harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_{l,c}=-p_{u}=\frac{1}{2}(E-p_{y})=\frac{p_{A}^{2}}{4p_{v}}+\frac{1}{4\alpha^{\prime}p_{v}}(\hat{N}_{L}+\hat{N}_{R})+f(\hat{J}_{L}+\hat{J}_{R})~.
\rho(x)\cong\frac{2}{x^{2}}+2\sum_{q\neq0}(\frac{x^{4}}{q^{6}}-\frac{12}{q^{8}}x^{2}(q\cdotx)^{2}+\frac{16}{q^{10}}(q\cdotx)^{4})+O(\frac{x^{6}}{q^{8}})~...~,
S_{f}=\sum_{rs}\intd^{D}x~h\psi_{r}^{B\dagger}\psi_{r}^{\dagger}U^{\dagger}\left(-\frac{1}{2m}\partial^{\mu}\partial_{\mu}-\frac{1}{2}mv_{\alpha}^{\mu}v_{\mu}^{\alpha}\right)U\psi_{s}\psi_{s}^{B}.
\prod_{a\inA\backslash\{1\}}\oint_{C_{a}}\frac{dw_{a}}{2\piiw_{a}}\left(\oint_{x^{-4}C_{1}}-\oint_{C_{1}}\right)\frac{dw_{1}}{2\piiw_{1}}\Psi_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})Q_{b}^{(n)}(w|\zeta_{1}^{-1},\tilde{\zeta})\frac{xz_{1}^{-1}-w_{1}}{x^{-3}z_{1}-w_{1}}R(w_{1}).
={\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}\Big(\delta(\epsilon_{12}^{\prime},t_{1})+\tilde{\delta}(g_{0}\epsilon_{12}^{\prime}g_{0}^{-1},t_{2})-\delta(\epsilon_{21}^{\prime},t_{2})-\tilde{\delta}(g_{0}\epsilon_{21}^{\prime}g_{0}^{-1},t_{1})\Big)g,
[det(\delta_{\mu\nu}+L_{\mu\nu})]=\frac{[1+\frac{1}{8}f_{ij}^{2}+\frac{1}{16}\frac{\epsilon^{ijkl}}{4!}\epsilon^{mnpq}f_{mi}f_{nj}f_{pk}f_{ql}]^{2}}{-det(\eta_{ab}+\frac{1}{2}F_{ab})}
\hat{G}_{MN}=\left(\begin{matrix}{e^{2\varphi}g_{{\mu}{\nu}}+G_{{m}{n}}A_{{\mu}}^{(1)\,m}A_{{\nu}}^{(1)\,n}}&{A_{{\mu}}^{(1)\,m}G_{{m}{n}}}\\{A_{{\nu}}^{(1)\,n}G_{{m}{n}}}&{G_{{m}{n}}}\\\end{matrix}\right),
{\calL}_{0}(J_{0})=\frac{1}{2}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}+\frac{J_{0}}{2}\phi_{0}^{2}+\frac{\lambda\mu^{2\varepsilon}}{4!}\phi_{0}^{4}-\zeta_{0}\;J_{0}^{2}
E=\int_{\Sigma}d^{3}\sigma{\frac{1}{\sqrt{1-v_{\perp}^{2}}}}=\int_{\Sigma}d^{3}\sigma{\frac{1}{|(e^{||}\cdote^{\psi})|}}=\int_{\Sigma}d^{3}\sigma{\frac{1}{|(e^{\perp}\cdotI.e^{\psi})|}}
\begin{array}{l}{(\partialb)^{*}=-qab^{-1}(\partiala),}\\{(\partialc)^{*}=-\frac{1}{q^{2}}c^{-1}(a\partiala),}\\{(\partiald)^{*}=q(a^{2}-1)(bc)^{-1}(\partiala).}\\\end{array}
\Omega^{-1}d\Omega\=\i\left[\omega^{a}(P)P_{a}+\omega^{\alpha}(Q)Q_{\alpha}+\bar{\omega}_{\dot{\alpha}}(\bar{Q})\bar{Q}^{\dot{\alpha}}+\omega^{\alpha}(S)S_{\alpha}+\bar{\omega}_{\dot{\alpha}}(\bar{S})\bar{S}^{\dot{\alpha}}\right]\,
R_{ik}-\frac{1}{2}g_{ik}R=-8\piG\left[\beta(\phi)\left(\phi^{i}\phi^{k}-\frac{1}{2}g^{ik}\phi^{\alpha}\phi_{\alpha}\right)+\frac{\Lambda}{8\piG}\alpha(\phi)g_{ik}+T_{ik}^{\mathrm{traceless}}\right]
C_{Q}=-2\pir_{+}^{2}\left[1-\frac{r_{-}}{r_{+}}\right]^{2q^{2}/(2q^{2}+3)}\left[1-\frac{3}{2q^{2}+3}\frac{r_{-}}{r_{+}}\right]\left[1-\frac{2q^{2}+9}{2q^{2}+3}\frac{r_{-}}{r_{+}}\right]^{-1}.
(A,B,C,D)=\left(\frac{\beta}{8\gamma^{2}},-\frac{\beta}{8\gamma^{2}},-\frac{1}{2\beta},-\frac{1}{2\gamma}\right),\quad\left(-\frac{\beta}{8\gamma^{2}},-\frac{\beta}{8\gamma^{2}},-\frac{1}{2\beta},-\frac{1}{2\gamma}\right).
v(\mathbf{p,}i)\simeq\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)\left[\begin{array}{c}{\frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdotp})}{2m}}\\{1}\\\end{array}\right]\chi_{i}=\left[\begin{array}{c}{\frac{(\overrightarrow{\mathbf{\sigma}}\mathbf{\cdotp})}{2m}}\\{\left(1+\frac{\mathbf{p}^{2}}{8m^{2}}\right)}\\\end{array}\right]\chi_{i}.
\frac{3}{2}(\partial^{\mu}\partial^{\nu}h_{\mu\nu}-\partial_{\mu}\partial^{\mu}\tilde{h})-\frac{15}{2}\partial_{4}\sigma\partial_{4}\tilde{h}-\frac{3}{2}\frac{\partial^{2}\tilde{h}}{\partial{x^{4}}^{2}}-\frac{3}{2}\partial_{\mu}\partial^{\mu}\phi+30k^{2}\phi-12k\phi\tilde{\delta}=0.
S={\frac{1}{2}}\int_{0}^{T}{dt\left({i{\frac{\xi^{*}\dot{\xi}-{\dot{\xi}}^{*}\xi}{1+\left|\xi\right|^{2}}}+{\frac{1-\left|\xi\right|^{2}}{1+\left|\xi\right|^{2}}}}\right)}\quad.
\bar{H}_{\mathrm{g}}=H_{\mathrm{g}}(q^{r},p^{r})+\lambda_{a}^{(1)r_{a}}\pi_{a}^{r_{a}}+\lambda_{a}^{(2)r_{a}}\psi_{a}^{r_{a}}+\lambda_{a}^{(3)}\theta_{a}+\lambda_{a}^{(4)}\gamma_{a},
Z_{\Lambda}^{ren}(\phi)=\frac{1}{192\pi^{2}}\frac{\lambda_{R}^{2}\phi^{2}}{m_{R}^{2}+\frac{1}{2}\lambda_{R}\phi^{2}}\left[1-\frac{\Lambda^{3}}{(\Lambda^{2}+m_{R}^{2}+\frac{1}{2}\lambda_{R}\phi^{2})^{3/2}}\right]
(E^{2}-c^{2}{\bfp}^{\,2})I^{(2)}\Psi=\left[EI^{(2)}-c{\bfp}\cdot{\bbox\sigma}\right]\left[EI^{(2)}+c{\bfp}\cdot{\bbox\sigma}\right]\Psi=\mu_{2}^{2}c^{4}\Psi\,.
e_{0}^{a}=\left(\begin{array}{c}{C(\eta)}\\{0}\\{0}\\{0}\\\end{array}\right)\qquade_{1}^{a}=\left(\begin{array}{c}{0}\\{C(\eta)}\\{0}\\{0}\\\end{array}\right)\qquade_{2}^{a}=\left(\begin{array}{c}{0}\\{0}\\{C(\eta)}\\{0}\\\end{array}\right)\qquade_{3}^{a}=\left(\begin{array}{c}{0}\\{0}\\{0}\\{C(\eta)}\\\end{array}\right)
u=\frac{\bar{y}_{1}y_{1}^{\prime}+\bar{y}_{2}y_{2}^{\prime}}{y_{1}y_{2}^{\prime}-y_{1}^{\prime}y_{2}},\quad\bar{u}=\frac{y_{1}\bar{y}_{1}^{\prime}+y_{2}\bar{y}_{2}^{\prime}}{\bar{y}_{1}\bar{y}_{2}^{\prime}-\bar{y}_{1}^{\prime}\bar{y}_{2}}.
{\mathcalL_{4}}={\frac{dz_{1}d\bar{z}_{1}[1+z_{2}\bar{z}_{2}]+dz_{2}d\bar{z}_{2}[1+z_{1}\bar{z}_{1}]}{[1+z_{1}\bar{z}_{1}+z_{2}\bar{z}_{2}]^{2}}}
\phi_{L}\rightarrowA^{\prime}e^{-\gammaax}e^{-\gammaaL}\frac{\Gamma(-2\gamma)\Gamma(1+2\sigma)}{\Gamma(1-\gamma+\sigma-\lambda)\Gamma(-\gamma+\sigma+\lambda)}+A^{\prime}e^{\gammaax}e^{\gammaaL}\frac{\Gamma(2\gamma)\Gamma(1+2\sigma)}{\Gamma(1+\gamma+\sigma-\lambda)\Gamma(\gamma+\sigma+\lambda)}
Q_{p}\equiv\frac{\kappaT_{p}}{(7-p)\Omega_{8-p}}\,(N_{1}+N_{2})~~,~~\Omega_{q}=\frac{2\pi^{(q+1)/2}}{\Gamma(\frac{p+1}{2})}~~,~~r^{2}=\sum_{i}(x^{i})^{2}
\hat{M}_{\mu\nu}=-\imath(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})-\frac{\imath}{4}\left(\begin{array}{cc}{[\Gamma_{\varsigma\mu},\Gamma_{\varsigma\nu}]}&{0}\\{0}&{[\Gamma_{\varsigma\mu},\Gamma_{\varsigma\nu}]}\\\end{array}\right),
{\calG}_{\omega}(r,r^{\prime},\theta)=\sum_{n=0}^{\infty}r^{1-D/2}\left[e_{n}H_{n-1+{\frac{D}{2}}}^{(1)}(|\omega|r)+f_{n}H_{n-1+{\frac{D}{2}}}^{(2)}(|\omega|r)\right]C_{n}^{(-1+{D/2})}(z)\quad(r^{\prime}>r>a).
\dot{x}^{-}=\frac{\sqrt{(\eta_{rs}A_{-}^{r}B^{s})^{2}-(\eta_{rs}A_{-}^{r}A_{-}^{s})(\eta_{tu}B^{t}B^{u})}-(\eta_{rs}A_{-}^{r}B^{s})}{\eta_{rs}A_{-}^{r}A_{-}^{s}}
\deltaN^{\pm}=\pm{u^{2}}\;\mathrm{sin}\theta_{r}\;\left[{\frac{3}{2}}\mathrm{cos}^{2}\theta_{r}\;\partial_{\mp}^{3}\xi^{\mp}-{\frac{1}{2}}\mathrm{sin}^{2}\theta_{r}\;\partial_{\pm}^{3}\xi^{\pm}\right]\,
\Psi_{j}(\nu)={\frac{\Big(\Gamma(1-g)\Big)^{-i\nu(1+\xi)}\Gamma(i\nu(\xi+1)/3)}{\Gamma(i\nu\xi/3)\Gamma((-1+i\nu)/3)\Gamma((1+i\nu)/3)}}\sum_{n=1}^{\infty}(-it_{n}^{(j)})^{-i\nu(1+\xi)/3}.
[P(\Omega_{p})]_{a_{1}\ldotsa_{p}}=\Omega_{\mu_{1}\ldots\mu_{p}}\left(\delta_{a_{1}}^{\mu_{1}}I_{N}+\lambda\frac{\partial\phi^{\mu_{1}}}{\partial\sigma^{a_{1}}}\right)\ldots\left(\delta_{a_{p}}^{\mu_{p}}I_{N}+\lambda\frac{\partial\phi^{\mu_{p}}}{\partial\sigma^{a_{p}}}\right)
B_{\mu\nu}(t)=\sum_{\rho=0}^{N}\frac{\eta_{\mu\rho}\eta_{\nu\rho}}{\sqrt{8\tilde{\omega}_{\mu}}}\left[\left(1+\frac{\tilde{\omega}_{\mu}}{\Omega_{\rho}}\right){e}^{i\Omega_{\rho}t}+\left(1-\frac{\tilde{\omega}_{\mu}}{\Omega_{\rho}}\right){e}^{-i\Omega_{\rho}t}\right]\;,
H_{0}\\equiv\(p_{1}^{2}-m_{1}^{2})\big\vert_{C}\=\(p_{2}^{2}-m_{2}^{2})\big\vert_{C}\=\\frac{P^{2}}{4}-\frac{1}{2}(m_{1}^{2}+m_{2}^{2})+\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{4P^{2}}+p^{T2}\.
y_{1}^{(\infty)}=\left(1+\big(y_{2}^{(\infty)}\big)^{-1}\right)^{-{\frac{1}{2}}},\qquady_{2}^{(\infty)}=\left(1+\big(y_{1}^{(\infty)}\big)^{-1}\right)^{-{\frac{1}{2}}},\qquad
T(0)=\prod_{j=l+1}^{P}\;\prod_{i=l}^{j-1}R_{ij}^{\iota^{P-j+i-l+1}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})\;\;\prod_{j=l}^{P-1}\;\prod_{i=j+1}^{P}R_{ij}^{\iota^{P-j+i}}((\epsilon^{-j}-\epsilon^{-i})\theta^{\prime})
\left[\left({\frac{H_{1}^{\prime}}{H_{1}}}+{\frac{H_{2}^{\prime}}{H_{2}}}\right)+{\frac{2}{r}}\right](h_{1}^{\prime}-h_{2}^{\prime})=-2\left[\left({\frac{H_{1}^{2}}{H_{1}^{2}}}+{\frac{H_{2}^{2}}{H_{2}^{2}}}\right)-{\frac{2}{r^{2}}}\right]h_{2}-4\left({\frac{H_{1}^{2}}{H_{1}^{2}}}-{\frac{H_{2}^{2}}{H_{2}^{2}}}\right)\varphi.
{\calM}_{ii\rightarrowjj}\simg^{2}\sum_{k}c_{iik}c_{jjk}\int_{0}^{\infty}dte^{-t(p^{2}+\lambda_{k})}=g^{2}\int_{0}^{\infty}dte^{-tp^{2}}\sum_{k}c_{iik}c_{jjk}e^{-t\lambda_{k}}
\Biggl[-{\frac{g^{2}}{(2\pi)^{2}}}{\frac{d^{2}}{dz^{2}}}+{\frac{g^{2}}{2}}\bigl[C(z)+D_{1/2}(z)\bigr]\biggr]\bar{\zeta}(z)=\epsilon_{1}\bar{\zeta}(z)\;.
{\calO}[\phi^{\prime}]\equiv{\calO}[\phi+(\tilde{\delta}_{1}\Theta_{1}+\tilde{\delta}_{2}\Theta_{2})\Theta^{\prime}]={\calO}[\phi]+(\tilde{\delta}_{1}\Theta_{1}+\tilde{\delta}_{2}\Theta_{2})\Theta^{\prime}{\frac{\delta{\calO}}{\delta\phi}}.
\left[a_{s}^{*-}(\mathbf{p}),a_{r}^{+}(\mathbf{p})\right]=\left[a_{s}^{-}(\mathbf{p}),a_{r}^{*+}(\mathbf{p})\right]=\varepsilon_{s}\delta_{sr}\delta(\mathbf{p}-\mathbf{p}^{\prime})
(\Phi^{\prime}{}^{n},\Phi_{m}^{\prime}{}^{*})=\delta_{m}^{n}\;,\qquad(\Phi^{\prime}{}^{n},\Phi^{\prime}{}^{m})=(\Phi_{n}^{\prime}{}^{*},\Phi_{m}^{\prime}{}^{*})=0\;,
S_{lc}^{F}=\frac{i}{4\pil_{s}^{2}}\intd\tau\int_{0}^{2\pil_{s}^{2}p^{+}}d\sigma\left[\left(\eta^{\alpha\beta}\delta_{AB}-\epsilon^{\alpha\beta}\sigma_{AB}^{3}\right)\partial_{\alpha}x^{+}\bar{\theta}^{A}\Gamma_{+}\left(\mathcal{D}_{\beta}\theta\right)^{B}\right],
X^{2}=b\left(\begin{array}{ccccc}{\ddots}&{\ddots}&{\ddots}&{}&{}\\{\ddots}&{0}&{e^{i\omegat}}&{\ddots}&{}\\{\ddots}&{e^{-i\omegat}}&{0}&{e^{i\omegat}}&{\ddots}\\{\ddots}&{\ddots}&{e^{-i\omegat}}&{0}&{\ddots}\\{}&{}&{\ddots}&{\ddots}&{\ddots}\\\end{array}\right)
(X)_{ll^{\prime}}=-\frac{\alpha}{n+\frac{\delta_{1}+\delta_{2}}{2}}\sum_{n_{1}=0}^{n-|m|-1}\left(2n_{1}-n+|m|+\frac{\delta_{1}-\delta_{2}}{2}+1\right)\widetildeW_{nlm}^{n_{1}}(\delta_{1},\delta_{2})\widetildeW_{nl^{\prime}m}^{n_{1}}(\delta_{1},\delta_{2}).
{\hat{\pi}}=-i\hbar\frac{\partial}{\partial{\vec{r}}}-{\hbars}\vec{A},\quad\vec{A}=\frac{{\vec{n}}_{3}{\vec{r}}}{r}\frac{\vec{n}_{3}\times\vec{r}}{r^{2}-({\vec{n}_{3}}{\vec{r}})^{2}}.
\stackrel{\smash{(0)}}{R}_{\alpha}^{i}\longrightarrowR_{\alpha}^{i}=\stackrel{\smash{(0)}}{R}_{\alpha}^{i}+g\stackrel{\smash{(1)}}{R}_{\alpha}^{i}+g^{2}\stackrel{\smash{(2)}}{R}_{\alpha}^{i}+....
\begin{array}{l}{\displaystyle\sigma(x^{i},x^{j})=\frac{1-{\bfx}{\bfx}^{\prime}}{1-\xi^{0}\xi^{0}}~,}\\{\displaystylex^{0}x^{0}=\frac{\xi^{0}\xi^{0}}{1-\xi^{0}\xi^{0}}(1-{\bfx}{\bfx}^{\prime})~,}\\\end{array}
S_{eff}({\mathbfn})=\frac{\mu^{2}}{2}\intd^{4}x\left[(\partial_{\mu}{\mathbfn})^{2}-\left({\mathbfn}(x)\cdot\mathrm{rot}\{\mathbfn}(x)\right)^{2}\right]-\frac{\theta\mu^{2}}{16\pi^{2}}\intd^{3}x\left({\mathbfn}(x)\cdot\mathrm{rot}\{\mathbfn}(x)\right)\,.
\left.\hat{A}_{\mu}^{I}(x)\right|_{\mathrm{sing}}=\hat{N}_{+}^{-1}\left.\hat{A}_{\mu}^{I}(x)\right|_{\mathrm{reg}}\hat{N}_{+}+i\,\hat{N}_{+}^{-1}\,\partial_{\mu}\hat{N}_{+},
L_{v}T_{i_{1}\ldotsi_{k},\bar{j}_{1},\ldots,\bar{j}_{l}}=v^{j}\frac{\partial}{\partial\Phi^{j}}T_{i_{1}\ldotsi_{k},\bar{j}_{1},\ldots,\bar{j}_{l}}+\sum_{a=1}^{k}T_{i_{1}\ldotsi_{a-1}i\ldotsi_{k},\bar{j}_{1},\ldots,\bar{j}_{l}}\frac{\partialv^{i}}{\partial\Phi^{i_{a}}},
(z_{0};z_{1},z_{2})=\left({\frac{1-x_{\mu}^{2}+h^{2}}{2h}}+i{\frac{x_{0}}{h}};{\frac{x_{1}+ix_{2}}{h}},x_{3}+i{\frac{1+x_{\mu}^{2}-h^{2}}{2h}}\right)
ds_{11}^{2}=\left(\frac{8\pi}{15}\right)^{1/3}l_{p}^{\prime2}\left(\frac{U^{2}}{Q^{2/3}}(-d\tau^{2}+dx_{8}^{2}+dx_{9}^{2})+Q^{1/3}(\frac{dU^{2}}{4U^{2}}+d\Omega_{(7)}^{2})\right)
(\omega_{1},\{\omega_{1},\omega_{1}\})+\{\omega_{1},(\omega_{1},\omega_{1})\}=(-1)^{\varepsilon_{A}(1+\varepsilon_{B})}\left(\omega_{1},\left.{\frac{\partial_{r}^{2}\Gamma}{\partial\Phi^{A}\partial\Phi^{B}}}\right|_{K}\right)\left.{\frac{\partial\omega_{1}}{\partialJ_{B}}}\right|_{K}\left.{\frac{\partial\omega_{1}}{\partialJ_{A}}}\right|_{K}.
{\calS}_{J_{H}^{-1}}^{(2k+1)}\cdot\left(2\varphi_{\bar{z}}\right)^{2k+l}e^{-k\varphi}=\lambda_{l}\left(2\varphi_{\bar{z}}\right)^{l-1}e^{(k+1)\varphi},\qquadl\in{\bfZ},
\frac{\delta\hat{\Gamma}[\vec{\varphi}_{1},\vec{\varphi}_{2}]}{\delta\vec{\varphi}_{j}(x)}=(-)^{j-1}\frac{\delta\Gamma_{C}[\vec{\varphi}]}{\delta\vec{\varphi}(x)}\;\;\mathrm{with}\;x_{0}\inC_{j}\;\;\;\;\;\;(j=1,2).
A^{\mu}(x)=\frac{\partialx^{\mu}}{\partialq^{\alpha}}\,A^{\alpha}(q)\quad\mathrm{and}\quadA^{a}=e_{\mu}^{a}(x)\,\frac{\partialx^{\mu}}{\partialq^{\alpha}}\,e_{b}^{\alpha}(q)\,\psi^{\dagger}(q)\sigma^{b}\psi(q),
\begin{array}{ll}{I_{eff}=+\frac{k}{8\pi}\intdz^{2}[\partial\varphi\overline{{\partial}}\varphi+\alpha\varphi-4e^{-2t}\partialx\overline{{\partial}}xe^{-2\varphi}-4e^{-2t}(x^{2}+4)e^{-2\varphi}}&{}\\{(\partial\varphi\overline{{\partial}}\varphi+\beta)],}&{}\\\end{array}
<A_{1}|n>=\left(\frac{k\tau_{2}}{2\pi^{2}}\right)^{1/4}\,\frac{1}{\sqrt{n!}}\,\left(\frac{i\bar{\tau}}{2\tau_{2}}\right)^{n/2}\,e^{\frac{ik\tau}{4\pi}A_{1}^{2}}\,P_{n}\left(\sqrt{\frac{ik\bar{\tau}}{2\pi}}\,A_{1};\frac{-i\tau}{i\bar{\tau}}\right)\\\.
\tilde{\calK}^{l}(p,q)=\frac{(p^{l}q^{+}-p^{+}q^{l})}{q^{+}}\tilde{\calK}(p,q)-i\pi^{2}\frac{q^{l}}{q^{+}}\,U(p,q)+{\calO}(2-\omega)\,\,,
\mathfrak{L}(x)\cdots\mathfrak{L}(x+(2M-10)i)v_{0}=\prod_{j=1}^{(M-3)/2}T_{M}(x-(4+4j)i)\left(\begin{array}{c}{T_{1}(x-i(M+3))}\\{T_{M-2}(x-2i)}\\\end{array}\right)
\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)_{\xi\eta}=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)_{\xi}\otimes\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)_{\eta}
a_{\omega}=\Sigma_{lm}\intd\omega^{\prime}(\alpha_{\omega\omega^{\prime}lm}^{b}b_{\omega^{\prime}lm}^{b}+\bar{\beta}_{\omega\omega^{\prime}lm}^{b}b_{\omega^{\prime}lm}^{b\dagger}+\alpha_{\omega\omega^{\prime}lm}^{a}b_{\omega^{\prime}lm}^{a}+\bar{\beta}_{\omega\omega^{\prime}lm}^{a}b_{\omega^{\prime}lm}^{a\dagger}).
{\frac{\partialW_{0}}{\partialm^{2}}}=-(i^{-\nu}){\frac{AT}{2}}\Gamma({\frac{1}{2}})\Gamma(\nu+{\frac{1}{2}})\sum_{n=1}^{\infty}\int\int{\frac{dk_{1}dk_{2}}{(2\pi)^{2}}}{\frac{1}{2\pi}}[k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+m^{2}]^{-(\nu+{\frac{1}{2}})}\,.
B^{(k)}=\left(\begin{array}{ccccc}{k-2}&{k-3}&{k-4}&{\cdots}&{1}\\{k-2}&{k-3}&{k-4}&{\cdots}&{1}\\{\vdots}&{}&{}&{}&{\vdots}\\{3}&{2}&{1}&{\cdots}&{1}\\{2}&{1}&{1}&{\cdots}&{1}\\{1}&{1}&{1}&{\cdots}&{1}\\{0}&{0}&{0}&{\cdots}&{0}\\\end{array}\right).
P_{u}(l)\;=\;-\frac{1}{il}\sum_{n=0}^{K-1}\left(\frac{1}{il}\right)^{n}\:\left.h_{u}^{(n)}(v)\:e^{-ivl}\right|_{\alpha_{u}}^{\beta_{u}}\:+\:\left(\frac{1}{il}\right)^{K}\int_{\alpha_{u}}^{\beta_{u}}h_{u}^{(K)}(v)\:e^{-ivl}\:dl\;.
ds^{2}=Md{\bfx}^{2}+{\frac{16\pi^{2}}{Me^{4}}}d\chi^{2}+\frac{4\pi\kappa}{e^{2}}\left[d\rho^{2}+{\frac{\rho^{2}}{4}}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2})\right].
\begin{array}{cl}{}&{F_{\sigma}^{(n)}(\cdots,\zeta_{j+1},\zeta_{j},\cdots)^{\cdotsk_{j-1}k_{j}k_{j+1}\cdots}}\\{=}&{\displaystyle\sum_{k_{j}^{\prime}}W\left(\left.\begin{array}{cc}{k_{j+1}}&{k_{j}}\\{k_{j}^{\prime}}&{k_{j-1}}\\\end{array}\right|\zeta_{j}/\zeta_{j+1}\right)F_{\sigma}^{(n)}(\cdots,\zeta_{j},\zeta_{j+1},\cdots)^{\cdotsk_{j-1}k_{j}^{\prime}k_{j+1}\cdots}.}\\\end{array}
B=\left(\begin{array}{cccccccc}{0}&{x}&{z}&{y}&{0}&{-y}\\{-x}&{0}&{x}&{z}&{y}&{0}\\{-z}&{-x}&{0}&{x}&{z}&{y}\\{-y}&{-z}&{-x}&{0}&{x}&{z}\\{0}&{-y}&{-z}&{-x}&{0}&{x}\\{y}&{0}&{-y}&{-z}&{-x}&{0}\\\end{array}\right),
\textrm{Tr}_{S}[\textrm{e}^{-\frac{i}{\hbar}HT}]=e^{-\frac{i}{\hbar}(M_{cl}+\deltaM)T}\underset{\{PID_{reg}\}}{\int{{\mathcal{D}}}\eta_{a}}\int_{\eta_{a},t^{\prime}}^{\eta_{a},t^{\prime\prime}}{{\mathcal{D}}}\eta\;e^{-\frac{i}{\hbar}\frac{1}{2}\underset{T\textrm{x}\tilde{L}}{\int}dtdz\frac{m}{l}\eta[\partial_{t}^{2}+O_{S}(z)]\eta}+O(\hbar^{2})
\left[D_{a}^{b},\left(D^{k}\right)_{c}^{d}\right]=\delta_{c}^{b}\left(D^{k}\right)_{a}^{d}-\delta_{a}^{d}\left(D^{k}\right)_{c}^{b}+\delta_{\bar{b}}^{d}\left(D^{k}\right)_{c}^{\bar{a}}-\delta_{c}^{\bar{a}}\left(D^{k}\right)_{\bar{b}}^{d}
{\frac{X}{M^{4}}}={\frac{P_{k}^{S}}{8L^{2}}}\left({\frac{M_{Pl}}{M}}\right)^{4}\approx3.5\times10^{-15}(10^{10}P_{k}^{S})\left({\frac{M_{Pl}}{M}}\right)^{4}
J_{\alpha}^{\mu{\mathbfi}}=192\left[-3i\sigma_{\alpha\dot{\alpha}}^{\mu}{\bar{D}}_{\mathbfj}^{\dot{\alpha}}D^{{\mathbfi}{\mathbfj}}T-3i{\bar{\sigma}}^{\mu\dot{\alpha}\beta}{\bar{D}}_{\dot{\alpha}}^{\mathbfi}D_{\alpha\beta}T-12\partial^{\mu}D_{\alpha}^{\mathbfi}T+a\sigma_{\alpha}^{\mu\nu\,\beta}\partial_{\nu}D_{\beta}^{\mathbfi}T\right]|\.
\left.{\frac{\partial^{2}V_{C}}{\partialq_{j}\partialq_{k}}}\right|_{\bar{q}}=\sum_{l=1}^{r}\left.{\frac{\partial^{2}W}{\partialq_{j}\partialq_{l}}}\right|_{\bar{q}}\left.{\frac{\partial^{2}W}{\partialq_{l}\partialq_{k}}}\right|_{\bar{q}}.
A_{n}=\intdx_{1}...\intdx_{n}\intdy_{1}...\intdy_{n}{\frac{\delta^{n}S}{\delta\sigma(x_{1},y_{1}).....\delta\sigma(x_{n},y_{n})}}|_{\sigma^{0}}\eta(x_{1},y_{1}).....\eta(x_{n},y_{n})
\frac{\partial^{2}S_{n}}{\partialt_{1}^{+2}}=-\frac{S_{n}}{1-S_{n}^{2}}(\frac{\partialS_{n}}{\partialt_{1}^{+}})^{2}-\frac{1}{t_{1}^{+}}\frac{\partialS_{n}}{\partialt_{1}^{+}}+\frac{(n+1)^{2}}{t_{1}^{+2}}\frac{S_{n}}{1-S_{n}^{2}}-4S_{n}(1-S_{n}^{2}).
\frac{1}{\pi^{d}}~\intd^{d}k_{1}~d^{d}k_{2}(k_{1}k_{2})~P_{k_{1},m_{1}}^{\nu_{1}}P_{k_{1}-k_{2},m_{2}}^{\nu_{2}}P_{k_{2},m_{3}}^{\nu_{3}+1}=\frac12\nu_{2}dJ_{\nu_{1}~\nu_{2}+1~\nu_{3}+1}^{(d+2)}.
f_{J_{1}n_{1}\sigma_{1};~J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}=(-1)^{2J_{1}(J_{2}-n_{2}-1)}{\tilde{f}}_{J_{1}n_{1}\sigma_{1};~J_{2}n_{2}\sigma_{2}}^{J_{3}n_{3}\sigma_{3}}.
\Gamma_{\underline{\alpha}\underline{\beta}}^{i}=\left(\begin{array}{cc}{0}&{\delta_{\beta}^{\alpha}\gamma_{qq^{\prime}}^{i}}\\{\delta_{\alpha}^{\beta}\tilde{\gamma}_{q^{\prime}q}^{i}}&{0}\\\end{array}\right)\,,\quadi=1,...,8\,,
\frac{\beta_{j^{\prime};j}}{\beta_{j^{\prime};j}^{(10)}}=b_{0}^{1/2}(\frac{2(s^{\prime}-s)-|m|+|m^{\prime}|b_{0}}{(2s^{\prime}+\frac{3}{2})+(|m^{\prime}|-|m|-2s-\frac{3}{2})b_{0}})\,,
P=\left(\begin{array}{cccccc}{0}&{1}&{}&{}&{}&{}\\{}&{0}&{1}&{}&{}&{}\\{}&{}&{\ddots}&{\ddots}&{}&{}\\{}&{}&{}&{}&{0}&{1}\\{1}&{}&{}&{}&{}&{0}\\\end{array}\right),\quadQ=\left(\begin{array}{cccccc}{1}&{}&{}&{}&{}&{}\\{}&{\omega}&{}&{}&{}&{}\\{}&{}&{\ddots}&{}&{}&{}\\{}&{}&{}&{}&{\omega^{N-2}}&{}\\{}&{}&{}&{}&{}&{\omega^{N-1}}\\\end{array}\right),
y^{2}D_{0}^{\prime\prime}+(1-y)yD_{0}^{\prime}+\left(-\frac{1}{4}(2\tilde{\beta}+\delta)^{2}+\frac{1}{4}\left(\tilde{\beta}(2+4\delta)-3(\tilde{\beta}^{2}-\delta^{2}-4)\right)y\right)D_{0}=0,
\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)\longrightarrow\left(\begin{array}{cc}{-1}&{2}\\{0}&{-1}\\\end{array}\right)\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right)=M_{\infty}\left(\begin{array}{c}{a_{D}}\\{a}\\\end{array}\right),
\psi_{_{(\alpha)}}^{2}(x)=(\frac{\mu}{2\pi})^{^{1/2}}{\calK}(\phi_{1})\,{\calK}(\phi_{3})\,:e^{-i\sqrt{\pi}\{\int_{-\infty}^{x}dz^{1}\pi_{_2}(z)+\gamma_{_{\alpha\alpha}}^{5}\phi_{_2}(x)\}}:,
\begin{array}{ll}{M_{1}=}&{\frac{1}{2}(\Box+2h^{2}\zeta\eta^{2}),}\\{M_{2}=}&{-\frac{1}{2}(\frac{1}{\alpha}\Box-2h^{2}\zeta\eta^{2}),}\\{M_{3}=}&{\frac{1}{2}(m+hg\zeta\eta^{2}),}\\{M_{4}=}&{\frac{1}{2}(1-g^{2}\zeta\eta^{2})\Box,}\\{M_{5}=}&{\frac{1}{8\beta}\Box,}\\\end{array}
R_{\mathrm{str}}={\frac{2(4\vec{P}_{F}\cdot\vec{P}_{H})^{2}-8M(4\vec{P}_{F}\cdot\vec{P}_{H})r+4(6M^{2}-(4\vec{P}_{F}\cdot\vec{P}_{H}))r^{2}}{\left((4\vec{P}_{F}\cdot\vec{P}_{H})+4Mr+r^{2}\right)^{3}}}\.
(3c)=e^{4}i^{3}\int{\calD}q\frac{1}{(p+q)^{2}}\;\varepsilon_{\mu\alpha\lambda}\frac{q^{\lambda}}{q^{2}}\;\varepsilon_{\beta\nu\rho}\frac{q^{\rho}}{q^{2}}\int{\calD}k\frac{(2k+q)^{\alpha}(2k+q)^{\beta}(2p+q)^{\mu}(2p+q)^{\nu}}{k^{2}(k+q)^{2}}
R_{~\mu\nu\rho}^{\lambda}[h]=R_{~\alpha\beta\sigma}^{\gamma}[g]h_{\mu}^{\alpha}h_{\rho}^{\sigma}h_{\gamma}^{\lambda}-A_{~\nu}^{\lambda}A_{\mu\rho}+A_{~\rho}^{\lambda}A_{\mu\nu}-2A_{~\mu}^{\lambda}A_{\nu\rho}~~~,
y(x)=A\\mathrm{e}^{-\frac{x^{2}}{4}}\,\mathrm{M}\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{x^{2}}{2}\right)+B\x\\mathrm{e}^{-\frac{x^{2}}{4}}\,\mathrm{M}\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},\frac{x^{2}}{2}\right)\,,
\xi=\sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}\left(\overline{{\eta}}_{a}D_{\mu}T_{b}^{a}\psi^{\mu}\eta^{b}\right)\,\overline{{\xi}}=\sqrt{\alpha^{\prime}}\frac{1}{\partial_{\tau}}\left(\overline{{\eta}}_{a}D_{\mu}T_{b}^{a}\psi^{\mu}\eta^{b}\right)\,
K_{l}(u)=\left(\begin{array}{ll}{2x_{0}(q^{-1}s^{-1}u-qsu^{-1})\;}&{x_{+}(q^{-1}u^{2}-qu^{-2})}\\{x_{-}(q^{-1}u^{2}-qu^{-2})\;}&{-2x_{0}(su-s^{-1}u^{-1})}\\\end{array}\right)\,,
\Psi^{kl}=\frac{\bar{\calD}^{2}F^{kl}}{4(\Phi^{k}-\Phi^{l})}\,,\qquad\bar{\Psi}^{kl}=\frac{{\calD}^{2}F^{kl}}{4(\bar{\Phi}^{k}-\bar{\Phi}^{l})}\;,
\omega=\omega_{{A_{1}\dotsA_{n}}}\,V^{{A_{1}}}\wedge\dots\wedgeV^{{A_{n}}}\quad\Leftrightarrow\quad\star\omega=\frac{1}{(10-n)!}\,\epsilon_{{A_{1}\dotsA_{10-n}B_{1}\dotsB_{n}}}\omega^{{B_{1}\dotsB_{n}}}\,V^{{A_{1}}}\wedge\dots\wedgeV^{{A_{10-n}}}
S=\intd^{3}x\left[\frac{1}{4g^{2}}\left(F_{\mu\nu}^{a}\right)^{2}+\frac12\left(D_{\mu}\Phi^{a}\right)^{2}+\frac{\lambda}{4}\left(\left(\Phi^{a}\right)^{2}-\eta^{2}\right)^{2}\right].
J=\frac{\partial(\ldots,\bar{A}_{-1}^{a},\bar{A}_{0}^{a},\bar{A}_{1}^{a},\ldots)}{\partial(\ldots,\,\bar{\varphi}_{-1}^{a},a_{1}^{a}\,,\bar{\varphi}_{1}^{a}\,,\ldots)}\;.
S(D,b)=\int\mathrm{d}^{D}x\;{\frac{\sqrt{-g}}{2\kappa_{D}^{2}}}\left[-R-{\textstyle{\frac{4}{D-2}}}(\partial\phi)^{2}-{\textstyle{\frac{1}{2D!}}}\,g_{s}^{2k}\left({\frac{e^{\phi}}{g_{s}}}\right)^{b}\,F_{D}^{2}\right]\,,
\eta_{ia}\eta_{jb}R^{a}{}_{m}{}^{b}{}_{n}=R^{a}{}_{i}{}^{b}{}_{j}\eta_{am}\eta_{bn},\quad\eta_{ai}R^{a}{}_{j}{}^{l}{}_{k}=\lambda^{-2}R^{-1}{}^{a}{}_{i}{}^{l}{}_{k}\eta_{ja}
L=\intd^{3}{x}\left[-\frac14(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})(\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu})+\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}V_{\mu}-m)\psi\right]\;.
c_{d}(f_{1},f_{2})=d\left\{m_{1}\lambda_{2}-m_{2}\lambda_{1}+\frac{1}{4\pi}\int_{0}^{2\pi}\biggl(\frac{d\tilde{f}_{1}}{d\sigma}\tilde{f}_{2}-\frac{d\tilde{f}_{2}}{d\sigma}\tilde{f}_{1}\biggr)d\sigma\right\}.
\left(A_{-1}^{2}-\frac{13-c_{A}+\sqrt{(25-c_{A})(1-c_{A})}}{12}A_{-2}\right)\Phi_{1,2}=0\left(A_{-1}^{2}-\frac{2+4\,a_{1,2}}{3}\,A_{-2}\right)\Phi_{1,2}=0\,.
\left[\lambda_{q}(R)-\left(l+\frac{\delta_{1}+\delta_{2}}{2}\right)\left(l+\frac{\delta_{1}+\delta_{2}}{2}+1\right)\right]T_{nqm}^{l}(R;\delta_{1},\delta_{2})=R\sum_{l^{\prime}=|m|}^{n-1}T_{nqm}^{l^{\prime}}(R;\delta_{1},\delta_{2})(X)_{ll^{\prime}},
D_{\eta}^{i}\equivD^{i}=\frac{\partial}{\partial\eta_{i}}-i\bar{\eta}^{i}\frac{\partial}{\partialt}\quad,\quad\bar{D}_{\eta\,i}\equiv\bar{D}_{i}=\frac{\partial}{\partial\bar{\eta}^{i}}-i\eta_{i}\frac{\partial}{\partialt}
\frac{\piN^{2}}{g_{nc}^{2}\theta}\left(\sum_{i_{1}}m_{i_{1}}^{2}+\frac{1}{2}\sum_{i_{2}}m_{i_{2}}^{2}+..+\frac{1}{k}\sum_{i_{k}}m_{i_{k}}^{2}-\frac{1}{N}\right);
S_{\mathrm{boson}}^{\mathrm{root}}=\intd^{4}x\left\{\frac1{4g^{2}}F_{\mu\nu}^{2}+\bar{\nabla}_{\mu}\bar{q}_{Af}\nabla_{\mu}q^{fA}+\frac{g^{2}}8[\mathrm{~Tr~}\bar{q}\tau^{a}q)]^{2}\right\},
G(k)=2\pi\left\{B\delta\left(k^{2}+m^{2}\right)+\left(\frac{g^{2}}2\right)h(-k^{2})\left|G_{ret}(k)\right|^{2}\theta\left(-k^{2}-4M^{2}\right)\right\}\mathrm{sign}(k^{0})
E_{0}\left(d\right)=-\alpha\left(d\right)\,\frac{L^{d-1}}{2}\int\frac{d^{d-1}p_{\bot}}{(2\pi)^{d-1}}\oint_{\Gamma}\frac{dq}{2\pi}\sum_{n}\frac{2q^{2}}{q^{2}+\Omega_{n}^{2}}.
\xi^{\prime(\pm)}=\mpJ_{0}\{\xi^{\pm}-\frac{1}{2}(\frac{M^{\mp}}{2}-\partial_{\pm}^{2})\xi^{\pm}\}\pmJ_{1}\partial_{\pm}\xi^{\pm}-J_{2}\{\xi^{\pm}+\frac{1}{2}(\frac{M^{\mp}}{2}-\partial_{\pm}^{2})\xi^{\pm}\}.
{\mathcalW}_{\mu_{1}\cdots\mu_{n}}^{(n)}(x_{1},\cdots,x_{n})\;=\;\left.\frac{\delta^{n}{\mathcalW}[b_{\mu}]}{\deltab_{\mu_{1}}(x_{1})\cdots\deltab_{\mu_{n}}(x_{n})}\right|_{b_{\mu}=0}\;.
\left(\left(-i\delta_{\alpha\beta}\:q_{A}\:-\:i\Gamma_{\alpha\beta}^{j}\:\frac{\partial}{\partialx_{jA}}\right)\frac{\partial}{\partial\lambda_{\alphaA}}\:+\:D_{\alphaA}^{(\beta)\:\dagger}\:\lambda_{\alphaA}\right)\tilde{\psi}\;=\;0\;\;\;,
\begin{array}{c}{\theta(x)\left(-\nabla_{x}+m^{2}\right)\varphi_{R}=0}\\{\theta(-x)\left(-\nabla_{x}+m^{2}\right)\varphi_{L}=0}\\{\left.\partial_{x}\left(\varphi_{R}-\varphi_{L}\right)\right|_{x=0}=\left.c\,\left(\varphi_{R}+\varphi_{L}\right)\right|_{x=0}}\\{\varphi_{R}(y,0)=\varphi_{L}(y,0)}\\\end{array}
\left[(1-u^{2})\partial_{u}^{2}-{\frac{1}{4}}\right]\phi=0=\left\{\left[1-({\calG}+B)^{2}\right]\left({\calG}^{\prime}\partial_{a}^{2}-{\calG}^{\prime\prime}\partial_{a}\right)-{\frac{1}{4}}{{\calG}^{\prime}}^{3}\right\}\phi=0,
{\Omega}_{ijk}^{u}=\frac{{\partial}{\omega}_{jk}^{u}}{{\partial}{\xi}^{i}}+\frac{{\partial}{\omega}_{ki}^{u}}{{\partial}{\xi}^{j}}+\frac{{\partial}{\omega}_{ij}^{u}}{{\partial}{\xi}^{k}}.
r_{s}^{d-1}=\frac{4|\tilde{\alpha}|/l^{2}}{1+4|\tilde{\alpha}|/l^{2}}r_{+}^{d-1}\left(1+\frac{l^{2}}{r_{+}^{2}}-\frac{|\tilde{\alpha}|l^{2}}{r_{+}^{4}}\right),