harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=\intd^{6}x\left[{\frac16}F_{{\mu}{\nu}{\sigma}}(A)F^{{\mu}{\nu}{\sigma}}(A)+\frac{1}{2({\partial}_{\lambda}a)({\partial}^{\lambda}a)}{\partial}^{\mu}a{\calF}_{{\mu}{\nu}{\sigma}}(A){\calF}^{{\nu}{\sigma}{\rho}}(A){\partial}_{\rho}a\right],
\frac{-2}{M^{2}}[x_{i}\tilde{\calL}_{i}/\rho+i\epsilon_{ijk}x_{k}(\tilde{\calL}_{i}a_{j})+2x_{i}a_{i}]+\frac{4}{M^{4}}x_{i}a_{j}\tilde{\calL}_{i}\tilde{\calL}_{j}.
\begin{array}{rcl}{\mathrm{integral}}&{=}&{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\overline{{D_{m}(k_{1}+p)}}}\\{}&{}&{\displaystyle\times2(4\pi)^{-2}\int_{0}^{1}dx~K_{0}\left(\sqrt{(1-x)(k_{1}^{2}x+m^{2})\tilde{k_{1}}^{2}}\right),}\\\end{array}
\mathbb{Z}_{2}\ni\alpha\,\hookrightarrow\,\left(\begin{array}{c|c}{{\bf1}_{2\times2}}&{0}\\\hline{0}&{\alpha\,{\bf1}_{4\times4}\}\\\end{array}\right)\,\otimes\,\left(\begin{array}{c|c}{{\bf1}_{n_{1}\timesn_{1}}}&{0}\\\hline{0}&{\alpha\,{\bf1}_{n_{4}\timesn_{4}}\}\\\end{array}\right)\,\in\,H_{\mathrm{hol}}\,.
\sqrt{(1-\frac{\betaB^{\prime}}{q_{\parallel}^{\prime2}})+(\frac{\alphaB^{\prime}}{2q_{\parallel}^{\prime2}})^{2}}=\frac{1}{q_{\parallel}^{\prime2}}\sqrt{q_{\parallel}^{\prime4}-(\frac{1}{B^{\prime}}+l_{1}+l_{2})B^{\prime}q_{\parallel}^{\prime2}+\frac{1}{4}(l_{1}-l_{2})^{2}B^{\prime2}},
i\left(\begin{matrix}{\langleU_{z},\bar{U}_{\bar{z}}\rangle}&{\langleU_{z},V\rangle}\\{\langle\bar{V},\bar{U}_{\bar{z}}\rangle}&{\langle\bar{V},V\rangle}\\\end{matrix}\right)=\left(\begin{matrix}{(z+\bar{z})^{-2}}&{\frac{a-1}{a+1}(z+\bar{z})^{-1}}\\{\frac{a-1}{a+1}(z+\bar{z})^{-1}}&{1}\\\end{matrix}\right)
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}g_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{k}
\displaystyle{\frac{2}{\kappa}}V_{\mu\nu}^{^1{scalar}}(k_{1};p_{1},p_{2})=\,p_{_1\mu}\,p_{_2\nu}+\,p_{_1\nu}\,p_{_2\mu}-p_{1}\cdotp_{2}\,\eta_{\mu\nu}-{m^{2}}\,\eta_{\mu\nu}+2\,\xi\,\left(\,k_{_1\mu}\,k_{_1\nu}-\,k_{1}^{2}\,\eta_{\mu\nu}\right)
\left(\begin{array}{c}{(19.34\mp2.834i)\times10^{-6}}\\{(2.653\mp2.310i)\times10^{-5}}\\{(-9.951\mp3.203i)\times10^{-3}}\\{(-17.81\pm6.820i)\times10^{-3}}\\{-0.3485\pm0.3392i}\\{0.8727}\\{0}\\{0}\\{\ldots}\\\end{array}\right).
\frac{\partial}{\partialt}\left(\begin{array}{c}{a({\bfk})}\\{a^{*}({\bfk})}\\\end{array}\right)=\mathrm{\cal~finite~terms}+\lambda^{2}\left(\begin{array}{cc}{A(0;t)}&{A^{*}(0;t)}\\{A(0;t)}&{A^{*}(0;t)}\\\end{array}\right)\\;\;\left(\begin{array}{c}{a({\bfk})}\\{a^{*}({\bfk})}\\\end{array}\right)
\psi_{n}(\tau+1)=e^{2\pii(h_{n}-1/24)}\psi_{n}(\tau)~,~~\psi_{n}(-\frac{1}{\tau})=\frac{1}{\sqrt{2k}}\sum_{n^{\prime}}e^{-\piinn^{\prime}/k}\psi_{n^{\prime}}(\tau)~,
m_{\phi_{S}^{0}}^{2}=p_{r}^{2}=a(p_{r})=4\left.\frac{(k_{r}+h)(\tilde{k}_{r}+\tilde{h})+(k_{i}+s^{I})(\tilde{k}_{i}+\tilde{s}^{I})}{(k_{r}+h)^{2}+(k_{i}+s^{I})^{2}}\right|_{p=p_{r}}.
\psi\left(\frac{w}{z}\right)\psi\left(\frac{x^{2}w}{z}\right)[u-v-\frac{1}{2}]=\frac{x^{\frac{(u-v-1/2)^{2}}{r}-(u-v-1/2)}(x^{2r};x^{2r})_{\infty}}{(xw/z;x^{2})_{\infty}(xz/w;x^{2})_{\infty}}.
A=D_{1}\,\frac{\Gamma(1-2\mu)\Gamma(-2\nu)}{\Gamma(-\mu-\nu-\lambda)\Gamma(1-\mu-\nu+\lambda)}e^{-i\pi\mu}+D_{2}\,\frac{\Gamma(1+2\mu)\Gamma(-2\nu)}{\Gamma(\mu-\nu-\lambda)\Gamma(1+\mu-\nu+\lambda)}e^{i\pi\mu}
\varepsilon_{\mu\nu\lambda\rho}\partial_{\lambda}\partial_{\rho}\theta_{a}^{\mathrm{sing}}(x)=2\pi\Sigma_{\mu\nu}^{a}(x)\equiv2\pi\int_{\Sigma_{a}}^{}d\sigma_{\mu\nu}\left(x^{(a)}(\xi)\right)\delta\left(x-x^{(a)}(\xi)\right).
J^{\mu}=\left[\left(1-\frac{\partial_{z}^{2}}{M_{r}^{2}}\right)\left(\not\!\partial\phi+iF\right)\right]\gamma^{\mu}\psi+\frac{1}{M_{r}^{2}}\,\delta_{1}^{\mu}\left\{-\partial_{z}(\!\not\!\partial\phi\not\!\partial\psi)+iF\,\partial_{z}\!\not\!\partial\psi-i\,(\partial_{z}F)\!\not\!\partial\psi\right\}\;.
{\calC}=\frac{1}{L}\int_{0}^{L}dx\left(\varphi^{\dagger}\gamma^{0}\varphi+\varphi\gamma^{0}\varphi^{\dagger}\right)=\frac{2i}{L}\sum_{k}\left(b_{k}^{\dagger}a_{k}-a_{k}^{\dagger}b_{k}\right)
\begin{array}{lll}{\deltax_{1}^{+}(0)=\deltay_{1}^{+}(0)=0}&{\deltax_{1}^{-}(0)=0}&{\deltay_{1}^{-}(0)=\deltax_{2}^{-}(0)=0}\\{\deltax_{N}^{-}(1)=\deltay_{N-1}^{-}(1)=0}&{\deltax_{N}^{+}(0)=0}&{\deltay_{N-1}^{+}(1)=\deltax_{N-1}^{+}(1)=0}\\\end{array}
\partial^{\alpha}\partial_{\alpha}\tilde{h}_{\mu\nu}-\partial_{\mu}\partial_{\alpha}\tilde{h}_{\nu}^{\alpha}-\partial_{\nu}\partial_{\alpha}\tilde{h}_{\mu}^{\alpha}+\frac{1}{3}\partial_{\nu}\partial_{\mu}\tilde{h}_{\alpha}^{\alpha}+\frac{1}{3}\eta_{\mu\nu}\left(\partial_{\alpha}\partial_{\beta}\tilde{h}^{\alpha\beta}-\partial^{\beta}\partial_{\beta}\tilde{h}_{\alpha}^{\alpha}\right)-\frac{2}{3}
\left(h_{\beta\gamma\delta}^{B}\right)\,\left(h_{\alpha\gamma\delta}^{B}\right)^{-1}\,\left(h_{\alpha\beta\delta}^{B}\right)\,\left(h_{\alpha\beta\gamma}^{B}\right)^{-1}\:=\:\left(h_{\alpha\beta\gamma\delta}^{F}\right)^{-1}\,\left(h_{\alpha\beta\gamma\delta}^{R}\right)
{D}_{s}=\hat{\gamma}^{\hat{\alpha}}\hat{\nabla}_{\hat{\alpha}}=i\sqrt{V}\vec{\hat{\gamma}}\cdot\vec{P}+\frac{i}{\sqrt{V}}\hat{\gamma}^{4}P_{4}+\frac{i}{2}V\sqrt{V}\hat{\gamma}^{4}\vec{\Sigma}^{*}\cdot\vec{B}
e^{-\frac{i}{\hbar}\theta\hat{\phi}}\,|z_{i}^{\pm}>=e^{i\theta[{\alpha_{i}^{+}}^{\dagger}\alpha_{i}^{+}-{\alpha_{i}^{-}}^{\dagger}\alpha_{i}^{-}]}\,|z_{i}^{\pm}>=|e^{\pmi\theta}z_{i}^{\pm}>\\\,
\partial_{\tau}f(\tau,\bbox{\phi})\,={\frac{\partial}{\partial\tau}}f(\tau,\bbox{\phi}(\tau,\bboxr))-{\frac{\partial}{\partial\phi^{s}}}f(\tau,\bbox{\phi}(\tau,\bbox{r})){\frac{\partial\phi^{s}(\tau,\bbox{r})}{\partial\tau}}
\frac{W_{\xi,\eta}(i)}{W_{\xi,\eta}(i-1)}=1\quad\frac{W_{\xi^{\prime},\eta^{\prime}}(i)}{W_{\xi^{\prime},\eta^{\prime}}(i-1)}=1\quad\frac{\overline{{W}}_{\xi^{\prime},\eta}(i)}{\overline{{W}}_{\xi^{\prime},\eta}(i-1)}=\omega\quad\frac{\overline{{W}}_{\xi,\eta^{\prime}}(i)}{\overline{{W}}_{\xi,\eta^{\prime}}(i-1)}=1\quad
\begin{array}{ccl}{\tilde{T}^{\,(+)}}&{=}&{i^{\,M}\,T\,\left(P_{\,(+)}g_{\,2M-2}^{\ast}g_{\,2M-1}^{\ast}\,-\,iP_{\,(-)}g_{\,2M-2}g_{\,2M-1}\right)}\\{\tilde{T}^{\,(-)}}&{=}&{i^{\,M}\,T\,\left(P_{\,(+)}g_{\,2M-2}g_{\,2M-1}\,+\,iP_{\,(-)}g_{\,2M-2}^{\ast}g_{\,2M-1}^{\ast}\right)}\\\end{array}
h^{\mu\nu}=-\frac{e}{2M^{2}}\left(\epsilon^{\alpha\beta\sigma\nu}\partial_{\sigma}T_{(\alpha\beta)}{}^{\mu}+\epsilon^{\alpha\beta\sigma\mu}\partial_{\sigma}T_{(\alpha\beta)}{}^{\nu}\right)+\frac{1}{3M^{2}}g^{\mu\nu}\left(e\epsilon^{\alpha\beta\rho\kappa}\partial_{\rho}T_{(\alpha\beta)\kappa}-\Theta_{\alpha}^{\alpha}\right)+\frac{1}{M^{2}}\Theta^{\mu\nu}\,,
w_{\mathrm{spinor}}=\frac{2(qE)^{2}}{(2\pi)^{3}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}e^{-ka_{f}}+\frac{2(qE)(qB)}{(2\pi)^{2}}\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}F_{\mathrm{spinor}}^{(2k-1)}(0),
Z_{MN}=\left(\begin{array}{lllllll}{0}&{-w_{9}}&{-k_{9}}&{0}&{0}&{\cdots}&{0}\\{w_{9}}&{0}&{-k_{10}}&{0}&{0}&{\cdots}&{0}\\{k_{9}}&{k_{10}}&{0}&{p_{0}}&{p_{1}}&{\cdots}&{p_{8}}\\{0}&{0}&{-p_{0}}&{0}&{0}&{\cdots}&{0}\\{0}&{0}&{-p_{1}}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}&{0}\\{0}&{0}&{-p_{8}}&{0}&{0}&{0}&{0}\\\end{array}\right)
\left.\sum_{b}S,_{b}\;(\delta_{\epsilon_{\phantom{}_{{Killing}}}}\,\phi^{b}),_{f}\right|_{\phi^{f}=0}=\left.\sum_{b}J_{b}\;(\delta_{\epsilon_{\phantom{}_{{Killing}}}}\,\phi^{b}),_{f}\right|_{\phi^{f}=0}=0\.
\mu_{I}^{3}=-i\Big((z_{+}T_{I}\bar{z}_{+})+(z_{-}T_{I}\bar{z}_{-})\Big)\,\quad\mu_{I}^{+}=(z_{+}T_{I}z_{-})\,\quad\mu_{I}^{-}=(\mu_{I}^{+})^{*}\.
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}+\frac{-{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=\left(\frac{E^{2}}{-{g^{\prime}}_{tt}}-V\right),
A_{N}^{m.s.}(a_{1},...,a_{N})=(\prod_{i=1}^{N-3}\frac{2}{s_{b_{i}}-s_{b_{i}}^{0}})f_{a_{1}a_{2}}^{b_{1}}f_{b_{1}a_{3}}^{b_{2}}...f_{b_{N-3}a_{n-1}}^{a_{n}}
\Psi^{k}(\sum_{(\gamma_{1},\gamma_{2})\in\mathbbN^{2}}W_{(-\gamma_{1},-\gamma_{2})}p^{\gamma_{1}}q^{\gamma_{2}})=\sum_{(\gamma_{1},\gamma_{2})\in\mathbbN^{2}}\Psi^{k}(W_{(-\gamma_{1},-\gamma_{2})})p^{k\gamma_{1}}q^{k\gamma_{2}}.
\begin{array}{c}{-\frac{1}{2}(k-1)k(k+1)<O_{k-2}O_{k_{1}}\dotsO_{k_{p-1}}P>}\\{+4t_{0}(k+1)<O_{k}O_{k_{1}}\dotsO_{k_{p-1}}P>+4t_{2}(k+2)<O_{k+2}O_{k_{1}}\dotsO_{k_{p-1}}P>}\\{+2\sum_{i=1}^{p-1}(2k+k_{i}+2)<O_{k+k_{i}}O_{k_{1}}...\hat{O}_{k_{i}}...O_{k_{p-1}}P>=0.}\\\end{array}
\prod_{m=0}^{(p-1)/2}\prod_{n_{1}=-\infty}^{\infty\prime}\left[{\frac{\omega_{n_{1}0}^{(m)}}{2\pi\alpha^{\prime}}}\right]^{-1}=\prod_{m=0}^{(p-1)/2}\prod_{n_{1}=1}^{\infty}\left[{\frac{\omega_{n_{1}0}^{(m)}}{2\pi\alpha^{\prime}}}\right]^{-2}=(2\pi\alpha^{\prime}l^{2})^{-(p+1)/2}\mathrm{det}({\bf1}+{\bfB})\quad,
{\alpha_{n}^{\mu}}^{*}=\alpha_{-n}^{\mu}\\,\\{\bar{\alpha}_{n}^{\mu*}}=\bar{\alpha}_{-n}^{\mu}\\,\alpha_{0}^{\mu}=\frac{1}{2}\sqrt{2\alpha^{\prime}}P^{\mu}=\bar{\alpha}_{0}^{\mu}\.
\begin{array}{c}{u(\sigma^{+},\sigma^{-})=u_{0}+\frac{p^{+}}2\left[(\sigma^{+}+\left|\sigma^{+}\right|_{per})+(\sigma^{-}+\left|\sigma^{-}\right|_{per})\right]}\\{v(\sigma^{+},\sigma^{-})=v_{0}+\frac{p^{-}}2\left[(\sigma^{+}-\left|\sigma^{+}\right|_{per})+(\sigma^{-}-\left|\sigma^{-}\right|_{per})\right]}\\\end{array}
\Gamma[\varphi_{0}]=\sum_{n=0}^{\infty}\frac{1}{n!}\intd^{4}x_{1}d^{4}x_{2}...d^{4}x_{n}~\Gamma^{(n)}(x_{1},x_{2},..,x_{n})~\varphi_{0}(x_{1})\varphi_{0}(x_{2})..\varphi_{0}(x_{n}).
\big|\pi_{i=1\simN_{c}}\big|=\Lambda_{T}^{2},\\,big|\pi_{i=N_{c}+1\simN_{f}}\big|=\xi\big|\pi_{i=1\simN_{c}}\big|,\\|\pi_{B}|\sim\Lambda_{T}^{N_{c}},\\,big|\pi_{\lambda}\big|\sim\Lambda^{3}\xi^{\frac{N_{f}-N_{c}+1}{N_{f}-N_{c}+2}},
S=\int(~V(J_{0}^{++}+\bar{J}_{0}^{++})(G_{0}^{+}+\bar{G}_{0}^{+})(\tilde{G}_{0}^{+}+\bar{\tilde{G}}_{0}^{+})V+[V,(G_{0}^{+}+\bar{G}_{0}^{+})V](G_{0}^{+}+\bar{G}_{0}^{+})V~)
E_{\chi}=32\pi^{2}f(\theta)N\quad\mathrm{with}\quadf(\theta):=|\theta^{12}\theta^{34}|\left[\left(\frac{1}{\theta^{12}}\right)^{2}+\frac{1}{\theta^{12}\theta^{34}}+\left(\frac{1}{\theta^{34}}\right)^{2}\right].
(u_{\omega{\bfk}_{\perp}},u_{\omega{\bfk}_{\perp}})=2\,(2\pi)^{2}\,\delta^{(2)}(0)\int_{-\infty}^{\infty}dx\,\left[\omega-qA(x)\right]\,\vertf_{\omega{\bfk}_{\perp}}(x)\vert^{2}.
H_{\mathrm{eff}}=\sum_{p}\epsilon_{p}a_{p}^{\dagger}a_{p}+\frac{3\lambda}2\frac1V\sum_{p_{1}+p_{2}=p_{3}+p_{4}}a_{p_{1}}^{\dagger}a_{p_{2}}^{\dagger}a_{p_{3}}a_{p_{4}}\;,
S^{(2)}=-\mathrm{tr}_{K}\,[S_{a},\deltaa_{n}^{\prime}]^{2}-\rho^{4}\mathrm{tr}_{K}\,[S_{a},\delta\chi_{b}]^{2}+\rho^{4}\mathrm{tr}_{K}[S_{a},\delta\chi_{a}]^{2}\.
W=\Lambda_{SU(7)}^{8}\left[\pm\frac{1}{v_{1}(v_{1}^{2}-v_{2}^{2})(v_{1}^{2}-v_{3}^{2})}\pm\frac{1}{v_{2}(v_{2}^{2}-v_{1}^{2})(v_{2}^{2}-v_{3}^{2})}\pm\frac{1}{v_{3}(v_{3}^{2}-v_{1}^{2})(v_{3}^{2}-v_{2}^{2})}\right],
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}(A_{\mu}+\partial_{\mu}\theta)(A^{\mu}+\partial^{\mu}\theta)+A^{\mu}\partial_{\mu}B-\partial_{\mu}\bar{\calC}\partial^{\mu}{\calC}-\frac{1}{2}\alphaB^{2}.
i\tilde{G}_{\mu\nu}^{^C}(k)=\left[g_{\mu\nu}-\frac{k_{\nu}}{a_{_C}-1}\left(\frac{4\pi}{e^{2}}-\frac{2}{k^{2}}\right)+\frac{k_{\mu}\tilde{k}_{\nu}+\tilde{k}_{\nu}}{(a_{_C}-1)\;k^{2}}\right]\frac{1}{k^{2}-m_{_C}^{2}},
{\calL}(V_{h},\phi_{h})=\frac{1}{16}\intd^{2}\theta\frac{1}{g_{h}^{2}}W^{a}(V_{h})W^{a}(V_{h})+\mathrm{h.c.}+\intd^{4}\thetaRe\left(\frac{2}{g_{h}^{2}}\right)\mathrm{Tr}\phi_{h}^{\dagger}e^{-2V_{h}}\phi_{h}e^{2V_{h}}
{\frac{1}{6}}\epsilon^{abcdef}\hat{\calD}_{c}H_{def}=-{\frac{1}{4!}}\epsilon^{abcdef}E_{c}^{\underline{D}}E_{d}^{\underline{C}}E_{e}^{\underline{B}}E_{f}^{\underline{A}}F_{\underline{{ABCD}}},
ds^{2}=-r_{0}^{2}W^{2}dUdV+r_{0}^{6}W^{4}\sigma^{2}\epsilon^{2}dU^{2}+\epsilon^{2}\sigmaW^{4}r_{0}^{4}dqdU+W^{4}r_{0}^{4}dq^{2}+r_{0}^{2}d\Omega_{3}^{2}+O(\epsilon^{4})+O({\frac{\deltap}{p}}),
\hat{T}_{\hat{\mu}\hat{\nu}}{}^{\hat{\rho}}=-2\hat{K}_{[\hat{\mu}\hat{\nu}]}{}^{\hat{\rho}}={\textstyle\frac{m}{2}}\left(i_{\hat{k}}\hat{C}\right)_{\hat{\mu}\hat{\nu}}\hat{k}^{\hat{\rho}}\,.
G=\left(\begin{array}{cc}{1}&{0}\\{{\calL}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{{\calG}_{1}^{-1}}&{0}\\{0}&{{\calG}_{2}}\\\end{array}\right)\left(\begin{array}{cc}{1}&{{\calR}}\\{0}&{1}\\\end{array}\right)
\Theta=\left(\begin{array}{ccccc}{0}&{-\theta_{1}}&{}&{}&{}\\{\theta_{1}}&{0}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}\\{}&{}&{}&{0}&{-\theta_{\frac{d}{2}}}\\{}&{}&{}&{\theta_{\frac{d}{2}}}&{0}\\\end{array}\right),
M_{ij}^{a}=\sum_{b,c,k,l}(g^{-1})_{ik}^{b-c}\eta_{kl}^{c}g_{lj}^{a-b}q^{a}-\sum_{b,c,k,l}(g^{-1})_{ik}^{b-c}\eta_{kl}^{c}g_{lj}^{-b}-\eta_{ij}^{a}+\eta_{ij}^{0}\,,\qquada\neq0,
{\frac{\Pi\cdotf_{s}}{(12)^{2}(34)^{2}Y}}={\frac{1}{x_{12}^{4}x_{34}^{4}}}\{\frac{a_{s}-{\frac{c_{s}}{s}}U+{\frac{b_{s}}{s^{2}}}U^{2}}{1+{\frac{1+s-t}{s}}U+{\frac{1}{s}}U^{2}}}\;,
S_{\mathrm{v}}=\intd^{4}x\bar{N}_{0}\left[-{\frac{1}{4}}(\partial_{\mu}v_{\nu}-\partial_{\nu}v_{\mu})(\partial_{\mu^{\prime}}v_{\nu^{\prime}}-\partial_{\nu^{\prime}}v_{\mu^{\prime}})\bar{g}^{\mu\mu^{\prime}}\bar{g}^{\nu\nu^{\prime}}-{\frac{1}{2}}a^{2}M_{v}^{2}v_{\mu}v_{\mu}\bar{g}^{\mu\nu}\right]~,
\hat{Y}_{\ellm}\hat{Y}_{\ell^{\prime}m^{\prime}}=\sum_{\ell^{\prime\prime}m^{\prime\prime},(\ell^{\prime\prime}\leq\mathrm{min}(\ell+\ell^{\prime},2j))}C_{\ellm,\ell^{\prime}m^{\prime}}^{\ell^{\prime\prime}m^{\prime\prime}}\hat{Y}_{\ell^{\prime\prime}m^{\prime\prime}}
S_{s}\sim{\frac{\alpha^{\prime}M}{\ell_{s}}}\simM/M_{s}~~~~,~~~~~~S_{\mathrm{BH}}\sim\frac{\mathrm{Area}}{G_{N}}\sim\frac{R_{\mathrm{BH}}^{d-1}}{G_{N}}\sim\frac{(g^{2}\,M/M_{s})^{\frac{d-1}{d-2}}}{g^{2}}\;,
{\tilde{t}}{\,}_{reg}^{00}(s)=(2\pi)^{1-d}\intd^{d-2}p\,\int_{0}^{\infty}dk\,k\left({\textbf{p}}^{2}+k^{2}+m^{2}\right)^{-s}\sum_{n\in\mathbbZ}J_{|n-\Phi|}^{2}(kr)\,.
\begin{array}{l}{[3(3,\bar{3},1)+(3,1,\bar{3})]_{1}+}\\{+[3(1,1,1+8)+(\bar{3},1,3)]_{2}+}\\{+[3(1,3,\bar{3})+(1,1+8,1)]_{3}+}\\{+[(2(1,1,1+8)+(1,\bar{3},3)+(\bar{3},1,3)]_{4}+}\\{+[2(1,1,1+8)+2(1,\bar{3},3)]_{5}+}\\{+[2(\bar{3},1,3)+(1,1,1+8)+(1,\bar{3},3)]_{6}+}\\{+[4(1,3,\bar{3})]_{7}}\\\end{array}
S_{Bose}\left[\mathbf{x;}z\right]\sim-\intd^{2}\sigma\left[\partial_{\alpha}\mathbf{x}\partial^{\alpha}\mathbf{x+}\partial_{\alpha}z_{i}\partial^{\alpha}\overline{{z}}_{i}+\nu^{2}(\mathbf{x}^{2}+\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2})\right]
{\calR}_{ij}=\frac{1}{m}\mathrm{Tr}\left\{\left(\partial_{(i}{\calE}\right){\calP}^{-1}\left(\partial_{j)}{\calE}^{\dagger}\right){\calP}^{-1}\right\}=-\frac{1}{2m}\mathrm{Tr}\left\{\left(\partial_{(i}{\calM}\right)\partial_{j)}{\calM}^{-1}\right\}.
{\mathrm{sTr}}(\omegaF_{L}^{p}[F_{L},\omega]({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q-2})=-{\frac{1}{p+1}}(2{\Pi}_{[p+1,q]}+q{\Gamma}_{[p+1,q]})\quad.
Z_{t}^{S_{0}}(\tau)=C\frac{1}{\eta^{8}(N\tau)}+\frac{1}{N}\frac{1}{\eta^{8}(\frac{\tau}{N})}+\frac{1}{N}\zeta_{N}^{\frac{1}{3}}\frac{1}{\eta^{8}(\frac{\tau+1}{N})}+\cdots+\frac{1}{N}\zeta_{N}^{\frac{N-1}{3}}\frac{1}{\eta^{8}(\frac{\tau+N-1}{N})},
T_{\lambda\rho\sigma\tau}=\delta_{\lambda\rho}\delta_{\sigma\tau}T+\delta_{\lambda\rho}(\gamma^{\alpha})_{\sigma\tau}T_{\alpha}+(\gamma^{\alpha})_{\lambda\rho}\delta_{\sigma\tau}T_{\alpha}^{\prime}+(\gamma^{\alpha})_{\lambda\rho}(\gamma^{\beta})_{\sigma\tau}T_{\alpha\beta},
{\frac{d{\bfx}}{d\sigma}}={\bfe}_{+},\;{\frac{d{\bfe}_{+}}{d\sigma}}={\bfe}_{1},\;{\frac{d{\bfe}_{1}}{d\sigma}}=\kappa{\bfe}_{+}+{\bfe}_{-},\;{\frac{d{\bfe}_{-}}{d\sigma}}=\kappa{\bfe}_{1}.
\begin{array}{l}{{\calW}(x_{1}^{0}-\mathrm{i}x_{1}^{4},{\bfx_{1}};\dots;x_{n}^{0}-\mathrm{i}x_{n}^{4},{\bfx_{n}})=}\\{\quad=\langle\,0|\phi(0,{\bfx_{1}})\mathrm{e}^{-\mathrm{i}(x_{1}^{0}-x_{2}^{0})H}\mathrm{e}^{-(x_{1}^{4}-x_{2}^{4})H}\phi(0,{\bfx_{2}})\dots\phi(0,{\bfx_{n}})|0\,\rangle\,.}\\\end{array}
\begin{array}{lcl}{E^{(J)}}&{=}&{{\displaystyle\frac{1}{2\pi}\int_{0}^{\lambda_{J}}\!d\sigma~{\dot{T}}^{(J)}}}\\{}&{}&{}\\{P_{X}^{(J)}}&{=}&{{\displaystyle\frac{1}{2\pi}\int_{0}^{\lambda_{J}}\!d\sigma{}~{\dot{X}}^{(J)}}}\\{}&{}&{}\\{P_{Y}^{(J)}}&{=}&{{\displaystyle\frac{1}{2\pi}\int_{0}^{\lambda_{J}}\!d\sigma{}~{\dot{Y}}^{(J)}}}\\\end{array}
W_{p,N_{r}}^{(o)}(t)=(-1)^{e(N_{r})}\sum_{N_{o},N_{o}^{\prime}}\left[E_{p,N_{o}}^{(-2)}\left(Y_{b,N_{r}}(t)\right)+\sum_{N}[\partialq_{N}Y_{p,N_{r}}(t)]B_{NN_{o}}\right]A_{N_{o}N_{o}^{\prime}}^{-1}\chi_{N_{o}^{\prime}}(t)
\Theta_{\,j}^{i\J=1}=Tr\left(\begin{array}{c}{-iU^{i}V_{j}}\\{\frac{\sqrt{2}}{2}\left(U^{i}\overline{U}_{j}+\overline{V}^{i}V_{j}\right)}\\{i\overline{V}^{i}\overline{U}_{j}}\\\end{array}\right)^{{\calN}\!=\!2}+{\calO}(\theta^{0})\,,
e^{i}e_{j}={\calP}_{jq}^{ip}\,e_{p}e^{q}\,,\\\\\\\\{\calP}_{jq}^{ip}\equiv\eta_{n}^{m}\,\tilde{c}{}_{mq}^{t}\,\tilde{c}{}_{ts}^{i}\,\chi_{r}^{s}\,f_{j}^{rl}f_{l}^{pn}\,.
V_{i}\cdot{\calN}_{\overline{{\alphaV}}}\equiv{\vec{U}}_{i}\cdot{\vec{P}}_{\overline{{\alphaV}}}-\sum_{r}V_{i}^{r}N_{\overline{{\alphaV}}}^{r}+\sum_{\ell:~right}T_{i}^{\ell}J_{\overline{{\alphaV}}}^{\ell}-\sum_{\ell:~left}T_{i}^{\ell}J_{\overline{{\alphaV}}}^{\ell}~.
W_{eff}=\frac{1}{h^{2F-1}\Lambda^{6F-4}}(M_{2})_{FF}(\mathrm{det}\,\tilde{M}_{2})\tilde{M}_{1}(\mathrm{cof}\,\tilde{M}_{2})(M_{2})_{FF}=m^{2}h^{2F-1}\Lambda^{6F-4}\frac{\tilde{M}_{1}\mathrm{cof}\,\tilde{M}_{2}}{(\mathrm{det}\,\tilde{M}_{2})^{3}}.
\begin{array}{l}{\qquad\bar{\theta}\Gamma^{+}\lambda_{\theta}=\bar{\theta}\Gamma^{i}\lambda_{\theta}=0,}\\{\qquad(\bar{\theta}\Gamma^{+\mu}\lambda_{\theta})P_{y\mu}=(\bar{\theta}\Gamma^{i\mu}\lambda_{\theta})P_{y\mu}=0,}\\{\qquad(\bar{\theta}\Gamma^{+\mu}\partial_{1}\theta)P_{y\mu}=(\bar{\theta}\Gamma^{i\mu}\partial_{1}\theta)P_{y\mu}=0,}\\\end{array}
\delta_{\gamma}^{\alpha}\delta_{\delta}^{\beta}=-\frac14\hat{C}_{\gamma\delta}(\hat{C}^{-1})^{\alpha\beta}-\frac14(\hat{C}\hat{\gamma}_{A})_{\gamma\delta}(\hat{\gamma}^{A}\hat{C}^{-1})^{\alpha\beta}-\frac12(\hat{C}\hat{S}_{AB})_{\gamma\delta}(\hat{S}^{AB}\hat{C}^{-1})^{\alpha\beta},
\varepsilon^{\mu}=-i\left(\begin{array}{c}{0}\\{a}\\{b}\\{c}\\\end{array}\right)~~~~~~~~{\calE}=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{c}&{-b}\\{0}&{-c}&{0}&{a}\\{0}&{b}&{-a}&{0}\\\end{array}\right)
\begin{array}{c}{{{\bfp}^{\bot}}_{i}\equiv{\bfp}_{i}-\sum_{j=1}^{N}({\bfp}_{i}{\bfe}_{j}){\bfe}_{j},\quad{\bfp}_{i}^{\bot}{\bfp}_{j}^{\bot}\equiv{\Phi}_{i.j},\quad{{\bfp}^{\bot}}_{i}{\bfe}_{j}=0,}\\{{\phi}_{i.j}\equiv{\bfp}_{i}{\bfe}_{j}-{\bfp}_{j}{\bfe}_{i},}\\{\chi_{ij}={\bfp}_{i}{\bfe}_{j},\quadi\geqj.}\\\end{array}
g=\frac{-q^{-1}}{\left[2\right]}\left[\begin{array}{cccc}{\lambdaq^{-1}}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\{0}&{q^{2}}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{array}\right]\quad;
0=F_{u}^{\prime\prime}+\frac{2}{r}F_{u}^{\prime}\left[1+rL_{+}^{\prime}(kr)\right]-\frac{s^{\prime}}{s+\frac{\omega}{m}}\left[F_{u}^{\prime}+F_{u}L_{+}^{\prime}(kr)\right]+V(r,m)F_{u}
C_{n}={\cal{\varepsilon}}^{2}\,\epsilon^{\underline{{\delta}}}\nabla_{\underline{{\delta}}}\,{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}\,{\calD}_{\alpha_{1}}\,{\calG}^{\alpha_{1}}{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}\ldots{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}\,{\calD}_{\alpha_{n}}\,{\calG}^{\alpha_{n}}{\mathrm{\small{~\frac12~}}}\bar{\calD}^{2}.
\left\{b({\bfk},i),b^{\dagger}({\bfk^{\prime}},j)\right\}=\left\{d({\bfk},i),d^{\dagger}({\bfk^{\prime}},j)\right\}=(2\pi)^{6}2\omega\delta^{6}({\bfk}-{\bfk^{\prime}})\delta_{ij}
\intd^{3}x{{\itF}_{n}^{a}({\bfr})}^{(l)}{{\eta_{n}^{a}({\bfr})}_{0}}^{(k)}=\frac{2H_{k}^{(ext)}}{3a}\intd^{3}xD_{m}\phi^{a}B_{m}^{a}=\frac{2}{3}gH_{k}^{(ext)},
H^{\prime}(t)=U_{A}^{\dagger}(t)V(t)U_{A}(t)=-\sum_{n,m,n\neqm}e^{-i(\alpha_{m}(t)-\alpha_{n}(t))}\langlem,t|i\hbar\partial_{t}|n,t\rangle|m,0\rangle\langlen,0|
\Theta^{0}=-\ell\,\frac{d\rho}{\rho}\,,\quad\Theta^{+}=-\frac\rho\ell\,L_{-}(x^{-})\,dx^{-}-\frac\ell\rho\,dx^{+}\,,\quad\Theta^{-}=-\frac\rho\ell\,L_{+}(x^{+})\,dx^{+}-\frac\ell\rho\,dx^{-}\,.
\begin{array}{|c|c|c|c|c|c|c|c|}\hline{{}~~1~~}&{~~2~~}&{~~3~~}&{~~4~~}&{~~5~~}&{~~6~~}&{~~7~~}&{~~8~~}\\\hline{-3}&{\frac{9}{4}}&{4}&{-\frac{1}{4}}&{\frac{9}{2}}&{\frac{9}{2}}&{-12}&{0}\\\hline{}\\\end{array}
p={\frac{1}{24}}(z_{1}^{24}+z_{2}^{24}+2z_{3}^{12}+8z_{4}^{3}+12z_{5}^{2})-\psi_{0}z_{1}z_{2}z_{3}z_{4}z_{5}-{\frac{1}{6}}\psi_{1}(z_{1}z_{2}z_{3})^{6}-{\frac{1}{12}}\psi_{2}(z_{1}z_{2})^{12}.
\frac{\partial}{\partial\xi}F=S(\frac{\partial}{\partial\xi}f)S^{-1}-\xi\frac{\partialS}{\partial\xi}S^{-1},\quad\frac{\partial}{\partial\bar{\xi}}F=S(\frac{\partial}{\partial\bar{\xi}}f)S^{-1}-\bar{\xi}\frac{\partialS}{\partial\bar{\xi}}S^{-1}.
=B^{\alpha}{\calT}_{\alpha}+\frac{1}{2}B^{\beta}B^{\alpha}U_{\alpha\beta}^{\gamma}\Pi_{\gamma}(-1)^{\varepsilon_{\beta}+\varepsilon_{\gamma}}+\Pi_{*}^{\alpha}\bar{\calP}_{\alpha}(-1)^{\varepsilon_{\alpha}}-\Pi_{*}^{\beta}B^{\alpha}U_{\alpha\beta}^{\gamma}B_{\gamma}^{*}(-1)^{\varepsilon_{\beta}},
(\mathrm{p}-e\mathrm{A}^{(n)\prime})e^{-ie\Theta}\psi=e^{-ie\Theta}(\mathrm{p}+e\nabla_{\bfy}\Theta-e\mathrm{A}^{(n)\prime})\psi=e^{-ie\Theta}(\mathrm{p}-e\mathrm{A}^{(n)})\psi
G_{a}^{(n)}=\sum_{m=0}^{n}\bigl\{T_{a}^{(n-m)},\,A^{(m)}\bigr\}_{(q,p)}+\sum_{m=0}^{n-2}\bigl\{T_{a}^{(n-m)},\,A^{(m+2)}\bigr\}_{(y)}+\bigl\{T_{a}^{(n+1)},\,A^{(1)}\bigr\}_{(y)}\,.
\begin{array}{llc}{}&{\Omega=+}&{SO(4)\mathrm{~rep.}}\\{\psi_{-1/2}^{\mu}|0,ij\rangle\lambda_{ji}\quad}&{\lambda=-\gamma_{\Omega,J}^{\prime}\lambda^{T}\gamma_{\Omega,J}^{-1}\}&{({\bf2},{\bf2})}\\{\psi_{-1/2}^{m}|0,ij\rangle\lambda_{ji}\quad}&{\lambda=\gamma_{\Omega,J}^{\prime}\lambda^{T}\gamma_{\Omega,J}^{-1}\}&{4({\bf1},{\bf1}).}\\\end{array}
\begin{array}{ccc}{g_{AB}}&{=}&{\tilde{g}_{AB}(y^{C}),}\\{}&{}&{}\\{g_{ij}}&{=}&{a_{k}(y^{C})a_{l}(y^{C})\eta_{ij}^{kl},}\\{}&{}&{}\\{g_{Ai}}&{=}&{0.}\\\end{array}
\frac{V_{p}^{\mu}}{V_{p}\cdotk}=\frac{(\eta+v)^{\mu}(\eta-v)\cdotk}{(k\cdot\eta)^{2}-({\boldsymbol{k}}\cdot{\boldsymbol{v}})^{2}}=\frac{(\eta+v)^{\mu}}{(\eta+v)\cdotk}=\frac{p^{\mu}}{p\cdotk}\,,
\left|R_{N}^{(2)}\right|_{\mathrm{(III)}}<{\frac{1}{\pix}}\left({\frac{4g}{x^{4}}}\right)^{N}\left[{\frac{4g}{x^{4}\beta}}N\Gamma(N+1)+\left({\frac{1}{\beta}}+{\frac{1}{2}}\right)\Gamma(N)\right].
{\frac{2B}{a^{3\gamma+2n}}}+{\frac{3(3\gamma+4n-2)kc_{0}^{2}}{4\piG(3\gamma+2n-2)a^{2}}}\leq{\frac{(3\gamma+4n)c_{0}^{2}\Lambda}{4\piG(3\gamma+2n)}}.
\mathrm{Tr}(F_{L}+{g^{-1}F_{R}g}+(\omega^{2}))^{m-1}=\sum_{p,q}{\frac{1}{p!q!(m-p-q-1)!}}\mathrm{sTr}(F_{L}^{p}({g^{-1}F_{R}g})^{q}(\omega^{2})^{m-p-q-1})\quad.
\Theta_{[1/2]}\,=\,\left[\begin{array}{ccc}{0}&{{}}&{-1}\\{1}&{{}}&{0}\\\end{array}\right]\,,\,\,\Theta_{[1]}\,=\,\left[\begin{array}{ccccc}{0}&{{}}&{0}&{{}}&{1}\\{0}&{{}}&{-1}&{{}}&{0}\\{1}&{{}}&{0}&{{}}&{0}\\\end{array}\right]\quad.
C_{1}^{(0)}(g,q_{1}^{+},k^{+})=-{\frac{g\phi_{c}(k)\sqrt{(q_{1}^{+}+k^{+})q_{1}^{+}}}{V(q_{1}^{+}+k^{+})q_{1}^{+}+{\frac{g}{2}}(2q_{1}^{+}+k^{+})}}.
\fracD{Dz^{\mu}(\sigma)}V^{\nu}(\sigma^{\prime})={\frac{\deltaV^{\nu}(\sigma^{\prime})}{\deltaz^{\mu}(\sigma)}}+\intd\sigma^{\prime\prime}\,{\bf\Gamma}_{\mu\rho}^{\nu}(\sigma,\sigma^{\prime},\sigma^{\prime\prime})V^{\rho}(\sigma^{\prime\prime})\,\,,
\langle\bar{P}\rangle_{1}={\frac{\partial}{\partial\bar{t}}}\,\bar{F}_{1}=-{\frac{\langleP\rangle_{1}}{\langlePP\rangle_{0}}}+{\frac{1}{2}}\,{\frac{\langlePPP\rangle_{0}}{\langlePP\rangle_{0}}}\.
y^{2}=x^{3}+x^{2}(\frac{\Lambda_{1}^{4}\Lambda_{3}^{4}}{M_{3}^{2}}+\Lambda_{2}^{4}M_{3}-M_{1}M_{2}M_{3}+T^{2})+x\Lambda_{1}^{4}\Lambda_{2}^{4}\Lambda_{3}^{4}.
\omega_{\pi}^{(2)}=\displaystyle{\frac{\lambda}{2L}\int_{-L}^{L}dx^{-}}\frac{1}{m_{\pi}^{2}}\Bigl\{\frac{1}{m_{\pi}^{2}}\Phi_{\pi}\varphi_{\sigma}^{2}+\frac{2}{m_{\sigma}^{2}}\Phi_{\sigma}+\frac{3}{m_{\pi}^{2}}\Phi_{\pi}\varphi_{\pi}^{2}\Bigr\}\quad,
Z_{\chi\mathrm{UE}}=\int_{2N\times2N\,\mathrm{\scriptsize~hermite}}\!\!\!\!\!\!\!\!\!\!dM\,\mathrm{e}^{-N\,\mathrm{tr}\,V(M^{2})}\,|\mathrm{det}\,M|^{\alpha},\\\\\\M=\left(\begin{array}{cc}{0}&{W^{\dagger}}\\{W}&{0}\\\end{array}\right)
(\bar{\alpha}^{I}\hat{\partial}^{I}+(s-1)\bar{\alpha}^{z})|\Lambda_{0}\rangle+\hat{\partial}^{+}|\Lambda_{1}\rangle=-(2s+d-4)\frac{\partial^{+}\eta^{I}}{\partial^{+}}\bar{\alpha}^{I}\bar{\alpha}^{z}|\Phi_{ph}\rangle\,.
r\widetilde{E}\left(\beta\right)\approx\frac{\pi^{4}}{15}\frac{r^{4}}{\beta^{4}}-\frac{1}{240}+\frac{\pi^{2}\left|\mu_{eff}^{2}\right|}{12}\frac{r^{2}}{\beta^{2}}-\frac{3\left|\mu_{eff}^{2}\right|}{16},
(G_{\alpha\beta})=\left(\begin{array}{cc}{2\dot{\zeta}-\stackrel{\stackrel{\cdot}{\rightharpoonup}^{2}}{x}}&{u_{1}\ldotsu_{M}}\\{{u_{1}}}&{}\\{\cdot}&{}\\{\cdot}&{-g_{rs}}\\{\cdot}&{}\\{u_{M}}&{}\\\end{array}\right)