{\rho}\epsilon,
{\frac{1}{2\ell}}\sum_{n}\,{\frac{1}{k^{2}+(k^{5})^{2}}}=\oint{\frac{dk^{5}}{2\pi}}{\frac{1}{e^{2ik^{5}\ell}-1}}{\frac{1}{k^{2}+(k^{5})^{2}}}\,
\left\{A,B\right\}=\frac{\partialA}{\partialq^{i}}\frac{\partialB}{\partialp_{i}}+\frac{\partialA}{\partialp_{i}}\frac{\partialB}{\partialq^{i}}+\frac{\partialB}{\partial\pi_{a}}\frac{\partialA}{\partial\theta^{a}}+\frac{\partialB}{\partial\theta^{a}}\frac{\partialA}{\partial\pi_{a}}\.
\frac{\partial^{3}\bar{f}_{\epsilon}}{\partial\bar{x}^{3}}|_{P_{0}}=\epsilon\frac{\partial^{3}\bar{f}_{\epsilon}}{\partial\bar{t}^{3}}|_{P_{0}}+\epsilon\frac{\partial^{2}\bar{\alpha}}{\partial\bar{t}^{2}}\frac{\partial\bar{f}_{\epsilon}}{\partial\bar{t}}|_{P_{0}}=0
Z_{N}=-{\frac{\beta^{N+1}}{\pi\omegax}}\int_{0}^{\infty}{\frac{ds}{s}}\,\Im\,\widetildeZ\oint_{C_{0}}{\frac{dw}{2\pii}}{\frac{1}{w^{N+1}(w-\beta)}}F_{0}(w,s),
{\calH}=\vec{\Lambda}\cdot\left(\frac{\lambda^{3}}{2},\frac{\lambda^{8}}{2}\right)={\frac{1}{2}}\mathrm{diag}\left({\frac{2}{3}},-{\frac{1}{3}},-{\frac{1}{3}}\right),\quad\mathrm{or}\quad{\frac{1}{2}}\mathrm{diag}\left({\frac{1}{3}},{\frac{1}{3}},-{\frac{2}{3}}\right).
\left.\frac{\delta^{2}X}{\delta\overline{{\eta}}(x)\delta\eta(y)}\right|_{\eta,\overline{{\eta}}=0}=-N_{f}\lambdae^{-i\phi(x,\delta/i\deltaJ)}\chi_{0}(x)\chi_{0}^{+}(y)e^{i\tilde{\phi}(y,\delta/i\deltaJ)}e^{i\intd^{2}x\frac{e^{2}}{2\pi}b^{(1)}\Boxb^{(1)}}
\begin{array}{rcl}{\mathrm{Htr}[T_{a},\cdots]=0,}&{{\qquad}}&{\mathrm{Htr}[\Sigma_{\alpha},\cdots]=0,}\\{\mathrm{Str}[T_{a},\cdots]=0,}&{{\qquad}}&{\mathrm{Str}\{\Sigma_{\alpha},\cdots\}=0,}\\\end{array}
\frac{\lambda(-z-w)}{\lambda(z)}=\frac{1}{\bar{\kappa}(u^{2})}\frac{(q^{2}u^{-2};t^{2})_{\infty}(t^{2}q^{-2}u^{2};t^{2})_{\infty}}{(u^{2};t^{2})_{\infty}(t^{2}u^{-2};t^{2})_{\infty}}.
Z_{1}=\int_{0}^{\infty}\frac{dT}{2T}\mathrm{Tr}e^{-2\piTL_{0}}=\frac{V_{26-D}}{(\sqrt{8\pi^{2}\alpha^{\prime}})^{26-D}}\int_{0}^{\infty}\frac{dT}{2T^{2}}T^{-(24-D)/2}e^{2\piT}f(\piT)^{-24}
ds^{2}={\frac{1}{\rho}}\left\{dy_{i}dy^{i}-{\frac{1}{\rho}}y_{i}dy^{i}y_{\bar{j}}dy^{\bar{j}}\right\}+\left\{\rho^{n}dwd{\bar{w}}+n\rho^{n-1}\left({\bar{w}}dwy_{\bar{i}}dy^{\bar{i}}+wd{\bar{w}}y_{i}dy^{i}\right)\right\}
{\calG}_{0}(\omega,{\bfk};z,z^{\prime})=\int_{-\infty}^{\infty}\frac{dq}{2\pi}\,\frac{e^{iq(z-z^{\prime})}}{\omega^{2}+{\bfk}^{2}+q^{2}+m^{2}}=\frac{e^{-\sigma|z-z^{\prime}|}}{2\sigma}\,,
f(\varphi-\varphi_{p})={\frac{e^{-i{\frac{\pi}{4}}}}{\sqrt{2\pip_{\perp}}}}e^{-i\varepsilonN(\varphi-\varphi_{p})}\sum_{l}\left[e^{i\pi(|l-\varepsilonN|-\nu)}\left[1-2\mu_{l-\varepsilonN}(\varphi_{p})\right]-1\right]e^{il(\varphi-\varphi_{p})}
\begin{array}{cl}{F(h)}&{=dA(h)+A(h)\otimesA(h)}\\{}&{=\frac{1}{2}F_{\mu\nu}(h){dx}^{\mu}\wedge{dx}^{\nu}+\frac{\lambda}{\mu}F_{\mur}(h){dx}^{\mu}\otimes{\chi}+\frac{{\lambda}^{2}}{{\mu}^{2}}F_{rr}(h){\chi}\otimes{\chi}.}\\\end{array}
\langle\hat{x}_{1}^{2}\rangle=\langle\hat{x}_{2}^{2}\rangle=\frac{1}{2}\langle\hat{x}_{1}^{2}+\hat{x}_{2}^{2}\rangle=\frac{1}{2}\langle\tilde{x}_{1}^{2}+\tilde{x}_{2}^{2}+\frac{\theta^{2}}{4\hbar^{2}}(\tilde{p}_{1}^{2}+\tilde{p}_{2}^{2})-\frac{\theta}{\hbar}\hat{L}\rangle;
\widetilde{S}_{V}(\widetilde{\Phi})\equivS_{W}(\Phi_{0}+\widetilde{\Phi})-S_{W}(\Phi_{0})=-\frac{1}{g_{o}^{2}}\left[\frac{1}{2}\langle\widetilde{\Phi},Q\widetilde{\Phi}\rangle+\frac{1}{3}\langle\widetilde{\Phi},\widetilde{\Phi}*\widetilde{\Phi}\rangle\right],
{\calL}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}\,+{\frac{\kappa}{4}}\epsilon^{\mu\nu\alpha}A_{\mu}F_{\nu\alpha}\,+{\frac{1}{2}}|D_{\mu}\Phi|^{2}-\frac{m^{2}}{2}|\Phi|^{2}\,,
\begin{array}{ccl}{d\cdotK_{2}}&{=}&{tr_{q(1)}(K_{2}R_{12}dK_{1}Y_{1}R_{12}^{-1})+q^{2}tr_{q(1)}(R_{21}^{-1}{\calP}_{12}dK_{2})}\\{}&{=}&{K_{2}tr_{q(1)}(R_{12}dK_{1}Y_{1}R_{12}^{-1})+q^{2}tr_{q(1)}(R_{21}^{-1}{\calP}_{12})dK_{2},}\\\end{array}
\sum_{t\sims}E_{t}^{(\beta)}(z)^{\prime}\otimesR_{t}^{(\beta)}\varphi(a)=z^{s}\otimes\varphi(a)+\sum_{t\sims,\;t\precs}z^{t}\otimes\bar{\varphi}_{t}(a),\quad(\bar{\varphi}_{t}(a)\in\otimes^{n}V).
{\calL}_{4}^{0\prime}=det(m)\sqrt{-det(\eta_{ab}+F_{ab})}\Big({\frac{1}{3\cdot3!}}E^{a^{\prime}}m^{-1}{}_{a^{\prime}}^{~a}\wedgee^{b}\wedgee^{c}\wedgee^{d}\epsilon_{abcd}-{\frac{1}{3\cdot4!}}e^{a}\wedgee^{b}\wedgee^{c}\wedgee^{d}\epsilon_{abcd}\Big)
F_{1}=\frac{1}{2}(m_{1}^{2}\phi_{1}+\lambda_{1}\phi_{1}^{3}+g\phi_{2}),\;\;\;\;\;\;F_{2}=\frac{1}{2}(m_{2}^{2}\phi_{2}+\lambda_{2}\phi_{2}^{3}+g\phi_{1})
0=\partial_{\mu}c\frac{\delta\Gamma}{\deltaA_{\mu}}+\frac{\delta\Gamma}{\delta\bar{K}}\frac{\delta\Gamma}{\delta\psi}+\frac{\delta\Gamma}{\deltaK}\frac{\delta\Gamma}{\delta\bar{\psi}}+B\frac{\delta\Gamma}{\delta\bar{c}}+\chi\frac{\partial\Gamma}{\partial\xi}
A_{i}^{\gamma}\,^{T}({\bfr})={\delta}_{{i}\,{\gamma}}\,{\calT}_{A}(r)+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,{\calT}_{B}(r)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\calT}_{C}(r)
\int_{R_{+}+h_{+}}^{R_{c}}\frac{r^{2}}{\Big(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{kr^{2}}{l^{2}}\Big)^{2}}dr=\int_{R_{+}+h_{+}}^{L}\frac{r^{6}}{f^{2}(r)}dr+\int_{L}^{R_{c}}\frac{r^{6}}{f^{2}(r)}dr.
\zeta^{(1/2)}(z)=4\sum_{m=0}^{\infty}\zeta_{R}\left(2z,\alpham+\gamma\right)={\frac{4}{\Gamma(2z)}}\int_{0}^{\infty}{\frac{y^{2z-1}}{1-e^{-y}}}e^{-\gammay}\left({\frac{1}{1-e^{-\alphay}}}\right)~dy~~~,
{\calL}_{\mathrm{scal}}^{(2)}[B]=-{(4\pi)}^{-D}{\frac{e^{2}}{2}}\int_{0}^{\infty}{\frac{dT}{T}}e^{-m^{2}T}T^{2-D}\int_{0}^{\infty}d\hat{T}\int_{0}^{1}du_{a}\,I(z,u_{a},\hat{T},D)\nonumber\,\nonumber\,
{\frac{d}{d\bar{t}}}\left(\bar{H}_{\mathrm{el}}+\bar{H}_{\mathrm{mag}}\right)=-{\frac{d}{d\bar{t}}}\left(\bar{H}_{\mathrm{kin}}+\bar{H}_{\mathrm{pot}}\right)=\sum_{x,i}\bar{E}_{x,i}^{a}\,{\bar{j}}_{x,i}^{a}
\phi({\bfx})=\int\frac{d^{3}{\bfk}}{\sqrt{(2\pi)^{3}2k^{+}}}[a({\bfk})e^{-ik^{+}x^{-}+i{\bfk}_{\perp}\cdot{\bfx}_{\perp}}+a^{\dagger}({\bfk})e^{ik^{+}x^{-}-i{\bfk}_{\perp}\cdot{\bfx}_{\perp}}],
0=\eta_{ab}[e_{\rho}^{a}\partial_{\mu}e_{\nu}^{b}+e_{\nu}^{a}\partial_{\mu}e_{\rho}^{b}-e_{\rho}^{a}e_{\sigma}^{b}\Gamma_{\mu\nu}^{\sigma}-e_{\nu}^{a}e_{\sigma}^{b}\Gamma_{\mu\rho}^{\sigma}]~.
\Psi=\left[\begin{array}{c}{0}\\{\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\{c_{4}\mathrm{L}_{N}^{m+1}F_{m-1}}\\{d_{4}\mathrm{L}_{N}^{\left|m\right|}F_{m}}\\\end{array}\right]
\partial_{t}\tilde{a}_{\varphi}^{0}(x,\tau)=-\frac{1}{\lambda}\\frac12\\frac{e}{4\piR}\left[e^{-\bigl|t-\tau-R/c\bigr|/\lambda}\epsilon(t-\tau-R/c)+e^{-\bigl|t-\tau+R/c\bigr|/\lambda}\epsilon(t-\tau+R/c)\right]
\widehat{S}_{V}(\Psi;\Lambda)=-\frac{1}{g_{o}^{2}}\left[\frac{a(\Lambda)}{2}\left\langle\Psi,c_{0}\left(1+\frac{L_{0}^{\mathrm{tot}}}{\Lambda}\right)\Psi\right\rangle+\frac{1}{3}\langle\Psi,\Psi*\Psi\rangle\right].
2\frac{\partial^{2}\mu}{\partialr^{2}}+\left(\frac{\partial\mu}{\partialr}\right)^{2}+\frac{2}{r}\left(\frac{\partial\mu}{\partialr}-\frac{\partial\nu}{\partialr}\right)-\frac{\partial\mu}{\partialr}\frac{\partial\nu}{\partialr}-e^{\nu-\mu}\left[2\frac{\partial^{2}\nu}{\partialt^{2}}+\left(\frac{\partial\nu}{\partialt}\right)^{2}-\frac{\partial\nu}{\partialt}\frac{\partial\mu}{\partialt}\right]=0.
\begin{array}{l}{R^{(1)}X_{1}Z_{2}=Z_{2}X_{1}\quad,}\\{\,}\\{Z_{1}^{\dagger}R^{(2)}Z_{2}=Z_{2}Z_{1}^{\dagger}\quad,}\\{\,}\\{X_{2}^{\dagger}R^{(3)}X_{1}=X_{1}X_{2}^{\dagger}\quad,}\\{\,}\\{X_{1}^{\dagger}Z_{2}^{\dagger}=Z_{2}^{\dagger}X_{1}^{\dagger}R^{(4)}\quad,}\\\end{array}
a_{n}^{R}=\sqrt{\frac{\alpha^{\prime}}{2n\Omega_{n}}}\alpha_{n}^{R},\;\;\;\;\tilde{a}_{n}^{R}=\sqrt{\frac{\alpha^{\prime}}{2n\Omega_{n}}}\tilde{\alpha}_{n}^{R},\;\;\;\;\;\;\mathrm{for~all}\;\;n>0
{\calL}=\frac{1}{2g}\left[\sum_{i=1}^{9}\mathrm{Tr}\dot{X}_{n}^{i}\dot{X}_{-n}^{i}-\sum_{j=2}^{9}\mathrm{Tr}\;S_{n}^{j}(S_{n}^{j})^{\dagger}-\frac{1}{2}\sum_{j,k=2}^{9}\mathrm{Tr}\;T_{n}^{jk}(T_{n}^{jk})^{\dagger}\right]
\left\{{\frac{\partial^{2}}{\partialr^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partialr}}-{\frac{l(l+4)}{r^{2}}}+(\omega^{2}-k_{1}^{2})f-(k_{2}^{2}+k_{3}^{2})\right\}\varphi_{B\to\infty}^{l}=0.
S_{1}=\frac{\sqrt{(y^{2}+x_{0}^{2})^{3}+(y^{2}+x_{0}^{2})^{2}I_{2}+(y^{2}+x_{0}^{2})\,(I_{2}^{2}-I_{4})/2+(I_{3})^{2}}}{y^{2}+x_{0}^{2}}\,,
\zeta(s_{1},\ldots,s_{k};\sigma_{1},\ldots,\sigma_{k})=\sum_{m_{j}\ge1}\int_{0}^{1}\prod_{j=1}^{k}\Omega^{s_{j}-1}\tau_{j}^{m_{j}}\Omega_{m_{j}}=\int_{0}^{1}\prod_{j=1}^{k}\Omega^{s_{j}-1}\omega_{j},
\begin{array}{lcr}{\nu=K_{11}^{-1}={\frac{1}{P_{1}-{\frac{1}{P_{2}-{\frac{1}{\ldots-{\frac{1}{P_{n}}}}}}}}}}\\{Q=-e{\sum}_{j}(K^{-1})_{i}^{j}{\ell}_{j}}\\\end{array}
Z_{\{\nu_{i}\}}=\frac{N^{\frac{3}{2}}\left[1^{\nu_{1}}2^{\nu_{2}}3^{\nu_{3}}...(N-1)^{\nu_{N-1}}\right]^{-\frac{3}{2}}}{\nu_{1}!\nu_{2}!\nu_{3}!...\nu_{N-1}!}.
\Phi^{(I)}=-\Phi^{(H)},~~~g_{\mu\nu}^{(I)}=e^{-\Phi^{(H)}}g_{\mu\nu}^{(H)},~~~B_{\mu\nu}^{(I)}=B_{\mu\nu}^{(H)}~~\mathrm{and}~~A_{\mu}^{(I)}=A_{\mu}^{(H)}.
C^{\mu,\nu}=\prod_{r=1}^{\mu+\nu}\Gamma\bigl(1+rh/\pi\bigr)\Bigl/\Bigl[\prod_{s=1}^{\mu}\Gamma\bigl(1+sh/\pi\bigr)\,\prod_{t=1}^{\nu}\Gamma\bigl(1+th/\pi\bigr)\Bigr],
{\bfG}(\Box)=\frac2{l^{2}\Box}\,\frac1{1-a^{2}}\left[\begin{array}{cc}{\,\,1\,\,}&{\,\,a^{2}\,\,}\\{\,\,a^{2}\,\,}&{\,\,a^{4}\,\,}\\\end{array}\right]+O(l^{2}\Box),
[p_{i},x_{j}]=i\,\delta_{ij}\,e^{-2p_{0}/\kappa}-\frac{i}{\kappa^{2}}\left(\vec{p}\,{}^{2}\,\delta_{ij}-2p_{i}p_{j}\right),\quad[p_{0},x_{i}]=-\frac{2i}\kappa\,p_{i}
\overline{{S}}(t_{j}^{i})=\overline{{\Omega}}{}_{m}^{i}\,\overline{{t}}{}_{n}^{m}\,\Omega_{j}^{n}\,,\\\\overline{{S}}(\overline{{t}}{}_{j}^{i})=t_{j}^{i}\,,\\\,overline{S}(E_{i})=-\overline{{S}}(t_{i}^{j})E_{j}\,.
\begin{array}{rl}{\mathrm{gravity}:}&{(3,3)+2(2,3)+(1,3)\nonumber}\\{\mathrm{tensor}:}&{(3,1)+2(2,1)+(1,1)\nonumber}\\{\mathrm{vector}:}&{(2,2)+2(1,2)\nonumber}\\{\mathrm{hyper}:}&{2(2,1)+4(1,1).}\\\end{array}
\begin{array}{cll}{}&{\{<U,0>,\;<0,V>:\;U,V\inD_{4}^{+}\}}&{(2\times24=48\mathrm{~elements)~}}\\{\cup}&{\{<U,V>:\;U,V\inD_{4}^{-},\;UV^{\dagger}=\pmq_{m}\}}&{(8\times24=192\mathrm{~elements)}}\\\end{array}
\begin{array}{ll}{V}&{=4k^{2}\phi\bar{\phi}\left(\sigma_{3}\bar{\sigma}_{3}+\gamma\bar{\gamma}+\delta\bar{\delta}\right)}\\{}&{+k^{2}\left[\sigma_{3}^{2}+\frac{\gamma\delta}{4}\right]\left[\bar{\sigma}_{3}^{2}+\frac{\bar{\gamma}\bar{\delta}}{4}\right].}\\\end{array}.
\psi_{0}(\vec{r})\,=\,\sum_{n=-\infty}^{\infty}T_{n}^{0}(r,\phi)\,=\,\sum_{n=-\infty}^{\infty}\int_{-\infty}^{\infty}d\lambda(-i)^{|\lambda|}J_{|\lambda|}(kr)e^{i\lambda(\phi+\pi-\theta+2n\pi)}.
\lbrack\delta_{\varepsilon_{1}},\delta_{\varepsilon_{2}}]\Phi=-i[\Phi,\varepsilon_{1}^{i}\varepsilon_{2}^{j}f_{ij}^{\\k}(\varphi)L_{k}]=-i\left[\Phi,\varepsilon_{3}^{k}L_{k}\right]\qquad,
f_{m}^{(0)}=Cm^{2}|m|^{1/2}\frac{1}{g_{\mathrm{YM}}^{12}}\left(\frac{g_{\mathrm{YM}}^{2}}{\phi}\right)^{5/2}K_{5/2}(|m|\phi/g_{\mathrm{YM}}^{2})e^{im\phi^{8}/g_{\mathrm{YM}}^{2}}
\hat{P}=-\mathrm{sign}(\lambda)\hbar\rho_{0}^{2}\left[2\mathrm{arctg}\sqrt{{\frac{v_{max.}}{v}}-1}+{\frac{4}{3}}\left({\frac{v}{v_{max.}}}\right)\sqrt{\left({\frac{v_{max.}}{v}}-1\right)^{3}}-\sqrt{{\frac{v_{max.}}{v}}-1}\right],
S=-\frac{1}{2\lambda_{s}^{2}}\intd^{4}x\sqrt{-g}e^{-\phi}\biggl[R+g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi-\frac{w\lambda_{s}^{2}}{4}\biggl(R_{GB}^{2}-(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi)^{2}\biggr)\biggr],
2\pi\alpha^{\prime}\partial_{\mu}(\tilde{\mathcal{G}}_{S}^{\mux}\partial_{x}T(x))=2\pi\alpha^{\prime}\partial_{\mu}(\tilde{\mathcal{G}}_{S}^{\mux})\partial_{x}T+2\pi\alpha^{\prime}\tilde{\mathcal{G}}_{S}^{xx}\partial_{x}\left(\partial_{x}T\right),
\Pi^{\underline{m}}=dX^{\underline{m}}-id\Theta^{1\underline{\mu}}\sigma_{\underline{\mu}\underline{\nu}}^{\underline{m}}\Theta^{1\underline{\nu}}-id\Theta_{\underline{\mu}}^{2}\tilde{\sigma}^{\underline{m}\underline{\mu}\underline{\nu}}\Theta_{\underline{\nu}}^{2}=\frac12u^{\underline{m}+2}\Pi^{-2}+\frac12u^{\underline{m}-2}\Pi^{+2}-u^{\underline{m}i}\Pi^{i},
W_{X}=(\eta_{A}-\mathrm{~\frac{1}{2}~}m^{2}(P_{+})_{A}^{B}\bar{\phi}_{B})\phi^{A}-\phi_{Aa}^{*}\pi^{Aa}-\bar{\phi}_{A}(\lambda^{A}-\mathrm{~\frac{1}{2}~}m^{2}(P_{-})_{B}^{A}\phi^{B}).
cK={\frac{x_{0}^{d-4}}{(x_{0}^{2}+|{\bfx}-{\bfx}^{\prime}|^{2})^{d-2}}}dx^{i}dx^{j}-{\frac{x_{0}^{d-5}x^{i}}{(x_{0}^{2}+|{\bfx}-{\bfx}^{\prime}|^{2})^{d-2}}}dx^{0}dx^{j}+{\frac{x_{0}^{d-5}x^{j}}{(x_{0}^{2}+|{\bfx}-{\bfx}^{\prime}|^{2})^{d-2}}}dx^{0}dx^{i},